71a1. All instruction given or received by way of argument proceeds from preexistent knowledge. This becomes evident upon a survey of all the species of such instruction.

71a3. The mathematical sciences and all other speculative disciplines are acquired in this way.

71a4. And so are the two forms of dialectical reasoning, syllogistic and inductive, for each of these latter make use of old knowledge to impart new. The syllogism assumes an audience that accepts its premises; induction exhibits the universal as implicit in the clearly known particular.

71a9. Again, the persuasion exerted by rhetorical arguments is in principle the same, since they use either example (a kind of induction) or enthymeme (a form of syllogism).

As the Philosopher says in Metaphysics 1.980b26: the human race lives by art and reasonings. In this statement, the Philosopher seems to touch on that property whereby man differs from the other animals. For the other animals are prompted to their acts by a natural impulse, but man is directed in his actions by a judgment of reason. And this is the reason why there are various arts devoted to the easy and orderly performance of human acts. For an art seems to be nothing more than a definite and fixed procedure established by reason, by which human acts reach their due end through appropriate means. Now, reason is not only able to direct the acts of the lower powers, but is also director of its own act, for what is peculiar to the intellective part of man is its ability to reflect upon itself; for the intellect knows itself. In like manner, reason is able to reason about its own act. Therefore, just as the art of building or carpentry, through which man is enabled to perform manual acts in an easy and orderly manner, arose from the fact that reason reasoned about manual acts, so in like manner an art is needed to direct the act of reasoning, so that by it a man might proceed in an orderly and easy manner and without error when performing the act of reasoning.

And this art is logic, or the science of reason. And it concerns reason not only because it is according to reason, for that is common to all arts, but also because it is concerned with the very act of reasoning as its proper matter.

Therefore, it seems to be the art of the arts, because it directs us in the act of reasoning, from which all arts proceed. Consequently, one should view the parts of logic according to the diversity among the acts of reason.

Now, there are three acts of the reason, the first two of which belong to reason regarded as an intellect. One action of the intellect is the understanding of indivisible or uncomplex things, and according to this action it conceives what a thing is. This operation is called by some the “informing of the intellect,” or “representing by means of the intellect.” To this operation of the reason is ordered the teaching which Aristotle hands down in the book of Categories.

The second operation of the intellect is its act of combining or dividing, in which the true or the false are for the first time present. And this act of reason is the subject of the teaching which Aristotle hands down in the book entitled On Interpretation.

But the third act of the reason is concerned with that which is peculiar to reason, namely, to advance from one thing to another in such a way that through that which is known a man comes to a knowledge of the unknown. And this act is considered in the remaining books of logic.

It should be noted that the acts of reason are in a certain sense not unlike the acts of nature. Thus, art imitates nature as far as it is able. Now, in the acts of nature we observe a threefold diversity.

For in some of them nature acts from necessity, namely, in such a way that it cannot fail;

in others, nature acts so as to succeed for the most part, although now and then it fails in its act.

Hence in this latter case there must be a twofold act:

one which succeeds in the majority of cases, as when from seed is generated a perfect animal;

the other when nature fails in regard to what is appropriate to it, as when from seed something monstrous is generated owing to a defect in some principle.

These three are found also in the acts of the reason.

For there is one process of reason which induces necessity, in which it is not possible to fall short of the truth. By such a process of reasoning, the certainty of science is acquired.

Again, there is a process of reason in which something true in most cases is concluded, but without producing necessity.

But the third process of reason is that in which reason fails to reach a truth because some principle which should have been observed in reasoning was defective.

Now, the part of logic which is devoted to the first process is called the judicative part, because it leads to judgments possessed of the certitude of science. And because a certain and sure judgment touching effects cannot be obtained except by analyzing them into their first principles, this part is called analytical, or resolvent. Now, the certitude obtained by such an analysis of a judgment is derived either from the mere form of the syllogism—and to this is ordered the book of the Prior Analytics, which treats of the syllogism as such—or from the matter along with the form, because the propositions employed are essential and necessaryCf. below, L. 10 & 13.—and to this is ordered the book of the Posterior Analytics, which is concerned with the demonstrative syllogism.

To the second process of reason, another part of logic, called investigative, is devoted. For investigation is not always accompanied by certitude. Hence in order to have certitude, a judgment must be formed, bearing on that which has been investigated. But just as in the works of nature, which succeed in the majority of cases, certain levels are achieved—because the stronger the power of nature, the more rarely does it fail to achieve its effect—so too in that process of reason which is not accompanied by complete certitude, certain levels are found according to how close one approaches to complete certitude.

Although science is not obtained by this process of reason, nevertheless belief or opinion is sometimes achieved (on account of the provability of the propositions one starts with), because reason leans completely to one side of a contradiction, but with trepidation about the other side. The Topics, or dialectics, is devoted to this. For the dialectical syllogism which Aristotle treats in the book of Topics proceeds from premises which are provable.

At times, however, belief or opinion is not altogether achieved, but suspicion is, because reason does not lean to one side of a contradiction unreservedly, although it is inclined more to one side than to the other. To this the Rhetoric is devoted.

At other times, a mere fancy inclines one to one side of a contradiction because of some representation, much as a man turns in disgust from certain food if it is described to him in terms of something disgusting. And to this is ordained the Poetics. For the poet’s task is to lead us to something virtuous by some excellent description.

And all these pertain to the philosophy of reason, for it belongs to reason to pass from one thing to another.

The third process of reasoning is served by that part of logic which is called sophistry, which Aristotle treats in the book On Sophistical Refutations.

Leaving aside the other parts of logic, we shall fix our attention on the judicative part as it is presented in the book of Posterior Analytics,

which is divided into two parts.

In the first, he shows the need for the demonstrative syllogism, with which this book is concerned.

In the second part he comes to a decision concerning that syllogism: we suppose ourselves (71b8; [28]).

Now the need for anything directed to an end is caused by that end. But the end of the demonstrative syllogism is the attainment of science. Hence if science could not be achieved by syllogizing or arguing, there would be no need for the demonstrative syllogism.

Plato, as a matter of fact, held that science in us is not the result of a syllogism but of an impression upon our minds of ideal forms from which, he said, are also derived the natural forms in natural things, which he supposed were participations of forms separated from matter. From this it followed that natural agents were not the causes of forms in natural things, but merely prepared the matter for participating in the separated forms. In like fashion, he postulated that science in us is not caused by study and training, but only that obstacles are removed, and man is brought to recall things which he naturally understands in virtue of an imprint of separated forms.

But Aristotle’s view is opposed to this on two counts.

For he maintains that natural forms are made actual by forms present in matter, that is, by the forms of natural agents.

He further maintains that science is made actual in us by other knowledge already existing in us. This means that it is formed in us through a syllogism or some type of argument. For in arguing we proceed from one thing into another.

Therefore, in order to show the need for demonstrative syllogism, Aristotle begins by stating that some of our knowledge is acquired from knowledge already existing.

Therefore, he does two things:

first, he states his thesis;

second, he explains the character of prior knowledge, at: the preexistent knowledge (71a11; [13]).

Concerning the first he does two things.

First (71a1), he asserts a universal proposition containing his thesis, namely, that the production of knowledge in us is caused from knowledge already existing. Hence he says: every doctrine and every discipline. He does not say, “all knowledge,” because not all knowledge depends on previous knowledge (for that would involve an infinite process), but the acquisition of every discipline comes from knowledge already possessed.

For the names “doctrine” and “discipline” pertain to the learning process, doctrine being the action exerted by the one who makes us know, and discipline the reception of knowledge from another.

Furthermore, “doctrine” and “discipline” are not taken here as pertaining only to the acquisition of scientific knowledge, but to the acquiring of any knowledge. That this is so is evident because he explains the proposition even in regard to dialectical and rhetorical disputations, neither of which engenders science. Hence this is another reason why he did not say, from preexistent "science" or "intuition," but knowledge universally.

However, he does add intellectual, in order to preclude knowledge acquired by sense or imagination. For reason alone proceeds from one thing into another.

Then, at: the mathematical sciences (71a3), he employs induction to prove his thesis.

First, he does this in regard to those demonstrations in which scientific knowledge is acquired. Of these, the best are the mathematical sciences, because of their most certain manner of demonstrating. After them come the other arts, because some manner of demonstrating is found in all of them; otherwise, they would not be sciences.

Second, at and so are the two forms (71a4), he proves the same thing in regard to disputative (that is, dialectical) arguments, because they employ syllogism and induction, in each of which the process starts from something already known. For in a syllogism, the knowledge of some universal conclusion is obtained from other universals already known; in induction, however, a universal is concluded from singulars made known in sense-perception.

Third, at again, the persuasion (71a9), he manifests the same thing in rhetorical arguments, in which persuasion is produced through an enthymeme or example, but not through a syllogism or complete induction because of the uncertainty attending the matters discussed—namely, the individual acts of men in which universal propositions cannot be truthfully assumed. Therefore, in place of a syllogism, in which there must be something universal, an enthymeme is employed, in which it is not necessary to have something universal. Similarly, in place of induction, in which a universal is concluded, an example is employed in which one goes from the singular to the singular, and not to the universal. Thus it is clear that just as the enthymeme is an abridged syllogism, so an example is an incomplete induction. Therefore, if in the case of the syllogism and induction one proceeds from knowledge already existing, the same must be granted in the case of the enthymeme and example.

71a11. The preexistent knowledge required is of two kinds. In some cases admission of the fact must be assumed, in others comprehension of the meaning of the term used, and sometimes both assumptions are essential. Thus, we assume that every predicate can be either truly affirmed or truly denied of any subject, and that “triangle” means so and so. As regards “unit,” we have to make the double assumption of the meaning of the word and the existence of the thing. The reason is that these several objects are not equally obvious to us.

71a17. Recognition of a truth may in some cases contain as factors both previous knowledge and also knowledge acquired simultaneously with that recognition—knowledge, this latter, of the particulars actually falling under the universal and therein already virtually known. For example, the student knew beforehand that the angles of every triangle are equal to two right angles, but it was only at the actual moment at which he was being led on to recognize this as true in the instance before him that he came to know “this figure inscribed in the semicircle” to be a triangle. For some things (namely, the singulars finally reached which are not predicable of anything else as subject) are only learnt in this way; that is, there is here no recognition through a middle of a minor term as subject to a major.

13. The preexistent knowledge required. After showing that every discipline is developed from knowledge already existing, the Philosopher shows what is the extent of this preexisting knowledge (71a11).

Concerning this he does two things.

First, he determines the extent of preexisting knowledge in regard to the things that must be known in order to attain knowledge of the conclusion, of which scientific knowledge is sought.

Second, he determines the extent of preexisting knowledge of the conclusion, of which scientific knowledge is sought through demonstration, at: before he was led on to recognition (71a24; [22]).

Now two things are included in preexisting knowledge, namely, the knowledge and the order of the knowledge.

First, therefore, he determines the extent of preexisting knowledge so far as the knowledge itself is concerned.

Second, he determines the extent of preexisting knowledge so far as the order of the knowledge is concerned: recognition of a truth (71a17; [20]).

14. In regard to the first, it should be noted that the object of which scientific knowledge is sought through demonstration is some conclusion in which a proper attribute is predicated of some subject, which conclusion is inferred from the principles. And because the knowledge of simple things precedes the knowledge of compound things, it is necessary that the subject and the proper attribute be somehow known before knowledge of the conclusion is obtained. In like manner it is required that the principle be known from which the conclusion is inferred, for the conclusion is made known from a knowledge of the principle.

15. Now, the extent of preexistent knowledge of these three items, that is, of the principle, of the subject, and of the proper attribute, is limited to knowing two things about them, namely, that each is and what each is.

But, as stated in Metaphysics 7, complex things are not defined. For there is no definition of white man, much less of an enunciation (proposition). Hence since a principle is an enunciation, there cannot be preexisting knowledge of what it is but only of the fact that it is true.

But in regard to the proper attribute, it is possible to know what it is, because, as is pointed out in the same book, accidents do have some sort of definition. Now, the being of a proper attribute and of any accident is being in a subject; and this fact is concluded by the demonstration. Consequently, it is not known beforehand that the proper attribute exists, but only what it is.

The subject, too, has a definition; moreover, its being does not depend on the proper attribute—rather its own being is known before one knows the proper attribute to be in it. Consequently, it is necessary to know both what the subject is and that it is, especially since the medium of demonstration is taken from the definition of the subject of the proper attribute.

16. This, therefore, is why the Philosopher says that the preexistent knowledge required is of two kinds (71a11); because two items are known beforehand concerning things of which we have preexisting knowledge, namely, that it is and what it is. There are some things concerning which it is necessary first to know that they are, such as principles, concerning which he then gives examples, citing as one example the first of all principles, namely that there is true affirmation or negation about everything.

Again, there are other things, namely, proper attributes, concerning which it is necessary to know what is said to be predicated, that is, what is signified by their name. And he does not say unqualifiedly, "what it is", but "what is said to be predicated", because one cannot properly know of something what it is before it is known that it is. (For there are no definitions of non-beings.) Hence the question “whether it is” precedes the question “what it is.” But whether a thing is cannot be shown unless it is known beforehand what is signified by its name. On this account the Philosopher teaches in Metaphysics 4 that in disputing against those who deny principles one must begin with the meanings of names.

An example of this is “triangle,” concerning which one must know beforehand that its name signifies such and such, namely, what is contained in its definition.

But since accidents are referred to their subjects in a definite order, it is not impossible for something which is an accident in relation to one thing to be a subject in relation to something else: for example, a surface is an accident in relation to a bodily substance, but in relation to color it is the first subject. However, that which is a subject in such a way as never to be an accident of anything else is a substance. Hence in those sciences whose subject is a substance, that which is the subject can never be a proper attribute, as in first philosophy and in natural science, which treats of mobile being. But in those sciences which bear upon accidents, nothing prevents a same thing from being taken as a subject in reference to one proper attribute, and as an attribute in reference to a more basic subject. Nevertheless, this must not develop into an infinite process, for one must arrive at something which is first in that science and which is taken as a subject in such a way that it is never taken as a proper attribute, as is clear in the mathematical sciences, which treat of continuous or discrete quantity. For in these sciences those things are postulated which are first in the genus of quantity; for example, unit and line and surface and the like. Once these are postulated, certain other things are sought through demonstration, such as the equilateral triangle and the square and so on in geometry. In these cases the demonstrations are said to be, as it were, operational, as when it is required to construct an equilateral triangle on a given straight line. But once it has been constructed, certain proper attributes are proved about it; for example, that its angles are equal, or something of that sort. It is clear, therefore, that in the first type of demonstration ‘triangle’ behaves as a proper attribute, and in the second type as a subject. Hence the Philosopher is using ‘triangle’ as a proper attribute and not as a subject when he says by way of example, we must assume that triangle means so and so (71a14).

He says, furthermore, that there are certain things about which we must know beforehand both what each is and whether it is. And he uses the example of ‘one,’ which is the principle in every genus of quantity. For although it is somehow an accident in reference to substance, yet in the mathematical sciences, which treat of quantity, it cannot be taken as a proper attribute but only as a subject, since in this genus [quantity] it has nothing prior to it.

The reason for this difference is shown by the fact that the manner in which the aforesaid, namely, principle, proper attribute and subject, are manifested is not the same. For the way in which they are known is not the same: for principles are known through the act of composing and dividing, but subject and proper attribute by the act of apprehending the essence. And this, too, does not belong in similar fashion to a subject and to a proper attribute, since a subject is defined absolutely, for nothing outside its essence is mentioned in its definition; but a proper attribute is defined with dependence on the subject which is mentioned in its definition. Therefore, since they are not known in the same way, it is not surprising if they are not foreknown in the same way.

Then, at: recognition of a truth (71a17), he determines the extent of foreknowledge on the part of the order which foreknowing implies.

For something is prior to another both in the order of time and in the order of nature. And this twofold order must be considered in regard to preexistent knowing. For something is known before something else in the sense of being known prior in time. Concerning such things he says that someone could know certain things by knowing them prior to the time when he knows the things to which they are said to be foreknown. But certain others are known at one and the same time, although one is prior by nature to the other.

Concerning these he says that one acquires a knowledge of some of these foreknown things at the same time that knowledge of the things to which they are foreknown is acquired.

He indicates what these are when he adds that they are the things contained under certain universals of which we have knowledge, that is, of which it is known that they are contained under such universals. Then he clarifies this with an example. For since two propositions are needed for inferring a conclusion, namely, a major and a minor; when the major proposition is known, the conclusion is not yet known. Therefore, the major proposition is known before the conclusion not only in nature but in time. Further, if in the minor proposition something is introduced or employed which is contained under the universal proposition which is the major, but it is not evident that it is contained under this universal, then a knowledge of the conclusion is not yet possessed, because the truth of the minor proposition will not yet be certain. But if in the minor proposition a term is taken about which it is clear that it is contained under the universal in the major proposition, the truth of the minor proposition is clear, because that which is taken under the universal shares in the same knowledge, and so the knowledge of the conclusion is had at once. Thus, suppose that someone should begin to demonstrate by stating that every triangle has three angles equal to two right angles. When this is known, the knowledge of the conclusion is not yet known. But when it is later assumed that this figure inscribed in a semicircle is a triangle, he knows at once that it has three angles equal to two right angles. However, if it were not clear that this figure inscribed in the semicircle is a triangle, the conclusion would not be known as soon as the minor was stated; rather, it would be necessary to search for a middle through which to demonstrate that this figure is a triangle.

In giving this example of things which are known at a time prior to the conclusion the Philosopher says (71a19) that a person obtaining a knowledge of the conclusion through demonstration foreknew this proposition even according to time, namely, that every triangle has the equal of two right angles. But inducing this assumption, namely, that this figure in the semicircle is a triangle, he knew the conclusion at the same time, because this induction shares in the evidence of the universal under which it is contained, so that there is no need to search for another middle.

He adds (71a21), therefore, that some things are only learnt in this way, that is, learnt in virtue of themselves, so that it is not necessary to learn them through some other middle which is the ultimate reached by analysis in which the mediate is reduced to the immediate. Or it can be read in such a way that the ultimate, that is, the extreme, which is subsumed under the universal middle does not need a further middle to show that it is contained under that universal. And he manifests what those things are which always share the knowledge of their universal, saying that they are the singulars, which are not predicated of any subject, since no middle can be found between singulars and their species.

71a24. Before he was led on to recognition or before he actually drew a conclusion, we should perhaps say that in a manner he knew, in a manner not. If he did not in an unqualified sense of the term know the existence of this triangle, how could he know without qualification that its angles were equal to two right angles? No: clearly he knows not without qualification but only in the sense that he knows universally.

71a29. If this distinction is not drawn, we are faced with the dilemma in the Meno: either a man will learn nothing or what he already knows.

71a30. For we cannot accept the solution which some people offer. A man is asked, ‘Do you, or do you not, know that every pair is even?’ He says he does know it. The questioner then produces a particular pair, of the existence, and so a fortiori of the evenness, of which he was unaware. The solution which some people offer is to assert that they do not know that every pair is even, but only that everything which they know to be a pair is even.

71a34. Yet what they know to be even is that of which they have demonstrated evenness, that is, what they made the subject of their premise, viz. not merely every triangle or number which they know to be such, but any and every number or triangle without reservation. For no premise is ever couched in the form ‘every number which you know to be such’, or ‘every rectilinear figure which you know to be such’: the predicate is always construed as applicable to any and every instance of the thing.

71b5. On the other hand, I imagine there is nothing to prevent a man in one sense knowing what he is learning, in another not knowing it. The strange thing would be, not if in some sense he knew what he was learning, but if he were to know it in that precise sense and manner in which he was learning it.

Before he was led on to recognition. Having shown the manner in which certain other things must be known before knowledge of the conclusion is obtained, the Philosopher now wishes to show how we know even the conclusion beforehand, that is, before knowledge of it is obtained through a syllogism or induction.

Concerning this he does two things:

First (71a24), he establishes the truth of the fact, saying that before an induction or syllogism is formed to beget knowledge of a conclusion, that conclusion is somehow known and somehow not known: for, absolutely speaking, it is not known; but in a qualified sense, it is known. Thus, if the conclusion that a triangle has three angles equal to two right angles has to be proved, the one who obtains science of this fact through demonstration already knew it in some way before it was demonstrated; although absolutely speaking, he did not know it. Hence in one sense he already knew it, but in the full sense he did not.

And the reason is that, as has been pointed out, the principles of the conclusion must be known beforehand. Now, the principles in demonstrative matters are to the conclusion as efficient causes in natural things are to their effects; hence in Physics II the propositions of a syllogism are set in the genus of efficient cause. But an effect, before it is actually produced, preexists virtually in its efficient causes but not actually, which is to exist absolutely. In like manner, before it is drawn out of its demonstrative principles, the conclusion is preknown virtually, although not actually, in its self-evident principles. For that is the way it preexists in them. And so it is clear that it is not preknown in the full sense, but in some sense.

Second, at: if this distinction (71a29), in virtue of this established fact, he settles a doubt which Plato maintained in the book Meno, which gets its title from the name of his disciple.

The doubt is presented in the following manner: A person utterly ignorant of the art of geometry is questioned in an orderly way concerning the principles known in themselves from which a geometric conclusion is concluded. By starting with principles that are known in themselves, to each of which this person ignorant of geometry gives a true answer, and leading him thus by questions to the conclusion, he gives the true answer step by step. From this, therefore, he would have it that even those who seem to be ignorant of certain arts really have a knowledge of them before being instructed in them.

And so it follows that either a man learns nothing or he learns what he already knew.

In dealing with this problem he does four things.

First, he suggests that it cannot be settled unless we grant the truth established above, namely, that the conclusion which a person learns through demonstration or induction was already known, not absolutely, but as it was virtually known in its principles concerning which a person ignorant of a science can give true answers. However, according to Plato’s theory the conclusion was preknown absolutely, so that no one learns afresh but is led to recall by some rational process of deduction. This is similar to Anaxagoras’ position on natural forms, namely, that before they are generated, they already preexisted in the matter absolutely, whereas Aristotle says that they preexist in potency and not absolutely.

Second, at: for we cannot accept the solution (71a30), he shows that the way some have answered Plato’s problem is false, namely, by saying that a conclusion is not in any sense known before it is demonstrated or learned by some method or other. For they might face the following objection based on Plato’s problem: If an unlearned person were asked by someone, "Do you know that every duo (pair) is an even number?" and, if upon answering that he does know this, he were presented with a duo which the person interrogated did not know existed, for example, the duo which is one third of six, the conclusion would be that he knew one third of six to be an even number, a fact which had not been known by him but which he learned through the demonstration proposed to him. And so it seems to follow that he either did not freshly learn this or that he learned what he already knew. To avoid this dilemma, they would answer that the person who was questioned and who answered that he knew every duo to be an equal number did not say that he knew every duo <to be even>; absolutely, but those he knew to be duos. Hence, since that duo which was proposed was utterly unknown to him, he did not in any sense know that this duo was an even number. And so it follows that when one knows the principles, the conclusion is not in any sense preknown, either absolutely or in a qualified sense.

Third, at: yet what they know to be even (71a34), he refutes this solution in the following way: That is known, concerning which a demonstration is had, or concerning which a demonstration is for the first time received. And this is said on account of those learners who begin to know scientifically. But learners do not obtain a demonstration touching every duo they happen to know but every duo absolutely; and the same applies to every number or every triangle. Therefore, it is not true that he knows something about every number which he knows to be a number, or of every duo which he knows to be a duo, but he knows it about every one absolutely.

And that he knows it not only of every number he happens to know is a number, but of every number absolutely, is proved, at: for no premise is ever couched (71b4), on the ground that the conclusion agrees with the premises in its terms. For the subject and predicate of the conclusion are the major and minor extremes in the premises. But in the premises no proposition concerning number or straight line is stated with the addition, "which you know", but it is stated of all without qualification. Neither, therefore, is the conclusion of the demonstration asserted with the aforesaid qualification, but it is asserted of all without reservation.

Fourth, at: on the other hand, I imagine (71b5), he presents the true solution of the problem under discussion in terms of the truth already established, saying that there is nothing to prevent a person from somehow knowing and somehow not knowing a fact before he learns it. For it is not a paradox if one somehow already knows what he learns, but it would be, if he already knew it in the same way that he knows it when he has learned it.

For learning is, properly speaking, the generation of science in someone. But that which is generated was not, prior to its generation, a being absolutely, but somehow a being and somehow non-being: for it was a being in potency, although actually non-being. And this is what being generated consists in, namely, in being converted from potency to act. In like fashion, that which a person learns was not previously known absolutely, as Plato preferred; but neither was it absolutely unknown, as they maintained whose answer was refuted above. Rather it was known in potency, that is, virtually, in the preknown universal principles; however, it was not actually known in the sense of specific knowledge. And this is what learning consists in, namely, in being brought from potential or virtual or universal knowledge to specific and actual knowledge.

71b9. We suppose ourselves to possess unqualified scientific knowledge of a thing, as opposed to knowing it in the accidental way in which the sophist knows,

71b10. when we think that we know the cause on which the fact depends, as the cause of that fact and of no other, and, further, that the fact could not be other than it is.

71b12. Now, that scientific knowing is something of this sort is evident—witness both those who falsely claim it and those who actually possess it, since the former merely imagine themselves to be, while the latter are also actually, in the condition described.

71b15. Consequently the proper object of unqualified scientific knowledge is something which cannot be other than it is.

71b16. There may be another manner of knowing as well—that will be discussed later.

71b17. What I now assert is that at all events we do know by demonstration. By demonstration I mean a syllogism productive of scientific knowledge, a syllogism, that is, the grasp of which is eo ipso such knowledge.

71b19. Assuming then that my thesis as to the nature of scientific knowing is correct, the premises of demonstrated knowledge must be true, primary, immediate, better known than and prior to the conclusion, which is further related to them as effect to cause.

71b22. Unless these conditions are satisfied, the basic truths will not be ‘appropriate’ to the conclusion.

71b23. Syllogism there may indeed be without these conditions, but such syllogism, not being productive of scientific knowledge, will not be demonstration.

71b25. The premises must be true: for that which is non-existent cannot be known—we cannot know, e.g., that the diagonal of a square is commensurate with its side.

71b26. The premises must be primary and indemonstrable; otherwise they will require demonstration in order to be known, since to have knowledge, if it be not accidental knowledge, of things which are demonstrable, means precisely to have a demonstration of them.

71b29. The premises must be the causes of the conclusion, better known than it, and prior to it; its causes, since we possess scientific knowledge of a thing only when we know its cause; prior, in order to be causes; antecedently known, this antecedent knowledge being not our mere understanding of the meaning, but knowledge of the fact as well. Now, ‘prior’ and ‘better known’ are ambiguous terms, for there is a difference between what is prior and better known in the order of being and what is prior and better known to man. I mean that objects nearer to sense are prior and better known to man; objects without qualification prior and better known are those further from sense. Now, the most universal causes are furthest from sense and particular causes are nearest to sense, and they are thus exactly opposed to one another.

72a5. In saying that the premises of demonstrated knowledge must be primary, I mean that they must be the ‘appropriate’ basic truths, for I identify primary premise and basic truth.

We suppose ourselves to possess. After indicating the need for the demonstrative syllogism, the Philosopher now begins to settle questions concerning the demonstrative syllogism itself.

And his treatment is divided into two parts.

In the first he determines concerning the demonstrative syllogism.

In the second, he determines concerning the middle from which the demonstrative syllogism proceeds in book 2, at the kinds of question we ask (90b21; [407]).

The first is divided into two parts.

In the first he determines concerning the demonstrative syllogism in itself.

In the second he compares demonstration to demonstration, at: since demonstrations may be either commensurately universal or particular (85a12; [323]).

The first is divided into two parts.

In the first he determines concerning the demonstrative syllogism.

In the second he shows that one does not proceed to infinity in demonstrations, at: every syllogism is effected by means of three terms (81b10; [255]).

The first is divided into two parts.

In the first he determines concerning the demonstrative syllogism through which we acquire science.

In the second he shows how we also acquire ignorance through a syllogism, at: ignorance, defined not as the negation of knowledge (79b23; [221]).

Concerning the first he does three things.

First, he determines concerning the demonstrative syllogism by showing what it is.

Second, he determines concerning the matter of the demonstrative syllogism, pointing out the nature and character of the matter out of which it is formed, at: since the object of pure scientific knowledge (73a21; [79]).

Third, he determines concerning the form of the syllogism, pointing out the figure in which it is chiefly presented, at: of all the figures the most scientific (79a17; [213]).

Concerning the first he does three things.

First, he shows what the demonstrative syllogism is.

Second, he clarifies certain terms that appear in the definition of the demonstrative syllogism, at: a basic truth in a demonstration (72a8; [44]).

Third, he excludes certain errors that could arise from his doctrine on the nature of demonstration, at: some hold that, owing to the necessity of knowing (72b5; [60]).

In regard to the first it should be noted that in all things which exist for an end, the definition which employs a final cause is both the explanation of the definition which expresses the material cause, and is the middle which proves the latter. For the reason why a house should made of stone and wood is that it is a structure protecting us from the cold and heat. Along these lines, therefore, he gives two definitions demonstration, one of which is expressed in terms of the end of demonstration, which is to know in a scientific manner. And from this one is concluded the other, which is drawn from the matter of a demonstration.

Hence he does three things in regard to this.

First, he defines what it is to know in a scientific manner.

Second, he defines demonstration in terms of its end, which is to know in a scientific manner, at: what I now assert is that at all events (71b18; [36]).

Third, from these two definitions he concludes to that definition of demonstration which is expressed in terms of the matter of demonstration, at: assuming then that my thesis (71b19; [37]).

Concerning the first he does five things.

First (71b9), he determines what the scientific knowing, which he intends to define, bears upon.

And in regard to this it should be recognized that we are said to know something in a scientific manner absolutely, when we know it in itself. On the other hand, we are said to know something in a scientific manner qualifiedly, when we know it in something else in which it exists either as a part in a whole (as we are said to know a wall through knowing the house), or as an accident in its subject (as in knowing Coriscus we are said to know who is coming toward us), or as an effect in its cause (as in the example given earlier, we know the conclusion in the principles), or indeed in any fashion similar to these. To know in these ways is to know incidentally, because we are said to know that which is somehow accidental to what is known of itself. However, what the Philosopher intends to define here is scientific knowing in the strict sense and not according to an accident. For this form of knowing is sophistical, since Sophists use a form of argument typified by the following: "I know Coriscus; Coriscus is coming toward me: therefore, I know the person coming toward me".

Then, at: when we think that we know the cause (71b10), he presents the definition of scientific knowing in the strict sense. Apropos of this it should be noted that to know something scientifically is to know it completely, which means to apprehend its truth perfectly. For the principles of a thing’s being are the same as those of its truth, as is stated in Metaphysics II. Therefore, the scientific knower, if he is to know perfectly, must know the cause of the thing known. But if he were to know the cause by itself, he would not yet know the effect actually—which would be to know it absolutely—but only virtually, which is the same as knowing in a qualified sense and incidentally. Consequently, one who knows scientifically in the full sense must know the application of the cause to the effect. Again, because science is also sure and certain knowledge of a thing, whereas a thing that could be otherwise cannot be known with certainty, it is further required that what is scientifically known could not be otherwise. To repeat: because science is perfect knowledge, he says, when we think that we know the cause; but because the knowledge through which we know scientifically in the full sense is actual, he adds, as the cause of that fact. Finally, because it is certain knowledge, he adds, and that the fact could not be other than it is (71b11).

Third, at: now, that scientific knowing (71b12), he explains the definition he laid down, appealing to the fact that both those who know scientifically and those who do not know in that way but believe that they do, take scientific knowing to be as above described. For those who do not know in a scientific manner but believe that they do, are convinced that they know in the manner described, whereas those who know in a scientific manner do know in the manner described.

Furthermore, this is the proper way to manifest a definition. For a definition is the notion which a name signifies, as it is stated in Metaphysics IV. But the signification of a name must be based on what is generally meant by those who employ the name. Hence it is stated in Topics II that names must be used as the majority of people use them. Again, careful consideration would indicate that this explanation seems rather to show what the name signifies than to signify something directly. For he does not explain science, concerning which a definition could, properly speaking, be formed, since it is a species of some genus; rather he explains scientific knowing. Hence at the very beginning he said, we suppose ourselves to possess unqualified scientific knowledge (71b8; [28]) and not that scientific knowledge is such and such.

Fourth, at: consequently the proper object (71b15), he draws a corollary from the definition, namely, that that of which there is unqualified scientific knowledge must be something necessary, that is, which cannot be otherwise.

Fifth, at: there may be another manner of knowing (71b16), he answers a tacit question, namely, whether there is another way of knowing scientifically in addition to the way described here. And he promises to discuss this later. For it is possible to know scientifically through an effect, as will be explained below [192]. Furthermore, there is a sense in which we are said to know scientifically the indemonstrable principles to which no cause is ascribed. But the proper and perfect manner of knowing scientifically is the one we have described.

Then, at: what I now assert is that at all events (71b17) he defines the demonstrative syllogism in terms of its end, which is to know in a scientific manner.

In regard to this he does three things.

First, he asserts that scientific knowing is the end of a demonstrative syllogism or is its effect, since to know scientifically seems to be nothing less than to understand the truth of a conclusion through demonstration.

Second, at: by demonstration I mean a syllogism (71b18), he defines demonstration in terms of the end, saying that a demonstration is a sciential syllogism, that is, producing scientific knowledge.

Third (71b18), he explains, sciential, saying that a sciential syllogism is one according to which we know scientifically insofar as we understand it, and not in the sense of a syllogism yielding knowledge to be put to use.

Then, at: assuming then that my thesis (71b19) he concludes from the foregoing a definition of the demonstrative syllogism that is based on its matter.

Concerning this he does two things.

First, he concludes it.

Second, he clarifies it, at: the premises must be true (71b24; [40]).

Concerning the first he does three things.

First (71b20), he sets forth the consequent in which the material definition of demonstration is concluded from the premises laid down above. And he says that if scientific knowing is what we have stated it to be, namely, knowing the cause of a thing, etc., then it is necessary that demonstrative science, that is, science acquired through demonstration, proceed from propositions which are true, first, and immediate, that is, not demonstrated by some other middle but clear in virtue of themselves (they are called immediate, inasmuch as they do not have a middle demonstrating them, but first, in relation to other propositions which are proved through them); and which, furthermore, are better known than, prior to, and causes of, the conclusion.

Second, at: unless these conditions are satisfied (71b22), he justifies himself for not adding another element which, it might seem, should be added, namely, that demonstration proceeds from proper principles. But he says that this is understood in virtue of the elements he did state. For since the propositions of a demonstration are causes of the conclusion, they must be its proper principles. For effects require proportionate causes.

Third, at: syllogism there may indeed (71b23), he manifests the necessity of the aforesaid consequence, saying that although a syllogism does not require these conditions in the premises from which it concludes, a demonstration does require them, for otherwise it would not produce science.

Then, at: the premises must be true (71b25) he explains this definition as well as the subsequent statement that unless these conditions are fulfilled in a demonstration it cannot beget science.

First, therefore, he shows that a demonstration must proceed from true principles in order to beget science, because there cannot be scientific knowledge of that which does not exist, for example, that the diagonal is symmetrical, that is, commensurable with the side of the square. For those quantities are said to be incommensurable which lack a common measuring unit. These are quantities whose ratio to one another cannot be expressed in terms of one number to another number. That this is the case with the diagonal of a square and its side is plain from Euclid’s sixteenth proposition. Now, what is not true does not exist, for to be and to be true are convertible. Therefore, anything scientifically known must be true. Consequently, the conclusion of a demonstration which does beget scientific knowing must be true, and a fortiori its premises. For the true cannot be known in a scientific way from the false, although something true can follow as a conclusion from something false, as he will show later [118].

Second, at: the premises must be primary and indemonstrable (71b26), he shows that the demonstration is composed of first and immediate or indemonstrable principles. For no one can possess scientific knowledge unless he possesses the demonstration of things that can be demonstrated, and I am speaking essentially and not accidentally. He says this because it would be possible to know some conclusion without having a demonstration of the premises, even were they demonstrable; because one would know it through other principles, and this would be accidental. Suppose, therefore, that a demonstrator syllogizes from demonstrable, that is, mediate, premises. Now, he either possesses a demonstration of those premises or he does not. If he does not, then he does not know the premises in a scientific way; nor consequently, the conclusion because of the premises. But if he does possess their demonstration, then, since one may not proceed to infinity in demonstrations, principles immediate and indemonstrable must be reached. And so it is required that demonstration proceed from principles that are immediate either straightway or through middles. Hence it is stated in Topics I that demonstration is composed of first and true statements or of statements made credible by these.

Third, at: the premises must be the causes of the conclusion (71b29), he proves that the propositions of a demonstration are the causes of the conclusion, because we know in a scientific manner when we know the causes. And in virtue of this he shows that they are prior and better known, because every cause is by nature prior and better known than its effect. However, the cause of a demonstrated conclusion must be better known not only with respect to the knowledge of what it is, but also with respect to the knowledge that it is. For in order to demonstrate that there is an eclipse of the sun, it is not enough to know, that the moon is interposed; in addition it is necessary to know that the moon is interposed between the sun and the earth. Again, because prior and better known are taken in two ways, namely, in reference to us and according to nature, he says that the things from which a demonstration, proceeds are prior and better known absolutely and according to nature, and not in reference to us. To elucidate this he says that those things are prior and better known absolutely, which are farthest from sense, as are universals; but the prior and better known in reference to us are nearest to sense, namely, the singulars, which are opposed to universals in the way that the prior and the later are opposite, or in the way that the nearest and the farthest are opposite.

However, it seems that the contrary of this is found in Physics I where it is stated that universals are prior in reference to us and later according to nature.

But it should be said that there [in the Posterior Analytics] he is speaking of the order of singular to universal absolutely; and this order must be taken according to the order of sensitive and intellectual knowledge in us. Now, in us sensitive knowledge is prior to intellectual, because intellectual knowledge in us proceeds from sense. For this reason the singular is prior and better known in relation to us than the universal. But in Physics I he is not speaking of the order of the universal to the singular absolutely but of the order of the more universal to the less universal, for example of animal to man. In this case the more universal is prior and better known in reference to us. For in every instance of generation, that which is in potency is prior in time but is later according to nature; whereas that which is complete in act is prior by nature but later in time. Now, one’s knowledge of a genus is, as it were, potential in comparison to one’s knowledge of the species in which all the essentials of a thing are actually known. Hence, too, in the generation of our science, knowledge of the more common precedes knowledge of the less common.

Again, in the Physics it is stated that it is natural for us to proceed from what is better known to us. Therefore, it seems that a demonstration is composed not of things that are prior absolutely but in reference to us.

But it must be said that here he is speaking according to the fact that what is in the sense is better known in reference to us than what is in the intellect; but there he was speaking according to the fact that what is better known in reference to us is also in the intellect. But demonstrations do not proceed from singulars which are in the sense but only from universals, which are in the intellect.

Or it might be said that in every demonstration one must proceed from things better known to us, provided they are not singulars but universals. For something is made known to us only by that which is more known to us. But sometimes that which is more known in reference to us is also more known absolutely and according to nature, as happens in mathematics where on account of abstraction from matter the demonstrations proceed from formal principles alone. In this case the demonstrations proceed from things which are more known absolutely. But sometimes that which is more known in reference to us is not more known absolutely, as happens in natural sciences where the essences and powers of things are hidden, because they are in matter, but are disclosed to us through the things which appear outwardly. Hence in these sciences the demonstrations are for the most part made through effects which are better known in reference to us but not absolutely. But he is not now speaking of this form of demonstration, but of the first.

Finally, because in his explanation he neglected to point out that demonstration should proceed from proper principles, he hastens to add that this fact is easily ascertainable from what he did say. For from the fact that he stated that demonstration is from things which are first, it follows that it is from proper principles, as he stated above. For first and principle seem to be the same: for that which is first and highest in each genus is the cause of all the things that are after it, as it is stated in Metaphysics 2.

72a7. A basic truth in a demonstration is an immediate proposition.

72a8a. An immediate proposition is one which has no other proposition prior to it.

72a8b. A proposition is either part of an enunciation, that is, it predicates a single attribute of a single subject

72a9. If a proposition is dialectical, it assumes either part indifferently; if it is demonstrative, it lays down one part to the definite exclusion of the other because that part is true.

72a11. The term ‘enunciation’ denotes either part of a contradiction indifferently. A contradiction is an opposition which of its own nature excludes a middle. The part of a contradiction which conjoins a predicate with a subject is an affirmation; the part disjoining them is a negation.

72a14. I call an immediate basic truth of syllogism a ‘thesis’ when, though it is not susceptible of proof by the teacher, yet ignorance of it does not constitute a total bar to progress on the part of the pupil: one which the pupil must know if he is to learn anything whatever is an axiom. I call it an axiom because there are such truths and we give them the name of axioms par excellence.

72a18. If a thesis assumes one part or the other of an enunciation, that is, asserts either the existence or the non-existence of a subject, it is a hypothesis; if it does not so assert, it is a definition. Definition is a ‘thesis’ or a ‘laying something down’, since the arithmetician lays it down that to be a unit is to be quantitatively indivisible; but it is not a hypothesis, for to define what a unit is is not the same as to affirm its existence.

A basic truth in a demonstration. Because the Philosopher had stated above that demonstration is from first and immediate principles, but had not yet identified them, he now sets out to identify them (72a7).

And this is divided into three parts.

In the first part he shows what an immediate proposition is.

In the second part he shows that such propositions must be better known than the conclusion, at: now, since the required ground of our knowledge (72a25; [53]).

In the third part he excludes certain errors which arose from the foregoing, at: some hold that, owing to the necessity (72b5; [60]).

Concerning the first he does two things.

First, he shows what an immediate principle is.

Second, he divides them, at: I call an immediate basic truth (72a15; [49]).

With respect to the first he proceeds this way.

First (72a7), he recall, what has been said above, namely, that a principle of demonstration is an immediate proposition, for he had also stated above that a demonstration is composed of things which are first and immediate.

Second, at: an immediate proposition (72a8a), he defines the immediate proposition.

He says that an immediate proposition is one which has no other one prior to it. The reason underlying this description is clear from what has been said. For it has been said above that demonstration is composed of things that, are prior. Accordingly, whenever a proposition is mediate, that is, has a middle through which the predicate is demonstrated of its subject, it is required that there be prior propositions by which this one is demonstrated. For the predicate of a conclusion is present in the middle previously to being present in the subject; in which, however, the middle is present before the predicate is. Therefore, it follows that that proposition which does not have some other one prior to it is immediate.

Third, at: a proposition is either part (72a8b), he shows what is the nature of the proposition which is mentioned in the definition of an immediate proposition.

Concerning this he does three things:

First, he defines absolutely what a proposition is, saying that it is one or the other part of an enunciation in which one thing is predicated of one thing. For the enunciation has two parts, namely, affirmation and negation. For anyone who syllogizes must propose one or the other of these parts but not both, for this latter procedure is characteristic of one who first raises a question. (Hence it is on this basis that proposition is distinguished from a problem). For just as one and only one thing is concluded in one syllogism, so the proposition which is a principle of the syllogism should be one—and it is one if one thing is stated of one thing. Hence in asserting that it is one of one, he distinguishes the proposition from the enunciation which is said to be of several, whether sundry things are said of one thing or of one thing sundry.

Second, at: if a proposition is dialectical (72a9), he lays down the difference between the dialectical and the demonstrative proposition, saying that whereas the demonstrative proposition takes one definite side of a question, the dialectical takes either side indifferently. For since dialectic begins with the probable, it can lead to each side of a contradiction. Hence when it lays down its propositions, it employs both parts of a contradiction and presents them in the form of a question.Is an animal that walks on its feet a man, or not? But a demonstrative proposition takes one side definitively, because a demonstrator never has any other alternative but to demonstrate the truth. Hence in forming its propositions he always assumes the true side of a contradiction.An animal which walks on two feet is a man, is it not? On this account he does not ask but posits something as known in the demonstration.

Third, at: the term "enunciation" (72a11), he defines the term "enunciation," which appeared in the definition of a proposition, saying that an enunciation embraces both sides of a contradiction, as is clear from what has been said. Then he shows what contradiction is, saying that contradiction is a form of opposition between whose parts there is of itself no middle. For although between privation and possession and between immediate contraries there is no middle in a given subject, nevertheless, absolutely speaking, there is one; for a stone is neither blind nor seeing, and something white is neither even nor odd. Furthermore, whatever immediacy they have in relation to a definite subject is traced to their participation in contradiction, for privation is negation in a definite subject; and of two things that are immediately contrary, one has some of the marks of privation. But contradiction in the full sense lacks a middle in all cases. And this belongs to it of its very nature and not in virtue of something else. Hence he says that of itself it has no medium.

He then explains what the parts of a contradiction are. For contradiction is an opposition of affirmation and negation; hence one of its parts is affirmation, which asserts something of something, and the other is negation, which denies something of something.

Then, at: I call an immediate basic truth of syllogism (72a14), he divides immediate principle.

Concerning this he does two things.

First, he divides.

Second, he subdivides, at: if a thesis assumes one part or the other (72a19; [51]).

He says therefore first, that there are two types of immediate principles of a syllogism: the first is called a "position" [thesis] and is said to be immediate because one does not demonstrate (neither is it required that the student, that is, the one being instructed in the demonstrative science, have it, that is, advert to it or assent to it); the other is called a "dignity" or "maxim," which anyone who is to be instructed must have in his mind and assent to. That there are such principles is clear from Metaphysics 4, where it is proved that one such is the principle that affirmation and negation are not simultaneously true, for no one can believe the contrary of this in his mind even though he should state it orally. To such principles we give the aforesaid name of ‘dignity’ or ‘maxim’ on account of their certainty in manifesting other things.

To clarify this division it should be noted that any proposition whose predicate is included within the notion of its subject is immediate and known in virtue of itself as it stands.

However, in the case of some of these propositions the terms are such that they are understood by everyone, as being and one and those other notions that are characteristic of being precisely as being: for being is the first concept in the intellect. Hence it is necessary that propositions of this kind be held as known in virtue of themselves not only as they stand but also in reference to us. Examples of these are the propositions that ‘It does not occur that the same thing is and is not’ and that ‘The whole is greater than its part’, and others like these. Hence all the sciences take principles of this kind from metaphysics whose task it is to consider being absolutely and the characteristics of being.

On the other hand, there are some immediate propositions whose terms are not known by everyone. Hence, although their predicate may be included in the very notion of their subject, yet because the definition of the subject is not known to everyone, it is not necessary that such propositions be conceded by everyone. Thus the proposition, ‘All right angles are equal’, is in itself a proposition which is immediate and known in virtue of itself, because equality appears in the definition of a right angle. For a right angle is one which a straight line form when it meets another straight line in such a way that the angles on each side are equal. Therefore, such principles are received as being posited or laid down.

There is yet another way, and according to it certain propositions are called ‘suppositions’. For there are some propositions which can be proved only by the principles of some other science; therefore, they must be supposed in the one science, although they are proved by the principles of the other science. Thus the geometer supposes that he can draw one straight line from one point to another, but the philosopher of nature proves it by showing that there is one straight line between any two points.

Then, at: if a thesis assumes one part or the other (72a18) he subdivides a member of the original division, namely, ‘position’, and says that there is one type of position which takes one side of an enunciation, namely, either affirmation or negation. He refers to this type when he says, that is, asserts either the existence or non-existence of a subject. Such a position is called a ‘supposition’ or ‘hypothesis’, because it is accepted as having truth. Another type of position is the one which does not signify existence or non-existence: in this way a definition is a position. For the definition of ‘one’ is laid down in arithmetic as a principle, namely, that one is the quantitatively indivisible. Nevertheless a definition is not called a supposition, for a supposition, strictly speaking, is a statement which signifies the true or the false. Consequently, he adds that the definition of ‘one’, inasmuch as it signifies neither the true nor the false, is not the same as ‘to be one’, which does signify the true or the false.

Now it might be asked how it is that definition is set down as a member of the subdivision of immediate proposition, if a definition is not a proposition signifying either existence or non-existence.

One might answer that in this subdivision he was not subdividing immediate proposition but immediate principle; and the principle of the syllogism can be said to be not only the proposition but also the definition.

Or one might answer that although a definition as such is not an actual proposition, it is one virtually, because once a definition is known, it becomes clear that it is truly predicated of the subject.

72a25. Now, since the required ground of our knowledge—that is, of our conviction—of a fact is the possession of such a syllogism as we call demonstration, and the ground of the syllogism is the facts constituting its premises, we must not only know the primary premises—some if not all of them—beforehand, but know them better than the conclusion.

72a29. For the cause of an attribute’s inherence in a subject always itself inheres in the subject more firmly than that attribute; e.g., the cause of our loving anything is dearer to us than the object of our love.

So since the primary premises are the cause of our knowledge—that is, of our conviction—it follows that we know them better—that is, are more convinced of them—than their consequences, precisely because of our knowledge of the latter is the effect of our knowledge of the premises.

72a32. Now, a man cannot believe in anything more than in the things he knows, unless he has either actual knowledge of it or something better than actual knowledge. But we are faced with this paradox if a student whose belief rests on demonstration has not prior knowledge.

72a36. A man must believe in some, if not in all, of the basic truths more than in the conclusion. Moreover, if a man sets out to acquire the scientific knowledge that comes through demonstration,

72a38. He must not only have a better knowledge of the basic truths and a firmer conviction of them than of the connexion which is being demonstrated: more than this, nothing must be more certain or better known to him than these basic truths in their character as contradicting the fundamental premises which lead to the opposed and erroneous conclusion. For indeed the conviction of pure science must be unshakable.

Now, since the required ground of our knowledge. After showing what immediate principles are, the Philosopher now determines concerning our knowledge of them.

Apropos of this he does two things.

First, he shows that immediate principles are better known than the conclusion.

Second, that the falsity of their contraries ought to be most evident, at: he must not only have a better knowledge (72a38; [59]).

Concerning the first he does three things.

First (72a25), he states his proposition and says that because we give our assent to a thing which has been concluded and we know it scientifically precisely because we have a demonstrative syllogism (and this insofar as we know the demonstrative syllogism in a scientific way), it is necessary not only to know the first principles of the conclusion beforehand, but also to know them better than we know the conclusion.

He adds, either all or some, because some principles require proof in order to be known; so that before they are proved, they are not better known than the conclusion. Thus the fact that an exterior angle of a triangle is equal to its two opposite interior angles is, until proved, as unknown as the fact that a triangle has three angles equal to two right angles. But there are other principles which, once they are posited, are better known than the conclusion. Or, in another way, there are some conclusions which are most evident; for example, those based on sense perception, as that the sun is eclipsed. Hence the principle through which this is proved is not better known absolutely—the principle being that the moon is between the sun and the earth—although it is better known within the reasoning process that goes from cause to effect. Or, in another way, he says this because he had said above that in the order of time certain principles are known before the conclusion, but others are known along with the conclusion at the same moment of time.

Second, at: for the cause of an attribute’s inherence (72a29), he proves his proposition in two ways:

First, with an ostensive argument, thus: that in virtue of which something is so, is itself more so; for example, if we love someone because of someone else, as a master because of his disciple, we love the disciple more. But we know conclusions and give our assent to them because of the principles. Therefore, we know the principles with more conviction and give them stronger assent than the conclusion.

Apropos of this reason it should be noted that a cause is always more noble than its effect. When, therefore, cause and effect have the same name, that name is said principally of the cause rather than of the effect; thus fire is primarily called hot rather than things heated by fire. But sometimes the name of the effect is not attributed to the cause. In that case, although the name the effect has does not belong to the cause, nevertheless, something more noble belongs to it. For example, although the sun does not possess heat, nevertheless, there is in it a certain power which is the principle of heat.

Then, at: now, a man cannot believe in anything more (72a32) he proves the same thing with a principle which leads to an impossibility. He reasons thus: Principles are known prior to the conclusion, as has been shown above; consequently, when the principles are known, the conclusion is not yet known. If, therefore, the principles were not more known than the conclusion, it would follow that a man would know things he does not know either as well as or better than the things he does know. But this is impossible. Therefore, it is also impossible that the principles not be better known than the conclusion.

Phrase by phrase this is explained in the following manner: a man who knows scientifically or even one who knows in a way superior to this, if such there be (he says this, having in mind the person who has the intuition of principles, a state he has not yet explained), cannot give more credence to things he does not know than to things he does know. But this will be the case if one who assents to a conclusion obtained through demonstration did not foreknow, that is, did not know the principles better.

In Greek it is stated more clearly: but no one, whether he has scientific knowledge or that form of knowledge which is better than the scientific (if there be such), can believe anything more firmly than the things he knows.

Third, at: a man must believe in some (72a36), he clarifies what he had said, saying that his statement to the effect that it is more necessary to believe the principles (either all or some) than the conclusion should be understood as referring to a person who is to acquire a discipline through demonstration. For if the conclusion were more known through some other source, such as sense-perception, nothing would preclude the principles not being better known than the conclusion in that case.

Then, at: he must not only have a better knowledge (72a38) he shows that it is not only necessary to know the principles more than the demonstrative conclusion, but nothing should be more certain than the fact that the opposites of the principles are false. And this because the scientific knower must not disbelieve the principles, but assent to them most firmly. But anyone who doubts the falseness of one of two opposites cannot assent firmly to the other, because he will always fear that the opposite one might be true.

Discussion of two errors—exclusion of the first one

72b5. Some hold that, owing to the necessity of knowing the primary premises, there is no scientific knowledge. Others think there is, but that all truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premises.

72b7. The first school, assuming that there is no way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other hand — they say — the series terminates and there are primary premises, yet these are unknowable because incapable of demonstration, which according to them is the only form of knowledge. And since thus one cannot know the primary premises, knowledge of the conclusions which follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premises are true.

72b15. The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.

72b18. Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premises is independent of demonstration. The necessity of this is obvious; for since we must know the prior premises from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.

72b23. Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.

Some hold that, owing to the necessity. After determining about the knowledge of the principles of demonstration, the Philosopher now excludes the errors which have arisen from these determinations.

Concerning this he does three things.

First, he states the errors.

Second, the reasons they erred, at: the first school (72b8; [62]).

Third, he removes the roots of these reasons, at: our own doctrine (72b18; [64]).

He says therefore first (72b5), that two contrary errors have arisen from one of the truths established above. For it has been established above that the principles of demonstration must be known and must be even better known. But the first of these is sufficient for our purpose. For some, basing themselves on this first statement, have come to believe that there is no science of anything, whereas others believe that there is science, even to the extent of believing that there is science of everything through demonstration. But neither of these positions is true and neither follows necessarily from their reasons.

Then, at: the first school (72b7) he presents the reasons why they have fallen into these errors.

And first of all he presents the reason given by those who say that there is no science, and it is this: The principles of demonstration either proceed to infinity or there is a halt somewhere. But if there is a process to infinity, nothing in that process can be taken as being first, because one cannot exhaust an infinite series and reach what is first. Consequently, it is not possible to know what is first. They are correct in thus arguing, for the later things cannot be known unless the prior ones are known. On the other hand, if there is a halt in the principles, then even so, the first things are still not known, if the only way to know scientifically is through demonstration. For first things do not have prior principles through which they are demonstrated. But if the first things are not known, it follows again that the later things are not known in the strict and proper sense, but only on condition that there are principles. For it is not possible for something to be known in virtue of something not known, except on condition that that unknown be a principle. So in either case, whether the principles stop or go on to infinity, it follows that there is no science of anything.

Second, at: the other party agree with them (72b15), he presents the reasoning of those who say that there is science of everything through demonstration, because to them there is science of everything through demonstration, because to their basic premise-the only way to know scientifically is by demonstration—they added another, namely, that one may demonstrate circularly. From these premises it followed that even if a limit is reached in the series of the principles of demonstration, the first principles are still known through demonstration, because, they said, those principles were demonstrated by previous ones. For a circular demonstration is one which is reciprocal, that is, something which was first a principle is later a conclusion, and vice versa.

Then, at: our own doctrine (72b18) he cuts away the false bases of these arguments.

First, their supposition that the only way to know scientifically is by demonstration.

Second, their statement that it is legitimate to demonstrate circularly, at: now, demonstration must be based (72b25; [68]).

He says therefore first (72b18), that not all scientific knowledge is demonstrative, that is, obtained through demonstration, but the scientific knowledge of immediate principles is indemonstrable, that is, not obtained by demonstration.

However, it should be noted that Aristotle is here taking ‘science’ in a wide sense to include any knowledge that is certain, and not in the sense in which science is set off against understanding, according to the dictum that science deals with conclusions and understanding [intuition] with principles.

But that it is necessary for some things to be held as certain without demonstration he proves in the following way: It is necessary that the prior things from which a demonstration proceeds be known in a scientific way. Furthermore, these must be ultimately reduced to something immediate; otherwise one would be forced to admit that there is an actual infinitude of middles between two extremes—in this case between the subject and predicate. Again, one would have to admit that no two extremes could be found between which there would not be an infinitude of middles. But as it is, the middles are such that it is possible to find two things which are immediate. But immediate principles, being prior, must be indemonstrable. Thus it is clear that it is necessary for some things to be scientifically known without demonstration.

Therefore, if someone were to ask how the science of immediate principles is possessed, the answer would be that not only are they known in a scientific manner, but knowledge of them is the source of a science. For one passes from the knowledge of principles to a demonstration of conclusion on which science, properly speaking, bears. But those immediate principles are not made known through an additional middle but through an understanding of their own terms. For as soon as it is known what a whole is and what a part is, it is known that every whole is greater than its part, because in such a proposition, as has been stated above, the predicate is included in the very notion of the subject. And therefore it is reasonable that the knowledge of these principles is the cause of the knowledge of conclusions, because always, that which exists in virtue of itself is the cause of that which exists in virtue of something else.

The second error is excluded by showing that circular demonstration is not acceptable

72b25. Now, demonstration must be based on premises prior to and better known than the conclusion. And the same things cannot simultaneously be both prior and posterior to one another: so circular demonstration is clearly not possible in the unqualified sense of ‘demonstration’,

72b26. but only possible if ‘demonstration’ be extended to include that other method of argument which rests on a distinction between truths prior to us and truths without qualification prior, that is, the method by which induction produces knowledge. But if we accept this extension of its meaning, our definition of unqualified knowledge will prove faulty; for there seem to be two kinds of it. Perhaps, however, the second form of demonstration, that which proceeds from truths better known to us, is not demonstration in the unqualified sense of the term.

72b32. The advocates of circular demonstration are not only faced with the difficulty we have just stated: in addition their theory reduces to the mere statement that if a thing exists, then it does exist—an easy way of proving anything. That this is so can be clearly shown by taking three terms, for to constitute the circle it makes no difference whether many terms or few or even only two are taken. Thus by direct proof, if A is, B must be; if B is, C must be; therefore if A is, C must be. Since then, by the circular proof, if A is, B must be, and if B is, A must be, A may be substituted for C above. Then ‘if B is, A must be’ equals ‘if B is, C must be’, which above gave the conclusion ‘if A is, C must be’: but C and A have been identified. Consequently the upholders of circular demonstration are in the position of saying that if A is, A must be—a simple way of proving anything.

73a6. Moreover, even such circular demonstration is impossible except in the case of attributes that imply one another, viz. ‘peculiar’ properties.

73a7. Now, it has been shown that the positing of one thing—be it one term or one premise—never involves a necessary consequent: two premises constitute the first and smallest foundation for drawing a conclusion at all and therefore a fortiori for the demonstrative syllogism of science. If, then, A is implied in B and C, and B and C are reciprocally implied in one another and in A, it is possible, as has been shown in my writings on the syllogism, to prove all the assumptions on which the original conclusion rested, by circular demonstration in the first figure. But it has also been shown that in the other figures either no conclusion is possible, or at least none which proves both the original premises. Propositions the terms of which are not convertible cannot be circularly demonstrated at all.

73a17. And since convertible terms occur rarely in actual demonstrations, it is clearly frivolous and impossible to say that demonstration is reciprocal and that therefore everything can be demonstrated.

Now, demonstration must be based. After excluding one false basis by showing that not all science depends on demonstration, the Philosopher now excludes another by showing that it is not possible to demonstrate circularly.

To understand this it should be noted that a demonstration is circular when the conclusion and one of the premises (in converted form) of a syllogism are used to prove the other premise. For example, we might form the following syllogism: Every rational mortal animal is risible; Every man is a rational mortal animal: Therefore, every man is risible. Now, if the conclusion were to be used as one principle and the minor in converted form as the other, we would get: Every man is risible; Every rational mortal animal is a man: Therefore, every rational mortal animal is risible—which was the major of the first syllogism.

Accordingly, he presents three arguments to show that it is not possible to demonstrate circularly.

The first of these (72b25) is this: In a circular syllogism the same thing is at once a conclusion and a principle. But a principle of a demonstration is prior to and better known than the conclusion, as has been shown above. Therefore, it follows that a same thing is both prior to and subsequent to one same thing, and also more known and less known. But this is impossible. Therefore, it is impossible to demonstrate circularly.

But someone might say that a same thing can be both prior and subsequent, although not in the same way. For example, this might be prior in reference to us, but that prior absolutely. Thus singulars are prior in reference to us and subsequent absolutely: and conversely for universals. Again, induction makes something known in one way and demonstration in another way. For demonstration proceeds from things that are prior absolutely, but induction from things that are prior in reference to us. Now, if a circular demonstration were so constructed that something is first concluded from things that are absolutely prior, and then from things that are prior in reference to us, it would follow that our doctrine on scientific knowing was not well established. For we stated that to know scientifically is to know the cause of a thing. From this it followed that a demonstration which causes scientific knowledge must proceed from the absolutely prior. But if demonstration were at one time to proceed from the absolutely prior and at another time from things which are prior in reference to us, we would be forced to admit that scientific knowing is not confined to knowing the cause of a thing, but that there is another, namely, that form of knowing which proceeds from what is later. Therefore, one must either admit both or admit that the second form, namely, the demonstration which proceeds from what is better known to us is not a demonstration in the absolute sense.

The aforesaid also reveals why a dialectical syllogism can be circular. For it proceeds from things which are probable. But things are said to be probable if they are better known to the wise or to a great number of persons. Consequently, a dialectical syllogism proceeds from things that are better known to us. However, it happens that a same thing is better known to some and less known to others. Consequently, there is nothing to hinder a dialectical syllogism from being circular. But a demonstration is formed from things that are absolutely prior. Therefore, as we have already stated, there cannot be circular demonstration.

Then he sets forth the second argument, at: the advocates of circular demonstration (72b33), and it is this: If there were circular demonstration, it would follow that a same thing is demonstrated by the same thing, as if I were to say: "If it is this, it is this." In this way it is easy for anyone to demonstrate everything, for anyone, wise or ignorant, will be able to do this. Accordingly, science is not acquired through demonstration. But this is against the definition of demonstration. Therefore, there cannot be circular demonstration.

He proves the truth of the first consequence in the following way. It is obvious, first of all, that with a circular demonstration the same thing is proved by a same thing, as has been stated above, that is, if only three terms are employed; although it makes no difference whether the reflexion be made with fewer terms or more. (By reflexion he means the process whereby one goes from principle to conclusion in a demonstration, and then from conclusion to principle). In such a reflexion it makes no difference, so far as the force of the argument is concerned, whether it involves several or fewer terms or even two. For an argument has the same force if one proceeds thus: “If it is A, it is B, and if it is B, it is C, and if it is C, it is D,” and then by reflecting continues, “If it is D, it is C, and if it is C, it is B, and if it is B, it is A”; or if he proceeds by reflecting at the very start, saying: “If it is A, it is B, and if it is B, it is A.” (Although he spoke above of three terms, he restricted himself to two terms in this example, because in the deduction he is about to make he will use a third term, which is the same as the first).

Then, at: if A is, B must be (72b38), he gives the form of the argument in three terms, namely: “If it is A, it is B, and if it is B, it is C; therefore, if it is A, it is of necessity C.”

Then, at: since then, by the circular proof (73a1), he shows by the aforesaid form of arguing that in a circular demonstration a same thing is proved by a same thing, using only two terms. For it consists in saying, “If it is A, it is B,” and then reflecting, “If it is B, it is A”—which is a circular demonstration. Now, according to the above given form it follows from these two, that “if it is A, it is A.” That it does follow is obvious: for just as in the first deduction which involved three terms’ C followed from B, so in the reflex deduction of two terms, A followed from B. Let us suppose, then, that the A of the second deduction, that is, the reflex, signifies the same thing that C signified in the first, that is, in the direct deduction which was composed of three terms. Therefore, to state in the second deduction that “if it is B it is A” is to state the same thing as was stated in the first deduction, namely, that “if it is B, it is C.” But when it was stated in the first deduction that “if it is B, it is C,” it followed that “if it is A, it is C.” Therefore, in the circular deduction it follows that “if it is A, it is A,” since C is assumed to be the same as A. In this way, it will be easy to demonstrate all things, as has been said.

Then he presents the third argument, at: moreover, even such circular demonstration (73a6), which is this: Those who suppose that everything can be known through demonstration on the ground that demonstration is circular, must grant that anything can be demonstrated by a circular demonstration and, as a consequence, grant that in a circular demonstration each of the premises can be concluded from the conclusion. However, the only cases in which this can be done are those in which mutual conversion is possible, that is, in things that are convertible, as properties. But not all things are so related. Therefore, it is ridiculous to say that everything can be demonstrated on the ground that there are such things as circular demonstrations.

Now the reason is obvious why in a circular demonstration all the propositions must be convertible. For it has been shown in the book of Prior Analytics that if one thing is laid down, another does not follow of necessity, whether the thing laid down be one term or one proposition. For every syllogism must start with three terms and two propositions as a minimum.

Therefore, in a circular demonstration three terms which are convertible must be taken, namely, A, B, C, such that A is in every B and in every C, and these, namely, B and C, must inhere in each other, so that every B is C and every C is B, and also inhere in A so that every A is B and every A is C. And so, the terms being thus related, it is possible, when using the first figure, to derive any one from any two circularly, that is, the conclusion from two premises and each premise from the conclusion and the remaining premise, as we pointed out in the Prior Analytics, where we treated the syllogism formally.

The way it is done is this: take the three convertible terms, “risible,” “rational mortal animal” and “man,” and form the syllogism: "Every rational mortal animal is risible; Every man is a rational mortal animal: Therefore, every man is risible". Then from the conclusion it is possible to conclude both the major and the minor; the major thus: "Every man is risible; But every rational mortal animal is a man: Therefore, every rational mortal animal is risible" and the minor thus: "Every risible is a rational mortal animal; But every man is risible: Therefore, every man is a rational mortal animal".

However, it has also been proved in the Prior Analytics that in figures other than the first, namely, in the second and third, one cannot form a circular syllogism, that is, one through which each of the premises can be syllogized from the conclusion; or if one is formed, it is done not by using the premises already used but by using propositions other than those which appear in the first syllogism.

That this is so is obvious. For the second figure always yields a negative conclusion. Consequently, one premise must be affirmative and the other negative. However, it is true that if both are negative, nothing can be concluded; and if both are affirmative, a negative conclusion cannot follow. Therefore, it is not possible to use the negative conclusion and the negative premise to obtain the affirmative premise as a conclusion. Hence, if this affirmative is to be proved, it must be proved through propositions other than the ones originally used.

Again, in the third figure the only conclusion ever obtained is particular. However, at least one premise must be universal; furthermore, if either premise is particular, a universal cannot be concluded. Hence it cannot occur that in the third figure each of the premises can be syllogized from the conclusion and the remaining premise.

For the same reasons it is obvious that such a circular syllogism (through which each premise could be concluded) cannot be formed in the first figure except in the first mode, which is the only one that concludes to a universal affirmative. Furthermore, even in this mode the only case in which a circular syllogism could be formed such that each of the premises could be concluded, is when the three terms employed are equal, that is, convertible.

The proof is this: The premise must be concluded from the conclusion and the converse of the other premise, as has been stated. But such a conversion of each premise is impossible (for each is universal), except when the terms happen to be equal.

a21. Since the object of pure scientific knowledge cannot be other than it is, the truth obtained by demonstrative knowledge will be necessary. And since demonstrative knowledge is only present when we have a demonstration, it follows that demonstration is an inference from necessary premises. So we must consider what are the premises of demonstration—that is, what is their character:

and as a preliminary, let us define what we mean by an attribute ‘true in every instance of its subject’, an ‘essential’ attribute, and a ‘commensurate and universal’ attribute.

a28. I call “true in every instance” what is truly predicable of all instances — not of one to the exclusion of others — and at all times, not at this or that time only; e.g., if animal is truly predicable of every instance of man, then if it be true to say ‘this is a man’, ‘this is an animal’ is also true, and if the one be true now the other is true now. A corresponding account holds if point is in every instance predicable as contained in line.

a32. There is evidence for this in the fact that the objection we raise against a proposition put to us as true in every instance is either an instance in which, or an occasion on which, it is not true.

Since the object of pure scientific knowledge. After showing what a demonstrative syllogism is, the Philosopher in this section begins to show the nature and characteristics of the things that comprise a demonstration.

Concerning this he does three things.

First, he connects this with what has already been established.

Second, he explains certain matters that must be understood first, at: and as a preliminary (73a25; [78]).

Third, he establishes what he had in mind, namely, to show what and of what sort are the things that constitute a syllogism, at: demonstrative knowledge must rest (74b5; [109]).

He says therefore first (73a21), that since the definition of scientific knowledge given above spoke of that which cannot be otherwise, that which is scientifically known through demonstration will be necessary. Then he explains what it is to know something in a scientific way through demonstration, saying that demonstrative science is what we possess in having a demonstration, that is, what we acquire through demonstration. Consequently, it follows that the conclusion of a demonstration is necessary. Now, although the necessary could be syllogized from the contingent, it is not possible through a contingent middle to obtain scientific knowledge of the necessary, as will be proved later. Furthermore, because the conclusion. of a demonstration is not only necessary, but, as has been said, is known through demonstration, it follows that a demonstrative syllogism proceeds from necessary things. Consequently, we must establish from what and from what sort of necessary things a demonstration proceeds.

Then, at: and as a preliminary (73a25) he interjects certain things that must be understood as preliminaries to the matters to be discussed.

Apropos of these he does two things:

First, he states his intention (73a25), saying that before determining specifically the nature and characteristics of the things that form a demonstrative syllogism, we must indicate what is meant when we say, “of all,” and “per se,” that is, in virtue of itself, and “commensurately universal.”

For if we are to understand the nature of the things that form a demonstration, we must know what these terms mean, because they describe things that must be observed in demonstrations. For in the propositions of a demonstration it is required that something be predicated universally~ which he signifies by the term 'said of all'—and 'per se,' that is, in virtue of itself, and 'first'—which he signifies by the words, “commensurately universal.” But these three things are related by adding something to the previous one. For whatever is predicated per se is predicated universally [that is, of all], but not vice versa. Again, whatever is predicated first is predicated per se, but not vice versa. This, therefore, shows why they are arranged as they are. But why there are three and wherein they differ are explained by the fact that something is said to be predicated “of all” or universally in relation to things contained under the subject. For, as it is stated in Prior Analytics, something is said “of all,” when there is nothing under the extension of ,the subject that does not receive the given predicate. But it is in relation to the subject that something is said to be predicated per se, because the subject is mentioned when this predicate is defined, or vice versa, as will be explained below. Finally, something is said to be predicated of another thing “first” in relation to items that are prior to the, subject and embrace or include it, as the more universal includes the less. Thus to have three angles equal to two right angles is not predicated “first” of isosceles, because it is previously predicated of something prior to isosceles, namely, of triangle.

Second, at: I call 'true in every instance' (73a28), he establishes his proposition.

And his treatment is divided into three parts.

First, he shows what is meant by “said of all.”

Second, what is meant by “said per se,” that is, in virtue of itself, at: essential attributes are (73a34; [84]).

Third, what is meant by “commensurately universal,” at: I term “commensurately universal” (73b27; [90]).

Concerning the first he does two things.

First (73a28), he states what it is to be 'said of all'.

And it should be noted that the phrase, 'said of all', is taken here in a sense somewhat different from the sense it has in Prior Analytics, where it is taken in a very general sense so as to accommodate both the dialectician and the demonstrator. Therefore, no more is mentioned in its definition than that the predicate be found in each of the things included under its subject. But that might be verified only at a given moment—which is the sense in which the dialectician sometimes uses it; or it might be verified absolutely and at all times—which is the sense to which the demonstrator must always limit himself.

Accordingly, two things are mentioned in the definition of 'said of all': one is that there is nothing within the extension of the subject that the predicate does not apply to. And he indicates this when he says, not of one to the exclusion of others; the other is that there is no time in which the predicate does not belong. And this he indicates when he says, not at this or that time only. And he gives the example of “man” and “animal,” saying that animal is predicated of every man; and of anything of which it is true to say that it is a man, it is true to say that it is an animal, and whenever it is a man, it is an animal. The same is true between line and point: for a point is in every line and always in every line.

Second, at: there is evidence for this (73a32), he explains this definition, using as evidence the techniques employed in rebuttals. For a universal proposition is not rebutted unless one or other of things it states is not verified. For when we are asked whether something is said “of all” in a demonstration, we can say, “No,” for two reasons, that is, either because it is not true of each instance of the subject, or because now and then it is not true. Hence it is clear that 'being said of all' signifies each of these.

a34. Essential attributes are (1) such as belong to their subject as elements in its essential nature (e.g., line thus belongs to triangle, point to line; for the very being or ‘substance’ of triangle and line is composed of these elements, which are contained in the formulae defining triangle and line):

a38. (2) such that, while they belong to certain subjects, the subjects to which they belong are contained in the attribute’s own defining formula. Thus straight and curved belong to line, odd and even, prime and compound, square and oblong, to number; and also the formula defining any one of these attributes contains its subject — e.g., line or number as the case may be. Extending this classification to all other attributes, I distinguish those that answer the above description as belonging per se to their respective subjects; whereas attributes related in neither of these two ways to their subjects I call accidents or ‘coincidents’; e.g., musical or white is a ‘coincident’ of animal.

b5. Further, that is essential which is not predicated of a subject other than itself: e.g., ‘the walking [thing]’ walks and is white in virtue of being something else besides; whereas substance, in the sense of whatever signifies a ‘this somewhat’, is not what it is in virtue of being something else besides. Things, then, not predicated of a subject I call essential; things predicated of a subject I call accidental or ‘coincidental’.

b10. In another sense again, a thing consequentially connected with anything is essential; one not so connected is ‘coincidental’. An example of the latter is ‘While he was walking it lightened’: the lightning was not due to his walking; it was, we should say, a coincidence. If, on the other hand, there is a consequential connexion, the predication is essential; e.g., if a beast dies when its throat is being cut, then its death is also essentially connected with the cutting, because the cutting was the cause of death, not death a ‘coincident’ of the cutting.

b16. So far, then, as concerns the sphere of connexions scientifically known in the unqualified sense of that term, all attributes which (within that sphere) are essential either in the sense that their subjects are contained in them, or in the sense that they are contained in their subjects, are necessary as well as consequentially connected with their subjects.

For it is impossible for them not to inhere in their subjects either simply or in the qualified sense that one or other of a pair of opposites must inhere in the subject; e.g., in line must be either straightness or curvature, in number either oddness or evenness. For within a single identical genus the contrary of a given attribute is either its privative or its contradictory; e.g., within number what is not odd is even, inasmuch as within this sphere even is a necessary consequent of not-odd. So, since any given predicate must be either affirmed or denied of any subject, essential attributes must inhere in their subjects of necessity.

b25. Thus, then, we have established the distinction between the attribute which is ‘true in every instance’ and the ‘essential’ attribute.

Essential attributes. After determining about 'said of all', the Philosopher now determines about 'said per se' [that is, said in virtue of itself] and does three things.

About this he does two things.

First, he shows the number of ways something is said per se.

Second, how the demonstrator makes use of these ways, at: so far then as concerns the sphere of connexions (73b16; [89]).

In regard to the first it should be noted that this preposition per [“in virtue of” or “by”] denotes a causal relationship, although sometimes it also signifies a state, as when someone is said to be per se, that is, by himself, when he is alone. But when it designates a relationship to a cause, sometimes the cause is formal, as when it is stated that the body lives in virtue of the soul; sometimes the relationship is to a material cause, as when it is stated that a body is colored in virtue of its surface, that is, because the surface is the subject of color; again, it might even designate a relationship to an extrinsic cause, particularly an efficient cause, as when it is said that water is made hot in virtue of fire. But just as this preposition per designates a relationship to a cause, when something extrinsic is the cause of that which is attributed to the subject, so also when the subject or something pertaining to the subject is the cause of that which is attributed to the subject. This latter is what per se, that is, in virtue of itself, signifies.

Therefore, the first way of saying something per se (73a34) is when that which is attributed to a subject pertains to its form. And because the form and essence of a thing are signified by its definition, the first mode of that which is per se is when the definition itself or something expressed in the definition is predicated of the thing defined.

This is what he means when he says, Essential attributes are such as belong to their subject as elements in its essential nature, that is, included in the definition which indicates what it is, whether those elements are stated in the nominative case or in one of the oblique cases. Thus, line is stated in the definition of triangle. Hence line is in triangle per se. Again, in the definition of line, point is mentioned; hence point is per se in line. And the reason why they are mentioned in the definition is stated when he says, for the very being or substance [that is, the essence, which the definition signifies] of triangle and line is composed of these elements, namely, of lines and points. However, this does not mean that a line is formed out of points, but that point is involved in the very notion of line, just as line is involved in the very notion of triangle. And he asserts this in order to exclude things which are part of a thing’s matter and not of its species: thus, semicircle is not mentioned in the definition of circle, or finger in the definition of man, as it is stated in Metaphysics VII. He states further that all those items which are found universally in the definition expressing what a thing is are attributed to it per se.

The second mode of saying per se is when this preposition 'per' implies a relationship of material cause, in the sense that that to which something, is attributed is its proper matter and subject. For it is required, when defining an accident, to mention its proper subject in one of the oblique, cases: thus when an accident is defined abstractly, we say that “aquilinity” is a curvature of a nose, but when it is defined concretely, the subject is put in the nominative case, so that we say that “the aquiline is a curved nose.” Now, the reason for this is that since the being of an accident depends on its subject, its definition—which signifies its being—must mention that subject. Hence it is the second mode of saying per se, when the subject is mentioned in the definition of a predicate which is a proper accident of the subject.

And this is what he means when he states (73a38), essential attributes are those such that while they belong to certain subjects, that is, to subjects of accidents, the subjects to which they belong are contained in the attribute’s own defining formula, that is, in the expression which describes what the accident is, that is, in the definition of the accident. Thus straight and curved belong to line per se. For line is mentioned in their definition. For the same reason odd and even belong per se to number, because number is mentioned in their definition. Again, prime and compound are predicated per se of number, and number is mentioned in their definition. (For a prime number, for example, seven, is one which is exactly divisible by no other number but one; but a compound number, for example, nine, is one which is exactly divisible by some number greater than one. Again, isoplural, that is, equilateral, and scalene, that is, having three unequal sides, belong per se to triangle, and triangle is mentioned in their definition. Accordingly, he adds that their respective subjects belong to each of the aforesaid accidents and are mentioned in the expression which states what each is, that is, in the definition: thus line belongs to some of them, and number to others. In each of these subjects that have been mentioned, I say that its accident is in it per se.

But those predicates which are neutral, that is, of such a nature as not to be mentioned in the definition of their subjects, nor the subjects in their definition, are accidents, that is, are predicated per accidens: for example, musical and white are predicated per accidens of animal.

Then, at: further, that is essential (73b5) he sets down another mode of that which is per se, that is, the sense in which it signifies something in isolation. Thus something which is a singular in the genus of substance and which is not predicated of any subject is said to be per se. The reason for this is that when I say, walking or white, I do not signify either of them as something isolated or apart, since something else which is walking or white is understood. But this is not the case with terms which signify a “this something,” that is, with terms that signify first substance. For when I say, “Socrates” or “Plato,” it is not to be supposed that there is something else, over and above what they really are, which would be their subject. Therefore, things which are thus not predicated of any subject are per se, but things which are predicated of a subject, as being in the subject, are accidents. However, not all things predicated of a subject, as universals of their inferiors, are accidents. It should be noted, however, that this mode is not a mode of predicating, but a mode of existing; hence at the very start he said that they exist per se and not that they are said per se.

Then, at: in another sense again (73b10) he gives the fourth mode, according to which the preposition 'per' designates a relationship of efficient cause or of any other. Consequently, he says that whatever is attributed to a thing because of itself, is said of it per se; but whatever is not so attributed is said per accidens, as when I say, “While he was walking, it lightened.” For it is not the fact that he walks that causes lightning, but this is said by coincidence. But if the predicate is in the subject because of itself, it is per se, as when we say, “Slaughtered, it died.” For it is obvious that because something was slaughtered, it died, and it is not a mere coincidence that something slaughtered should die.

Then, at: so far, then, as concerns the sphere (73b16) he shows how the demonstrator uses the aforesaid modes.

But first it should be noted that, since science bears on conclusions, and understanding [intuition] bears on principles, the scientifically knowable are, properly speaking, the conclusions of a demonstration wherein proper attributes are predicated of their appropriate subjects. Now, the appropriate subjects are not only placed in the definition of attributes, but they are also their causes. Hence the conclusions of demonstrations involve two modes of predicating per se, namely, the second and the fourth.

And this is what he means when he says that the predications in the scientifically knowable in the strict sense, that is, in the conclusions of demonstrations are per se in the sense of something contained in the predicates, that is, in the way that subjects are contained in the definition of accidents which are predicated of the former; or are present on account of them, that is, in the way that predicates are in a subject by reason of the subject itself, which is the cause of the predicate.

Then he shows that such scientifically knowable things are necessary, because it is impossible for a proper accident not to be predicated of its subject. But this can occur in two ways: sometimes it is absolute, as when the accident is convertible with its subject, as “having three angles equal to two right angles” is convertible with triangle, and “risible” with man. At other times, two opposites stated disjunctively are of necessity in the subject, as straight or oblique in line, and odd or even in number. He shows that the reason for this is the fact that contrariety, privation and contradiction are in the same genus. For privation is nothing more, than a negation in a determinate subject. Again, a contrary is equivalent to a negation in some genus, as in the genus of numbers, odd is the same as not even by way of consequence. Therefore, just as it is necessary, either to affirm or deny, so it is necessary that one of two things that belong per se, be in its proper subject.

Then, at: thus, then, we have established (73b25) he summarizes, and the text is clear.

b27. I term “commensurately universal” an attribute which belongs to every instance of its subject, and to every instance per se and as such;

b28. from which it clearly follows that all commensurate universals inhere necessarily in their subjects.

b29. The essential attribute, and the attribute that belongs to its subject as such, are identical. e.g., point and straight belong to line per se, for they belong to line as such; and triangle as such has two right angles, for it is per se equal to two right angles.

b32. An attribute belongs commensurately and universally to a subject when it can be shown to belong to any random instance of that subject and when the subject is the first thing to which it can be shown to belong.

b34. Thus, e.g., (1) the equality of its angles to two right angles is not a commensurately universal attribute of figure. For though it is possible to show that a figure has its angles equal to two right angles, this attribute cannot be demonstrated of any figure selected at haphazard, nor in demonstrating does one take a figure at random — a square is a figure but its angles are not equal to two right angles. On the other hand, any isosceles triangle has its angles equal to two right angles, yet isosceles triangle is not the primary subject of this attribute but triangle is prior. So whatever can be shown to have its angles equal to two right angles, or to possess any other attribute, in any random instance of itself and primarily — that is the first subject to which the predicate in question belongs commensurately and universally,

a1. and the demonstration, in the essential sense, of any predicate is the proof of it as belonging to this first subject commensurately and universally: while the proof of it as belonging to the other subjects to which it attaches is demonstration only in a secondary and unessential sense. Nor again (2) is equality to two right angles a commensurately universal attribute of isosceles; it is of wider application.

I term “commensurately universal." After determining about 'said of all' and 'said per se', the Philosopher here determines concerning the “universal.”

This treatment falls into two parts. In the first he shows what the universal is. Second, how error occurs in our understanding of it, at: we must not fail to observe (74a4; [98]).

Concerning the first he does two things.

First, he shows what the universal is.

Second, how the demonstrator uses the universal, at: and the demonstration (74a1; [97]).

Concerning the first he does two things.

First, he shows that the universal contains within itself the attributes of 'being said of all' and of being said per se'.

Second, he shows what the universal adds to them, at: an attribute belongs commensurately (73b33; [95]).

To understand what is being said here it should be noted that “universal” is not to be taken here in the sense that anything predicated of several is a universal, as when Porphyry treats of the five universals; rather “universal” is taken here according to a certain correspondence or commensurateness of the subject with the predicate, so that the predicate is not found outside the subject nor is the subject without the predicate.

With this in mind, it should be noted that he does three things with respect to the first point.

First (73b27), he says that the universal, namely, the predicate, is both verified of all, that is, is predicated universally of its subject, and is said per se, that is, is in and belongs to the subject according to the essential nature of the subject. For many things are said universally of certain things to which they do not belong per se and as such. Thus, every stone is colored, but not precisely as stone, but as it has a surface.

Second, at: from which it clearly follows (73b28), he draws a corollary from this and says that since the universal is something which is per se in a thing, and since it has been shown that whatever things are in something per se are in it of necessity, it is obvious that universal predicates, as they are being taken here, are necessarily present in the things of which they are predicated.

Third, at: the essential attribute (73b29), lest anyone suppose that 'per se' and 'precisely as such', both of which were mentioned in the definition of the universal, are different, he shows that they are the same. Thus, point is per se in line in the first way, and straight in the second way. For each is in line precisely as it is a line. In like manner, two right angles belongs to triangle precisely as triangle, that is, its angles are equal to two right angles, which is per se in triangle.

Then, at: an attribute belongs commensurately (73b33), he shows what “universal” adds to the notions, 'being said of all' and 'being said per se'.

In regard to this he does two things:

First, he says that a predicate is universal, when it is not only in each thing of which it is asserted, but it is demonstrated to be first or primarily in the thing which receives that predicate.

Second, at: the equality of its angles (73b34), he clarifies this with an example and says that “having three angles equal to two right angles” is not found in just any figure in general, although this could be demonstrated of some figure, because it is demonstrated of triangle, which is a figure; yet it is not found in any random figure, nor is just any figure used when it is demonstrated. For a rhombus is a figure, but it does not have three angles equal to two right angles. But an isosceles, that is, a triangle with two equal sides, always has its three angles equal to two right angles. Nevertheless, isosceles is not the primary thing to which this belongs, for it belongs basically to triangle, and belongs to isosceles precisely as it is a triangle. Therefore, whatever is demonstrated basically to have its three angles equal to two right angles (or whatever else be thus demonstrated), the universal predicate is present in it primarily, as in triangle.

Then, at: and the demonstration (74a1), he shows how a demonstrator uses the “universal,” saying that demonstration is concerned per se with such a universal, but with other things qualifiedly and not per se. For a demonstrator demonstrates a proper attribute of its proper subject; and if he demonstrates it of anything else, he does so only insofar as it pertains to that subject. Thus, he proves that some property of triangle belongs to a figure and to an isosceles precisely as some figure is a triangle, and as the isosceles is a triangle. But the reason why “having three” is not in isosceles primarily is not because it is not predicated of it universally, but because it is found more frequently, that is, in more things than in isosceles, since this is common to every triangle.

How error occurs in taking the universal

a4. We must not fail to observe that we often fall into error because our conclusion is not in fact primary and commensurately universal in the sense in which we think we prove it so.

a6. We make this mistake (1) when the subject is an individual or individuals above which there is no universal to be found: (2) when the subjects belong to different species and there is a higher universal, but it has no name: (3) when the subject which the demonstrator takes as a whole is really only a part of a larger whole; for then the demonstration will be true of the individual instances within the part and will hold in every instance of it, yet the demonstration will not be true of this subject primarily and commensurately and universally. When a demonstration is true of a subject primarily and commensurately and universally, that is to be taken to mean that it is true of a given subject primarily and as such.

a13. Case (3) may be thus exemplified. If a proof were given that perpendiculars to the same line are parallel, it might be supposed that lines thus perpendicular were the proper subject of the demonstration because being parallel is true of every instance of them. But it is not so, for the parallelism depends not on these angles being equal to one another because each is a right angle, but simply on their being equal to one another.

a17. An example of (1) would be as follows: if isosceles were the only triangle, it would be thought to have its angles equal to two right angles qua isosceles.

a18. An instance of (2) would be the law that proportionals alternate. Alternation used to be demonstrated separately of numbers, lines, solids, and durations, though it could have been proved of them all by a single demonstration. Because there was no single name to denote that in which numbers, lengths, durations, and solids are identical, and because they differed specifically from one another, this property was proved of each of them separately. Today, however, the proof is commensurately universal, for they do not possess this attribute qua lines or qua numbers, but qua manifesting this generic character which they are postulated as possessing universally.

a25. Hence, even if one prove of each kind of triangle that its angles are equal to two right angles, whether by means of the same or different proofs; still, as long as one treats separately equilateral, scalene, and isosceles, one does not yet know, except sophistically, that triangle has its angles equal to two right angles, nor does one yet know that triangle has this property commensurately and universally, even if there is no other species of triangle but these. For one does not know that triangle as such has this property, nor even that ‘all’ triangles have it — unless ‘all’ means ‘each taken singly’: if ‘all’ means ‘as a whole class’, then, though there be none in which one does not recognize this property, one does not know it of ‘all triangles’.

a33. When, then, does our knowledge fail of commensurate universality, and when it is unqualified knowledge? If triangle be identical in essence with equilateral, that is, with each or all equilaterals, then clearly we have unqualified knowledge: if on the other hand it be not, and the attribute belongs to equilateral qua triangle; then our knowledge fails of commensurate universality.

a35. ‘But’, it will be asked, ‘does this attribute belong to the subject of which it has been demonstrated qua triangle or qua isosceles? What is the point at which the subject to which it belongs is primary? (that is, to what subject can it be demonstrated as belonging commensurately and universally?) Clearly this point is the first term in which it is found to inhere as the elimination of inferior differentiae proceeds. Thus the angles of a brazen isosceles triangle are equal to two right angles: but eliminate brazen and isosceles and the attribute remains. ‘But’ — you may say — ‘eliminate figure or limit, and the attribute vanishes.’ True, but figure and limit are not the first differentiae whose elimination destroys the attribute. ‘Then what is the first?’ If it is triangle, it will be in virtue of triangle that the attribute belongs to all the other subjects of which it is predicable, and triangle is the subject to which it can be demonstrated as belonging commensurately and universally.

We must not fail to observe. After specifying what the universal is, the Philosopher here shows how one might err in understanding the universal.

In regard to this he does three things.

First, he says that sometimes one might err in this matter.

Second, he tells in how many ways, at: we make this mistake (74a6; [100]).

Third, he gives the criterion for knowing whether the universal is being employed correctly, at: “but,” it will be asked, “does this attribute” (74a35; [108]).

He says therefore first (74a4), that in order to avoid mistakes in demonstrating, one should be aware of the fact that quite often something universal seems to be demonstrated, which is not being demonstrated.

Second, at: we make this mistake (74a6), he indicates the ways in which this mistake can occur.

And in regard to this he does two things.

First, he enumerates these ways and says that there are three possible errors in understanding a universal.

The first is likely to occur when under some common genus there is nothing else to take as the thing to which the universal initially applies than this singular, to which it is incorrectly applied. For example, if man were the only animal existing, and “sensible,” which is initially and per se in animal, were to be assigned as a primary universal to man.

(It should be noted that 'singular' is being used here in a wide sense for any inferior, in the way that a species might be called a 'singular' contained under a genus.) Or we might say that it is not possible to find a genus with only one species: for a genus is divided into species through opposing differences. But if one contrary is found in nature, so must the other, as the Philosopher explains in On the Heavens II. Therefore, if one species is found, another will be found. However, one species is divided into distinct individuals by the division of matter. But it sometimes happens that all the matter proportionate to a given species is comprehended under one individual, so that in that case there is only one individual under one species. Hence it is significant that he did say, 'singular'.

The second way is when it is possible to take several inferiors under something common which is verified in things that differ in species, but that common item has no name. For example, if “animality” had no name, and “sensibility,” which is proper to animal, were to be assigned to the inferiors of animal (either collectively or distributively) as their primary universal.

The third way is when that of which something is demonstrated to be its primary universal is related to what is demonstrated of it, as a whole is related to a part. For example, if the power to see were assigned as a primary universal to animal: for not every animal can see. In this case the demonstration, that is, what should have been demonstrated, is “in those things which are in part,” that is, in some but not all of the things included under the subject; furthermore, it will be a demonstration “of all,” but not of all that the demonstration mentions. For the power to see can indeed be demonstrated universally of something, but not of animal universally, as of that to which it belongs primarily. And he explains why he says, “primarily,” namely, because a demonstration bears on what is both universal and first.

Second, at: if a proof were given (74a13), he gives examples of each of these ways.

First, of the third way, saying that if someone were to demonstrate of two straight lines that they do not intersect, that is, that they do not meet, it might seem that we have a demonstration of this sort, that is, one that bears on a primary universal, on the ground that “not to meet” is true of certain straight lines, and not that this happens only because the straight lines are equal, that is, equally distant. But if the lines should be equal, that is, equidistant, then “not to meet” belongs to any and all of them, because it is universally true that lines which are straight and equally distant, even should they be lengthened ad infinitum, will not meet.

Second, at: if isosceles were the only triangle (74a17), he gives an example of the first way, saying that if there were no triangle but the isosceles which is a triangle having two equal sides, it might seem that what is true of triangle as triangle should be true of isosceles as isosceles. But this would not be so.

Third, at: the law that proportionals alternate (74a18), he gives an example of the second way. Now, it seems that he saved this for last because he wished to spend more time on it.

Concerning it he does three things.

First, he gives the example.

Second, he draws a corollary from what he has said, at: hence, even if one prove (74a25; [106]).

Third, he gives a reason for the aforesaid, at: when, then, does our knowledge (74a33; [107]).

Concerning the first it should be noted that a ratio is a relation of one quantity to another, as 6 is related to 3 in the ratio of 2 to 1. The comparing of one ratio to another is a proportion, which, if it is disjoint, has four terms: for example, "as 4 is to 2, so 6 is to 3". But if it is joint, it has three terms, one of which is used twice: for example, "as 8 is to 4, so 4 is to 2". Now, it is obvious that in a proportion two terms are antecedents and two are consequents. For example, "in the proportion that 6 is to 3 as 4 is to 2", 6 and 4 are the antecedents, 3 and 2 the consequents. Again, a proportion is alternated by bringing the antecedents together and the consequents together. For example, when I say: "As 4 is to 2, so 6 is to 3"; therefore, "as 4 is to 6, so 2 is to 3".

What he is saying, therefore, is that alternate proportion is verified in lines, numbers, solids, (that is, bodies), and times. But just as this is established separately for each of them, namely, for numbers in arithmetic, for lines and solids in geometry, and for times in natural philosophy or astronomy, so it is possible to prove it of all of them in a single demonstration. But the reason why a separate demonstration was employed to prove alternation for each was that the one feature they had in common was unnamed.

For although quantity is a feature common to all of them, quantity includes other things besides them; for example, speech and other things that are accidentally quantitative. Or, better still, it was because alternate proportion does not belong to quantity precisely as quantity, but as compared to another quantity according to a fixed ratio. That is why at the very start he spoke of alternate proportion. But there is no general name for the aforesaid things precisely as they are proportional.

Furthermore, when alternate proportion is demonstrated separately for each of them, it is not a universal that is being demonstrated. For “to be alternately proportional” is not in numbers and lines according to what each of them is, but according to something common. Besides, those who demonstrate alternate proportion of lines look upon this attribute as a universal predicate of line precisely as it is a line, or of number precisely as it is number.

Then, at: hence, even if one prove (74a25), he draws a corollary from the above and says that for the same reason that something universal is not being demonstrated when a common unnamed attribute is proved to be a universal predicate of each species, so too when something is demonstrated this way of a common attribute which does have a name. For example, if someone uses the same demonstration for each species of triangle and proves that each has three angles equal to two right angles, or if he uses different demonstrations, say one for isosceles and another for scalene, even then he does not on that account know that a triangle has three angles equal to two right angles except in a sophistical way, that is, per accidens: for he does not know it of triangle as triangle, but as equilateral or as having two equal sides or as having three unequal sides. Furthermore, one who demonstrates in this way does not know the universal of triangle, that is, he does not have a knowledge of triangle in a universal way (even if there happens to be no other triangle besides those of which he knew this: and this because he had not the knowledge of triangle as triangle but under the aspect of its various species). Again, strictly speaking, he did not know every triangle: for although he knew every triangle according to number (if there was none he did not know), yet according to species he did not know each one. For something is known universally according to species, when it is known according to the notion of the species, but according to number and not universally, when it is known according to the multifarious things contained under the species. And in this matter there is no difference whether we compare species to individuals or genera to species. For triangle is the genus of equilateral and isosceles.

Then, at: when, then, does our knowledge (74a33) he assigns the reason for the aforesaid, first asking when does one know universally and absolutely, if one who knew in the aforesaid way did not know universally. And he answers that obviously if the essence of triangle in general and of each of its species (each taken separately or all taken together) were the same, then one would know universally and absolutely about triangle, when he knew about any one species of it separately or about all of them together. But if the essence is not the same, then it will not be the same but something different to know triangle and to know its several species. And by knowing something of the species one does not know it of triangle precisely as triangle.

Then, at: “but,” it will be asked, “does this attribute” (74a35), he gives the rule on how the universal can be properly understood, saying that whether something belongs to triangle precisely as triangle or to isosceles precisely as isosceles, and when it is that what is demonstrated is first and universal in the given subject precisely as this, that is, precisely according to the subject—all this will be clear from what I shall say. For whenever some item is removed from the subject and the original universal still applies to what remains, then it is not the first universal of that subject. For example, upon the removal of isosceles or of brazen there still remains the triangle which has three angles equal to two right angles; hence the possession of three angles equal to two right angles is not a first universal of isosceles or of brazen triangle. But if the item “figure” be removed, nothing which has three angles equal to two right angles remains, nor again if you remove “bounded” which is more universal than figure, since a figure is something enclosed by a bound or bounds. Nevertheless, the attribute in question [namely, having three angles equal to two right angles] does not belong first to figure or to bounded, because it does not belong to either of them universally. To what then will it be first? Obviously to triangle, because it belongs to the others (both superiors and inferiors) precisely as they are triangles For it belongs to figure to have three angles equal to two right angles only because a triangle is some figure, and similarly to isosceles, only because it is a triangle, and it is of triangle that “having three...” is universally demonstrated. Hence, it is to triangle that it belongs as a first universal.

Demonstration proceeds from necessary things

b5. Demonstrative knowledge must rest on necessary basic truths; for the object of scientific knowledge cannot be other than it is.

b6. Now, attributes attaching per se to their subjects attach necessarily to them: for essential attributes are either elements in the essential nature of their subjects, or contain their subjects as elements in their own essential nature. (The pairs of opposites which the latter class includes are necessary because one member or the other necessarily inheres.) It follows from this that premises of the demonstrative syllogism must be connexions essential in the sense explained: for all attributes must inhere essentially or else be accidental, and accidental attributes are not necessary to their subjects.

b13. We must either state the case thus, or else premise that the conclusion of demonstration is necessary and that a demonstrated conclusion cannot be other than it is, and then infer that the conclusion must be developed from necessary premises. For though you may reason from true premises without demonstrating, yet if your premises are necessary you will assuredly demonstrate — in such necessity you have at once a distinctive character of demonstration.

b18. That demonstration proceeds from necessary premises is also indicated by the fact that the objection we raise against a professed demonstration is that a premise of it is not a necessary truth — whether we think it altogether devoid of necessity, or at any rate so far as our opponent’s previous argument goes.

b22. This shows how naive it is to suppose one’s basic truths rightly chosen if one starts with a proposition which is (1) popularly accepted and (2) true, such as the sophists’ assumption that to know is the same as to possess knowledge. For (1) popular acceptance or rejection is no criterion of a basic truth, which can only be the primary law of the genus constituting the subject matter of the demonstration; and (2) not all truth is ‘appropriate’.

b27. A further proof that the conclusion must be the development of necessary premises is as follows. Where demonstration is possible, one who can give no account which includes the cause has no scientific knowledge. If, then, we suppose a syllogism in which, though A necessarily inheres in C, yet B, the middle term of the demonstration, is not necessarily connected with A and C, then the man who argues thus has no reasoned knowledge of the conclusion, since this conclusion does not owe its necessity to the middle term; for though the conclusion is necessary, the mediating link is a contingent fact.

b32. Or again, if a man is without knowledge now, though he still retains the steps of the argument, though there is no change in himself or in the fact and no lapse of memory on his part; then neither had he knowledge previously. But the mediating link, not being necessary, may have perished in the interval; and if so, though there be no change in him nor in the fact, and though he will still retain the steps of the argument, yet he has not knowledge, and therefore had not knowledge before. Even if the link has not actually perished but is liable to perish, this situation is possible and might occur. But such a condition cannot be knowledge.

a1. When the conclusion is necessary, the middle through which it was proved may yet quite easily be non-necessary. You can in fact infer the necessary even from a non-necessary premise, just as you can infer the true from the not true. On the other hand, when the middle is necessary the conclusion must be necessary; just as true premises always give a true conclusion. Thus, if A is necessarily predicated of B and B of C, then A is necessarily predicated of C. But when the conclusion is nonnecessary the middle cannot be necessary either. Thus: let A be predicated non-necessarily of C but necessarily of B, and let B be a necessary predicate of C; then A too will be a necessary predicate of C, which by hypothesis it is not.

a13. To sum up, then: demonstrative knowledge must be knowledge of a necessary nexus, and therefore must clearly be obtained through a necessary middle term; otherwise its possessor will know neither the cause nor the fact that his conclusion is a necessary connexion. Either he will mistake the non-necessary for the necessary and believe the necessity of the conclusion without knowing it, or else he will not even believe it — in which case he will be equally ignorant, whether he actually infers the mere fact through middle terms or the reasoned fact and from immediate premises.

Demonstrative knowledge must rest. Having finished his treatment of that which is said 'of all' and 'per se' and 'universally,' all of which we use in demonstration, the Philosopher now begins to discuss the items from which demonstration proceeds.

And there are two parts.

In the first he shows what demonstration of the reasoned fact [demonstration 'propter quid'] proceeds from.

In the second what demonstration of the fact [demonstration 'quia'] proceeds from, at: knowledge of the fact (78a22; [192]).

The first is divided into two parts.

In the first he shows the sort of things from which a demonstration proceeds.

In the second, what the principles of demonstration are, at: as regards both these primary truths (76a32; [152]).

The first is divided into three parts.

In the first he shows that demonstration issues from necessary things.

In the second that it issues from things that are per se, at: of accidents that are not essential (76a26; [120]).

In the third that it proceeds from proper principles, at: it follows that we cannot (75a38; [127]).

Concerning the first he does two things.

First, he shows that demonstration should proceed from necessary things.

Second, he proves certain things he had presupposed, at: a further proof that the conclusion (74b27; [116]).

In regard to the first he does three things.

First, he makes a connection with what went before.

Second, he proves his proposition, at: now, attributes attaching per se (74b6; [110]).

Third, he draws a conclusion from the aforesaid, at: this shows how naive it is (74b22; [114]).

He says therefore first (74b5), as an inference from what has already been established, that if there be demonstrative science, that is, if science is acquired through demonstration, it must issue from necessary principles. The necessity of this inference is clear, because that which is known scientifically cannot be otherwise than it is, as was pointed out in the definition of scientific knowing.

Then, at: now, attributes attaching per se (74b6), he shows that demonstration issues from necessary things.

First, with a reason.

Second, with a sign, at: that demonstration proceeds (74b19; [113]).

With respect to the first he gives two reasons.

The first (74b6) is this: Things predicated per se are in a thing necessarily. (And he manifests this in two of the modes of predicating per se: in the first mode, because the things predicated per se are “in that which is what,” that is, in the definition of the subject. But whatever is put in the definition of anything is predicated of it necessarily. In the second mode, because certain subjects are put in “that which is the what of things predicated of them,” that is, in the definition of their predicates. And if these are opposites, one or other of them is necessarily in the subject: thus, either “odd” or “even” is in number, as we showed above). Now, it is clear that it is from principles of this kind, namely, per se, that a demonstrative syllogism proceeds. (And this is proved by the fact that whatever is predicated, is predicated either per se or per accidens, and that things predicated per accidens are not necessary). Therefore, since it is not a demonstration but rather a sophistic syllogism that issues from things that are per accidens, it follows that demonstration proceeds from necessary things.

Furthermore, it should be noted that, since in a demonstration a proper attribute is proved of a subject through a middle which is the definition, it is required that the first proposition (whose predicate is the proper attribute, and whose subject is the definition which contains the principles of the proper attribute) be per se in the fourth mode, and that the second proposition (whose subject is the subject itself and the predicate its definition) must be in the first mode. But the conclusion, in which the proper attribute is predicated of the subject, must be per se in the second mode.

Then, at: we must either state the case thus (74b14) he sets forth the second reason and it is this: “Demonstration is concerned with the necessary and the demonstrated,” that is, the conclusion of the demonstration cannot be other than it is. This statement is to be taken as the principle proving our proposition that demonstration proceeds from necessary things. And from what has been said, the truth of this principle is obvious. From this principle one argues thus: a necessary conclusion cannot be scientifically known save from necessary principles; but a demonstration makes a necessary conclusion scientifically known: therefore, it must proceed from necessary principles. Herein lies the difference between a demonstration and other syllogisms. For in the latter it is enough if one syllogizes from true principles. Nor is there any other type of syllogism in which it is required to proceed from the necessary: in a demonstration alone must this be observed. And this is proper to a demonstration, that is, to proceed from the necessary.

Then, at: that demonstration proceeds (74b18) he proves the same thing in the following way, using a sign: Suit is brought against an argument solely on the ground that something is wanting that should have been observed in that argument. But against one who believes that he has demonstrated, we bring the charge that the things from which he proceeded are not necessarily true, whether we are convinced that they could be otherwise than stated, or whether we bring a charge of reason, that is, for the sake of disputing. Therefore, demonstration should proceed from necessary things.

Then, at: this shows how naive it is (74b22) he draws a conclusion from the foregoing, saying that "this," namely, the fact that a demonstration must conclude from necessary things, "shows" that they are obtuse who assumed that they rightly chose the principles of demonstration, if the proposition they chose was probable or true, as the Sophists do, that is, those who appear wise but are not. For scientific knowing consists in nothing less than having science, namely, from demonstration. But when something is probable or improbable, it is not certain whether it is first or not first, whereas that on which a demonstration bears must be first in some genus and be true. This does not mean that a demonstrator may take anything that is first: it must be first in that genus with respect to which he is demonstrating. Thus, arithmetic does not choose what is first in regard to magnitude, in regard to number.

It should be noted that Sophists are not taken here as they are in the book of Sophistical Refutations, that is, who proceed from things that seem probable but are not, or seem to syllogize but do not. For just as some are called Sophists, that is, seeming to be wise but not really so, inasmuch as they appear by their arguments to be scientific knowers but are not, so dialectical arguments, if they seem to prove demonstratively but do not, are sophistic.

Then, at: a further proof that the conclusion (74b27), he proves something he had presupposed.

And concerning this he does two things.

First, he shows that a necessary conclusion cannot be scientifically known from non-necessary principles.

Second, that although the necessary cannot be scientifically known from the non-necessary, it can nevertheless be syllogized, at: when the conclusion is necessary (75al; [118]).

He proves the first with two reasons, one of which (74b27) is this:

If one does not have an argument which shows the cause why [that is, propter quid], he does not know scientifically, even though a syllogism be had; because to know scientifically is to know the cause of a thing, as we stated above. But an argument which infers a necessary conclusion from non-necessary principles does not show the cause why. To clarify this he gives an example, using general terms. Assume that the conclusion, “Every C is A,” is necessary, and that it is demonstrated by a middle, B, which is not a necessary middle but a contingent one, so that, for example, one of the propositions, “Every B is A,” and “Every C is B,” or both, are contingent. Now, it is obvious that through this contingent middle, namely, B, one cannot obtain scientific knowledge propter quid of the necessary conclusion, “Every C is A.” For if the cause on which something depends is removed, the effect must be removed. But this middle, being contingent, can be withdrawn, whereas the conclusion, since it is necessary, cannot be withdrawn. It follows, therefore, that a necessary conclusion cannot be scientifically known through a contingent middle.

The second reason, at: or again, if a man (74b32) is this: “If a man is without scientific knowledge now, then even though he possesses the same argument he had up to now, and he has been kept sound,” that is, has not ceased to be, “and the thing known has not changed; and if he has not forgotten, then obviously he has not known scientifically.” In this passage the Philosopher mentions four ways in which a person loses the science he once had: one way is when there slips from his mind the argument through which he formerly knew scientifically. Another is through the destruction of the knower. A third is by a change occurring in the thing known: thus if, while you are sitting, I know that you are sitting, this knowledge perishes when you are not sitting. The fourth way is by forgetting. Hence if none of these ways has occurred, then, if a person does not know a thing scientifically now, he never did know it. Now, one who holds a necessary conclusion in virtue of a contingent middle, no longer knows it scientifically when the middle ceases to be, for the middle no longer exists, even though he retains the same argument and the fact remains the same, and he has not forgotten anything. Therefore, even when the middle had not ceased to be, he did not know scientifically. But that a contingent middle is liable to perish he proves by the fact that whatever is not necessary must at some time perish. And even though the middle has not perished, nevertheless, because it is not necessary, it is obvious that it is liable to perish. Now, when something contingent is put forward, its result is not impossible but possible and contingent. But what followed in our case was impossible, namely, that one should have had science of something and later not have had it, all the conditions described above prevailing. And this follows from the fact that a contingent middle has perished, which, although it may not be true [that it has], is nevertheless liable to, as has been said.

Then, at: when the conclusion is necessary (75a1) he shows that although a necessary conclusion cannot be scientifically known through a contingent middle, nevertheless a necessary conclusion can be syllogized from a non-necessary middle.

Hence he says that nothing hinders a necessary conclusion from being obtained through a non-necessary middle, as witness the case in which something is syllogized with a dialectical syllogism, although not with a demonstrative syllogism which causes science. For the necessary happens to be syllogized from the non-necessary, just as the true happens to be syllogized from the untrue—although the converse does not occur. For when the middle is necessary, the conclusion too will be necessary; similarly, from true premises something true is concluded. But that the necessary is always concluded from necessary premises he proves thus: “Let A be necessary of B,” that is, let this proposition, “Every B is A,” be necessary, “and the latter of C,” that is, let “Every C is B” also be necessary. Now, from these two necessary things follows a third necessary thing, namely, the conclusion, “Every C is A.” For it has been proved in the Prior Analytics that “from two propositions of necessity follows a conclusion of necessity.” Then by way of consequence he shows that if the conclusion were not necessary, the middle could not be necessary. For example, suppose that this conclusion, “Every C is A,” is not necessary, but that the two premises are necessary. According to our previous doctrine it follows that the conclusion is necessary—which is contrary to the supposition in our example, namely, that the conclusion not be necessary.

Then, at: to sum up, then (75a13) he infers from the aforesaid the conclusion originally intended, saying that since a thing must be necessary if it is made known by way of demonstration, it is clear from the foregoing that a demonstration must rest on a necessary middle. For otherwise it would not be scientifically known that the conclusion is necessary, neither propter quid nor quia, since the necessary cannot be known through the non-necessary, as we have shown. But if someone rests on an argument based on a non-necessary middle, he will be in one of two states. For since he does not actually know in a scientific way, he will either believe that he does know in a scientific way, if he assumes a non-necessary middle as necessary, or he will not presume that he knows in a scientific way, that is, if he believes that he does not have a necessary middle. And this is to be universally understood both of scientific knowledge 'quia', in which something is known through mediate principles, and of science 'propter quid', in which something is known through immediate principles. The difference between these two will be explained later.

Demonstration bears upon and proceeds from things which are per se

a18. Of accidents that are not essential according to our definition of essential there is no demonstrative knowledge; for since an accident, in the sense in which I here speak of it, may also not inhere, it is impossible to prove its inherence as a necessary conclusion.

a21. A difficulty, however, might be raised as to why in dialectic, if the conclusion is not a necessary connexion, such and such determinate premises should be proposed in order to deal with such and such determinate problems. Would not the result be the same if one asked any questions whatever and then merely stated one’s conclusion?

a24. The solution is that determinate questions have to be put, not because the replies to them affirm facts which necessitate facts affirmed by the conclusion, but because these answers are propositions which if the answerer affirm, he must affirm the conclusion and affirm it with truth if they are true.

a28. Since it is just those attributes within every genus which are essential and possessed by their respective subjects as such that are necessary it is clear that both the conclusions and the premises of demonstrations which produce scientific knowledge are essential; for accidents are not necessary. Further, since accidents are not necessary, one does not necessarily have reasoned knowledge of a conclusion drawn from them (this is so even if the accidental premises are invariable but not essential, as in proofs through signs; for though the conclusion be actually essential, one will not know it as essential nor know its reason); but to have reasoned knowledge of a conclusion is to know it through its cause. We may conclude that the middle must be consequentially connected with the minor, and the major with the middle.

Of accidents that are not essential. After showing that demonstration proceeds from necessary things, the Philosopher then shows that it is concerned with things that are per se and proceeds from things that are per se.

In regard to this he does three things.

First, he shows that demonstration is concerned with things that are per se, that is, the conclusions of demonstrations are per se.

Second, he raises a problem, at: a difficulty, however (75a21).

Third, he shows that demonstration proceeds from things that are per se, that is, that the principles of demonstrations must be per se, at: since it is just those attributes (75a28).

He says therefore first (75a18) that demonstrative science cannot bear on accidents that are not per se in the way that 'per se' was explained above, namely, that a 'per se' accident is one in whose definition the subject is mentioned, as “even” or “odd” is a 'per se' accident of number. But “white” is not a 'per se' accident of animal, because animal is not mentioned in its definition. That there cannot be demonstration bearing on accidents that are not per se he proves in the following way: An accident which is not per se might happen not to be present (and this is the accident under discussion). Therefore, if a demonstration were to bear on an accident which is not per se, it would follow that the conclusion of the demonstration would not be necessary: which is contrary to what has been established.

That an accident which is not 'per se' does not inhere necessarily can be obtained from the following: If an accident inheres necessarily and always in a subject, it must have its cause in the subject—in which case the accident cannot but inhere. Now, this can occur in two ways: in one way, when it is caused from the principles of the species, and such an accident is called a per se attribute or property; in another way, when it is caused from the principles of the individual, and this is called an inseparable accident. In either case every accident which is caused from the principles of the subject must, if it is defined, be defined in such a way as to mention the subject in its definition: for a thing is defined in terms of its proper principles. Thus it is clear that an accident which necessarily inheres in its subject must be a per se accident. Therefore, those that are not per se do not inhere of necessity.

It seems, however, that Aristotle is using the circular demonstration, which he previously repudiated. For he had proved earlier that a demonstration is concerned with necessary things, because it is concerned with, things which are 'per se'; but now he shows that demonstration is concerned with things which are 'per se,' because it is concerned with things which are necessary.

The answer is that above Aristotle proved that demonstration is concerned with necessary things not only because it is concerned with things which are per se but also because of the definition of scientific knowing—and this was a true way to demonstrate. But the proof that demonstration is concerned with necessary things because it is concerned with things which are per se, is not a demonstration but an indication directed to a person who already knows that demonstration is concerned with things which are per se.

Then, at: a difficulty, however (75a21) he raises a problem.

In regard to this he does two things.

First, he states the problem, saying that someone may perhaps, object that if the conclusion which follows from contingent things or things which are per accidens is not necessary, why are inquiries made concerning contingent things or things which are per accidens, so that from such data one may reach a conclusion, since it is required of the’ syllogism that the conclusion follow of necessity. That inquiries touching contingent things or things which are per accidens do occur is evident from his next statement, namely, that it makes no difference whether one inquires into contingent things and then gives his conclusion. As if to say: A conclusion can be drawn from contingent things that have been investigated and verified, just as from necessary things. For each employs the same syllogistic form.

Second, at: the solution is that determinate questions (75a24), he solves the difficulty and says that questions solved by contingent premises are not of such a nature that the conclusion is necessary absolutely, on the basis of the things investigated, that is, in virtue of the contingent premises, but because it is necessary for the one who admits the premises to admit the conclusion and to admit truth in the conclusion, if the things premised are true. It is as though he were saying that although a conclusion which is necessary with absolute necessity does not follow from contingent premises, yet it follows with the necessity of consequence according to which the conclusion follows from premises.

Then, at: since it is just those attributes (75a28) he shows that demonstration proceeds from things which are 'per se,' using the following argument: Demonstration proceeds from necessary things and bears on necessary things—and this because it is scientific, that is, makes one know scientifically. But things which are not 'per se' are not necessary, for they are per accidens and, being such, are not necessary, as has been stated above. On the other hand, those things bear with necessity on a genus which are 'per se' and belong to it according to what it is. It follows, therefore, that demonstration cannot be from anything or of anything but what is 'per se.'

He further shows that even if the premises were always both necessary and true but not 'per se,' one would not know the cause why [propter quid] of the conclusion. This is clear in syllogisms which prove through signs, for although the conclusion be 'per se,' one does not know it 'per se' nor 'propter quid.' For example, if someone were to prove that every element is corruptible on the ground that it is seen to grow old. This would be a proof through a sign but neither 'per se' nor 'propter quid,' because to know 'propter quid' one must know through the cause. Therefore, the middle must be the cause of that which is concluded in the demonstration. And this is obvious from the premises: for the middle must inhere in the third causatively, that is, 'per se,' and likewise the first in the middle. Here he calls the two extremes the first and the third.

Demonstration does not skip from one genus to an alien genus

a38. It follows that we cannot in demonstrating pass from one genus to another. We cannot, for instance, prove geometrical truths by arithmetic.

a39. For there are three elements in demonstration: (1) what is proved, the conclusion — an attribute inhering essentially in a genus; (2) the axioms, that is, axioms which are premises of demonstration; (3) the subject — genus whose attributes, that is, essential properties, are revealed by the demonstration.

b2. The axioms which are premises of demonstration may be identical in two or more sciences: but in the case of two different genera such as arithmetic and geometry you cannot apply arithmetical demonstration to the properties of magnitudes unless the magnitudes in question are numbers. How in certain cases transference is possible I will explain later. Arithmetical demonstration and the other sciences likewise possess, each of them, their own genera;

b8. so that if the demonstration is to pass from one sphere to another, the genus must be either absolutely or to some extent the same.

b10. If this is not so, transference is clearly impossible, because the extreme and the middle terms must be drawn from the same genus: otherwise, as predicated, they will not be essential and will thus be accidents.

b13. That is why it cannot be proved by geometry that opposites fall under one science, nor even that the product of two cubes is a cube. Nor can the theorem of any one science be demonstrated by means of another science, unless these theorems are related as subordinate to superior (e.g., as optical theorems to geometry or harmonic theorems to arithmetic).

b17. Geometry again cannot prove of lines any property which they do not possess qua lines, that is, in virtue of the fundamental truths of their peculiar genus: it cannot show, for example, that the straight line is the most beautiful of lines or the contrary of the circle; for these qualities do not belong to lines in virtue of their peculiar genus, but through some property which it shares with other genera.

It follows that we cannot. After showing that demonstration is from things that are per se, the Philosopher here concludes that demonstration is from principles which are proper and not from extraneous or common principles.

His treatment falls into two parts.

In the first he shows that demonstration proceeds from proper principles.

In the second he establishes which principles are proper and which common, at: it is hard to be sure (76a26; [149]).

The first is divided into two parts.

In the first he shows that demonstration does not proceed from extraneous principles;

in the second that it does not proceed from common principles, at: it is clear that if the conclusion (75b37; [143]).

The first is divided into two parts.

In the first, from the preceding, he shows that demonstration is not from extraneous principles.

In the second, also from the preceding, he shows that demonstrations do not bear on corruptible things but on eternal, at: it is also clear that if the premises (75b21; [135]).

Concerning the first he does three things.

First, he states his proposition.

Second, he proves it, at: for there are three elements (75a39).

Third, he concludes to what he intended, at: that is why it cannot be proved (75b13).

He says therefore first (75a38) that inasmuch as demonstration is from things that are 'per se', it is plain that demonstration does not consist in descending or skipping from one genus to another, as geometry, demonstrating from its own principles, does not descend to something in arithmetic.

Then, at: for there are three elements (75a39) he proves this proposition.

About this he does three things.

First, he lays down the things that are necessary for a demonstration, saying that “three things are necessary in demonstrations. One is that which is demonstrated,” namely, the conclusion which, as a matter of fact, contains within itself that which inheres in its genus per se. For through demonstration a proper attribute is concluded of its proper subject. Another is the dignities [maxims or axioms] from which demonstration proceeds. The third is the generic subject whose proper attributes and per se accidents demonstration reveals.

Second, at: the axioms which are premises (75b3), he shows which of the three aforesaid can be common to various sciences and which cannot, saying that one of the three, namely, the dignities, from which demonstration proceeds, happens to be the same in diverse demonstrations and even in diverse sciences. But in those sciences whose respective generic subject is diverse, as in arithmetic which is concerned with numbers, and in geometry which is concerned with magnitudes, it does not occur that a demonstration which starts with the principles of one science, say, of arithmetic, descends to the subjects of another science, say to magnitudes, which pertain to geometry; unless perchance the subject of one science should be contained under the subject of the other, for example, if magnitudes should be contained under, numbers. (How this occurs, namely, how the subject of one science may be contained under the subject of another, will be discussed later. For magnitudes are not contained under numbers except perhaps in the sense that magnitudes are numbered.) In any case, the subjects of diverse demonstrations or sciences are diverse. For an arithmetical demonstration always has its proper genus in respect to which it demonstrates, and so do the other sciences.

Third, at: so that if the demonstration (75b8), he proves his proposition.

Here he does two things.

First he brings in the main intent after the manner of a conclusion on the ground that it can be obtained from the aforesaid, saying: Hence it is clear that it is necessary either that the genus with which the principles and conclusions deal be absolutely the same (in which case there is no descent or skipping from one genus to another); or if the demonstration is to descend from one genus to another, the two must be one thus, that is, somehow. For otherwise it is impossible for a conclusion to be demonstrated from principles, since there is not the same genus either absolutely or in a qualified sense.

However, it should be noted that a genus absolutely the same is being taken when on the part of the subject no determinate difference is admitted which is alien to the nature of that genus: for example, if someone using principles verified of triangle should proceed to demonstrate something about isosceles or about any other species of triangle. But a genus is one in a qualified sense when a difference alien to the nature of the genus is admitted of the subject, as “visual” is alien to the genus of line, and “sound” to the genus of number. Therefore, number absolutely, which is the generic subject of arithmetic, and sonant number, which is the generic subject of music, are not one genus absolutely speaking. The same goes for line absolutely, which geometry considers, and the visual line considered in optics. Hence it is clear that when matters pertaining absolutely to the line are applied to the visual line, a descent is being made to another genus—which is not the case when matters pertaining to triangle are applied to isosceles.

Second, at: if this is not so (75b10), he manifests the proposition in this way. In demonstration the middles and the extremes must belong to the same genus. Now, the extremes are contained in the conclusion: for the major extreme in the conclusion is its predicate, and the minor its subject. But the middle is contained in the premises. It is required, therefore, that the principles and conclusion be taken with respect to the same genus. When we add to this the fact that diverse sciences are of necessity concerned with subjects generically diverse, it follows that from the principles of one science, something in another science not under it may not be concluded. That the middles and extremes in a demonstration must be of one genus he proves thus: Suppose that a middle belongs to a genus other than that of the extremes which might be, for example, 'triangle' and 'have three angles equal to two right angles.' Now, a proper attribute concluded of triangle is in it per se but is not in "brazen" per se; or, conversely, if the proper attribute is per se in “brazen,” say, “high-sounding” or the like, it is obviously in triangle per accidens. This example makes it plain that if the subject of the conclusion and the middle are in entirely different genera, then the proper attribute is either not in the middle per se or else not in the subject per se. Consequently, it is in one of them per accidens. If it is in the middle per accidens, something will be in the premises per accidens. But if it is in the subject per accidens, something will be per accidens in the conclusion: and this on the part of a proper attribute. But either way something will be per accidens in the premises, inasmuch as the subject is subsumed under the middle, for example, triangle under brazen, or vice versa. However, it has been previously established that in a demonstration both the conclusion and the premises are per se and not per accidens. It is required, therefore, in demonstrations that the middle and the extremes be of the same genus.

Then, at: that is why it cannot be proved (75b13) he infers two conclusions from the foregoing.

The first conclusion is that no science demonstrates anything about the subject of another science, when this other science is more common than or entirely disparate from the first. Thus, geometry does not demonstrate that one and the same science deals with the both of two contraries: for contraries pertain to the common science, namely, to first philosophy or to dialectics. In like manner, geometry does not demonstrate that “two cubes are one cube,” that is, that the product of one cube and another cube is a cube. “Cube” here means the number which results from multiplying a number by itself twice: for example, 8 is a cube, for it is the result of multiplying 2 by itself twice, for 2 times 2 times 2 are 8. Similarly,, 27 is a cube, whose root is 3, because 3 times 3 times 3 make 27. Now, if 8 be multiplied by 27 the product is 216, which is a cube whose root is 6, because 6 times 6 times 6 make 216. Therefore, arithmetic, not, geometry has: the power to prove this. Similarly, that which pertains to one science cannot be proved by another science unless the one happens, to be under the other, as optics is under geometry, and consonance (or harmony, that is, music,) is under arithmetic.

The second conclusion, at: geometry again cannot prove (75b17) states that a science cannot prove just any random accident of its subject, but the accidents proper to its genus Thus, if something belongs to lines not as lines or not according to the proper principles of lines, geometry does not demonstrate it of lines: for example, that a straight line is the most beautiful of lines, or whether straight line is contrary or not to the curved. For these matters are outside the proper genus of line and belong to something more general. For beauty and contrary transcend the genus of line.

Demonstration is not of perishable but of eternal matters

b21. It is also clear that if the premises from which the syllogism proceeds are commensurately universal, the conclusion of such that is, in the unqualified sense — must also be eternal. Therefore, no attribute can be demonstrated nor known by strictly scientific knowledge to inhere in perishable things.

b24. The proof can only be accidental, because the attribute’s connexion with its perishable subject is not commensurately universal but temporary and special.

If such a demonstration is made, one premise must be perishable and not commensurately universal (perishable because only if it is perishable will the conclusion be perishable; not commensurately universal, because the predicate will be predicable of some instances of the subject and not of others); so that the conclusion can only be that a fact is true at the moment—not commensurately and universally.

b30. The same is true of definitions, since a definition is either a primary premise or a conclusion of a demonstration, or else only differs from a demonstration in the order of its terms.

b32. Demonstration and science of merely frequent occurrences—e.g., of eclipse as happening to the moon—are, as such, clearly eternal: whereas so far as they are not eternal they are not fully commensurate. Other subjects too have properties attaching to them in the same way as eclipse attaches to the moon.

It is also clear that if the premises. After concluding from the above that demonstration does not conclude from extraneous principles, the Philosopher intends to conclude something else from the above, namely, that demonstration is not of destructible things.

Concerning this he does two things.

First, he shows that demonstration is of eternal and not of destructible things.

Second, he shows how it is of things that occur now and then, at: demonstration and science (75b32; [142]).

Concerning the first he does two things.

First, he shows that demonstration is not of perishable but of eternal things.

Second, he shows that the same is true of definition, at: the same is true of definitions (75b30; [138]).

In regard to the first he does two things.

First, he proposes the intended conclusion.

Second, he sets down the reason proving it, at: if such a demonstration is made (75b24; [137]).

Therefore, first (75b21), he sets down two conclusions, one of which follows from the other. The first is that the conclusion of this demonstration which we are now discussing and which we can call demonstration in the full sense must be eternal: which, of course, follows from what has been stated so far, namely, that the propositions of a syllogism should be universal: which he signified by 'said of all.' The second conclusion is that neither demonstration nor science is of destructible things, that is, absolutely speaking, but only accidentally.

Then, at: if such a demonstration is made (75b24) he presents an argument to prove the proposed conclusions.

It is this: It is not the character of a destructible non-eternal conclusion to contain what is universally so, but what is so for a time and in certain instances. (For it has been established above that 'said of all' contains two things, namely, that it is not such as to be so in one case and not in another, or so at one time and not at another. But in destructible things we find that at one time something is so and at another not so. Hence it is clear that 'said of all' or “said universally” are not found in destructible things.) But where a conclusion is not universal, at least one of the premises is not universal. Therefore, a destructible conclusion must have followed from premises, one of which is not universal. Accordingly, when we add to this the fact that demonstration absolutely must be from what is universal, it follows that a demonstration cannot have a destructible conclusion, but must have an eternal one.

Then, at: the same is true of definitions (75b30) he shows that definition, too, is not of destructible but of eternal things.

The reason for this is that demonstration, both as to its principles and conclusions, is of eternal and not of destructible things. But a definition is either a principle of a demonstration, a conclusion of a demonstration, or a demonstration with a different ordering of its terms. Therefore, a definition is not of destructible, but of eternal things.

For a better understanding of this passage it should be noted that it is possible to give different definitions of the same thing, depending on the different causes mentioned. But causes are arranged in a definite order to one another: for the reason of one is derived from another. Thus the reason of matter is derived from the form, for the matter must be such as the form requires. Again, the agent is the reason for the form: for since an agent produces something like unto itself, the mode of the form which results from the action must be according to the mode of the agent. Finally, it is from the end that the reason of the agent is derived, for every agent acts because of an end. Consequently, a definition which is formulated from the end is the reason and cause proving the other definitions which are formulated from the other causes. Therefore, let us lay down two definitions, one of which is formulated from the material cause, for example, that a house is a shelter composed of stones, cement and wood; the other being formulated from the final cause, namely, that a house is a shelter protecting us from the rain and heat and cold. Now, the first definition can be demonstrated from the second in the following way: Every shelter protecting us from rain, heat and cold should be composed of wood, cement and stones; but a house is such a thing: therefore... Thus it is clear that the definition formulated from the end is the principle of the demonstration, whereas the one formulated from the matter is the conclusion of the demonstration. However, the two can be combined in the following way to form one definition: A house is a shelter composed of the aforesaid, protecting us from rain, cold and heat. But such a definition contains all the elements of a demonstration, namely, a middle and a conclusion. Accordingly, such a definition is a demonstration differing merely in the arrangement of its terms, because the only way it differs from the demonstration is that it is not arranged in a syllogistic mode and figure.

Here we might remark that because demonstration, as well as definition, is not of destructible but of eternal things, Plato was led to posit “Ideas.” For since sensible things are destructible, it appeared that there could be neither demonstration nor definition of them. As a consequence it seemed necessary to postulate certain indestructible substances concerning which demonstrations and definitions could be given. These eternal substances are what he calls “Forms” or “Ideas.”

However, Aristotle opposed this opinion above, when he said that demonstration is not of destructible things except per accidens. For although those sensible things are destructible as individuals, nevertheless in the universal they have a certain everlasting status. Therefore, since demonstration is made about those sensible things universally and not individually, it follows that demonstration is not of destructible things except per accidens, but of eternal things per se.

Then, at: demonstration and science (75b32) he shows how there can be demonstration of things that occur now and then, saying “that there are science and demonstrations of things that occur frequently, as the eclipse of the moon,” which is not always. For the moon is not always being eclipsed, but only now and then. Now, things that occur frequently, so far as they are such, that is, so far as demonstrations are given concerning them, are always; but they are particular, so far as they are not always. But demonstration cannot be of particulars, as we have shown, but only of universals. Hence it is clear that these things, insofar as there is demonstration of them, are always. And as in the case of the eclipse of the moon, so in all kindred matters.

However, there are certain differences to be noted among them. For some are not always with respect to time, but they are always in respect to their cause, because it never fails that under given conditions the effect follows, as in the eclipse of the moon. For the moon never fails to be eclipsed when the earth is diametrically interposed between sun and moon. But others happen not to be always even in respect to their causes, that is, in those cases where the causes can be impeded. For it is not always that from a human seed a man with two hands is generated, but now and then a failure occurs, owing to a defect in the efficient cause or material cause. However, in both cases the demonstration must be so set up that a universal conclusion may be inferred from universal propositions by ruling out whatever can be an exception either on the part of time alone, or also of some cause.

Demonstration does not proceed from common principles, but from principles proper to the thing demonstrated

b37. It is clear that if the conclusion is to show an attribute inhering as such, nothing can be demonstrated except from its ‘appropriate’ basic truths. Consequently a proof even from true, indemonstrable, and immediate premises does not constitute knowledge.

b40. Such proofs are like Bryson’s method of squaring the circle; for they operate by taking as their middle a common character — a character, therefore, which the subject may share with another — and consequently they apply equally to subjects different in kind. They therefore afford knowledge of an attribute only as inhering accidentally, not as belonging to its subject as such: otherwise they would not have been applicable to another genus. Our knowledge of any attribute’s connexion with a subject is accidental unless we know that connexion through the middle term in virtue of which it inheres, and as an inference from basic premises essential and ‘appropriate’ to the subject — unless we know, e.g., the property of possessing angles equal to two right angles as belonging to that subject in which it inheres essentially, and as inferred from basic premises essential and ‘appropriate’ to that subject: so that if that middle term also belongs essentially to the minor, the middle must belong to the same kind as the major and minor terms.

a8. The only exceptions to this rule are such cases as theorems in harmonics which are demonstrable by arithmetic. Such theorems are proved by the same middle terms as arithmetical properties, but with a qualification — the fact falls under a separate science (for the subject genus is separate), but the reasoned fact concerns the superior science, to which the attributes essentially belong. Thus, even these apparent exceptions show that no attribute is strictly demonstrable except from its ‘appropriate’ basic truths, which, however, in the case of these sciences have the requisite identity of character.

a17. It is no less evident that the peculiar basic truths of each inhering attribute are indemonstrable; for basic truths from which they might be deduced would be basic truths of all that is, and the science to which they belonged would possess universal sovereignty.

a19. This is so because he knows better whose knowledge is deduced from higher causes, for his knowledge is from prior premises when it derives from causes themselves uncaused: hence, if he knows better than others or best of all, his knowledge would be science in a higher or the highest degree.

a23. But, as things are, demonstration is not transferable to another genus, with such exceptions as we have mentioned of the application of geometrical demonstrations to theorems in mechanics or optics, or of arithmetical demonstrations to those of harmonics.

It is clear that if the conclusion. Above the Philosopher showed that demonstration does not proceed from extraneous principles; here he shows that it does not proceed from common principles.

Concerning this he does two things.

First, he states his proposition.

Second, he draws a conclusion from what he has said, at: it is no less evident (76a17; [146]).

Concerning the first he does two things.

First (75b37), he states his intention, saying that since it is clear that it is not just through anything at random that something is demonstrated, but it is required that demonstration be from one or another of a thing’s principles in such a way that what is demonstrated be in line with what the thing as such is, that is, it is required that the principles of the demonstration belong per se to that which is demonstrated; if, I say, this is so, then in order that something be scientifically known, it is not enough that it be demonstrated from true and immediate principles, but it is further required that it be demonstrated from proper principles.

Second, at: such proofs are like Bryson’s method (75b40), he proves his proposition, namely, that it is not enough to demonstrate something from true and immediate principles, because then something could be demonstrated in the way that Bryson proved squaring, that is, the squaring of a circle. For he showed that some square is equal to a circle, using common principles in the following way: In any matter in which it is possible to have something greater and something less than something else, one can find something equal to it. But in the genus of squares it is possible to find one which is less than a given circle, namely, one inscribed in the circle, and another which is greater, namely, one circumscribed about the circle. Therefore, one can be found which is equal to the circle. But this proof is according to something common: for equal and greater and less transcend the genus of square and circle. Hence such characteristics demonstrate according to something common, because the middle is present in other things besides the one with which the demonstration is concerned. Therefore, such characteristics, since they belong to other things as well, do not belong to the things of which they are said as to proximate subjects. Consequently, one who knows through such characteristics does not know according to what the thing is as such, that is, per se, but only per accidens. For if it were based on what the thing is according to itself, the demonstration would not apply to something of another genus. For we know a thing according to what it accidentally is, when we do not know it according to what it is in virtue of its own principles, that is, according to what comes per se from its principles, as “having three angles equal to two right angles” belongs per se to triangle, that is, according to what it is in virtue of its principles. Consequently, if the middle which one used belonged per se to the conclusion, it would necessarily have to be in the same proximity, that is, proximate according to genus, to the conclusion.

Third, at: the only exceptions to this rule (76a8), he excludes a doubt.

For it sometimes happens that the middle of a demonstration is not in the same genus as the conclusion. How this can happen he shows when he suggests that if the middle is not in the same proximate genus as the conclusion, but is present in the way that theorems in harmonics, (that is, in music), are demonstrable by arithmetic, it is nevertheless true that such things are also demonstrated. (For in a lower science there is demonstration through the principles of a higher science, as we have established, just as in the higher science there is demonstration through the principles of that higher science.) But there is this difference: on the part of the other science, namely, the lower one, there is only knowledge quia [that is, knowledge of the fact], (for the generic subject of the lower science is different from the generic subject of the higher science, from which the principles are borrowed.) But knowledge propter quid [that is, of the cause why] is proper to the higher science to which those proper attributes belong per se.

For since it is in virtue of the middle that a proper attribute is in a subject, that science will consider the 'propter quid' to which pertains the middle in which the proper attribute being demonstrated inheres per se. But if the subject belongs to another science, it will not pertain to that science to know 'propter quid' but only 'quia'. Nor will the proper attribute demonstrated of such a subject belong per se to it, but it will be through an alien middle. However, if the middle and the subject pertain to the same science, then it will be characteristic of that science to know 'quia' and 'propter quid'.

Having settled this doubt, he draws the conclusion chiefly intended, saying that in the light of the foregoing it is clear that one should not demonstrate haphazardly, that is, in any random way, but in such a way that a demonstration is made from the principles proper to each thing. But even the principles proper to the particular sciences have something common prior to them.

Then, at: it is no less evident (76a17) he draws a conclusion that follows from the aforesaid.

And in regard to this he does three things.

First, he draws the conclusion that if demonstrations in the particular sciences do not proceed from common principles, and if the principles of those sciences have something prior to them which is common, then it is clear that it is not the business of each science to prove its proper principles. For those prior principles through which the proper principles of the particular sciences can be proved are principles common to all; and the science which considers such common principles is proper to all, that is, is related to things which are common to all, as those other particular sciences are related to things respectively proper to each. For example, since the subject of arithmetic is number, arithmetic considers things proper to number. In like manner, first philosophy, which considers all principles, has for its subject “being,” which is common to all. Therefore, it considers the things proper to being (which are common to all) as proper to itself.

Second, at: this is so because he knows (76a19), he shows the pre-eminence of this science which considers common principles, namely, first philosophy, over the others. For that through which something is proved must always be scientifically more known or at least more known. For one whose science of something proceeds from higher causes must understand those causes even better, because his science proceeded from the absolutely prior, since it is not through something caused that he knows these causes. For when one’s knowledge of causes proceeds from caused things, he does not derive his knowledge from the absolutely prior and better known, but from what is prior and better known in reference to us. But when the principles of a lower science are proved by the principles of a higher science, the process is not from caused things to causes, but conversely. Therefore, such a process must go from the absolutely prior and from the absolutely more known. Thus, those items of a higher science that are used in proving matters of a lower science must be more known. Furthermore, that by which all other things are proved and which is itself not proved by anything prior must be the most known. Consequently, a higher science will be science in a fuller sense than a lower; and the highest science, namely, first philosophy, will be science in the fullest sense.

Third, at: but, as things are (76a23), he returns to the principal conclusion.

He says that demonstration does not cross over into another genus except, as already mentioned, when a demonstration from geometry is applied to certain subordinate sciences, as the mechanical arts, which employ measurements, or the perspective arts, such as the sciences which deal with vision, as optics which deals with the visual. The same applies to arithmetic in relation to harmonics, that is, music.

a26. It is hard to be sure whether one knows or not; for it is hard to be sure whether one’s knowledge is based on the basic truths appropriate to each attribute — the differentia of true knowledge. We think we have scientific knowledge if we have reasoned from true and primary premises. But that is not so: the conclusion must be homogeneous with the basic facts of the science.

a31. I call the basic truths of every genus those elements in it the existence of which cannot be proved.

a32. As regards both these primary truths and the attributes dependent on them the meaning of the name is assumed. The fact of their existence as regards the primary truths must be assumed; but it has to be proved of the remainder, the attributes. Thus we assume the meaning alike of unity, straight, and triangular; but while as regards unity and magnitude we assume also the fact of their existence, in the case of the remainder proof is required.

a37. Of the basic truths used in the demonstrative sciences some are peculiar to each science, and some are common, but common only in the sense of analogous, being of use only in so far as they fall within the genus constituting the province of the science in question.

a40. Peculiar truths are, e.g., the definitions of line and straight; common truths are such as ‘take equals from equals and equals remain’.

a42. Only so much of these common truths is required as falls within the genus in question: for a truth of this kind will have the same force even if not used generally but applied by the geometer only to magnitudes, or by the arithmetician only to numbers.

b2. Also peculiar to a science are the subjects the existence as well as the meaning of which it assumes, and the essential attributes of which it investigates, e.g., in arithmetic units, in geometry points and lines. Both the existence and the meaning of the subjects are assumed by these sciences; but of their essential attributes only the meaning is assumed. For example arithmetic assumes the meaning of odd and even, square and cube, geometry that of incommensurable, or of deflection or verging of lines, whereas the existence of these attributes is demonstrated by means of the axioms and from previous conclusions as premises. Astronomy too proceeds in the same way. For indeed every demonstrative science has three elements: (1) that which it posits, (the subject genus whose essential attributes it examines); (2) the so-called axioms, which are primary premises of its demonstration; (3) the attributes, the meaning of which it assumes.

b16. Yet some sciences may very well pass over some of these elements; e.g., we might not expressly posit the existence of the genus if its existence were obvious (for instance, the existence of hot and cold is more evident than that of number); or we might omit to assume expressly the meaning of the attributes if it were well understood. In the way the meaning of axioms, such as ‘Take equals from equals and equals remain’, is well known and so not expressly assumed. Nevertheless in the nature of the case the essential elements of demonstration are three: the subject, the attributes, and the basic premises.

It is hard to be sure. After showing that demonstration does not proceed from common but from proper principles, the Philosopher, to elucidate this point, decides questions concerning proper and common principles.

With respect to this he does two things.

First (76a26), he shows the need for such a determination, saying that it is difficult to discern whether we know from proper principles, which alone is truly scientific knowing, or do not know from proper principles. For many believe that they know scientifically if they possess a syllogism composed of things true and first. But this is not so; indeed, to know in a scientific manner it is required that the principles be proximate to the things to be demonstrated (here they are called 'first', just as above they were called “extremes”) or proximate to the first indemonstrable principles.

Second, at: I call the basic truths (76a3l), he determines concerning proper and common principles.

And in regard to this he does two things.

First, he determines concerning common and proper principles.

Second, he shows how the demonstrative sciences are related to such principles, at: the law that it is impossible to affirm (77a10; [167]).

Concerning the first he does two things.

First, he distinguishes principles from non-principles.

Second, he distinguishes the principles from each other, at: of the basic truths used (76a37; [153]).

In regard to the first he does two things.

First (76a3l), he shows what the principles are and says that the principles in any genus are those which, since they are true, happen not to be demonstrated, either not at all, if they are first principles, or at least not in that science in which they are accepted as principles. And he says, "the existence of which", (that is, whose truth), cannot be proved, to distinguish from the false which are not demonstrated in any science.

Second, at: as regards both these primary truths (76a32), he shows the points of agreement and of difference between principles and non-principles.

For principles agree with non-principles in the fact that with respect to each, that is, with respect both to the first, namely, the principles, and to the things that proceed from them, that is, things assumed from the principles, one must accept the supposition signify, because the quod quid est [that which the definition of what they signifies] of a thing pertains properly to the science concerned with substance, that is, to first philosophy, from which all other sciences accept this. But they are unlike in the fact that in regard to the principles one must accept the supposition that they are, whereas in regard to things from principles one is required to demonstrate that they are. Thus in mathematics one accepts by supposing both what unity is (which is a principle) and what straight and triangle are (which are not principles but proper attributes): but the fact that unity is or that magnitude is, the mathematician accepts as principles; but the other things, namely, things that are from principles, he demonstrates. For he demonstrates that a triangle is equilateral, or that an angle is right, or even that this line is straight.

Then, at: of the basic truths used (76a37), he distinguishes principles from one another.

First, the proper from the common.

Second, the common, one from the other, at: that which expresses necessary (76b23; [159]).

The first is divided into two parts.

First, he divides proper and common principles.

Second, he settles something which might be doubtful, at: yet some sciences (76b16; [158]).

In regard to the first he does three things.

First (76a37), he lays down a division, saying that of the principles which we use in demonstrative sciences, some are proper to each single science but others common. And because this statement might seem contrary to what he established above, namely, that demonstrative sciences do not proceed from common principles, he hastens to add that the common principles are taken in each demonstrative science according to an analogy, that is, as proportionate to that science. And this is what he means when by way of explanation he further states that it is useful to accept such principles in the sciences insofar as they pertain to the genus of the subject which is investigated in that science.

Second, at: peculiar truths (76a40), he gives examples of each, saying that proper principles are, for example, that a line or a right angle is such and such. For in the sciences the definitions, both of the subject and of the proper attribute, are held as principles. But common principles are, for example, that 'if equals be subtracted from equals, the remainders are equal', and other common conceptions in the mind.

Third, at: only so much of these common truths (76a42), he shows how the demonstrative sciences use the aforesaid principles.

First, in regard to the common principles, he says that it suffices to accept each of those common ones, so far as it pertains to the generic subject with which the science is concerned. For geometry does this if it takes the above-mentioned common principle not in its generality but only in regard to magnitudes, and arithmetic in regard to numbers. For the geometer will then be able to reach his conclusion by saying that 'if equal magnitudes be taken from equal magnitudes, the remaining magnitudes are equal', just as if he were to say that 'if equals are taken from equals, the remainders are equal'. The same must also be said for numbers.

Second, at: also peculiar to a science (76b2), he shows how the demonstrative sciences employ proper principles. And he says that proper principles are things supposed in the sciences as existing, namely, the subjects, whose proper attributes are investigated in the sciences. In this way arithmetic considers unities, and geometry considers signs, that is, points, and lines. For they suppose these things to be and to be this, that is, they suppose of them that they are and what they are. But in regard to the proper attributes they suppose what each signifies. Thus arithmetic supposes what even and odd are, and what a square or cubic number is. Similarly, geometry supposes what is rational in lines. (For a rational line is one about which we can reason, the line being given. For example, a rational line is any line commensurable with the given lines; but one which is incommensurable with it is called irrational and surd). In like manner, geometry supposes what a reflex or what a curved line are. However, these sciences demonstrate concerning all the above-mentioned proper attributes that they are, and they do so through common principles and principles demonstrated from the common principles. And what has been said of geometry and arithmetic should also be understood of astronomy. For every demonstrative science is concerned with three things: one is the generic subject whose per se attributes are investigated; another is the common dignities (axioms) from which, as from basic truths, it demonstrates; the third are the proper attributes concerning which each science supposes what their names signify.

Then, at: yet some sciences (76b16) he clarifies something about which there might be doubt.

For since he had said that the sciences suppose concerning the principles that they are and concerning the proper attributes what they are, but concerning the subject both that it is and what it is, someone might believe that he should have made special mention of all these. Hence he removes this by saying that nothing hinders certain sciences from neglecting some of the aforesaid, that is, from making express mention of them, as for example, not mentioning that it takes the existence of its generic subject for granted, if it is already obvious that it does exist. For we do not have the same evidence in all cases that they do exist, as we do in the case of number and in the case of hot and cold, the one being close to reason and the other to sense. Again, certain sciences do not suppose what the proper attributes signify in the sense of making express mention of them, just as they do not think it necessary always to make express mention of the common principles, because they are known. Be that as it may, the three above-mentioned items are naturally to be supposed in each science.

b23. That which expresses necessary self-grounded fact, and which we must necessarily believe, is distinct both from the hypotheses of a science and from illegitimate postulate — I say ‘must believe’, because all syllogism, and therefore a fortiori demonstration, is addressed not to the spoken word, but to the discourse within the soul, and though we can always raise objections to the spoken word, to the inward discourse we cannot always object.

b27. That which is capable of proof but assumed by the teacher without proof is, if the pupil believes and accepts it, hypothesis, though only in a limited sense hypothesis — that is, relatively to the pupil; if the pupil has no opinion or a contrary opinion on the matter, the same assumption is an illegitimate postulate. Therein lies the distinction between hypothesis and illegitimate postulate: the latter is the contrary of the pupil’s opinion, demonstrable, but assumed and used without demonstration.

The definition, namely, those which are not expressed as statements that anything is or is not, are not hypotheses: but it is in the premises of a science that its hypotheses are contained. Definitions require only to be understood, and this is not hypothesis — unless it be contended that the pupil’s hearing is also an hypothesis required by the teacher. Hypotheses, on the contrary, postulate facts on the being of which depends the being of the fact inferred.

b39. Nor are the geometer’s hypotheses false, as some have held, urging that one must not employ falsehood and that the geometer is uttering falsehood in stating that the line which he draws is a foot long or straight, when it is actually neither. The truth is that the geometer does not draw any conclusion from the being of the particular line of which he speaks, but from what his diagrams symbolize.

a3. A further distinction is that all hypotheses and illegitimate postulates are either universal or particular, whereas a definition is neither.

a5. So demonstration does not necessarily imply the being of Forms nor a One beside a Many, but it does necessarily imply the possibility of truly predicating one of many; since without this possibility we cannot save the universal, and if the universal goes, the middle term goes witb. it, and so demonstration becomes impossible. We conclude, then, that there must be a single identical term unequivocally predicable of a number of individuals.

That which expresses necessary self-grounded fact. After dividing common principles from proper principles, Aristotle now distinguishes among the common principles.

His treatment is divided into three parts.

In the first he lays down a distinction among the common principles.

In the second he shows the difference between a definition and a certain genus of common principles, at: the definition, namely, those (76b35; [163]).

In the third he excludes an error, at: so demonstration does not (77a5; [166]).

Concerning the first he does two things.

First, he distinguishes the common conceptions in the mind from postulates or suppositions.

Second, he distinguishes among the latter, at: that which is capable of proof (76b27; [162]).

He says the following about the first.

It should be noted that the common. conceptions in the mind have something in common with the other principles of demonstration and something proper: something common, because both they and the others must be true in virtue of themselves. But what is proper to the former is that it is necessary not only that they be true of themselves but that they be seen to be such. For no one can think their contraries.

He says, therefore, that that principle of which it is not only required that it be in virtue of itself but further required that it be seen (76b23), namely, a common conception in the mind or a dignity, is neither a postulate nor a supposition. He proves this in the following way: A postulate and a supposition can be confirmed by a reason from without, that is, by some argumentation; but a common conception in the mind does not bear on a reason from without (because it cannot be proved by any argument), but bears on that reason which is in the soul, because it is made known at once by the natural light of reason. That it does not bear on any reason from without is shown by the fact that a syllogism is not formed to prove such common conceptions of the mind. Furthermore, that these are not made known by an outward reason but by the inward he proves by the fact that it is possible to contest an outward reason, either truly or apparently, but it is not always possible to do so with the inward reason.

This is so because nothing is so true that it cannot be denied orally. (For even this most evident principle that the same thing cannot be and not be has been orally denied by some). On the other hand, some things are so true that their opposites cannot be conceived by the intellect. Therefore, they cannot be challenged in the inward reason but only by an outward reason which is by the voice. Such are the common conceptions in the mind.

Then, at: that which is capable of proof (76b27) he distinguishes suppositions and postulates from one another.

Here, too, it should be noted that they have something in common and something in which they differ. What is common to them is that although they can be demonstrated, the demonstrator assumes them without demonstrating, (chiefly because they are not demonstrable by his own science but by another, as explained above. Hence they are reckoned among the immediate principles because the demonstrator uses them without a middle, since they do not have a middle in that science).

Yet they do differ, because if such a proposition is accepted as reasonable by the pupil to whom a demonstration is being made, it is called a “supposition.” In that case it is not called a supposition absolutely, but relative to him. But if the pupil has no reason for or against the proposition, the demonstrator must request him to admit it as reasonable, and then it is called a “postulate.” However, if the pupil has a contrary opinion, then it will be a “question,” which must be settled between them. At any rate, what is common to all of them is that the demonstrator uses each of thein without demonstrating, although they are demonstrable.

Then, at: the definition, namely (76b35) he distinguishes definitions from suppositions with two reasons, the second of which begins at: a further distinction (77a3; [165]).

In regard to the first he does two things.

First, he presents this reason: Every postulate or supposition declares something to be or not to be; but terms, that is, definitions, do not declare that something is or is not. Terms, therefore, taken by themselves, are neither postulates nor suppositions. But when they are employed in propositions, they are suppositions, as in the statement, “Man is a rational mortal animal.” But terms taken by themselves are only understood; and understanding is not supposing, any more than hearing is. Rather supposition bears on things that exist such that a conclusion is made from them in regard to what they are, that is, in virtue of the premises.

Second, at: nor are the geometer’s hypotheses (76b39), he excludes a doubt.

For there were some who said that a geometer uses a false supposition when he says that a line not one foot long is one foot long, or that a line traced in the sand is straight when it really is not. But he says that a geometer does not on this account suppose something false. For since the geometer demonstrates nothing of singulars but of universals, as explained above, and these lines are singulars, it is obvious that he is not demonstrating anything about these lines or from these lines; rather he is using them as examples of the universals (which are understood by examples) from which and about which he demonstrates.

Then, at: a further distinction (77a3) he presents the second reason which is this: every supposition or postulate is in whole or in part, that is, is either a universal proposition or a particular. But definitions are neither of these, because nothing is placed or predicated in them universally or particularly. Therefore...

Then, at: so demonstration does not (77a5) he shows from the foregoing that it is not necessary to posit “Ideas,” as Plato did. For it was shown above that demonstrations are concerned with universals and in that sense with eternal things. Therefore, it is not necessary for the validity of demonstration that there be forms, that is, “Ideas,” or a one outside the many, as Plato posited separated mathematical beings along with Ideas in order that thereby demonstrations might bear on eternal things. What is required is that there be one in many and about many, if there is to be demonstration, because it will not be universal unless it is one about many. And if it is not universal, there will not be a middle of demonstration and, consequently, no demonstration. That the middle of a demonstration must be universal is plain from the fact that the middle of a demonstration must be some one same thing predicated of many not equivocally but according to the same aspect, which is a universal aspect. But if it should happen to be equivocal, a defect in reasoning would occur.

Relation between demonstrative sciences and common principles

a10. The law that it is impossible to affirm and deny simultaneously the same predicate of the same subject is not expressly posited by any demonstration except when the conclusion also has to be expressed in that form; in which case the proof lays down as its major premise that the major is truly affirmed of the middle but falsely denied. It makes no difference, however, if we add to the middle, or again to the minor term, the corresponding negative. For grant a minor term of which it is true to predicate man — even if it be also true to predicate not-man of it — still grant simply that man is animal and not not-animal, and the conclusion follows: for it will still be true to say that Callias — even if it be also true to say that not-Callias — is animal and not not-animal. The reason is that the major term is predicable not only of the middle, but of something other than the middle as well, being of wider application; so that the conclusion is not affected even if the middle is extended to cover the original middle term and also what is not the original middle term.

a22. The law that every predicate can be either truly affirmed or truly denied of every subject is posited by such demonstration as uses reductio ad impossibile, and then not always universally, but so far as it is requisite; within the limits, that is, of the genus — the genus, I mean (as I have already explained), to which the man of science applies his demonstrations.

a26. In virtue of the common elements of demonstration — I mean the common axioms which are used as premises of demonstration, not the subjects nor the attributes demonstrated as belonging to them — all the sciences have communion with one another,

a29. and in communion with them all is dialectic and any science which might attempt a universal proof of axioms such as the law of excluded middle, the law that the subtraction of equals from equals leaves equal remainders, or other axioms of the same kind. Dialectic has no definite sphere of this kind, not being confined to a single genus. Otherwise its method would not be interrogative; for the interrogative method is barred to the demonstrator, who cannot use the opposite facts to prove the same nexus. This was shown in my work on the syllogism.

The law that it is impossible. After determining about proper and common principles, the Philosopher now shows how the demonstrative sciences behave in regard to the common and to the proper principles.

His treatment falls into two parts.

In the first he shows how the demonstrative sciences are related to common principles.

In the second he shows how they are related to proper principles, at: if a syllogistic question (77a36; [173]).

Concerning the first he does two things.

First, he shows how the demonstrative sciences are related to the first of the common principles.

Second, how they are related generally to all the common principles, at: in virtue of the common elements (77a29; [170]).

Concerning the first he does two things.

First, he shows how the demonstrative sciences behave relative to the principle that one should not affirm and deny the same thing.

Second, how they behave relative to the principle that "there is either true affirmation or true negation of each thing", at: the law that every predicate (77a22; [169]). For, as is proved in Metaphysics IV, these two principles are the first of all principles.

He says therefore first (77a10) that no demonstration makes use of the principle that one does not affirm and deny at one and the same time. For if a demonstration were to use it to show some conclusion, it would have to use it in such a way as to assert that the first, that is, the major, extreme is affirmed of the middle and not denied; because if it were to admit affirmation and negation on the part of the middle, it would make no difference whether it was this way or that. And the same reasoning holds for the third, that is, minor, extreme in relation to the middle. For example, take 'animal' as the first, 'man' as the middle’ and 'Callias' as the third. If someone wished to use this principle in a demonstration, he would have to argue in the following way: "Every man is an animal and is not a non-animal; But Callias is a man: Therefore, Callias is an animal and is not a non-animal". For when we say that every man is an animal, it matters not whether it is also true that a non-man is an animal, or whether it is not true. Similarly, in the conclusion it matters not, Callias being an animal, whether non-Callias is an animal or not. Now, the cause of this is that the first is not limited to being said only of the middle, but can also be said of something else diverse from the middle and described by a negation of the middle (since the first is sometimes said of many things other than the middle, as 'animal' is said of things other than 'man'; for it is said of horse, which is not man). Hence if the middle is taken the same and not the same, that is, if an affirmative and a negative middle be taken, as when I say, “man" and "non-man” are animals, it contributes nothing to the conclusion.

However, if the affirmation and negation are taken on the part of the major extreme, it does make a difference both as to the conclusion and as to the truth of the premises. For if man were not an animal, it would not be true that man is an animal, nor would it follow that Callias is an animal. Yet nothing more is verified by stating that man is an animal and is not a non-animal than by merely stating that man is an animal, for the same thing is conveyed by each. And thus it is clear that demonstrations do not use the principle that affirmation and negation are not simultaneously true, either on the part of the predicate or on the part of the subject.

Then, at: the law that every predicate (77a22) he shows how demonstrative sciences use the principle that "of anything there is either true affirmation or negation".

And he says that this principle is utilized in a demonstration leading to the impossible. For in this demonstration something is proved to be true by the fact that its opposite is false. (This of course would never happen, if it were possible for the two opposites to be false). Nevertheless such a demonstration does not always employ this principle, for sometimes that opposite which is shown to be false is not a negation but an immediate contrary. For example, if a number were shown to be even on the ground that its opposite, namely, “it is odd,” is false, and this were done by leading to the impossible. Neither does it use this principle universally, that is, in its universality, namely, under the terms 'being' and 'non-being', but only so far as is sufficient for a genus or so far as it is narrowed to a generic subject. And I mean the first genus involved in the demonstration. For example, in leading to the impossible in geometry the terms would be “straight” and “non-straight” in the case where it is shown that some line is straight on the ground that it is false to state that it is not straight.

Then, at: in virtue of the common elements (77a26) he shows how the sciences as a community function relative to all common principles.

In regard to this he does two things.

First, he says that all the sciences share alike in the common principles in the sense that they all use them as items from which they demonstrate-which is to use them as principles. But they do not use them as things about which they demonstrate something, that is, as subjects, or as things which they demonstrate, that is, as conclusions.

Second, at: and in communion with them (77a29), he shows that certain sciences employ the common principles in a manner other than has been described. For dialectics is concerned with common things, and a certain other science is concerned with common things, namely, first philosophy, whose subject is being and which considers the things which follow upon being as the proper attributes of being.

Yet it should be noted that dialectics is concerned with common things under an aspect different from logic and first philosophy. For first philosophy is concerned with common things because its consideration is focused on those common things, namely, on being and on the parts and attributes of being.

But because reason occupies itself with all things that are, and logic studies the operations of reason, logic will also be concerned with matters common to all things, that is, with reason’s intentionalities which bear on all things, but not in such a way that logic has these common things as its subject-for logic considers as its subject the syllogism, enunciation, predication, and things of that type. But although the part of logic which is demonstrative is engaged in teaching about common intentionalities, the use of a demonstrative science does not consist in proceeding from common intentionalities to show anything about the things which are the subjects of the other sciences.

But dialectics does this, for it goes from common intentions and argues to things that pertain to other sciences, whether they be proper or common things, but mainly common things. Thus, it argues that hatred is in the concupiscible appetite, because love is, on the ground that contraries are concerned with a same thing. Consequently, dialectics is concerned with common things not only because it treats concerning the common intentionalities of reason, which is common to all logic, but also because it argues about common characteristics of things.

But any science that argues about the common characteristics of things must argue about the common principles, because the truth of the common principles is made manifest from the knowledge of common terms, as “being” and “non-being,” “whole” and “part” and the like. It is significant that he says, and any science which might attempt, because first philosophy does not demonstrate the common principles, since they are absolutely indemonstrable, although some have unwittingly attempted to demonstrate them, as is stated in Metaphysics IV. Or else, because even though they cannot, strictly speaking, be demonstrated, the first philosopher attempts to uphold them in a way that is possible, namely, by contradicting those who deny them, appealing to things that must be conceded by them, though not to things which are more known. It should also be noted that the first philosopher demonstrates them not only in this way, but also shows something about them as about subjects: for example, that “it is impossible for the mind to think their opposites,” as is clear from Metaphysics IV.

Therefore, although both the first philosopher and the dialectician debate about these first principles, yet one does so in one way and the other in another way. For the dialectician neither proceeds from demonstrative principles nor takes only one side of a contradiction but is open to both. For each side might happen to be probable or be upheld by probable statements, which the dialectician utilizes: and that is why he asks [his questions in terms of two alternatives]. But the demonstrator does not ask [in that way], because he is not open to opposites. And this is the difference between the two, as was laid down in the treatment of the syllogism, namely, in the beginning of the Prior Analytics. Therefore, first philosophy in treating common principles proceeds after the manner of a demonstration and not after the manner of a dialectical disputation.

a36. If a syllogistic question is equivalent to a proposition embodying one of the two sides of a contradiction, and if each science has its peculiar propositions from which its peculiar conclusion is developed, then there is such a thing as a distinctively scientific question, and it is the interrogative form of the premises from which the ‘appropriate’ conclusion of each science is developed. Hence it is clear that not every question will be relevant to geometry, nor to medicine, nor to any other science:

a42. only those questions will be geometrical which form premises for the proof of the theorems of geometry or of any other science, such as optics, which uses the same basic truths as geometry. Of the other sciences the like is true.

b2. Of these questions the geometer is bound to give his account, using the basic truths of geometry in conjunction with his previous conclusions; of the basic truths the geometer, as such, is not bound to give any account. The like is true of the other sciences.

b6. There is a limit, then, to the questions which we may put to each man of science; nor is each man of science bound to answer all inquiries on each several subject, but only such as fall within the defined field of his own science.

b8. If, then, in controversy with a geometer qua geometer the disputant confines himself to geometry and proves anything from geometrical premises, he is clearly to be applauded; if he goes outside these he will be at fault, and obviously cannot even refute the geometer except accidentally. One should therefore not discuss geometry among those who are not geometers, for in such a company an unsound argument will pass unnoticed. This is correspondingly true in the other sciences.

If a syllogistic question. After showing how demonstrative sciences function in relation to common principles, the Philosopher now shows how they employ proper principles.

And his treatment falls into two parts.

In the first he shows that in each science there are questions, responses and disputations peculiar to each.

In the second he shows how in each science there are deceptions, peculiar to each, at: since there are “geometrical” questions (77b16; [180]).

Concerning the first he does two things.

First, he, shows that in each science there are its own questions.

Second, that each science employs the responses and disputations proper to it, at: there is a limit (77b6; [178]).

Concerning the first he does two things.

First, he shows that in each science there are questions peculiar to each.

Second, what these are, at: only those questions will be geometrical (77a42; [176]).

He shows the first (77a36) in the following way. A syllogistic question and a proposition which takes one definite side of a contradiction are the same as to content, but they have not the same function. (For the answer which one gives to the question is made to serve as a proposition in some syllogism). But in each of the sciences, the propositions from which a syllogism is formed, are peculiar to it, for it has been shown that every science proceeds from things proper to it. Therefore, in each science there are questions peculiar to it. Hence not any random question is pertinent to geometry or to medicine or to some one of the other sciences.

However, it should be noted that questioning occurs one way in demonstrative sciences and another way in dialectics. For in dialectics not only the conclusion but also the premises are open to question; but in demonstrative sciences the demonstrator takes premises as per se known or proved by such principles. Hence, he asks only about the conclusion. And when he has demonstrated it, he uses it as a proposition to demonstrate some other conclusion.

Then, at: only those questions will be geometrical (77a42) he explains which questions are peculiar to each science.

First, insofar as they are taken as propositions from which the demonstrator proceeds.

Second, when taken as conclusions, at: of these questions the geometer (77b2; [177]).

He says therefore first (77a42) that geometric questions are ones from which something is demonstrated pertaining to matters of geometry or pertaining to matters demonstrated from the principles of geometry—for example, the ones from which something is demonstrated in optical science, that is, perspective, which proceeds from the principles of geometry. And what is said of geometry applies to other sciences, namely, that a proposition or question is peculiar to a science, if it is one from which a demonstration in that science or in a subalternate science proceeds.

Then, at: of these questions the geometer (77b2) he analyzes geometric questions insofar as they are conclusions, saying that solutions to geometric questions must be proved by demonstrating their truth from geometric principles and from conclusions already demonstrated by those principles. (For sometimes the proofs of certain geometric demonstrations are not drawn from the first principles of geometry, but from matters concluded from those first principles). Hence when it is a matter of questions which are always conclusions in demonstrative sciences, the proof can be obtained in the science itself, but the proof of the principles cannot be drawn from geometry qua geometry. The same also applies to other sciences. For no science proves its own principles, as we have explained above. And he says, "from geometry qua geometry", because it may happen that a science proves its own principles, insofar as that science assumes the principles of another science, as a geometer proves his own principles insofar as he assumes the role of first philosopher, that is, of metaphysician.

Then, at: there is a limit (77b6), he shows that each has its own responses and disputations.

First, that each has its own responses.

Second, its own disputations, at: if, then, in controversy (77b8).

He says therefore (77b6) that from the foregoing it is clear that each scientific knower does not ask just any question whatever. Hence it is also clear that he will not answer just any random question but only those which conform to his own science on the ground that a question and its answer pertain to the same science.

Then (77b8) because disputations are concerned with a question and its answer, he shows that there are disputations—peculiar to each science, saying that if a geometer disputes with a geometer precisely as geometer, that is, in matters pertaining to geometry, then obviously the disputation goes well if the disputation not only concerns a point of geometry but proceeds from the principles of geometry. But it does not go well, if the disputation in geometry does not proceed along these lines. For if someone disputes with a geometer in matters alien to geometry, it is clear that he does not argue, that is, does not convince, except accidentally: for example, if the dispute concerns music, and the geometer accidentally happens to be a musician. Hence it is clear that one should not dispute in geometry about matters not geometric, because it will not be possible to judge by the principles of that science whether the dispute went favorably or unfavorably: and the same holds for other sciences.

Each science has its own deceptions and areas of ignorance

b16. Since there are “geometrical” questions, does it follow that there are also distinctively “ungeometrical” questions?

b17. Further, in each special science—geometry for instance—what kind of error is it that may vitiate questions, and yet not exclude them from that science?

b19. Again, is the erroneous conclusion one constructed from premises opposite to the true premises, or is it formal fallacy though drawn from geometrical premises?

b21. Or, perhaps, the erroneous conclusion is due to the drawing of premises from another science; e.g., in a geometrical controversy a musical question is distinctively ungeometrical,

b23. whereas the notion that parallels meet is in one sense geometrical, being ungeometrical in a different fashion: the reason being that ‘ungeometrical’, like ‘unrhythmical’, is equivocal, meaning in the one case not geometry at all, in the other bad geometry? It is this error, that is, error based on premises of this kind — ‘of’ the science but false — that is the contrary of science.

b27. In mathematics the formal fallacy is not so common, because it is the middle term in which the ambiguity lies, since the major is predicated of the whole of the middle and the middle of the whole of the minor (the predicate of course never has the prefix ‘all’); and in mathematics one can, so to speak, see these middle terms with an intellectual vision, while in dialectic the ambiguity may escape detection. e.g., ‘Is every circle a figure?’ A diagram shows that this is so, but the minor premise ‘Are epics circles?’ is shown by the diagram to be false.

b34. If a proof has an inductive minor premise, one should not bring an ‘objection’ against it. For since every premise must be applicable to a number of cases (otherwise it will not be true in every instance, which, since the syllogism proceeds from universals, it must be), then assuredly the same is true of an ‘objection’; since premises and ‘objections’ are so far the same that anything which can be validly advanced as an ‘objection’ must be such that it could take the form of a premise, either demonstrative or dialectical.

b40. On the other hand, arguments formally illogical do sometimes occur through taking as middles mere attributes of the major and minor terms. An instance of this is Caeneus’ proof that fire increases in geometrical proportion: ‘Fire,’ he argues, ‘increases rapidly, and so does geometrical proportion’. There is no syllogism so, but there is a syllogism if the most rapidly increasing proportion is geometrical and the most rapidly increasing proportion is attributable to fire in its motion.

a5. Sometimes, no doubt, it is impossible to reason from premises predicating mere attributes: but sometimes it is possible, though the possibility is overlooked.

a8. If false premises could never give true conclusions ‘resolution’ would be easy, for premises and conclusion would in that case inevitably reciprocate. I might then argue thus: let A be an existing fact; let the existence of A imply such and such facts actually known to me to exist, which we may call B. I can now, since they reciprocate, infer A from B.

a10. Reciprocation of premises and conclusion is more frequent in mathematics, because mathematics takes definitions, but never an accident, for its premises—a second characteristic distinguishing mathematical reasoning from dialectical disputations.

a13. A science expands not by the interposition of fresh middle terms, but by the apposition of fresh extreme terms. e.g., A is predicated of B, B of C, C of D, and so indefinitely. Or the expansion may be lateral: e.g., one major A, may be proved of two minors, C and E. Thus let A represent number — a number or number taken indeterminately; B determinate odd number; C any particular odd number. We can then predicate A of C. Next let D represent determinate even number, and E even number. Then A is predicable of E.

Since there are “geometrical” questions. After showing that each science has its own questions, responses and disputations, the Philosopher shows that each science has its own deceptions and errors.

And his treatment is divided into two parts.

In the first he raises certain questions.

In the second he solves them, at: or, perhaps, the erroneous conclusion (77b21; [182]).

Accordingly, he poses three questions.

The first (77b16) is this: Since there are geometric questions, as we have shown, are there not also non-geometric ones? And what is asked of geometry can be asked of every other science.

Then, at: further, in each special science (77b17) he poses the second question, namely: May questions which arise from ignorance bearing on some particular science be called geometric; and likewise for questions proper to any other science? (Questions arising from ignorance bearing on some science are those which ask about matters contrary to the truths of that science).

Then, at: again, is the erroneous conclusion (77b19) he poses the third question, namely: In each science it is possible to be deceived by a syllogism which he calls according to ignorance. But deception through a syllogism can occur in two ways: in one way when it fails as to form, not observing the correct form and mode of a syllogism. In another way, when it fails in matter, proceeding from the false. Now, there is a difference between these two ways, because one that fails in matter is still a syllogism, since everything is observed that pertains to the form of a syllogism. But one that fails in form is not even a syllogism, but a paralogism, that is, an apparent syllogism. In dialectics, deception can occur in both these ways. Hence in Topics I Aristotle speaks of the contentious, which is a syllogism, and of the one defective in form which is not a syllogism but an apparent one. Hence the question is this: Whether or not a syllogism of ignorance which is used in the demonstrative sciences is a syllogism based on matters opposed to the science, that is, one that proceeds from false premises, or a paralogism, namely, one that fails in form, which is not a syllogism, but an apparent one?

Then, at: or, perhaps, the erroneous conclusion (77b21) he solves these questions.

First, he solves the first.

Second, the second, at: whereas the notion (77b23; [183]).

Third, the third, at: in mathematics the formal fallacy (77b27; [184]).

He says therefore first (77b21) that a completely non-geometric question is one which is formed entirely from another art, say, music. For example, if one asks in geometry whether a tone could be divided into two equal semi-tones, such a question would be entirely non-geometric, because it concerns matters which do not pertain at all to geometry.

Then, at: whereas the notion (77b23) he solves the second question.

He says that a question about geometry, that is, about matters pertinent to geometry, when a person is asked about something which is against the truth of geometry (for example, if the questions concerned parallels meeting, that is, equidistant lines coming together), would be geometric in one sense and non-geometric in another. For just as arrythmon, that is, without rhythm or sound, can be taken in two senses: in one sense for that which has no sound at all, as wool, and in another sense for that which does not give a good sound, as a poorly-sounding bell, so a question is called non-geometric in two ways. In one way, because it is completely alien to geometry, as a question about music. In another way, because it mistakenly holds something in the field of geometry, namely, because it holds something contrary to geometric truth. Therefore, the question about the convergence of parallel lines is non-geometric not in the first way, since it touches on a point of geometry, but in the second way, because it is mistaken about some point of geometry. Such ignorance, namely, which consists in wrongly using the principles of geometry is contrary to the truth of geometry.

Then, at: in mathematics (77b27) he solves the third question.

And he does two things.

First, he shows that in demonstrative sciences there is no paralogism in language.

Second, nor apart from language, at: if a proof has an inductive minor premise (77b34; [185]).

Now although paralogism in language occurs in any of six sophistical ways, he takes one of them, namely, the paralogism which proceeds by way of equivocation, and shows that such a paralogism cannot occur in demonstrative sciences, being easier to detect.

He says, therefore, that formal paralogism, that is, a syllogism defective in form, as in dialectics, does not occur in the disciplines. For in a demonstrative syllogism the middle must always be the same in two ways, that is, the same middle must be compared to the two extremes: for the major extreme is predicated universally of the middle, and the middle is predicated universally of the minor extreme, even though when it is predicated, we do not say “every,” that is, the sign of universality is not applied to the predicate. But in the fallacy of equivocation the middle is the same according to vocal sound but not according to reality. Consequently, it escapes notice when it is proposed orally; but if it is demonstrated to the senses, the deception cannot succeed. For example, the name 'circle' is said equivocally of a figure and a poem. Therefore, in reasons, that is, in argumentations, the ambiguity may go unnoticed, that is, deception is possible, as for example, if one were to say: “Every circle is a figure; Homer’s poem is a circle: therefore, Homer’s poem is a figure.” But if a circle is drawn for someone to look at, there cannot be deception. For it will be obvious that songs are not circles. Now, just as in this case the deception is prevented by presenting the middle to the senses, so in demonstrative syllogisms, deception is prevented by the fact that the middle is shown to the intellect. For when something is defined, it is as plain to the intellect as something sensibly drawn is plain to the sight. Hence he says that in demonstrative syllogisms these, that is, things defined, are seen by the intellect. But in demonstrations one always proceeds from definitions. Hence, deception through equivocation has no place there, much less through any of the other fallacies of language.

Then, at: if a proof has an inductive minor premise (77b34), he shows that in demonstrative syllogisms paralogism according to fallacy outside of language cannot occur.

And because a paralogism of this kind is frequently challenged by citing an objection, through which the defect in the form of syllogizing is shown.

First, he shows how an objection should be presented in demonstrative matters.

Second, he shows that in demonstrative matters there cannot be paralogism according to fallacy outside the language, at: on the other hand, arguments (77b40; [186]).

He says therefore first (77b34) that one should not bring an objection against it, that is, against a paralogism, by citing an inductive, that is, particular, proposition. (For an induction proceeds from particulars as a syllogism proceeds from universals). The reason for this is that in demonstrative matters no proposition is admitted unless it is verified in the greater number of cases, for if it is not verified in the greater number it will not be in all. But a demonstrative syllogism must proceed from universals. Therefore, it is obvious that in demonstrative matters an objection must be universal, because the propositions and the objections must be the same. For in dialectical and in demonstrative matters that which is taken as an objection is later used as a proposition to syllogize against the one who proposed.

Then, at: on the other hand, arguments (77b40) he shows that in demonstrative matters deception through paralogisms outside of language does not occur.

And just as above he showed that there was no paralogism in language in demonstrative matters by showing it for one, namely, for the paralogism which employs the fallacy of equivocation, so now he shows that in demonstrative matters there is not paralogism outside of language by showing it of the one which relies on fallacy of consequent. For it is obvious that paralogism. in demonstrative matters cannot occur according to the other fallacies outside of language: not the fallacy according to accident, because demonstration proceeds from things that are per se; nor according to qualified and absolute, because the statements used in demonstrations are taken universally and always, and without qualification.

Therefore, he does two things in regard to his thesis.

First, he shows how paralogism according to fallacy of consequent works.

Second, that deception does not take place in this manner in demonstrative matters, at: sometimes, no doubt (78a5; [188]).

He says therefore first (77b40) that illogical arguments do sometimes occur, that is, do not observe syllogistic form, because they admit both inherences, that is, they accept a middle predicated affirmatively of each extreme, which is the same as syllogizing in the second figure but employing two affirmative propositions, thus committing the fallacy of consequent. This is what the philosopher Caeneus did in order to show that fire is in multiple proportion, that is, that fire is generated in greater quantity than was the body from which it is generated, on the ground that fire, being the most rarified of bodies, is generated from other bodies through rarefaction. Hence it is required that the matter of the previous body be spread out under larger dimensions when assuming the form of fire. To prove this he used the following syllogism: “Whatever is generated in multiplied proportion is generated quickly; but fire is generated quickly: therefore, fire is generated in multiplied proportion.”

Then, at: sometimes, no doubt (78a5) he shows that deception arising from this form of reasoning does not occur in demonstrative sciences.

In regard to this he does two things.

First, he makes it clear that this form of syllogizing does not always end in deception, saying that according to this form of arguing there are cases when one cannot syllogize from the premises, namely, when the terms are not convertible-for it does not follow, if every man is an animal, that every animal is a man. But now and then there are cases when one can syllogize, namely, when the terms are convertible. For just as it follows that if a thing is a man, it is a rational mortal animal, so conversely, if a thing is a rational mortal animal, it is a man. However, it does not seem to follow syllogistically, because the due form of a syllogism is not observed.

Second, at: if false premises could never give (78a7), he shows that it is possible to syllogize in the abovementioned manner without deception.

And he shows this in three ways.

The first is this: According to the above-mentioned manner of syllogizing, deception occurs because the consequence which was assumed convertible is not converted. But in this situation deception would not occur, if to the extent that the conclusion is true, so are the premises true: for there will then be no deception in converting. For if it is stated of Socrates that he is a man, therefore he is an animal; no deception or falsity follows: but it does if it is argued conversely that he is an animal, therefore he is a man. But if the premise is true and the conclusion false, then deception occurs in converting. For example, if I were to say: “If an ass is a man, it is an animal; therefore, if it is an animal, it is a man.” Consequently, if it were impossible to deduce the true from the false, and it were always necessary to deduce the true from the true, then it would be easy to analyze a conclusion into its principles without deception, because there would be no falsity if either of the premises were to be inferred from the conclusion. On this supposition the conclusion and premises would be converted of necessity as to truth. For just as the conclusion is true if the premise is true, so vice versa. For suppose that A is, and granting this, suppose that the same things follow as are known by me to be true, say B. Hence since both are true, I can then also infer A from B.

And so, one reason why deception through fallacy of consequent does not occur in demonstrative sciences is that it is impossible in demonstrative sciences for the true to be syllogized from the false, as we have explained above.

Then, at: reciprocation of premises (78a10) he gives the second reason. For in cases of convertible terms, deception through fallacy of consequent does not occur, because in those cases the consequent is converted. Now, the things which are used in mathematical, that is, in demonstrative sciences, are for the most part convertible, because these sciences do not admit as a middle anything predicated accidentally, but only definitions which are principles of demonstration, as we have explained. And this is their point of difference from the statements in dialogues, that is, in dialectical syllogisms, in which accidens are frequently admitted.

Then, at: a science expands not by the interposition (78a13) he gives the third reason and it is this: In demonstrative sciences there are determinate principles from which one proceeds to the conclusions. Hence it is possible to return from the conclusions to the principles as from something determinate to something determinate. That demonstrations do proceed from determinate principles he shows by the fact that demonstrations are not increased by middles, that is, in demonstrations one does not use a series of different middle terms to demonstrate one conclusion. (This, of course, is to be understood of demonstrations propter quid, of which he is speaking). For of one effect there can be but one sole cause why it is. But although demonstrations are not increased by using several middles of demonstration, nevertheless they are increased in two ways:

in one way, by assuming one after another, that is, subsuming one middle under another, say B under A, C under B, D under C, and so on. For example, when “having three angles equal to two right angles” is proved of triangle on the ground that it is a figure having an external angle equal to the two opposite interior angles, and is then proved of isosceles on the ground that it is a triangle.

Another way they are increased is laterally, as when A is proved of C and of E. For example, every quantified number is either finite or infinite. Let A be this predicate, that is, “to be finite or infinite.” But an odd number is a quantified number. Let B represent “quantified number,” and C, “odd number.” It follows therefore, that A (to be finite or infinite) is predicated of C (odd number), that is, that an odd number is either a finite or an infinite number. And he says that it is possible along the same lines to conclude concerning even number, and this through the same middle.

The passage: a science expands (78a13) can be introduced in another way. Since he had just said that in demonstrative matters, definitions are used for middles, and since there is one sole definition of one thing, it follows that it is not through middles that, demonstrations are increased.

a22. Knowledge of the fact differs from knowledge of the reasoned fact. To begin with, they differ within the same science and in two ways: (1) when the premises of the syllogism are not immediate (for then the proximate cause is not contained in them — a necessary condition of knowledge of the reasoned fact): (2) when the premises are immediate, but instead of the cause the better known of the two reciprocals is taken as the middle; for of two reciprocally predicable terms the one which is not the cause may quite easily be the better known and so become the middle term of the demonstration.

a30. Thus (2) (a) you might prove as follows that the planets are near because they do not twinkle: let C be the planets, B not twinkling, A proximity. Then B is predicable of C; for the planets do not twinkle. But A is also predicable of B, since that which does not twinkle is near — we must take this truth as having been reached by induction or sense-perception. Therefore, A is a necessary predicate of C; so that we have demonstrated that the planets are near. This syllogism, then, proves not the reasoned fact but only the fact; since they are not near because they do not twinkle, but, because they are near, do not twinkle.

a39. The major and middle of the proof, however, may be reversed, and then the demonstration will be of the reasoned fact. Thus: let C be the planets, B proximity, A not twinkling. Then B is an attribute of C, and A — not twinkling — of B. Consequently A is predicable of C, and the syllogism proves the reasoned fact, since its middle term is the proximate cause.

b3. Another example is the inference that the moon is spherical from its manner of waxing. Thus: since that which so waxes is spherical, and since the moon so waxes, clearly the moon is spherical. Put in this form, the syllogism turns out to be proof of the fact, but if the middle and major be reversed it is proof of the reasoned fact; since the moon is not spherical because it waxes in a certain manner, but waxes in such a manner because it is spherical. (Let C be the moon, B spherical, and A waxing.)

b10. Again (b), in cases where the cause and the effect are not reciprocal and the effect is the better known, the fact is demonstrated but not the reasoned fact.

Knowledge of the fact. After determining about demonstration propter quid, the Philosopher here shows the difference between demonstration quia and demonstration propter quid.

And he does two things about this.

First, he shows how they differ in the same science.

Second, in diverse sciences, at: but there is another way (78b33; [207]).

Concerning the first he does two things.

First, he states the twofold difference between the two in the same science.

Second, he clarifies this with examples, at: thus you might prove (78a30; [196]).

He says therefore first (78a22) that, as said above, demonstration is a syllogism causing scientific knowledge and proceeds from the causes both first and immediate of a thing. Now, this is to be understood as referring to demonstration 'propter quid'. But there is a difference between knowing 'that' a thing is so and 'why' it is so. Therefore, since demonstration is a syllogism causing scientific knowledge, as has been said, it is necessary that a demonstration 'quia' which makes one know that a thing is so should differ from the demonstration 'propter quid' which makes one know why. Consequently, this difference must be considered first in the same science and later in sciences that are diverse.

In one and the same science each of the above is said to differ in regard to the two things required for demonstration in the strict sense—which causes knowledge of the why—namely, that it be from causes and from immediate causes. Hence one way that scientific knowledge 'quia' differ from 'propter quid' is that it is the former if the syllogism is not through, immediate principles but through mediate ones. For in that case the first cause will not be employed, whereas science propter quid is according to the first cause; consequently, the former will not be science 'propter quid'.

It differs in another way, because it is science 'quia' when the syllogism, although not through middles, that is, mediate, but through immediate things, is not through the cause but through “convertence,” that is, through effects convertible and immediate. Hence a demonstration of this kind is through the better known, namely, to us; otherwise it would not effect scientific knowledge. For we do not reach a knowledge of the unknown except through something better known. However in the case of two things equally predicable, that is, convertible, one of which is the cause and the other the effect, there is nothing to preclude that now and then the better known will not be the cause but the effect. For sometimes the effect is better known than the cause both in respect to us and according to sense-perception, although absolutely and according to nature the cause is the better known. Consequently, through an effect better known than the cause there can be demonstration which does not engender 'propter quid' knowledge but only 'quia'.

Then, at: thus you might prove (78a30) he clarifies these differences by examples.

And this is divided into two parts.

In the first he develops an example of the demonstration 'quia' which is through an effect.

In the second of demonstration 'quia' which is through a mediate cause, at: this also occurs when the middle (78b13; [201]).

The first is divided into two parts.

In the first he gives an example of a syllogism which is through a convertible effect.

In the second of a syllogism through a nonconvertible effect, at: again, in cases where the cause (78b10; [200]).

The first is divided into two parts according to the two examples he gives, the second of which begins at: another example is the inference (78b3; [199]).

Concerning the first he does two things.

First, he gives an example of demonstration 'quia' which is through an effect.

Second, he states when it can be converted into a demonstration 'propter quid', at: the major and middle of the proof (78a39; [198]).

He says therefore first (78a30) that demonstration 'quia' is through an effect if one concludes for example that the planets are near because they do not twinkle. For 'non-twinkling' is not the cause why the planets are near, but vice versa: for the planets do not twinkle because they are near. For the fixed stars twinkle because in gazing at them the sight is beclouded on account of the distance. Therefore, the syllogism might be formed in the following way: “Whatever does not twinkle is near; but the planets do not twinkle: therefore, they are near.” Here we let C be the planets, that is, let 'planets' be the minor extreme, and let B consist in not twinkling, and A “to be near” be the major extreme. Then the proposition, “Every C is B,” is true, namely, the planets do not twinkle. Also it is true that “Every B is A,” that is, every star that does not twinkle is near. Rowever, the truth of such a proposition must be obtained through induction or through sense perception, because the effect here is better known than the cause. And so, the conclusion, “Every C is A,” follows. In this way, then, it has been demonstrated that the planets, that is, the wandering stars, are near. Consequently, this syllogism is not “propter quid” but “quia.“ For it is not because they do not twinkle that planets are near but rather, because they are near, they do not twinkle.

Then, at: the major and middle of the proof (78a39) he teaches how a demonstration 'quia' is changed to a demonstration 'propter quid'. And he says that it is possible to demonstrate the one through the other, that is, to demonstrate that they do not twinkle, because they are near. Then the demonstration will be 'propter quid'. Thus let C be the wanderers, that is, let 'wandering star' be the minor extreme; let B consist in being near, that is, let 'to be near', which was the major extreme above, be the middle term; and let A consist in not twinkling, that is, let 'not to twinkle', which above was the middle term, now be the major term. Therefore, B is in C, that is, “Every planet is near”; and A is in B, that is, “Any planet which is near does not twinkle.” Wherefore, it follows that A is in C, that is, “A planet does not twinkle.” In this way we have a syllogism 'propter quid', since it rests on the first and immediate cause.

Then, at: another example is the inference (78b3) he presents another example of this, saying that in this way (that is, by means of a demonstration 'quia'), one demonstrates that the moon is round because of its phases, according to which it waxes and wanes every month. They argue thus: “Everything which waxes thus circularly is circular; but the moon waxes thus: therefore, it is circular.” Put in this form it is a syllogism demonstrating 'quia'. But if the middle be interchanged, that is, if 'circular' be made the middle term and 'waxes' the major term, “it becomes a demonstration 'propter quid'. For the moon is not circular because it waxes in that way, but because it is circular it undergoes such phases. Therefore, let C be the moon, that is, the minor extreme; let waxing be B, that is, the major extreme, and let circular be B, the middle term. This will be the situation in the syllogism propter quid.

Then, at: again, in cases where the cause (78b10) he shows that a demonstration through a non-convertible effect is quid. He says, therefore, that even in those syllogisms in which the middles are not converted with the extremes, and in which an effect rather than a cause is taken as the middle better known in reference to us, even in those cases the demonstration is “quia” and not “propter quid."

If the middle be such that it can be converted with the major extreme and it exceeds the minor, then obviously it is a fitting syllogism; for example, if one proves that Venus is near because it does not twinkle. On the other hand, if the minor exceeded the middle, it would not be a fitting syllogism: for one cannot conclude universally of stars that they are near because they do not twinkle. Quite the contrary is true in comparison to the major term: for if the middle is in less things than is the major term, the syllogism is fitting, as when it is proved that someone has a sensible soul on the ground that he is capable of progressive local motion. But if it is in more, than the syllogism is not fitting, for from an effect which can proceed from several causes, one of them cannot be concluded. Thus, one cannot conclude from a rapid pulse that he has a fever.

b13. This also occurs (1) when the middle falls outside the major and minor, for here too the strict cause is not given, and so the demonstration is of the fact, not of the reasoned fact.

b14. For example, the question ‘Why does not a wall breathe?’ might be answered, ‘Because it is not an animal’; but that answer would not give the strict cause, because if not being an animal causes the absence of respiration, then being an animal should be the cause of respiration, according to the rule that if the negation of causes the non-inherence of y, the affirmation of x causes the inherence of y; e.g., if the disproportion of the hot and cold elements is the cause of ill health, their proportion is the cause of health; and conversely, if the assertion of x causes the inherence of y, the negation of x must cause y’s non-inherence. But in the case given this consequence does not result; for not every animal breathes.

b24. A syllogism with this kind of cause takes place in the second figure. Thus: let A be animal, B respiration, C wall. Then A is predicable of all B (for all that breathes is animal), but of no C; and consequently B is predicable of no C; that is, the wall does not breathe.

b27. Such causes are like far-fetched explanations, which precisely consist in making the cause too remote, as in Anacharsis’ account of why the Scythians have no flute-players; namely because they have no vines.

b32. Thus, then, do the syllogism of the fact and the syllogism of the reasoned fact differ within one science and according to the position of the middle terms.

This also occurs when the middle. After clarifying with examples how there is demonstration 'quia' through effect, the Philosopher here shows how there is demonstration 'quia' through things not immediately connected.

About this he does two things.

First, he amplifies his proposition.

Second, he shows how the middles relate themselves to the conclusions in this type of demonstration, at: such causes are like far-fetched (78b27; [205]).

Concerning the first he does three things.

First, he states his intention.

Second, he clarifies it with examples, at: for example, the question (78b14; [203]).

Third, he puts it in syllogistic form, at: a syllogism with this kind (78b24; [204]).

He says therefore first (78b13) that there is demonstration 'quia' and not 'propter quid' not only in those matters that are proved through an effect but also in matters in which the middle is set outside.

Now, a middle is said to be set outside when it is diverse from the major term, as in negative syllogisms; or a middle is said to be set outside when it is outside the genus as being something more common and not convertible with the major term. That something cannot be demonstrated 'propter quid' through such a middle he proves on the ground that demonstration 'propter quid' is through a cause. But the middle in question is not, properly speaking, a cause.

Then, at: for example, the question (78b14) he clarifies what he had said with an example, saying that if someone attempted to prove that a wall does not breathe because it is not an animal, he would not be demonstrating 'propter quid' or giving the cause. Because if the fact of not being an animal were the cause of not breathing, it would be required that being an animal would be the cause of breathing-which is false. For there are many animals which do not breathe. For it is required, if a negation is the cause of a negation, that the affirmation be the cause of the affirmation, as the fact of not being warm and cold in due measure is the cause of someone’s not getting well, and to be warm and cold in due measure is the cause of someone’s getting well. The converse is also true, namely, that if an affirmation is the cause of an affirmation, the negation is the cause of the negation. But this does not occur in the case at hand, because the affirmation is not the cause of an affirmation, since not every animal breathes.

Then, at: a syllogism with this kind (78b24) he arranges the aforesaid example in syllogistic form and says that it should be arranged in the second figure.

And this is so because in the first figure there cannot be a negative conclusion such that the major would be affirmative, as our example requires. For “to breathe,” which is the major extreme, must be joined with “animal,” which is the middle term, according to affirmation. But “wall,” which is the minor extreme, must be joined with “animal,” which is the middle, according to negation. Consequently, the major will be affirmative and the minor negative. But such a thing never occurs in the first figure, but only in the second.

Let “animal,” then, be A, that is, the middle term, “breathe” be B, that is, the major extreme, and “wall” be C, that is, the minor extreme. it therefore follows that A is in every B, because everything that breathes is an animal; but A is in no C, because no wall is an animal. Hence it follows that B, too, is in no C, that is, that no wall breathes.

Furthermore, if the most proximate middle were used, it would be demonstration 'propter quid': for example, if it were shown that a wall does not breathe because it does not have lungs. For whatever has lungs breathes, and conversely.

Then, at: such causes are like far-fetched (78b27) he shows how these middles are related to the conclusion, saying that such far-away causes are compared to what they explain as being too remote, because they go beyond the pale of the conclusion to be proved. Furthermore, a middle of this sort happens to assign what is far-fetched, as in Anacharsis’ proof that there are no flutists among the Scythians because there are no vines there. For this is quite a far-fetched middle. A nearer one would be that they have no wine, and nearer still, that they do not drink wine, from which follows a merry heart which moves one to sing-if 'fluting' is taken to mean singing. Or it might be better to say that 'fluting' is not taken for just any singing but for the song of the grape harvesters, called “celeuma.”

Then, at: thus, then, do the syllogism (78b32) he summarizes what he had said and declares that these are the differences between the syllogism 'quia' and the syllogism 'propter quid' in the same science, and according to the position of the same, that is, of those that have the same order. This he says to exclude what he will discuss later, namely, that one science is under another.

b34. But there is another way too in which the fact and the reasoned fact differ, and that is when they are investigated respectively by different sciences.

b35. This occurs in the case of problems related to one another as subordinate and superior, as when optical problems are subordinated to geometry, mechanical problems to stereometry, harmonic problems to arithmetic, the data of observation to astronomy.

a1. Some of these sciences bear almost the same name; e.g., mathematical and nautical astronomy, mathematical and acoustical harmonics.

a3. Here it is the business of the empirical observers to know the fact, of the mathematicians to know the reasoned fact; for the latter are in possession of the demonstrations giving the causes, and are often ignorant of the fact: just as we have often a clear insight into a universal, but through lack of observation are ignorant of some of its particular instances. These connexions have a perceptible existence though they are manifestations of forms. For the mathematical sciences concern forms: they do not demonstrate properties of a substratum, since, even though the geometrical subjects are predicable as properties of a perceptible substratum, it is not as thus predicable that the mathematician demonstrates properties of them.

a10. As optics is related to geometry, so another science is related to optics, namely the theory of the rainbow. Here knowledge of the fact is within the province of the natural philosopher, knowledge of the reasoned fact within that of the optician, either qua optician or qua mathematical optician.

a13. Many sciences not standing in this mutual relation enter into it at points; e.g., medicine and geometry: it is the physician’s business to know that circular wounds heal more slowly, the geometer’s to know the reason why.

But there is another way. After showing how demonstration 'quia' differs from demonstration 'propter quid' in the same science, the Philosopher shows how they differ in sciences that are diverse.

And he does two things.

First, he states his proposition, saying (78b34) that in a way other than the above the 'propter quid' differs from the 'quia' due to the fact that they are considered in diverse sciences, that is, that the 'propter quid' pertains to one science and the 'quia' to another.

Second, at: this occurs in the case (78b34) he elucidates his proposition.

First, he elucidates it in sciences one of which is under the other.

Second, in sciences one of which is not under the other, at: many sciences not standing (79a13; [212]).

Concerning the first he does two things.

First, he shows how those sciences are related, one of which is under the other and to one of which pertains the 'quia' and to the other the 'propter quid'.

Second, he shows how in these sciences the 'quia' pertains to one and the 'propter quid' to the other, at: here it is the business (79a3; [210]).

Concerning the first he does two things.

First, he shows how such sciences relate to one another as to order.

Second, how they relate to one another as to agreement, at: some of these sciences (79a1; [209]).

He says therefore first (78b35) that these sciences (that is, to one of which pertains 'quia' and to the other 'propter quid') are the ones so related that one is under the other.

But this occurs in two ways: in one way, when the subject of one science is a species of the subject of the higher science, as animal is a species of natural body—consequently, the science of animals is under natural science; in another way, when the subject of the lower science is not a species of the subject of the higher science but is compared to the latter as material to formal.

An example of this latter way of one science being under another is the way specular, that is, optics, is under geometry. For geometry is concerned with lines and other magnitudes, whereas optics is concerned with a line determined to matter, that is, the visual line. Now, the visual line is not, strictly speaking, a species of line any more than wooden triangle is a species of triangle: for wooden is not a difference in respect to triangle. In like manner mechanical engineering, that is, the science of making machines, is related to stereometry, that is, the science of measuring bodies. This science is said to be under the other as applying the formal to the material. For the measures of bodies’ absolutely are compared to the measures of wood and other material required for machines as the application of the formal to the material. In like manner, harmonics, that is, music, is related to arithmetic: for music applies formal number, which arithmetic considers, to matter, that is, to sounds. And the same is true of appearance, that is, nautical science (which considers the signs indicative of calm or storm), as compared to astronomy, which considers the motions and positions of the stars.

Then, at: some of these sciences (79a1) he shows how the aforesaid sciences relate to one another in point of agreement. And he says that these sciences are almost mutually univocal. And he says "almost" because they agree in the name of their genus and not in the name of their species. Thus, all the sciences mentioned above are called 'mathematical': some, indeed, because they are concerned with a subject which is abstracted from matter, as geometry and arithmetic, which are absolutely mathematical; but others are so through applying mathematical principles to material things, as astronomy is called mathematical and as nautical science is. Similarly, harmonics, that is, music, is called mathematical and so is acoustics, that is, practical music, which knows sounds through experience based on hearing. Or it might be said that they are univocal because they also agree in the name of their species, for we speak of nautical astronomy as astronomy and of practical music as music. But because this is not so in all but only in some, he says "almost".

Then, at: here it is the business (79a3) he shows how in these sciences the 'quia' pertains to the one science and the 'propter quid' to the other.

In regard to this there are two points.

First, he shows how those sciences which contain others under them state the 'propter quid'.

Second, how sciences which are contained under another one state the 'propter quid' of some others, at: as optics is related to geometry (79a10; [211]).

One should note, therefore, with respect to the first point that in all the above-mentioned sciences, the ones which are contained under others apply mathematical principles to sensible things, while the ones which contain others under them are more mathematical. Accordingly, the Philosopher first of all says (79a3) that to know the 'quia' pertains to the sensible, that is, to the lower sciences, which make application to sensible things; but to know the 'propter quid' pertains to the mathematical, that is, to those sciences whose principles are not applied to sensible things. For these latter sciences are concerned with demonstrating matters which are accepted as causes in the lower sciences.

But because someone might suppose that whoever knows the 'propter quid' also necessarily knows the 'quia', he dismisses this, saying that very frequently those who know the 'propter quid' do not know the 'quia'. And he gives as an example of this those who, when considering the universal often take no account of the singulars precisely because their speculation does not consider them: thus, one who knows that every mule is sterile, does not know this of the one he does not consider. In like manner, a mathematician who demonstrates 'propter quid', now and then does not know the quia, because he does not apply the principles of the higher sciences to matters demonstrated in the lower science.

And because he had said that it belongs to mathematics to know the 'propter quid', he proposes to indicate which genus of cause is used by mathematics. Hence he says that those sciences which receive the 'propter quid' from mathematics are something else, that is, they differ from the mathematical according to subject, that is, insofar as they make application to matter. Hence these latter use forms, that is, formal principles, which they receive from mathematics: for the mathematical sciences concern forms. For their considerations do not bear on the subject, that is, on matter, because although the items which geometry considers exist in matter, for example, the line, plane and so on, nevertheless geometry does not consider them precisely as they are in matter, but as abstracted. For those things that are in matter according to existence, geometry abstracts from matter according to consideration. Conversely, the sciences subalternated to it accept those things which were considered in the abstract by geometry and apply them to matter. Hence it is plain that it is according to the formal cause that geometry states the 'propter quid' in those sciences.

Then, at: as optics is related to geometry (79a10), he shows that even the subalternated sciences state the 'propter quid', not of its subalternating science but of some other science. Thus, optics is subalternated to geometry, so that if we compare the one with the other, optics states the 'quia' and geometry the 'propter quid'. But just as optics is subalternated to geometry, so the science of the rainbow is subalternated to optics, for it applies to a determinate matter the principles which optics hands down absolutely. Hence it belongs to the naturalist who treats of the rainbow to know the 'quia', but to the expert in optics to know the 'propter quid'. For the naturalist says that the cause of the rainbow is the convergence of a visual line at a cloud arranged in some relation to the sun; but the 'propter quid' he takes from optics.

Then, at: many sciences not standing (79a13) he shows how 'quia' and 'propter quid' differ among sciences that are diverse but not subalternate. And he says that many sciences which are not subalternate are nevertheless related, that is, in such a way that one states the 'quia' and the other the 'propter quid'. This is true of medicine and geometry. For the subject of medicine is not subsumed under the subject of geometry as the subject of optics is. Nevertheless, the principles of geometry are applicable to certain conclusions reached in medicine: for example, it belongs to the man of medicine who observes it to know quia that circular wounds heal rather slowly; but to know the 'propter quid' belongs to the geometer, whose business it is to know that a circle is a figure without corners. Hence the edges of a circular wound are not close enough to each other to allow them to be easily joined.

It should also be noted that this difference of 'quia' and 'propter quid' between sciences that are diverse is contained under one of the modes previously discussed, namely, when the demonstration is made through a remote cause.

Demonstrative syllogisms best made in the first figure; on mediate and immediate negative propositions

Of all the figures the most scientific is the first.

Thus, it is the vehicle of the demonstrations of all the mathematical sciences, such as arithmetic, geometry, and optics, and practically all of all sciences that investigate causes: for the syllogism of the reasoned fact is either exclusively or generally speaking and in most cases in this figure—a second proof that this figure is the most scientific; for grasp of a reasoned conclusion is the primary condition of knowledge.

a23. Third, the first is the only figure which enables us to pursue knowledge of the essence of a thing. In the second figure no affirmative conclusion is possible, and knowledge of a thing’s essence must be affirmative; while in the third figure the conclusion can be affirmative, but cannot be universal, and essence must have a universal character: e.g., man is not two-footed animal in any qualified sense, but universally.

a30. Finally, the first figure has no need of the others, while it is by means of the first that the other two figures are developed, and have their intervals closepacked until immediate premises are reached. Clearly, therefore, the first figure is the primary condition of knowledge.

a33. Just as an attribute A may (as we saw) be atomically connected with a subject B, so its disconnexion may be atomic. I call ‘atomic’ connexions or disconnexions which involve no intermediate term; since in that case the connexion or disconnexion will not be mediated by something other than the terms themselves.

a36. It follows that if either A or B, or both A and B, have a genus, their disconnexion cannot be primary. Thus: let C be the genus of A. Then, if C is not the genus of B — for A may well have a genus which is not the genus of B — there will be a syllogism proving A’s disconnexion from B thus: all A is C; no B is C; therefore no B is A. Or if it is B which has a genus D, we have: all B is D; no D is A; therefore no B is A, by syllogism; and the proof will be similar if both A and B have a genus.

b5. That the genus of A need not be the genus of B and vice versa, is shown by the existence of mutually exclusive coordinate series of predication. If no term in the series ACD... is predicable of any term in the series BEF..., and if G — a term in the former series — is the genus of A, clearly G will not be the genus of B; since, if it were, the series would not be mutually exclusive. So also if B has a genus, it will not be the genus of A.

b12. If, on the other hand, neither A nor B has a genus and A does not inhere in B, this disconnexion must be atomic. If there be a middle term, one or other of them is bound to have a genus, for the syllogism will be either in the first or the second figure. If it is in the first, B will have a genus — for the premise containing it must be affirmative: if in the second, either A or B indifferently, since syllogism is possible if either is contained in a negative premise, but not if both premises are negative.

b21. Hence it is clear that one thing may be atomically disconnected from another, and we have stated when and how this is possible.

Of all the figures the most scientific. After determining about the material of the syllogism, the Philosopher here determines about its form, showing in which figure chiefly the, demonstrative syllogism is formed.

His treatment is divided into two parts.

In the first he shows that the demonstrative syllogism is formed first and foremost in the first figure.

In the second, because it is possible in the first figure to proceed from negatives and because a demonstration must proceed from things immediate, he shows how a negative proposition happens to be immediate, at: just as an attribute A (79a33; [217]).

He shows the first with three reasons, the first of which (79a17) is this.

In whatever figure the 'propter quid' syllogism is best made, that figure is the best for causing scientific knowledge and for that reason is most suitable for demonstrations, since demonstration is a syllogism which causes scientific knowledge. But a syllogism 'propter quid' is best made in the first figure, and this is evidenced by the fact that the mathematical sciences of arithmetic and geometry and all other sciences which demonstrate 'propter quid' employ the first figure in most cases. Therefore, the first figure is the one which first and foremost causes scientific knowledge and is most suitable for demonstrations.

But the reason why demonstration 'propter quid' is best made in the first figure is this: in the first figure the middle term is both subjected, to the major extreme, which is the predicate of the conclusion, an(I predicated of the minor extreme, which is the subject of the conclusion. Now, in demonstrations 'propter quid' the middle must be the cause of the proper attribute which is predicated of the subject in the conclusion. Furthermore, one of the modes of “saying per se” is when the subject is the cause of the predicate, as “being butchered, he died,” as has been explained above; and this is verified in the first figure in which the middle is subject to the major extreme, as has been said.

The second reason, at: third, the first is the only figure (79a23) is this: the quod quid est [that is, the definition as signifying the essence] plays a most important role in demonstrative sciences because, as has been said, a definition is either the principle of a demonstration or the conclusion or a demonstration differing in the position of its terms. Now, when it is a question of formulating a definition the first figure alone is suitable. For that is the only figure in which a universal affirmative is concluded, which alone produces science of the quod quid est. (For the quod quid est is known through an affirmation. Furthermore, the definition is predicated affirmatively and universally of the defined: for it is not some man that is a two-legged animal, but every man.) Therefore, the first figure is foremost in causing scientific knowledge and is best accommodated to demonstrations.

Then, at: finally, the first figure (79a30) he gives the third reason which is this: For purposes of demonstration the other figures need the first, but the first does not need them. Therefore, the first figure is better equipped for causing scientific knowledge than the others. That the others need the first is evidenced by the fact that if complete scientific knowledge is to be had, the mediate propositions present in demonstrations must be reduced to immediate ones. But this reduction is made in two ways, namely, by condensing middles [that is, by inserting fresh middles, the movement proceeding from the extremes toward the middle first employed] and by expanding outwards [that is, by going from the middle toward the remote extreme]. It is done by condensing when the middle actually employed is joined mediately to each of the extremes or only to the minor extreme. Hence when other middles are introduced between the first middle and the extremes, the middles, as it were, close in. Thus, if someone first says, “Every E is C,” [minor], “Every C is A,” [major], and then a middle, say D, is introduced between C and E, and a middle B between C and A. It is done by expanding when the middle is immediate to the minor extreme and mediate to the major: for then it is necessary to introduce several other middles more general than the middle first taken. Thus, if one says, “Every E is D,” “Every D is A,” and the later other middles more general than D are introduced [between D and A]. Now, these processes of condensing and expanding can be performed only in the first figure, both because the first figure is the only one which yields a universal affirmative conclusion, and because it is only in the first figure that the middle lies between both extremes. For in the second figure the middle lies outside the extremes, as being predicated of them; in the third figure [the middle] is below each of the extremes, as being subjected to them.

Then, at: just as an attribute A (79a33) he teaches how a negative proposition can be immediate.

In regard to this he does two things.

First (79a33), he states his intention, saying that just as it is possible for every A to be in B atomically, that is, immediately, so too not to be, that is, so too it is possible for a proposition signifying that A is not in B to be immediate. Then he goes on to explain what it is to be or not to be atomically, namely, when the affirmation or negation does not have a middle through which it might be proved.

Second, at: it follows that if either A or B (79a36), he elucidates his proposition.

In regard to this he does two things.

First, he shows when a negative proposition is mediate.

Second, when immediate, at: if, on the other hand (79b12; [220]).

Concerning the first he does two things.

First, he clarifies his proposition.

Second, he proves something he had presupposed, at: that the genus of A need not (79b5; [219]).

He says therefore first (79a36), that when A, that is, the major term, or B, that is, the minor term, is in some whole as a species in its genus, or both are under diverse genera or predicaments, it does not occur that A is not in B first, that is, it does not occur that this proposition, “No B is A,” is immediate. First, then, he manifests this when A is in some whole, say in C, and B is in no whole. For example, if A is 'man', C is 'substance', and B is 'quantity', it is possible to form a syllogism to prove that A is in no B on the ground that C is in every A and in no B. Thus we get a syllogism in the second figure: "Every man is a substance; No quantity is a substance: Therefore, no quantity is a man". In like manner, if B, that is, the minor term, is in some whole, say D, but A is not in that whole, it is possible to syllogize that A is in no B. For example if A is 'substance', B 'line', and D 'quantity', we get a syllogism in the first figure: "No quantity is a substance; Every line is a quantity; Therefore, no line is a substance." In the same way, a negative conclusion could be demonstrated if either is in some whole. For example, if A is 'line', C 'quantity', B 'whiteness' and D 'quality', it is possible to form a syllogism in the first and in the second figure. In the second figure thus: "Every line is a quantity; No whiteness is a quantity: Therefore, no whiteness is a line." And in the first figure thus: "No quality is a line; Every whiteness is a quality: Therefore, no whiteness is a line."

We should understand, however, that a negative proposition is mediate, when both terms exist in some whole which is not the same but different for each. For if they are in the same whole, the proposition will be immediate, as “No rational being is irrational,” or “No biped is a quadruped.”

Then, at: that the genus of A (79b5) he explains something he had presupposed, namely, “on condition that one of the extremes exist in some whole and that the other be not in the same,” saying that it is clear from the orderings of the various predicaments, which are not mutually interchangeable. In other words, because that which is in one predicament is not in another, it is plain that B happens not to be in the whole in which A is, or vice versa, because one of the terms happens to be taken from one predicament in which the other is not found. Thus, let one ordering of the predicaments be ACD, say the predicament of substance, and another ordering be BEF, say the predicament of quantity. Then if none of those in the ordering ACD be predicated of none in the ordering BEF, while A is in P as in that most general item which is the principle of the whole first ering, it is plain that B is not in P, because then the orderings, that is, the predicaments, would be interchanged. Similarly, if B is in some whole, say in E, it is plain that A is not in E.

Then, at: if, on the other hand (79b12) he indicates how a negative proposition may be immediate, saying that if neither is in some whole, that is, neither A nor B, and A is not in B, it is necessary that this proposition, “No B is A,” be immediate. Because if a middle were taken to syllogize it, then one of them would have to be in some whole, for the syllogism would have to be made either in the first figure or in the second, (since in the third figure a universal negative cannot be concluded, as is required for an immediate proposition). However, if it is made in the first, B would have to be in some whole, because B is the minor extreme, and in the first figure the minor proposition must always be affirmative. For a syllogism with the major affirmative and the minor negative cannot be formed in the first figure. But if it be in the second figure, either, that is, A or B, may be in a whole, because in the second figure the first proposition may be negative in some moods and the minor in other moods. Of course it is never permitted, neither in the first nor the second, to have both propositions negative. And so it is required that when either proposition is affirmative, one of the extremes must be in some whole. Thus it is clear that a negative proposition is immediate, when neither of its terms is in some whole.

This does not mean, however, that although neither is in some whole, a middle could be found to conclude it, namely, if one were to take a convertible middle, because it is necessary that such a middle be prior and better known. And this is either the genus itself or the definition, which is not without a genus.

Then, at: hence it is clear that one thing (79b21) he concludes and summarizes what has been said. Here the text is sufficiently clear.

b23. Ignorance, defined not as the negation of knowledge but as a positive state of mind, is error produced by inference. (1) Let us first consider propositions asserting a predicate’s immediate connexion with or disconnexion from a subject.

Here, it is true, positive error may befall one in alternative ways; for it may arise where one directly believes a connexion or disconnexion as well as where one’s belief is acquired by inference. The error, however, that consists in a direct belief is without complication; but the error resulting from inference—which here concerns us—takes many forms.

Thus, let A be atomically disconnected from all B: then the conclusion inferred through a middle term C, that all B is A, will be a case of error produced by syllogism. Now, two cases are possible. Either (a) both premises, or (b) one premise only, may be false. (a) If neither A is an attribute of any C nor C of any B, whereas the contrary was posited in both cases, both premises will be false.

(C may quite well be so related to A and B that C is neither subordinate to A nor a universal attribute of B: for B, since A was said to be primarily disconnected from B, cannot have a genus, and A need not necessarily be a universal attribute of all things. Consequently both premises may be false.)

On the other hand, (b) one of the premises may be true, though not either indifferently but only the major A-C since, B having no genus, the premise C-B will always be false, while A-C may be true. This is the case if, for example, A is related atomically to both C and B; because when the same term is related atomically to more terms than one, neither of those terms will belong to the other. It is, of course, equally the case if A-C is not atomic.

Error of attribution, then, occurs through these causes and in this form only — for we found that no syllogism of universal attribution was possible in any figure but the first.

Ignorance, defined not as the negation. After determining about the demonstrative syllogism through which science is acquired, the Philosopher here determines concerning the syllogism through which ignorance or deception is produced in us.

In regard to this he does two things.

First, he shows what sort of ignorance can be induced by a syllogism.

Second, he shows the mode in which such a syllogism proceeds, at: here, it is true, positive error (79b28; [225]).

He distinguishes therefore first (79b23) a twofold ignorance, one of which is negative and the other a positive state.

There is ignorance in a negative way, when a man has no scientific knowledge at all about a thing. This ignorance consists in not attaining, as the Philosopher says in Metaphysics IX, and is exemplified in a peasant who knows absolutely nothing about whether a triangle has three angles equal to two right angles. But ignorance is present as a positive state, when one does have a definite opinion but it is unsound. For example, when he falsely thinks something about a thing, either because he thinks something to be which is not, or something not to be which is. And this ignorance is the same as error. Now the first ignorance is not produced by a syllogism, but the second can be, and then it is called deception.

Such ignorance or deception can concern two things.

First, those things which are first and immediate principles, namely, when a person opines things opposed to the principles. And although he cannot so opine inwardly in the mind, as has been stated above, because these things do not fall under apprehension, nevertheless he can contradict them orally and according to a false imagination, as is said in Metaphysics IV of some who deny the principles.

Second, it might be concerned with conclusions which are not first and immediate. Furthermore, the first of these ignorances or deceptions is opposed to the knowledge which is understanding, and the second to the knowledge which is science.

Each of these states of ignorance, whether concerned with things that are first or things not first, can befall a man in two ways.

In one way, simply, when independently of any process of reasoning he thinks a falsehood by affirming or denying.

In another way, when he is brought to his opinion through some syllogized reason.

Hence the Philosopher says in Metaphysics IV that some contradict the principles as though persuaded by reasons; but others, not as though persuaded by reasons, but through lack of erudition or through wilfullness which demands to have a demonstration in all matters.

Then, at: here, it is true, positive error (79b28) he shows how these ignorances are caused.

First, how the ignorance produced by a syllogism is caused.

Second, how ignorance befalls a man without syllogizing, at: it is also clear that the loss (81a38; [248]).

Concerning the first he does two things.

First, he states how ignorance in regard to first and immediate principles is engendered by a syllogism.

Second, how it is caused in regard to things not first and immediate, at: in the case of attributes (80b17; [240]).

Concerning the first he does two things.

First, he states how the ignorance is caused whereby that which is not is believed to be.

Second, that whereby that which is is believed not to be, at: on the other hand, an error (80a8; [233]).

Regarding the first he does three things.

First, he states how the aforesaid ignorance is generally caused.

Second, he assigns its various possible forms, at: it may occur when both premises (79b31; [227]).

Third, he answers a tacit question, at: error of attribution (80a6; [232]).

He says therefore first (79b28) that the deception according to false opinion which above was said to occur straightway is simple, that is, is engendered in only one way. For it is not caused by a reason, which might vary, but by the lack of a reason which [lack] is not diversified into various modes each having its own characteristics any more than other negations are. Yet because false reasons can be various and many, this ignorance, when it is engendered by a syllogism can occur in many ways according to the several ways in which a syllogism can be false. The common way is set forth when he says, Let A be in no B individually (79b29), that is, let this proposition be immediately true, “No B is A.” For two negations are stated in the place of one— for instance if we should say, “No quality is a substance”— according to the doctrine on immediate negative propositions presented above. Therefore, if someone were to conclude the opposite of this with a syllogism, taking C as a middle to show that every B is A, there will be a deception through syllogism.

Then, at: it may occur when both premises (79b31) he shows in how many ways this can vary.

And first of all it should be noted that a false conclusion is not concluded except when the syllogism is false. But a syllogism can be false in two ways: first, because it lacks syllogistic form, in which case it is not a syllogism but appears to be one. In a second way, because it employs false propositions, in which case it is a syllogism as to form, but it is a false one because of the false propositions used. Therefore, in a dialectical disputation, which bears on probables, use could be made of both types of false syllogism, because such a disputation proceeds from common premises. Consequently, error can arise in it both as regards the matter employed, which is common, and also as regards the form, which is common. But in a demonstrative disputation, which bears on necessary things, the type of syllogism used is the one which could be false only on account of the matter, because as it is stated in Topics I, a paralogism in a given discipline proceeds from things proper to the discipline, but not from true things. Hence, since syllogistic form must be counted among the common things, paralogism within a discipline-which is the matter under discussion-is not defective in form but only in matter, and furthermore in regard to proper and not in regard to common things.

Hence he first tells how such a syllogism might proceed from two false premises.

Second, how it might proceed from one or the other premise being false, at: on the other hand, one of the premises (79b40; [230]).

But since the first can occur in two ways, namely, the false proposition might be contrary to the true or contradictory to it, therefore:

First, he shows how such a syllogism might proceed from two false statements that are contrary to true statements.

Second, how a contradiction is accepted, at: C may quite well be so related to A and B (79b34; [229]).

First, then (79b30) he says that in a syllogism causing deception it sometimes happens that both premises are false, and sometimes only one. Let us take the case of two false premises, each contrary to what is true. For example, suppose that C is so related to A and to B. that no C is A and no B is C. Now, if we take the contraries of these, namely, that every C is A and every B is C, these two propositions will be completely false. For example, if I should say: “Every quality is a substance; every quantity is a quality: therefore, every quantity is a substance.”

Then, at: C may quite well be so related to A and B (79b34) he shows how both premises can be false and not contrary, but contradictory, to what is true. For example, if C is so related to A and to B, that it is neither contained totally under A nor is universally in B. Thus one might take C as 'perfect' or 'actual being,' and proceed in the following way: “Every perfect thing is a substance; every quantity is perfect: therefore, every quantity is a substance. Now, although both are false, they are not entirely so, for their contradictories are true, namely, “Some perfect things are not substances,” and “Some quantity is not perfect”; but their contraries are false, namely, “No perfect thing is a substance,” and “No quantity is perfect.”

But that C is not in B universally (that is, that the statement, “Every quantity is perfect,” which was the minor, is not true), that is, that “Every B is C,” is not true he proves by the fact that B cannot be contained under any whole that might be predicated universally of it. And this is so because the proposition, “No B is A,” was said to be immediate, which means that A is universally not in B. But, as has been stated above, those negative propositions are immediate, neither of whose terms is under a whole.

Nevertheless, this proof does not seem sufficient, because something can be predicated universally even of that which is not under some whole as a species under a genus. For the genus and the difference are not the only things predicated universally, but a property is too.

But it must be said that although the proof in question is not efficacious generally speaking, it is in this case. Because, as it is stated in Topics I, a paralogism within a discipline— with which we are concerned here— proceeds from items that accord with the discipline. Hence he intends to use such middles as a demonstrator uses. But the middle of a demonstration is the definition, as has been stated above. Therefore, even in the syllogism of which he is speaking, he intends to use a definition as the middle. Now, a definition contains the genus and difference. Hence that which is predicated universally in this syllogism should contain that in which the subject is, as in a whole.

Again, that A is not in C universally, that is, that the statement, “Every perfect thing is a substance,” is not universally true is proved by the fact that it is not required of every universal that it be universally in all things that are: because no predicament is predicated of things contained under another predicament nor universally predicated of those items that commonly follow upon being, namely, act and potency, perfect and imperfect, prior and posterior, and the like.

Then, at: on the other hand, one of the premises (79b40) he shows how the aforesaid syllogism might proceed from one true and one false.

And he says that in the aforesaid syllogism we might take one true, that is, the major, which is AC, and the other false, namely, the minor, which is BC. That the minor proposition, which is BC, is always the false one he proves, as he did above, on the ground that B is in nothing as in a whole. But that this proposition AC could be true, while the other is false he proves in terms. Let A be in B and in C individually, that is, immediately, as a genus in its species; for example, as color is to whiteness and blackness. Now, under these conditions it is obvious that the major will be true, namely, “Every C is A,” for example, “Every whiteness is a color,” and the minor false, namely, “Every blackness is whiteness”: for when something is predicated first of several, none of the several is predicated of any other of them. For the first predication of a genus is of species which are opposite.

Here, too, a doubt arises, because from the terms used here, not a false but a true conclusion follows. For the conclusion will be that A is in B, it having also been assumed that it is in B individually.

But it must be answered that this example was used merely to illustrate how the major could be true and the minor false; although it is of no use in a case where a false conclusion is sought. Hence the Philosopher at once adds: it is equally the case of AC if not atomic. However, we could take terms such that A is not in B either individually or in any way, but is rather immediately removed from B. Neither is it necessary that it be in C individually, because it is not necessary that a demonstrator employ only immediate propositions: for he may use ones which are supported by immediate propositions. We can, therefore, take other terms pertaining to the present case, for example, “intellectual substance,” as the middle, for “Every intelligence is a substance,” and take as the false minor, “Every quantity is an intelligence.” Hence a false conclusion follows.

Then, at: error of attribution (80a6) he answers a tacit question. For someone might request that he exemplify the diversity of such a syllogism in the other figures. But he answers that a deception, which bears on being, that is, through which someone opines a false affirmative proposition, can be derived only by the first figure, because in the next figure, that is, in the second, an affirmative syllogism cannot be formed. As for the third figure, it has no bearing on the case, because it cannot conclude a universal, which is principally intended in demonstration and in this syllogism.

a8. On the other hand, an error of non-attribution may occur either in the first or in the second figure. Let us therefore first explain the various forms it takes in the first figure and the character of the premises in each case.

a11. (c) It may occur when both premises are false; e.g., supposing A atomically connected with both C and B, if it be then assumed that no C is and all B is C, both premises are false.

a14. (d) It is also possible when one is false. This may be either premise indifferently. A-C may be true, C-B false — A-C true because A is not an attribute of all things, C-B false because C, which never has the attribute A, cannot be an attribute of B; for if C-B were true, the premise A-C would no longer be true, and besides if both premises were true, the conclusion would be true.

a20. Or again, C-B may be true and A-C false; e.g., if both C and A contain B as genera, one of them must be subordinate to the other, so that if the premise takes the form No C is A, it will be false. This makes it clear that whether either or both premises are false, the conclusion will equally be false.

a27. In the second figure the premises cannot both be wholly false; for if all B is A, no middle term can be with truth universally affirmed of one extreme and universally denied of the other: but premises in which the middle is affirmed of one extreme and denied of the other are the necessary condition if one is to get a valid inference at all. Therefore, if, taken in this way, they are wholly false, their contraries conversely should be wholly true. But this is impossible. On the other hand, there is nothing to prevent both premises being partially false; e.g., if actually some A is C and some B is C, then if it is premised that all A is C and no B is C, both premises are false, yet partially, not wholly, false. The same is true if the major is made negative instead of the minor.

a38. Or one premise may be wholly false, and it may be either of them. Thus, supposing that actually an attribute of all A must also be an attribute of all B, then if C is yet taken to be a universal attribute of all but universally non-attributable to B, C-A will be true but C-B false. Again, actually that which is an attribute of no B will not be an attribute of all A either; for if it be an attribute of all A, it will also be an attribute of all B, which is contrary to supposition; but if C be nevertheless assumed to be a universal attribute of A, but an attribute of no B, then the premise C-B is true but the major is false.

b6. The case is similar if the major is made the negative premise. For in fact what is an attribute of no A will not be an attribute of any B either; and if it be yet assumed that C is universally non-attributable to A, but a universal attribute of B, the premise C-A is true but the minor wholly false. Again, in fact it is false to assume that that which is an attribute of all B is an attribute of no A, for if it be an attribute of all B, it must be an attribute of some A. If then C is nevertheless assumed to be an attribute of all B but of no A, C-B will be true but C-A false.

b14. It is thus clear that in the case of atomic propositions erroneous inference will be possible not only when both premises are false but also when only one is false.

On the other hand, an error of non-attribution. After showing how a false affirmative conclusion contrary to an immediate negative is obtained by syllogizing, the Philosopher here shows how by syllogizing a false negative is concluded contrary to an immediate affirmative.

First, in the first figure.

Second, in the second, at: in the second figure the premises (80a27; [237]).

Concerning the first he does two things.

First (80a8), he states his intention and says that since a universal negative may be concluded in the first and in the second figure, we must first show in which moods a syllogism of ignorance is formed in the first figure and under which conditions of truth and falsity in the propositions.

Second, at: thus, let A be atomically disconnected (80a11), he establishes his proposition.

First, he shows how such a proposition is formed from two false premises in the first figure.

Second, how it is formed from one false and one true premise, at: it is also possible when one is false (80a14; [235]).

He says therefore first (80a11) that the aforesaid syllogism can be formed from premises, both of which are false. This is clear if A is both in B and in C individually, that is, immediately. It is thus that a genus is immediately in the proximate species into which it is first divided, as color into blackness and whiteness. For the genus is predicated per se of the species, because the former is placed first in the definition of the latter; and it is predicated immediately of a proximate species, because it is put in its definition immediately and not in the way that a remote genus-which is put in the definition of a defining part-is related to an ultimate species. Therefore, let the terms be 'color', 'whiteness' and 'blackness'. If, then, we assume that A is in no C, for example, if we say, “No whiteness is a color,” but C is in every B, say “All blackness is whiteness,” both propositions are false, as is the conclusion, “No blackness is a color.”

Then, at: it is also possible when one is false (80a14) he shows how there can be one false and one true premise in the syllogism under discussion.

First, he shows how the major can be true and the minor false.

Second, how it might be the reverse, at: or again, C-B may be true (80a20; [236]).

He says therefore first (80a14) that a negative syllogism of ignorance can be formed in the first figure no matter which one of the propositions happens to be false. For it might happen that the proposition AC, which is the major, is true, and the proposition BC, which is the minor, is false. That the major proposition could be true he proves by the fact that the term A, whatever it be, need not be in all things, as color is not predicated of all beings. That the minor would be false he proves on the ground that it is not possible to assume a term of which A would be universally denied and which would also be predicated of B: for we are supposing that the proposition, “Every B is A,” is true and immediate. Therefore, if something were universally predicated of B, so that “Every B is C” would be true, then A cannot be universally denied of C. Consequently, this proposition, “No C is A,” which was the major, will not be true. For if every B is A, as we supposed, and every B is C, as we are now assuming, it follows in the third figure that some C is A, which contradicts the major. Therefore, the proposition, “No C is A,” will be false. Hence if this is true, which is the major, it is required that this be false, which is the minor, that is, “Every B is C.” Then he proves the same thing on the ground that if both premises are true, then as has been proved above, a false conclusion cannot follow—which is out of place in a syllogism of ignorance, which ought to conclude to a false conclusion. But it was given that this is true, namely, “No C is A”’: if then it is also true that “Every B is C,” it follows that the conclusion, “No B is A,” is true, whereas it is supposed to be false, being contrary to this immediate proposition, “Every B is A.”

Then, at: or again, C-B may be true (80a20) he shows how the minor can be true, the major being false.

And he says that the proposition CB, namely, the minor, can be true, while the major is false. For since this proposition, “Every B is A,” whose contrary is to be concluded, is immediate, it is necessary that B exist in A as a part in a whole, as “whiteness” in “color.” But it is possible to take something else in which B also exists as in a whole, though not immediately—let this other thing be 'quality', that is, C. It is necessary, therefore, according to the aforesaid, that as between these two, namely, A and C, one should be under the other, that is, color under quality. Now, if someone assumes that A is in no C and says, “No quality is a color,” the proposition will be false. But the minor will be true, namely, “Every whiteness is a quality.” The conclusion, however, “No whiteness is a color,” will be false and contrary to an immediate proposition. And so it is clear that a negative syllogism of ignorance can be formed in the first figure when either one or both of the premises are false.

Then, at: in the second figure the premises (80a27) he shows how a negative syllogism of ignorance is formed in the second figure.

First, when both are false.

Second, when one or the other is false, at: or one premise may be wholly false (80a38; [238]).

He says therefore first (80a27) that in the second figure it does not happen that both propositions are entirely false. And he calls those propositions entirely false which are contrary to true propositions. He proves this: For since we are trying to conclude a false negative contrary to an immediate affirmative, we must assume that this proposition, “Every B is A,” is true and immediate, say, “Every whiteness is a color.” But with terms so related it is impossible to find a middle term which would be predicated universally of one and universally removed from the other. For suppose that the term C could be universally removed from A and universally predicated of B. Then the proposition, “No A is C,” will be true; consequently, its converse, “No C is A,” will also be true. But every B is C. Therefore, no B is A, the contrary of which was supposed. Similarly, it cannot be universally removed from B and universally predicated of A. For if it is true that every A is C, the converse, “Some C is A,” will be true. But if it is true that no B is C, its converse, “No C is B,” will be true. So, then, from these two propositions, “Some C is A” and “No C is B,” there follows, “Some B is not A,” which is the contradictory of what was supposed, namely, that “Every B is A.” What remains, therefore, is that it is impossible to find any middle which, A and B being related in the way we have supposed, can be predicated of one and removed from the other. Yet if a syllogism is to be formed in the second figure, the middle must be predicated of one of the extremes and denied of the other. Therefore, if both are totally false, the contraries would have to be true, which is impossible as has been proved. However, nothing prevents both from being false particularly: thus we may take some middle which is predicated particularly of A and of B, say 'male', which is predicated particularly of animal and of man. Now, if C is taken in every A, say “Every animal is male,” and in no B, say “No man is male,” each proposition will be false, not entirely but particularly. And the same holds if, conversely, the major is negative and the minor affirmative, that is, if we should say, “No animal is male” and “Every man is male.”

Then, at: or one premise may be wholly false (80a38) he shows how it happens when one is false.

First, in the second mode of the second figure.

Second, in the first mode, at: the case is similar if the major (80b6; [239]).

He says therefore first (80a38) that in this figure it occurs that either proposition may be false. This is clear from the fact that if A is supposed to be predicated per se and immediately of B, whatever is in every A is in every B, as whatever is predicated universally of animal is predicated universally of man. Therefore, if some middle, C, be taken which is universally predicated of A, say “Every animal is living,” and universally removed from B, say “No man is living,” it is evident that AC, which is the major proposition, will be true, but CB, which is the minor, will be false. Similarly, he proves that the converse occurs when the major is false. For it cannot be that something be universally removed from B and universally predicated of A, when the terms have that position. For it has been stated that if something is in A universally, it follows that it is also in B. Consequently, if something be removed universally from B, it cannot be that it is predicated universally of A. For example, anything universally removed from “man” cannot be universally predicated of “animal.” Therefore, if something is taken which is universally removed from man, say 'irrational', and you state that “Every animal is irrational” and “No man is irrational,” it follows that the minor proposition is true and the major false. But in these terms the major premise is not totally false. However, one can take a term in which it is totally false, for example, if we should take 'inanimate' as the middle.

Then, at: the case is similar if the major (80b6) he shows the same in the first mode of the second figure, where the major is negative. For it is clear that with the terms A and B so related, as was said, something universally removed from A cannot be in any B. Therefore, if a middle, C, be taken which is universally removed from A and universally predicated of B, the major proposition will be true and the minor false. For example, if the terms are 'inanimate', 'animal' and 'man.' Similarly, he shows that the minor can be true and the major false. For it is clear, according to the aforesaid, that that which is universally predicated of B cannot be universally removed from every A, because what is universally predicated of B must be in some A at least. Therefore, if C be taken as a middle which is universally predicated of B, say, 'rational' or 'living,' and universally denied of A, there will be a true minor proposition, namely, “Every man is rational” or “living.” But the major, namely, “No animal is rational,” is false in part, while “No animal is living,” is false entirely.

Then summarizing (80b14) he concludes that a deceptive syllogism can be formed in immediates, when both propositions are false or only one is false.

b17. In the case of attributes not atomically connected with or disconnected from their subjects, (a) (i) as long as the false conclusion is inferred through the ‘appropriate’ middle, only the major and not both premises can be false. By ‘appropriate middle’ I mean the middle term through which the contradictory—that is, the true-conclusion is inferrible. Thus, let A be attributable to B through a middle term C: then, since to produce a conclusion the premise C-B must be taken affirmatively, it is clear that this premise must always be true, for its quality is not changed. But the major A-C is false, for it is by a change in the quality of A-C that the conclusion becomes its contradictory—that is, true.

b27. Similarly (ii) if the middle is taken from another series of predication; e.g., suppose D to be not only contained within A as a part within its whole but also predicable of all B. Then the premise D-B must remain unchanged, but the quality of A-D must be changed; so that D-B is always true, A-D always false. Such error is practically identical with that which is inferred through the ‘appropriate’ middle.

b32. On the other hand, (b) if the conclusion is not inferred through the ‘appropriate’ middle — (i) when the middle is subordinate to A but is predicable of no B, both premises must be false, because if there is to be a conclusion both must be posited as asserting the contrary of what is actually the fact, and so posited both become false: e.g., suppose that actually all D is A but no B is D; then if these premises are changed in quality, a conclusion will follow and both of the new premises will be false. When, however, (ii) the middle D is not subordinate to A, A-D will be true, D-B false—A-D true because A was not subordinate to D, D-B false because if it had been true, the conclusion too would have been true; but it is ex hypothesi false.

a5. When the erroneous inference is in the second figure, both premises cannot be entirely false; since if B is subordinate to A, there can be no middle predicable of all of one extreme and of none of the other, as was stated before. One premise, however, may be false, and it may be either of them. Thus, if C is actually an attribute of both A and B, but is assumed to be an attribute of A only and not of B, C-A will be true, C-B false: or again if C be assumed to be attributable to B but to no A, C-B will be true, C-A false. We have stated when and through what kinds of premises error will result in cases where the erroneous conclusion is negative.

a16. If the conclusion is affirmative, (a) (i) it may be inferred through the ‘appropriate’ middle term. In this case both premises cannot be false since, as we said before, C-B must remain unchanged if there is to be a conclusion, and consequently A-C, the quality of which is changed, will always be false.

a20. This is equally true if (ii) the middle is taken from another series of predication, as was stated to be the case also with regard to negative error; for D-B must remain unchanged, while the quality of A-D must be converted, and the type of error is the same as before.

a25. (b) The middle may be inappropriate. Then (i) if D is subordinate to A, A-D will be true, but D-B false; since A may quite well be predicable of several terms no one of which can be subordinated to another. If, however, (ii) D is not subordinate to A, obviously A-D, since it is affirmed, will always be false, while D-B may be either true or false; for A may very well be an attribute of no D, whereas all B is D, e.g., no science is animal, all music is science. Equally well A may be an attribute of no D, and D of no B. It emerges, then, that if the middle term is not subordinate to the major, not only both premises but either singly may be false.

a35. Thus we have made it clear how many varieties of erroneous inference are liable to happen and through what kinds of premises they occur, in the case both of immediate and of demonstrable truths.

In the case of attributes not atomically. After showing how a syllogism of ignorance in regard to immediate propositions is made, the Philosopher here shows how it is made in regard to mediate propositions.

First, how a false negative proposition is concluded which is opposed to a true affirmative.

Second, how a false affirmative is concluded which is opposed to a true negative, at: if the conclusion is affirmative (81a16; [245]).

Concerning the first he does two things.

First, he shows this in the first figure.

Second, in the second, at: when the erroneous inference (81a4; [244]).

In regard to the first he does three things.

First, he shows how a syllogism of ignorance is constructed in mediate propositions through a proper middle.

Second, how one is constructed through a middle which although not proper has a relationship to the terms, a relationship akin to that of a proper middle, at: similarly, if the middle (80b27; [242]).

Third, he shows how such a syllogism is constructed through an extraneous middle, at: on the other hand, if the conclusion (80b32; [243]).

He says therefore first (80b17) that when a syllogism is constructed which concludes something false in propositions which are not individual, that is, not immediate, if the proper middle through which the syllogism is formed be taken, then both propositions cannot be false, but only the major. He explains what he means by “proper middle.” For since the proposition whose contrary is to be syllogized is mediate, it is required that the predicate be syllogized of the subject through some middle. Therefore, the same middle can be employed to conclude the opposite. Say that the mediate proposition is “Every triangle has three angles equal to two right angles.” The middle through which the predicate is syllogized of the subject is, “A figure having an exterior angle equal to the two opposite interior angles.” Now, if we would prove through the same middle that no triangle has three angles equal to two right angles, it will be a syllogism of falsity through a proper middle. Hence he says that a proper middle is one through which the syllogism of contradiction is made, that is, leading to the opposite conclusion. In the above example, A would be 'triangle', B, 'having three...' and C, the middle, 'such a figure'. Now, in the first figure the minor must be affirmative; therefore, that which was the minor in the true syllogism must remain unconverted and not changed into its opposite in the syllogism of falsity. Hence it must always be true. But the major proposition of the true syllogism is changed into its contrary negative; hence the major must be false. For example, we might say: “No figure having an exterior angle equal to the two opposite interior angles has three angles equal to two right angles; but the triangle is such a figure: therefore, no triangle has three angles equal to two right angles.”

Then, at: similarly, if the middle is taken (80b27) he shows how the aforesaid syllogism is constructed through a middle which is extraneous but like the proper.

And he says that it will be syllogized in like fashion if the middle is taken from another ordering. For example, if A had been demonstrated of B through C, and we were to take in the syllogism of falsity not C but D as the middle, in such a way, however, that D is also contained universally under A and predicated universally of B: say if we took for the middle, 'a closed figure of three lines', because here too the minor proposition DB must remain as it was in the syllogism which concluded the true, although through a proper middle. But the major proposition will have to be changed into its contrary. And so the minor will always be true and the major always false. But as to the mode of the argument, this deception is similar to that which is formed through the proper middle.

Then, at: on the other hand, if the conclusion (80b32) he shows how a syllogism of falsity is made through a middle which is extraneous and unlike the proper.

For a middle can be taken such that it is contained universally under A but is predicated of no B. In this case both propositions will have to be false, because in order that the syllogism be formed in the first figure, it will be necessary to take propositions to the contrary, namely, a major which is negative, for example, “No D is A,” and the minor affirmative, for example, “Every B is D.” Clearly then both are false.

Now this relationship of terms cannot be found in things convertible, say in a subject and its proper attribute which is concluded of the subject through some middle. For it is obvious that no middle can be taken such that the proper attribute would be universally predicated of it, and that middle be removed universally from the subject. But this relationship can be found when the proposition is mediate, for example, when a higher genus or the proper attribute of a higher genus is predicated of an ultimate species, as when we say, every man is living. For 'living' can be concluded of man through the middle, 'animal'. Therefore, if we should take something of which 'living' would be universally predicated, say 'olive', but would be universally removed from man, the relationship of terms that we are seeking will result. For this will be false, “No olive is living”; and the minor, too, will be false, “Every man is an olive.” Similarly, the conclusion will be false, “No man is living,” which is contrary to the true mediate proposition.

It also happens that the major may be true and the minor false. For example, if we take as middle something which is not contained under A, say "stone"; then the major, AD, will be true, namely, “No stone is living,” because 'stone' is not contained under 'living', but the minor will be false, namely, “Every man is a stone.” For if the minor remained true, while the first was true, then the conclusion would be true, whereas it has been said that it is to be false.

However, the converse does not occur, that is, that the minor be true when the middle is extraneous, because such a middle cannot be predicated universally of B. But in the first figure the minor taken must always be an affirmative statement.

Then, at: when the erroneous inference (81a5) he shows how a negative syllogism of ignorance is made in the second figure.

And he says that in the second figure it cannot occur that both propositions be totally false. For if we are to conclude the false proposition, “No B is A,” contrary to the true, it would have been required that A be predicated universally of B. Hence nothing will be able to be found which would be universally predicated of one and universally denied of the other, as has been established above when we treated concerning the syllogism of ignorance in immediates. But one or the other of them can be totally false. And he manifests this first in the second mood of the second figure where the major is affirmative and the minor negative. Thus, let the middle be related to the extremes so that it is predicated universally of both, as 'living' is predicated universally of man and of animal. Then if we take the major affirmative, say “Every animal is living,” and the minor negative, say “No man is living,” the major will be true, the minor false, and the conclusion false. In like manner also, if in the first mood of the second figure we take the major negative, say “No animal is living,” and the minor affirmative, say “Every man is living,” the major will be false, the minor true, and the conclusion false.

Having said these things he sums up and concludes that he has stated when and through which kinds of premises deception can occur, if the deceptive syllogism is privative [negative].

Then, at: if the conclusion is affirmative (81a16) he shows how an affirmative syllogism of deception is formulated in mediate propositions.

First, when it is formulated through a proper middle.

Second, when it is formulated through a middle similar to a proper middle, at: this is equally true if the middle (81a20; [246]).

Third, when it is formulated through an extraneous middle, at: the middle may be inappropriate (81a25; [247]).

He says therefore first (81a16), that if an affirmative syllogism of deception is to be formulated in mediate propositions, if a proper middle such as explained above be taken, it is impossible that both propositions be false. For since a syllogism of this kind can be formed only in the first figure, both propositions being affirmative, it is required that the minor proposition remain as it was in the true syllogism. Hence the major will have to be changed, namely, from negative to affirmative, so that it will have to be false. For example, if we desire to conclude that “Every man is a quantity,” which is the contrary of the statement, “No man is a quantity,” whose proper middle is 'substance', we will take the false proposition, “Every substance is a quantity,” and the true proposition, “Every man is a substance.”

Then, at: this is equally true if the middle (81a20) he shows how a syllogism of ignorance is formed when a non-proper middle is taken which is not of the same order but from some other ordering. For example, if I say, “Every agent is a quantity; every man is an agent: therefore, every man is a quantity.” For it is necessary in this case for the minor to remain, but the major will have to be changed from negative to affirmative. Hence this deception is similar to the previous deception, as was stated in the privative syllogism.

Then, at: the middle may be inappropriate (81a25) he shows how an affirmative syllogism of deception is formulated through an extraneous middle.

And he says that if an extraneous middle be taken such that it is contained under the major extreme, then the major will be true and the minor false. For A, the major extreme, can be predicated universally of many things that are not under one another, say 'habit' of grammar and virtue. For this is mediate, “No grammar is a virtue.” There we can conclude the contrary of this, namely, “All grammar is virtue,” through a middle which is contained under virtue. Then the major will be true and the minor false. For example, we might say: “All temperance is a virtue; grammar is temperance: therefore, all grammar is a virtue.” But if a middle is taken which is not under the major extreme, the major will always be false, because it will be affirmative. But the minor may sometimes be false with such a major; then both will be false. For example, if we should say: “Every whiteness is a virtue; all grammar is whiteness: therefore....” But sometimes it can be true. For when the terms are so related, there is nothing to hinder A from being removed from every D, and D from being in every B, as happens in these terms, namely, 'animal,' 'science,' 'music.' For the major extreme, “animal,” is removed universally from all science; hence this proposition which is taken as the major in the syllogism of ignorance is false, namely, “Every science is an animal.” But the minor, namely, “All music is science,” is true. And the conclusion will be false, being contrary to the true mediate negative. It can also happen that A is in no D, and D in no B, as has been said. Therefore, it is evident that when the middle is not contained under the major extreme, they may both be false or just one of them, because the major and the minor may be false. However, with the three terms so related, as we have said above, the major cannot be true.

Finally (81a35), he summarizes and concludes that it is plain from the foregoing how many ways and through which alignment of true and false propositions it is possible to construct deceptions through syllogisms, both in immediate propositions and in mediate propositions, which are proved by demonstration.

a38. It is also clear that the loss of any one of the senses entails the loss of a corresponding portion of knowledge, and that,

a39. since we learn either by induction or by demonstration, this knowledge cannot be acquired. Thus demonstration develops from universals, induction from particulars; but since it is possible to familiarize the pupil with even the so-called mathematical abstractions only through induction — that is, only because each subject genus possesses, in virtue of a determinate mathematical character, certain properties which can be treated as separate even though they do not exist in isolation — it is consequently impossible to come to grasp universals except through induction. But induction is impossible for those who have not sense-perception. For it is sense-perception alone which is adequate for grasping the particulars: they cannot be objects of scientific knowledge, because neither can universals give us knowledge of them without induction, nor can we get it through induction without sense-perception.

It is also clear that the loss. After the Philosopher has determined concerning ignorance through deception which is caused by a syllogism, he now determines concerning the ignorance of simple negation which is produced without a syllogism.

First, he shows in which type of person this ignorance is had of necessity.

Second, he proves his proposition, at: since we learn either by induction (81a39; [251]).

He says therefore first (81a38) that if a person lacks any of the senses, say, sight or hearing, then necessarily the science of the sensible objects proper to those senses will be lacking. Thus, if a person lacks the sense of sight, then of necessity the science of colors will be lacking in him. And so he will have ignorance of negation in regard to colors, being entirely ignorant of colors. However, this must be understood of persons who never had the sense of sight, as a person born blind. But if someone loses the sight he once had, he does not on that account necessarily lack a science of colors, because the memory of colors previously sensed remains in him.

But it is possible that ignorance of negation be had of certain things which can nevertheless be known through a sense we possess.. For example, if someone with sight were always in the dark, he would de facto lack a science of colors, but not of necessity, because he could acquire it by sensing colors-which does not occur in one who lacks the sense of sight. Hence he adds that it is impossible to receive it because one who lacks the power of sight cannot even acquire a knowledge of colors.

Then, at: since we learn either by induction (81a39) he proves his proposition on the ground that there are two ways of acquiring science: one is through demonstration, and the other is through induction, as we have stated in the beginning of this book. But these two ways differ, because demonstration proceeds from universals, but induction from particulars. Therefore, if any universals from which demonstration proceeds could be known without induction, it would follow that a person could acquire science of things of which he does not have sense experience. But it is impossible that universals be known scientifically without induction. This is quite obvious in sensible things, because we receive the universal aspect in them through the experience which we have in regard to sensible things, as is explained in Metaphysics I.

But this might be doubted in things which are abstract, as in mathematics. For although experience begins from sense, as it is stated in Metaphysics 1, it seems that this plays no role in studying things already isolated or abstracted from sensible matter.

Therefore, to exclude this he says that even those things which are said abstractly happen to be made known through induction, because in each genus of abstract things are certain particulars which are not separable from sensible matter, so far as each of them is a “this something.” For although “line” is studied in isolation from its sensible matter, nevertheless “this line,” which is in sensible matter, so far forth as it is individualized, cannot be so isolated, because its individuation is from this matter. Furthermore, the principles of abstracted [isolated] things, from which demonstrations regarding them are formed, are not made manifest to us except from certain particulars which we perceive by sense. Thus, from the fact that we see some single sensible whole we are led to know what a whole is and what a part, and we know that every whole is greater than its part by considering this in many. Thus the universals from which demonstration proceeds are made known to us only through induction.

Now men who lack any of the senses cannot make an induction from singulars pertaining to that sense, because sense is the sole knower of the singulars from which induction proceeds. Hence such singulars are utterly unknown, because it does not occur that anyone lacking a sense receives science of such singulars: first, because he cannot demonstrate from universals without induction through which universals are known, as has been said; second, because nothing can be known through induction without the sense which is concerned with the singulars from which induction proceeds.

It should be noted that by these words of the Philosopher two positions are excluded.

The first is Plato’s, who stated that we do not have science of things except through Forms participated from ideas. If this were so, universals could be made known to us without induction, and we would be able to acquire a science of things of which we have no sense. Hence Aristotle also uses this argument against Plato at the end of Metaphysics I.

The second is the position of those who claim that in this life we can know separated substances by understanding their quiddities, which however cannot be known through sensible objects which we know and which are entirely transcended by them. Hence if they were known according to their essences, it would follow that some things would be known without induction and sense perception, which the Philosopher here denies even in regard to abstracted things.

Three questions about proceeding to infinity in confirming demonstrations

b10. Every syllogism is effected by means of three terms. One kind of syllogism serves to prove that A inheres in C by showing that A inheres in B and B in C; the other is negative and one of its premises asserts one term of another, while the other denies one term of another.

b14. It is clear, then, that these are the principles and so-called hypotheses of syllogism. Assume them as they have been stated, and proof is bound to follow — proof that A inheres in C through B, and again that A inheres in B through some other middle term, and similarly that B inheres in C.

b18. If our reasoning aims at gaining credence and so is merely dialectical, it is obvious that we have only to see that our inference is based on premises as credible as possible: so that if a middle term between A and B is credible though not real, one can reason through it and complete a dialectical syllogism. If, however, one is aiming at truth, one must be guided by the real connexions of subjects and attributes.

b24. Thus, since there are attributes which are predicated of a subject essentially or naturally and not coincidentally — not, that is, in the sense in which we say ‘That white (thing) is a man’, which is not the same mode of predication as when we say ‘The man is white’: the man is white not because he is something else but because he is man, but the white is man because ‘being white’ coincides with ‘humanity’ within one substratum — therefore there are terms such as are naturally subjects of predicates.

b30. Suppose, then, C is such a term not itself attributable to anything else as to a subject, but the proximate subject of the attribute B — that is, so that B-C is immediate; suppose further E related immediately to F, and F to B. The first question is, must this series terminate, or can it proceed to infinity?

b34. The second question is as follows: Suppose nothing is essentially predicated of A, but A is predicated primarily of H and of no intermediate prior term, and suppose H similarly related to G and G to B; then must this series also terminate, or can it too proceed to infinity? There is this much difference between the questions: the first is, is it possible to start from that which is not itself attributable to anything else but is the subject of attributes, and ascend to infinity? The second is the problem whether one can start from that which is a predicate but not itself a subject of predicates, and descend to infinity?

a2. A third question is, if the extreme terms are fixed, can there be an infinity of middles? I mean this: suppose for example that A inheres in C and B is intermediate between them, but between B and A there are other middles, and between these again fresh middles; can these proceed to infinity or can they not?

a7. This is the equivalent of inquiring, do demonstrations proceed to infinity, that is, is everything demonstrable? Or do ultimate subject and primary attribute limit one another?

a9. I hold that the same questions arise with regard to negative conclusions and premises: viz. if A is attributable to no B, then either this predication will be primary, or there will be an intermediate term prior to B to which a is not attributable — G, let us say, which is attributable to all B — and there may still be another term H prior to G, which is attributable to all G.

The same questions arise, I say, because in these cases too either the series of prior terms to which a is not attributable is infinite or it terminates. a15. One cannot ask the same questions in the case of reciprocating terms, since when subject and predicate are convertible there is neither primary nor ultimate subject, seeing that all the reciprocals qua subjects stand in the same relation to one another, whether we say that the subject has an infinity of attributes or that both subjects and attributes — and we raised the question in both cases — are infinite in number. These questions then cannot be asked — unless, indeed, the terms can reciprocate by two different modes, by accidental predication in one relation and natural predication in the other.

Every syllogism is effected. After the Philosopher has determined concerning the demonstrative syllogism by showing from what and what sort of things it proceeds and in which figure demonstrations can be formed, he now inquires whether demonstrations can proceed to infinity.

First, he raises the question.

Second, he settles it, at: now, it is clear that if the predications (82a21; [268]).

Concerning the first he does two things.

First, he sets down certain prefatory remarks needed for understanding the question.

Second, he raises the question, at: suppose, then, C is such a term (81b30; [260]).

Concerning the first he does two things.

First, he prefaces something about the syllogistic form one must observe in demonstrations.

Second, he reviews what the matter of demonstration should be, at: it is clear, then, that these (81b14; [257]).

With respect to the first (81b10) he touches on three things, the first of which is common to every syllogism, namely, that every syllogism is formed in three terms, as is indicated in Prior Analytics I. The second however, pertains to an affirmative syllogism whose form is such that it concludes A to be in C, because A is in B and B in C. And this is the form of a syllogism in the first figure in which alone can be concluded a universal affirmative, the chief quest in demonstration. The third pertains to a negative syllogism which of necessity has one affirmative proposition and one negative, but differently in the first figure and in the second, as is clear from what has been shown in the Prior Analytics.

Then, at: it is clear, then, that these (81b14) he reviews what the matter of demonstration should be.

In regard to this he does three things.

First, he states what this matter is.

Second, he shows the difference between this matter and the matter of a dialectical syllogism, at: if our reasoning aims (81b18; [258]).

Third, he clarifies this difference, at: thus, since there are attributes (81b24; [259]).

He says therefore first (81b14) that since a syllogism has three terms from which are formed the two propositions which conclude the third, it is clear that these propositions, from which one proceeds in a demonstrative syllogism according to the aforesaid form, are the principles and suppositions we discussed earlier. For one who accepts such principles thus demonstrates through them in the syllogistic form we have mentioned, namely, that A is in C is proved through B; and if the proposition AB is again mediate, another middle is used to demonstrate that A is in B. The like is done if the minor proposition, BC, is mediate.

Then, at: if our reasoning aims (81b18), apropos of what has been said, he shows the difference between a demonstrative and a dialectical syllogism.

For since the latter aims at producing opinion, the sole intent of a dialectician is to proceed from things that are most probable, and these are things that appear to the majority or to the very wise. Hence if a dialectician in syllogizing happens upon a proposition which really has a middle through which it could be proved, but it seems not to have a middle because it appears to be per se known on account of its probability, this is enough for the dialectician: he does not search for a middle, even though the proposition is mediate. Rather he syllogizes from it and completes the dialectical syllogism satisfactorily. The demonstrative syllogism, on the other hand, is ordained to scientific knowledge of the truth; accordingly, it pertains to the demonstrator to proceed from truths which are really immediate. Hence if he happens upon a mediate proposition, he must prove it through its proper middle until he reaches something immediate, because he is not content with the probability of a proposition.

Then, at: thus, since there are attributes (81b24) he elucidates what he has said, asserting that his claim that the demonstrator proceeds to the truth from things that are is supported by the fact that it is possible to find something which is not predicated of a thing accidentally. What this means he explains by showing how in the case of affirmative statements something is predicated accidentally. For something is predicated accidentally in two ways: in one way, when the subject is predicated of an accident, as when we say, “The white thing is a man”; in another way, when the accident is predicated of the subject, as when we say, “The man is white.” Now, this way differs from the first, because when the accident is predicated of the subject, it is stated that the man is white not because something else is white but because the man himself is white. Yet it is a per accidens proposition, because “white” does not belong to man according to the specific nature of man. For neither is placed in the definition of the other. But when it is stated that the white thing is a man, this is not so stated because being a man is in the whiteness, but because being a man is in the subject of whiteness, which subject happens to be white. Hence this way is further removed from per se predication than the first. But there are certain things which are not predicated per accidens in either of these ways: these are said to be per se. Such are the things from which the demonstrator proceeds. But the dialectician is not so demanding; consequently, the question concerning such things as are predicated per se is not relevant to the dialectical syllogism but only to the demonstrative syllogism.

Then, at: suppose, then, C is such a term (81b30) he raises the questions he intended.

Concerning this he does two things.

First, he raises the questions in regard to things to which they are relevant.

Second, he shows the cases in which they are not relevant, at: the same questions arise (82a15; [266]).

Concerning the first he does two things.

First, he raises questions in affirmative demonstrations.

Second, he shows that these questions also have relevance in negative demonstrations, at: I hold that the same questions (82a9; [266]).

In regard to the first he does two things.

First, he raises the questions.

Second, he shows where such questions are relevant, at: this is the equivalent of inquiring (82a7; [265]).

In regard to the first (81b30) he raises three questions corresponding to the three terms of a syllogism.

First, he raises a question concerning the major extreme, namely, whether one can go to infinity in ascending order? In this question an ultimate subject is supposed which is not predicated of any other, but other things are predicated of it. Let this subject be C, and let B be in C first and immediately, and let E be in B, as universally predicated of B; furthermore, let Z be in E as universally predicated of it. The question is this: Should this ascending process come to a halt somewhere, so that something is reached which is predicated universally of other things but nothing else is predicated of it, or is that not necessary but a process to infinity occurs?

Second, at: the second question is as follows (81b34) he raises the question on the part of the minor term, namely, whether one can go to infinity in descending. In this question some first universal predicate is supposed which is predicated of other things and nothing is more universal than it so as to be predicated of it. Thus let A be such that nothing is predicated of it as a universal whole of a part, but A is predicated of H both first and immediately, and H of G, and G of B. The question then is this: Is it necessary to come to a halt in this descending process, or may it proceed to infinity?

Then he shows the difference between these two questions. For in the first one we asked: If someone begins from a most particular subject which is in nothing else the way a universal whole is in a part but other things are in it, does an infinite ascending process occur? But in the second we are asking: If someone begins with a most universal predicate, which is predicated of other things as a universal whole of its parts but nothing is predicated of it in this way, does an infinite descending process occur?

Third, at: a third question is (82a2) he raises the third question on the part of the middle term. In this question two determinate extremes are supposed, namely, a most universal predicate and a most particular subject. The question is whether under these conditions there can be an infinity of middles. Thus, if A is the most universal predicate and C the most particular subject, and if between A and C there is the middle, B, and again between A and B another middle, and likewise between B and C; furthermore, if there are other middles of these middles between them and the extremes both in ascending and in descending order. The question then is this: May these processes go on to infinity or is that impossible?

Then, at: this is the equivalent of inquiring (82a7) he shows what is the tenor of these questions. And he says that these questions pertain to the matter now under discussion, namely, to demonstrations.

He says, therefore, that the attempt to reach true answers to these questions is the same as trying to settle the question whether demonstrations proceed to infinity by ascending or descending. By ascending, that is, so that each proposition from which a demonstration proceeds would be demonstrable by another prior proposition. (This is what he means when he asks: is there a demonstration of everything, that is, of every proposition? By so thinking, some have erred in regard to principles, as is stated in Metaphysics IV.) And by descending, that is, whether it is possible from any demonstrated proposition to proceed again to another demonstration subsequent to it. Thus, one element of the doubt is whether demonstrations proceed to infinity either by ascending or by descending. The other element is whether demonstrations are mutually limiting, so that one demonstration may be confirmed by another in an ascending process, and from one demonstration another may proceed by a descending process: and this until a limit is reached.

Then, at: I hold that the same questions arise (82a9) he shows that these questions are also relevant to negative demonstrations, because a negative demonstration must employ an affirmative proposition in which the subject of the conclusion is contained under the middle and from which the predicate of the conclusion is removed. Therefore, to the extent that there is ascent and descent in affirmative, there must be ascent and descent in negative syllogisms and propositions. For example, if the conclusion of a demonstrative syllogism is, “No C is A,” and the middle taken is B, from which A is removed: the first thing to be considered, therefore, is whether A is removed from B first and immediately, or whether there is another middle I to be taken, from which A would be removed before it would be removed from B. In that case it would again be necessary to consider whether A would be removed from something else before I, namely, from T which is predicated universally of I. Therefore, in these also the question arises whether one can proceed to infinity in removing (so that something would always remain from which it would have to be removed), or must one stop somewhere.

Then, at: the same questions arise (82a15) he shows the cases in which these questions have no relevance.

In cases in which there is mutual predication and mutual conversion there is no prior and subsequent to be taken in the sense in which the prior [notion] is that with which a subsequent [notion] is not convertible, as universals are prior; because no matter whether the predicates be infinite, so that one might proceed to infinity in predicating, or whether there be infinity on both sides, that is, on the side of the predicate as well as of the subject, all such infinites bear a like relationship to all, because any of them could be predicated of any other and be the subject of any of the convertibles. However, there can be this difference: one of them might be predicated as an accident and another as a predicament, that is, as a substantial predicate.

And this is the difference between a property, and a definition: although the two are convertible with the subject, nevertheless the definition is an essential predicate and therefore naturally prior to the property, which is an accidental predicate. That is why in demonstrations we use the definition as the middle to demonstrate a proper attribute of the subject.

Solution of some of these doubts hinges upon solution of others

a21. Now, it is clear that if the predications terminate in both the upward and the downward direction (by ‘upward’ I mean the ascent to the more universal, by ‘downward’ the descent to the more particular), the middle terms cannot be infinite in number.

a24. For suppose that A is predicated of Z, and that the intermediates — call them B B’ B..”. — are infinite, then clearly you might descend from and find one term predicated of another ad infinitum, since you have an infinity of terms between you and Z; and equally, if you ascend from Z, there are infinite terms between you and A. It follows that if these processes are impossible there cannot be an infinity of intermediates between A and Z.

a30. Nor is it of any effect to urge that some terms of the series AB...F are contiguous so as to exclude intermediates, while others cannot be taken into the argument at all: whichever terms of the series B...I take, the number of intermediates in the direction either of A or of Z must be finite or infinite: where the infinite series starts, whether from the first term or from a later one, is of no moment, for the succeeding terms in any case are infinite in number.

a37. Further, if in affirmative demonstration the series terminates in both directions, clearly it will terminate too in negative demonstration. Let us assume that we cannot proceed to infinity either by ascending from the ultimate term (by ‘ultimate term’ I mean a term such as was, not itself attributable to a subject but itself the subject of attributes), or by descending towards an ultimate from the primary term (by ‘primary term’ I mean a term predicable of a subject but not itself a subject). If this assumption is justified, the series will also terminate in the case of negation.

b4. For a negative conclusion can be proved in all three figures. In the first figure it is proved thus: no B is A, all C is B. In packing the interval B-C we must reach immediate propositions — as is always the case with the minor premise — since B-C is affirmative. As regards the other premise it is plain that if the major term is denied of a term D prior to B, D will have to be predicable of all B, and if the major is denied of yet another term prior to D, this term must be predicable of all D. Consequently, since the ascending series is finite, the descent will also terminate and there will be a subject of which A is primarily non-predicable.

b13. In the second figure the syllogism is, all A is B, no C is B, no C is A. If proof of this is required, plainly it may be shown either in the first figure as above, in the second as here, or in the third. The first figure has been discussed, and we will proceed to display the second, proof by which will be as follows: all B is D, no C is D..., since it is required that B should be a subject of which a predicate is affirmed. Next, since D is to be proved not to belong to C, then D has a further predicate which is denied of C. Therefore, since the succession of predicates affirmed of an ever higher universal terminates, the succession of predicates denied terminates too.

b23. The third figure shows it as follows: all B is A, some B is not C. Therefore, some A is not C. This premise, that is, C-B, will be proved either in the same figure or in one of the two figures discussed above. In the first and second figures the series terminates. If we use the third figure, we shall take as premises, all E is B, some E is not C, and this premise again will be proved by a similar prosyllogism. But since it is assumed that the series of descending subjects also terminates, plainly the series of more universal non-predicables will terminate also.

b28. Even supposing that the proof is not confined to one method, but employs them all and is now in the first figure, now in the second or third — even so the regress will terminate, for the methods are finite in number, and if finite things are combined in a finite number of ways, the result must be finite. Thus it is plain that the regress of middles terminates in the case of negative demonstration, if it does so also in the case of affirmative demonstration.

Now, it is clear that if the predications. After raising the questions, the Philosopher here begins to settle them.

And his treatment is divided into two parts. In the first he shows that the solution of some of the doubts is reduced to the solution of others. In the second he settles the doubt as to those items in which the difficulty lies per se as in its source, at: that in fact the regress (82b34) [L. 33].

Concerning the first he does two things. First, he shows that the doubt bearing on the middles is reduced to the one which is concerned with the extremes and is solved by the solution of the latter. Second, he shows that the doubt bearing on negative demonstrations is reduced to the one which is concerned with affirmative demonstrations, at: further, if in affirmative demonstration (82a37).

In regard to the first he does three things. First, he states his intended proposition. Second, he proves this proposition, at: for suppose that A is predicated of Z (82a24). Third, he excludes a subterfuge, at: for is it of any effect to urge (82a30).

He says therefore first (82a21) that it will be plain to anyone who considers the following reason that an infinity of middles does not occur, if the predications both upwards and downwards stop at certain terms, namely, at the highest predicate and the lowest subject. And he explains what upward and downward predication consists in, saying that one proceeds upwards when there is movement to the more universal, one of whose marks it is that it be predicated; but one proceeds downwards when there is movement to the most particular, one of whose marks it is that it functions as a subject.

Then, at: for suppose that A is predicated of Z (82a24) he shows what he has proposed in the following way: Let the case be that A is the highest predicate and Z the lowest subject, and that there is an infinitude of middles, each of which we shall call B. Now, since A was the first predicate, it will be predicated of some middle near it, and that middle of another middle below it. Since the middles are infinite, it follows that the predication will proceed downwards to infinity—which is contrary to what we are assuming. For it was assumed that the predications do not proceed downwards to infinity. The result is the same if we start at Z, which is the lowest subject, and proceed upwards to infinity before A is reached—this too would be against our assumption. Therefore, if these are impossible, namely, that one may proceed to infinity by ascending or by descending, it will be impossible for the middles to be infinite. Thus it is clear that the question of an infinity of middles is reduced to a question of the infinity of the extremes.

Then, at: nor is it of any effect (82a30) he excludes an objection. For someone might object saying that the aforesaid proof would hold if ABC, that is, the middle and the extremes, were so related as to be had to one another, that is, so that there would be no middle between them: for this is the way the Philosopher defines 'had' in Physics V, namely, that it is next to something without anything between. And this seemed to be supposed in the above proof, namely, that A is predicated of some middle as though had to it, that is, following it immediately. But one who posits infinite middles will say that this cannot be supposed. For he will say that between any two terms that are taken there is a middle.

But the philosopher says that it makes no difference whether the infinitude be of middles that are “had” to one another, the way discrete things are (for example, in a city. house is consecutive to house, and in numbers, unity to unity), or whether something “had” cannot be found in the middles although between any two middles it is always possible to find another, as happens in continua, in which, between any two signs, that is, between two points, another can be found between them. That this makes no difference one way or the other to the matter at hand he manifests in the following way: Granted that between A and Z there is an infinitude of middles, each of which is called B, yet no matter which of these I employ, there is either an infinitude of middles between it and A and Z, or there is not an infinitude of them between it and one or the other of the extremes. (For example, let us suppose that the middles are mutually “had,” as happens in discrete things, and let us take a middle which is “had” to A; then it will be necessary that between that middle and Z there is still an infinitude of middles; and similarly, if we assume a certain finitude of middles between that middle and A. And the same reasoning holds if the middle which is taken be joined immediately to Z or is distant from it by a finitude of middles.) From the fact, therefore, that from any given middle one must take an infinitude of middles to one or other of the extremes, it makes no difference whether it is joined to either extreme immediately, that is, without a middle, or not immediately, that is, through other middles: because even if it be joined to one extreme without a middle, it will still be necessary later to find an infinitude of middles in relation to the other. Consequently, it will always be required, if there is an infinitude of middles, to proceed to infinity in predications either by ascending or by descending, as the above proof showed.

Then, at: further, if in affirmative demonstration (82a37) he shows that if there is no process to infinity in affirmadve demonstrations, then neither in negative demonstrations: and thus the question of negative demonstrations is reduced to the question of affirmative ones.

He does three things in regard to this point. First, he proposes what he intends. Second, he proves what he proposed, at: for a negative conclusion can be proved (82b4). Third, he excludes an objection, at: even supposing that the proof (82b28).

He says therefore first (82a37) that it will be clear from what follows that if in the predicative, that is, in the affirmative demonstration, a stop is made at both, that is, upwards and downwards, it will be necessary that a stop be made in the negative demonstration. To elucidate what he is proposing he says: Let the case be such that from the ultimate, that is, from the lowest subject one cannot go in ascending order to infinity toward universal predicates. And he explains that 'ultimate' means that which is not in any other as in a less particular, but something else is in it, and let it be Z. And let the case also be that one does not go to infinity when proceeding from the first to the ultimate. And he explains that the 'first' means that which is predicated of others but nothing else is predicated of it as more universal than it. Thus the 'first' is understood to be the most universal, and the 'ultimate' the most particular. If, therefore, on both sides there be a stop in affirmative demonstrations, he says that as a consequence, there is also a stop in negative demonstrations.

Then, at: for a negative conclusion can be proved (82b4) he proves his proposition.

First, in the first figure. Second, in the second, at: in the second figure (8513). Third, in the third, at: the third figure shows (82b23). For a negative can be concluded in three figures.

He says therefore first (82b4) that there are three ways of demonstrating a negative proposition through which something is signified not to be.

In one way in the first figure according to the mode that B is universally in C in the universal affirmative minor, but A is in no B in the universal negative major. Now, since we are supposing that there is a stop in affirmatives, both upwards and downwards, it is necessary that the proposition, BC, which is affirmative, if it is not immediate but a space exists with middles between B and C, be reduced to immediates, because that space which exists between the middle term and the extreme is affirmative, in which a stop is supposed. But if we take the other space, which is between B and A, it is clear that if this proposition, “No B is A,” is not immediate, it is necessary that A be removed from something else before being removed from B. Let this be D. But if this D be taken as a middle between A and B, it is necessary that it be universally predicated of B, because the minor must be affirmative. And if this too is not immediate, that is, “No D is A” then A has to be denied of something prior to D, say E, which again will be predicated universally of D for the same reason. Therefore, since there is a stop in affirmatives when we ascend, as supposed, it follows that something is reached of which A should be denied first and immediately; otherwise one would go still further in affirmatives, as is clear from the foregoing.

Then, at: in the second figure (82b13) he proves the same thing for the negative which is concluded in the second figure. For let the case be that B, which is the middle, is predicated universally of A and denied universally of C, so that the conclusion is “No C is A.” Now, if the negative premise again needs to be demonstrated because it is mediate, it must be demonstrated either in the first figure in the mode of demonstrating concerning which we have shown that there is a stop, if there is a stop in affirmatives; or it must be demonstrated through this mode, that is, in the second figure, or through a third mode, that is, in the third figure. Now, it has been established that there is a stop in negatives of the first figure, if there is a stop in affirmatives. Consequently, the same will now be demonstrated as to the second figure. Therefore, let this proposition, “No C is B,” be demonstrated in such a way that D is universally predicated of B in the universal affirmative major and denied universally of C in the universal negative minor. Now, again, if the proposition, “No C is D,” is mediate, it will be required to take some other middle which will be predicated universally of D and universally removed from C. Continuing thus, it will be necessary to proceed in negative demonstrations just as we do in affirmatives, namely, B will be predicated of A and D of B, and something else of D, and so on to infinity in affirmatives. But because we are supposing an upward stop in the affirmatives, it is also necessary to come to a stop in the negatives according to this mode in which a negative is demonstrated in the second figure.

Then, at: the third figure shows it (82b23) he shows the same thing in the third figure. Therefore, let B be a middle of which A is universally predicated, but C is universally denied of it: the conclusion will be a particular negative, namely, C is denied of some A. Now, that there is a stop in the affirmative premise, “Every B is A,” is granted by our supposition. Furthermore, that there must be a stop in the negative, “No B is C,” which is the major, is evident, because if it had to be demonstrated, it would be done either through what was said above, that is, through the first and second figure or in the way that the conclusion was concluded, namely, through the third figure, in which case this minor is not affirmed as universal but as particular. But there is a stop in that way, that is, if one proceeds in the first and second figure. But if one proceeds in the third figure to conclude that “Some B is not C,” let a middle, E, be taken such that B is universally affirmed of it but C is particularly denied of it. Then this happens once more in like manner, that is, according to this, one will always proceed in the negative demonstration by accumulating affirmative predications in descending order, because B which was the first middle will be predicated of E, and E of something else, and so on to infinity. But since we are supposing that there is a stop in the descending order in affirmatives, it is clear that there will be a stop in the negatives on the part of C.

Then, at: even supposing that the proof (82b28) he excludes an objection. For someone could say that it is necessary to stop in negative propositions when there is a stop in the affirmatives, provided that one always syllogizes according to the same figure; but if one demonstrates now in one figure and now in another, one can go to infinity. And he says that it is obvious that if one does not limit himself to one figure in demonstrating but uses all, proceeding now in the first figure and now in the second and third, there must still be a stop in the negatives if there is one in the affirmatives. For these various ways of demonstrating are finite, and each of them will be enlarged not to infinity but finitely by ascending or descending, as was shown. Now, if, finite things be taken a finite number of times, the result must be finite. Hence it remains that in all the modes there must be a stop in negative demonstrations, if there is a stop in the affirmatives.

That one does not proceed to infinity in essential predicates is shown logically

b34. That in fact the regress terminates in both these cases may be made clear by the following dialectical considerations. In the case of predicates constituting the essential nature of a thing, it clearly terminates, seeing that if definition is possible, or in other words, if essential form is knowable, and an infinite series cannot be traversed, predicates constituting a thing’s essential nature must be finite in number.

a1. But as regards predicates generally we have the following prefatory remarks to make. (1) We can affirm without falsehood ‘the white (thing) is walking’, and that big (thing) is a log’; or again, ‘the log is big’, and ‘the man walks’. But the affirmation differs in the two cases. When I affirm ‘the white is a log’, I mean that something which happens to be white is a log — not that white is the substratum in which log inheres, for it was not qua white or qua a species of white that the white (thing) came to be a log, and the white (thing) is consequently not a log except incidentally. On the other hand, when I affirm ‘the log is white’, I do not mean that something else, which happens also to be a log, is white (as I should if I said ‘the musician is white,’ which would mean ‘the man who happens also to be a musician is white’); on the contrary, log is here the substratum — the substratum which actually came to be white, and did so qua wood or qua some wood and qua nothing else. If we must lay down a rule, let us entitle the latter kind of statement predication, and the former not predication at all, or not strict but accidental predication. ‘White’ and ‘log’ will thus serve as types respectively of predicate and subject. We shall assume, then, that the predicate is invariably predicated strictly and not accidentally of the subject, for on such predication demonstrations depend for their force.

a21. It follows from this that when a single attribute is predicated of a single subject, the predicate must affirm of the subject either some element constituting its essential nature, or that it is in some way qualified, quantified, essentially related, active, passive, placed, or dated.

a24. (2) Predicates which signify substance signify that the subject is identical with the predicate or with a species of the predicate. Predicates not signifying substance which are predicated of a subject not identical with themselves or with a species of themselves are accidental or coincidental;

a27. For example, white is a coincident of man, seeing that man is not identical with white or a species of white, but rather with animal, since man is identical with a species of animal. These predicates which do not signify substance must be predicates of some other subject, and nothing can be white which is not also other than white.

a33. The Forms we can dispense with, for they are mere sound without sense; and even if there are such things, they are not relevant to our discussion, since demonstrations are concerned with predicates such as we have defined.

That in fact the regress terminates. After showing that if there is a stop in the extremes there must be a stop in the middles, and if there is a stop in affirmatives there must be a stop in negatives, the Philosopher now shows that there is a stop in affirmatives both upwards and downwards.

And his treatment is divided into two parts. In the first part he shows his thesis “logically,” that is, through characteristics common to every syllogism, which are based on predicates considered commonly. In the second he shows the same thing analytically, that is, through characteristics proper to demonstration, which are based on per se predicates which are proper to demonstration, at: but an analytic process will show (84a8) [L. 35].

The first is divided into two parts. In the first he shows that one does not proceed to infinity in predicates which are predicated in eo quod quid (that is, pertaining to the essence of the subject]. In the second he shows universally that one does not proceed to infinity in affirmative predicates, at: but as regards predicates (83a1).

He says therefore first (82b34) that since we have established that an infinite process does not occur in negatives if there is a stop in affirmatives, our present task will be to show how one speculates through logical reasons that there is a stop in those, that is, in affirmatives. (These reasons are called “logical,” because they proceed from certain common notions that pertain to the considerations of logic). Now, this truth is clear in regard to things predicated as constituting the essence of a thing, namely, the predicates from which the quod quid est, that is, the definition, is formed. For if such predicates were agreed to be infinite, the result would be that nothing can be defined, and that if something is defined, its definition cannot be known: and all because the infinite cannot be traversed. For a thing cannot be defined, or its definition known, except by reaching the ultimate through descent and the first through ascent. Therefore, if something can be defined, or if a thing’s definition can be known, then in either case this consequence will follow, namely, there is no infinite process in the aforementioned predicates, but there is a stop in them.

Then, at: but as regards predicates generally (83a1) he shows universally that there is not a process to infinity in affirmative predicates.

In regard to this he does two things. First, he prefaces certain things needed for establishing the thesis. Second, he establishes it, at: if A is a quality of B (83a36) [L. 34].

In regard to the first he does two things. First, he distinguishes per accidens from per se predicates. Second, he distinguishes among the per se predicates, at: it follows from this (83a21).

He says therefore first (83a1) that since it has been established in regard to certain predicates that there is no infinite process in them, namely, in those which are predicated as pertaining to the essence, our task is to show that this is universally so in all affirmative predicates.

And he begins his consideration with per accidens predicates in which there are three modes of true predication. One mode is when an accident is predicated of an accident, as when we say, “Something white is walking.” The second mode is when the subject is predicated of an accident, as when we say, “Something white is wood.” The third mode is when an accident is predicated of a subject, as when we say, “Wood is white,” or “Man walks.” Now, these modes of predicating are mutually other and diverse, because when its subject is predicated of an accident, say “Something white is wood,” what is signified is that the universal predicate, “wood,” is being predicated of a subject which happens to be wood, that is, this particular wood in which there is whiteness. For when I say, “Something white is wood,” the meaning is the same as “This wood which happens to be white is wood”; in other words, the sense is not that “white” is the subject of wood. And he proves this on the ground that it is either according to its totality or according to a part that a subject comes to be that which is predicated of it as of a subject, as a man comes to be white. But neither the white nor its white part, which is really white, that is, which pertains to the very essence of whiteness, becomes wood: for an accident is not the subject of change whereby wood comes to be from non-wood. But whatever begins to be such and such, comes to be it; therefore, if it does not come to be this, it is not it-unless it is granted that it always was this. But it was not always true to say that the white [object] is wood, because at some time the whiteness and the wood were not together. Therefore, since it is not true to say that the white [object] becomes wood, it is obvious that the white [object], properly and per se speaking, is not wood. Yet if it be granted that something white is wood, it is understood per accidens, namely, because that particular subject, which happens to be white, is wood. This, therefore, is the sense of any predication in which a subject is predicated of an accident. But when I say, “Wood is white,” predicating an accident of the subject, I do not signify, as I did in the previous mode of predication, that there is something else substantially white, such that to be wood happens to it: which, of course, is signified both in the previous mode, where a subject is predicated of an accident, and in the other mode, where an accident is predicated of an accident, as when I say, “The musician is white”: for this signifies nothing but the fact that this particular man, say Socrates, who happens to be a musician, is white. But when I say, “The wood is white,” I signify that the wood itself has become the subject of white, and not that something other than the wood or other than a section of the wood (which is some wood) has become white.

Therefore, there is a difference among these three modes of predicating, because when an accident is predicated of a subject, it is not predicated in virtue of some other subject; but when the subject is predicated of an accident or an accident of an accident, the predication is made in virtue of that which is subjected to the terms acting as the subject, of which another accident is predicated accidentally in the second case, and the species of the subject is predicated essentially in the first case.

Now since in each of the above modes we use the name, 'predication', and since it lies within our power to employ names as well as to restrict them, let us agree so to use our names in the following proof that only those things are said to be predicated which are said in this way, namely, not in virtue of some other subject. Consequently, whatever is said in another way, namely, by reason of another subject, as when a subject is predicated of an accident or an accident of an accident, shall not be said to be predicated—or if it is said to be predicated, it shall be said to be predicated not absolutely but per accidens. Furthermore, that which is after the manner of “white,” let us always take on the part of the predicate, and that which is after the manner of “wood,” let it be taken on the part of the subject. Therefore, in the following proof let us suppose this always to be predicated which is predicated not per accidens, but absolutely, of that of which it is predicated. And the reason why we should use the word “predication” in this way is that we are speaking of demonstrative matters, and demonstrations use only such predications.

Then, at: it follows from this (83a21) he shows the differences among per se predicates.

In regard to this he does two things. First, he distinguishes these predicates into diverse genera. Second, he shows the differences among them, at: predicates which signify substance (83a24).

He says therefore first (83a21) that because we are saying that those things alone are being predicated which are predicated not in virtue of some other subject, and this is diversified according to the ten predicaments, it follows that whatever is thus predicated is predicated either in quod quid est, that is, after the manner of an essential predicate, or after the manner of quality or quantity or one of the other predicaments discussed in the Categories. And he adds, when one is predicated of one, because if a predicate is not one but several, it cannot be said to be predicated precisely as quid or quale, but perhaps jointly as quale quid, as when I say “Man is a white animal.”

Now it was necessary to add this because if several things are predicated of one in such a way that they function as one predicate, predications will be multipliable to infinity according to the infinite modes of combining predicates one to another. Hence when there is question of a stop in things that are predicated, it is necessary to take one thing predicated of one thing.

Then, at: predicates which signify substance (83a24) he indicates the difference among the aforesaid predicates.

In regard to this he does three things. First, he proposes the difference. Second, he clarifies it with examples, at: for example, white is a coincident (83a27). Third, he excludes an objection, at: the Forms we can dispense with (83a33).

He says therefore first (83a24) that those which signify substance must signify, in respect to that of which they are predicated, what it truly is or something that it truly is.

This can be understood in two ways: in one way, as showing a distinction on the part of the predicate which might signify either the entire essence of the subject, as a definition does (and he signifies this when he says, "what it truly is"), or part of the essence, as a genus or a difference does (and he signifies this when he says, "or something that it truly is".) In another way, and better, as showing a distinction on the part of the subject, which is sometimes convertible with an essential predicate, as the definitum with the definition (and he signifies this, when he says, "what it truly is"), and sometimes is a subjective part of the predicate, as man is of animal (and he signifies this when he says, "or something that it truly is", for man is a certain animal).

But those which do not signify substance but are said of some subject (which subject is not truly, that is, essentially, that predicate nor something of it), all such predicates are accidental.

Then, at: for example, white is a coincident (83a27) he clarifies the aforesaid difference with examples, saying that when we say, “Man is white,” the predicate is accidental, because man is not that which white truly is, that is, to be white is not the essence of man or anything of what white truly is, as was explained above. But, when it is stated, “Man is an animal,” perhaps man is that which animal truly is. For animal signifies the essence of man, because that which is man is essentially animal. And although predicates that do not signify substance are accidents, they are not predicated per accidens. For they are not predicated of some subject in virtue of some other subject; for when I say, “Man is white,” 'white' is not predicated of man by reason of some other subject’s being white in virtue of which man is called white as was explained above in regard to things predicated per accidens.

Then, at: the Forms we can dispense (83a33) he excludes an objection.

For someone could say that predicates which signify the substance are not truly and essentially that which they are predicated of, nor are they something of it; and that accidents which exist in individuals as in subjects, do not correspond to any common essential predicates, because such universal predicates signify certain separated essences always subsisting by themselves, as the Platonists say.

But he answers that if Forms, that is, Ideas, are assumed to exist, they should be happy, because according to the Platonists they have a nobler existence than the material things known to us. For the latter are particular and material, but the former universal and immaterial. For they are premonstrations, that is, certain exemplars, of material things (taking 'premonstrations' here as above, when we spoke of something being shown beforehand in order to prove something). Therefore, since they are the premonstrations or exemplars of natural things, it is necessary that in these natural things there be found certain participations of those Forms which pertain to the essences of these natural things. And thus, if such separated Forms exist, as the Platonists contend, they have nothing to do with the present matter. For we are concerned with things, the science of which is produced in us through demonstration. And these are things existing in matter and known to us and concerning which demonstrations deal. Consequently, if it be granted that “animal” is something separated, an existing premonstration, as it were, of natural animals, then when I say, “Man is an animal,” in the sense that we use this preposition in demonstrating, 'animal' signifies the essence of the natural thing concerning which the demonstration is made.

Logical reasons why one does not proceed to infinity in predicates

a36. (3) If A is a quality of B, B cannot be a quality of A — a quality of a quality.

a37. Therefore, A and B cannot be predicated reciprocally of one another in strict predication: they can be affirmed without falsehood of one another, but not genuinely predicated of each other.

a39. For one alternative is that they should be substantially predicated of one another, that is, B would become the genus or differentia of A — the predicate now become subject.

b1. But it has been shown that in these substantial predications neither the ascending predicates nor the descending subjects form an infinite series; e.g., neither the series, man is biped, biped is animal, etc., nor the series predicating animal of man, man of Callias, Callias of a further. subject as an element of its essential nature, is infinite. For all such substance is definable, and an infinite series cannot be traversed in thought: consequently neither the ascent nor the descent is infinite, since a substance whose predicates were infinite would not be definable.

b8. Hence they will not be predicated each as the genus of the other; for this would equate a genus with one of its own species.

b10. Nor can a quale be reciprocally predicated of a quale, nor any term belonging to an adjectival category of another such term, except by accidental predication; for all such predicates are coincidents and are predicated of substances.

b13. On the other hand, in proof of the impossibility of an infinite ascending series, every predication displays the subject as somehow qualified or quantified or as characterized under one of the other adjectival categories, or else is an element in its substantial nature: these latter are limited in number, and the number of the widest kinds under which predications fall is also limited, for every predication must exhibit its subject as somehow qualified, quantified, essentially related, acting or suffering, or in some place or at some time. I assume first that predication implies a single subject and a single attribute, and second that predicates which are not substantial are not predicated of one another. We assume this because such predicates are all coincidents, and though some are essential coincidents, others of a different type, yet we maintain that all of them alike are predicated of some substratum and that a coincident is never a substratum — since we do not class as a coincident anything which does not owe its designation to its being something other than itself, but always hold that any coincident is predicated of some substratum other than itself, and that another group of coincidents may have a different substratum.

b24. Subject to these assumptions then, neither the ascending nor the descending series of predication in which a single attribute is predicated of a single subject is infinite. For the subjects of which coincidents are predicated are as many as the constitutive elements of each individual substance, and these we have seen are not infinite in number, while in the ascending series are contained those constitutive elements with their coincidents — both of which are finite. We conclude that there is a given subject (D) of which some attribute (C) is primarily predicable; that there must be an attribute (B) primarily predicable of the first attribute, and that the series must end with a term (A) not predicable of any term prior to the last subject of which it was predicated (B), and of which no term prior to it is predicable.

b33. The argument we have given is one of the so-called proofs; an alternative proof follows. Predicates so related to their subjects that there are other predicates prior to them predicable of those subjects are demonstrable; but of demonstrable propositions one cannot have something better than knowledge, nor can one know them without demonstration. Second, if a consequent is only known through an antecedent (viz. premises prior to it) and we neither know this antecedent nor have something better than knowledge of it, then we shall not have scientific knowledge of the consequent. Therefore, if it is possible through demonstration to know anything without qualification and not merely as dependent on the acceptance of certain premises — that is, hypothetically — the series of intermediate predications must terminate. If it does not terminate, and beyond any predicate taken as higher than another there remains another still higher, then every predicate is demonstrable. Consequently, since these demonstrable predicates are infinite in number and therefore cannot be traversed, we shall not know them by demonstration. If, therefore, we have not something better than knowledge of them, we cannot through demonstration have unqualified but only hypothetical science of anything.

As dialectical proofs of our contention these may carry conviction.

If A is a quality of B. Having set forth the distinction of predicates from one another as a necessary preliminary to demonstrating his proposition, the Philosopher now undertakes to show his proposition, namely, that there is no infinite process in predicates.

And his treatment falls into two parts, according to the two ways in which he shows his proposition, the second part beginning at: the argument we have given (83b33).

Concerning the first he does two things. First, he shows that one does not proceed to infinity in predicates after the manner of circularity (see Lect. 8). Second, that one does not proceed to infinity in a direct line upwards or downwards, at: on the other hand, in proof (83b13).

Concerning the first he does three things. First, granting what has gone before, he adds certain things required for showing his proposition. Second, from these and other established facts he concludes the proposition, at: therefore, A and B cannot be (83a37). Third, he proves it, at: for one alternative (8309).

First, therefore (83a6) he proposes two things: one of these is that since a predicate which signifies an accident signifies some genus of accident, such as quality, it cannot occur that two things be so related to one another that the first is a quality of the second and the second a quality of the first: for the nature of a quality and that of which it is the quality are diverse. The second is that universally it is not possible that a quality have some other quality inhering in it, because no accident is the subject of another accident, absolutely speaking. For to substance alone does the notion of subject properly belong.

Then, at: therefore, A and B cannot be (83a37) he proposes, as though concluding from these antecedents, what he intends to prove.

He says, if our preliminary rules are true, it is impossible that there be equal reciprocal predication, that is, according to any of the above-mentioned ways. But this does not exclude the possibility of one thing’s being truly predicated of another and conversely. For we say truly that man is white and that something white is a man. However, this is not done equally, that is, according to an equal manner of predicating. And it is the same in essential predicates.

Then, at: for one alternative is that (83a39) he shows the proposition.

First, in essential predicates. Second, in accidental ones, at: nor can a quale be reciprocally (83b10).

In regard to the first he does three things. First, he lays down a division of essential predicates. Second, he recalls something established above, at: but it has been shown (83b1). Third, he proves the proposition, at: hence they will not be predicated (83b8).

He says therefore first (83a39) that in order to show that they are not reciprocally predicated equally, it will be necessary to consider this in essential predicates. For that which is predicated equally will be predicated either as a substance or some other way: and if as a substance, then either as a genus or as a difference. For these are the two parts of a definition which signify the essence.

Then, at: but it has been shown (83b1) he recalls what he had proved above at the beginning of the previous lecture, namely, that predicates of this kind are not infinite; because if they were to proceed to infinity, reciprocity or circularity would find no place therein.

He says, therefore, that as has been shown above, an infinite process does not take place in such predicates either by ascending or by descending: for example, if two-footed be predicated of man, and animal of two-footed, something else of animal, there is no process to infinity either upwards or downwards. Thus, if animal were said essentially of man, and “man” of Callias, and this of somone else (supposing that there were a genus containing under it species, one of which would be Callias), it would not be possible to go on in this way to infinity. And he recalls the reason he used earlier to prove this: for every such substance which has something more universal that can be predicated of it and which can be predicated of some inferior is capable of being defined; but the most general genera of which other more universal things are not predicated, and singulars which are not predicated of any inferiors, are not capable of definition. Only the substance between these can be defined. But that substance of which an infinite number of things is predicated turns out to be indefinable, because one who would define it must go through and understand all the items which are substantially predicated of the defined, since all of them occur in the definition either as a genus or as a difference. But the infinite cannot be gone through. Therefore, it is required that neither a universal substance, which is not a supreme genus, nor a lowest subject, can have an infinitude of predicates which are predicated of it substantially. Consequently, there is no infinite process either upwards or downwards.

Then, at: hence they will not be predicated (83b8) he shows that there is no process to infinity after the manner of circularity in substantial predicates.

And he says that if certain substantial predicates are predicated as genera of something, they are not predicated of one another equally, that is, convertibly, that is, so that one would be the genus of another and vice versa. To prove this he continues, for the one will be what something truly is. As if to say: if something is predicated of something as a genus, that of which it is predicated is something which truly is that, that is, something particular which receives that predication substantially. Therefore, if this be predicated of that as a genus, it will follow that something which belongs to something particularly would conversely receive the predication of it, which is tantamount to saying that a same thing is both a whole and a part in relation to the same thing—which is impossible. And the same reasoning applies to differences. Hence in Topics I it is stated that problems concerning a difference are reduced to problems concerning the genus.

Then, at: nor can a quale be reciprocally (83b10) he shows that there cannot be an infinite circular process in predications in which an accident is predicated of a subject.

And he says that there cannot be conversion of a quality with its subject; furthermore, none of the others which are predicated accidentally can have that sort of predication, unless the predication be made per accidens in the sense described above, namely, that accidents are not predicated of their subjects except per accidens. For quality and all these others are accidental to the subject; hence they are predicated of their substances as an accident of a subject.

Then, at: on the other hand, in proof (83b13) he shows universally that in no genus of predication is there an infinite process upwards or downwards.

And he says that not only is there no infinite process in predications according to circulation, but also that in proceeding upwards or downwards the predicates will not be infinite.

To prove this: First, he recalls certain things which were established above. Second, from these he concludes his proposition, at: subject to these assumptions (83b24).

In regard to the first he reaffirms (83b13) that certain things can be predicated of a subject, whether they signify quality or quantity or any other genus of accident, or even items which constitute the substance of a thing, that is, essential predicates. Second, he reiterates, that the latter, that is, the substantial predicates, are finite. Third, he reaffirms that the genera of predicaments are finite, namely, quality, quantity and so on. For if someone were to say that quantity is predicated of substance, and quality of quantity, and so on to infinity, he excludes this on the ground that the genera of predicaments are finite. Fourth, he reaffirms that, as stated above, one thing is predicated of one thing in simple predication.

And he mentions this because someone might say that one thing may well be first predicated of one thing, as “animal” of man, and this predication will be multiplied until something else can be found predicable of man, and when this is found, two things, will be predicated of one, so that it will be said that man is a white animal. Thus, many more predicates might be found according to various combinations of predicates. And so by continually adding to this number, the predicates will be increased more and more, so that there will be a process to infinity in predicates, just as there is in the succession of numbers. But he excludes this by predicating one of one.

Fifth, he repeats that we should not say that certain items are predicated absolutely of things which are not something, that is, of accidents, none of which is a subsistent being. For, as shown above, neither the, subject nor an accident is properly predicated of an accident. For all things of this sort that are not substantial are accidents, and nothing is predicated, simply speaking, of such things. Yet whether they be substantial predicates or accidental, they are predicated per se of their subjects; but if they are subjects or accidents being predicated of an accident, they are predicated in another way, that is, per accidens. For it belongs to the very notion of all accidents that they be said of a subject; and since an accident is not a subject, nothing can, properly speaking, be predicated of it, because none of such, that is, no accidents, are stated to be such that they are said to be that which is said, that is, stated to be such that they themselves, rather than something else distinct from them, receives the predication of that which is predicated of them, as happens in the case of substances. For man is not called “animal” or “white” because something else is “animal” or “white,” but because the very thing which is a man is animal or white; but a white thing is called “man” or “musician” because something else, namely, the subject of white is the man or musician. But the accident itself is in something distinct from it; and others which are predicated of an a accident are predicated of something other than that accident, namely, of the subject of the accident: it is on this ground that they are predicated of the accident, as has been stated. (Now he mentioned this because if an accident is predicated of a subject and vice versa, and all accidents of a subject can be mutually predicated of one another, it will follow that predication could proceed to infinity, because an infinitude of things can happen to one thing).

Then, at: subject to these assumptions (83b24) from these premises he shows his proposition, namely, that in a predication in which one thing is predicated of one thing there is no upward or downward process to infinity, because, as the fifth supposition states, all accidents are predicated of items which pertain to the substance of the thing. Furthermore, according to the second supposition, substantial predicates are not infinite; consequently, on the part of the subjects there is no infinite process downwards in these predications. Again, neither of these is infinite in the upward movement, that is, neither the substantial nor the accidental predicates, both because the genera of accidents are finite, and because there is no infinite process upwards or downwards in any of these genera any more than there is in substantial predicates, because in each predicament the genus is predicated of a species in regard to something essential. Hence we can conclude universally that there must be some first subject of which something is predicated, thus establishing a stop in downward predication; then something else will be predicated of this, but it will come to a stop in the ascending process, so that something will be found which is not found predicated of another, either as the subsequent is predicated of its prior per accidens, or as the prior is predicated per se of its subsequent. This, therefore, is one way of demonstrating the proposition logically, and it is based on the diverse modes of predicating.

Then, at: the argument we have given (83b33) he sets forth the second way of proving.

And he says that when a proposition in which something is predicated of a subject is such that certain things can be predicated per prius of that subject, that proposition will be demonstrable: (for example, this proposition, “Man is a substance,” is demonstrated by the proposition, “Animal is a substance,” because “substance” is predicated of animal before it is predicated of man.) But if a proposition is demonstrable, there is no better way of knowing it than to know it by demonstration, just as we know indemonstrable principles better than by way of demonstration because we know them as self-evident. Besides, we cannot scientifically know such demonstrable propositions except through demonstration, because a demonstration is a syllogism productive of scientific knowledge, as we showed above. Furthermore, if a proposition is known through another one, and if we do not scientifically know the one through which it is known, or know it in the way which is better than knowing scientifically, then we do not scientifically know that proposition which is made known through it.

With these three suppositions in mind he proceeds thus: If one does know something purely through demonstration and not from something or from supposition, it is necessary that there be a stop in the predicates which are taken as middles.

He says: "purely and not from something", to exclude demonstrations leading to the impossible, in which one proceeds, against certain positions by arguing from propositions that have been agreed upon. Furthermore, he says: "or from supposition", to exclude such demonstrations as are formed in subalternate sciences, which suppose the conclusion of higher sciences, as explained above. Therefore, one knows purely through demonstration when each one of the premised propositions, if it is demonstrable, is known through demonstration; and if it is not demonstrable, is known in virtue of itself.

Under these suppositions it is necessary that there be a stop in predications, because if there is no stop but something prior can always be taken, it follows that there is demonstration of everything, as was said above. Therefore, if a. conclusion is demonstrated, each of the premises must be demonstrable. But if we can have knowledge of it in no better way than by knowing it through demonstration, and if it will be necessary to demonstrate it through other propositions, and those through others again, and so on to infinity, then, since it is not possible to go through those infinites, we shall not be able to make it known through demonstration or through the method better than demonstration, since all things are demonstrable. The consequence will be that one knows nothing purely through demonstration, but only from supposition.

Finally, by way of summary he concludes the main proposition.

That there is not an infinite process upward or downward in predicates is shown analytically

a8. But an analytic process will show more briefly that neither the ascent nor the descent of predication can be infinite in the demonstrative sciences which are the object of our investigation.

a10. Demonstration proves the inherence of essential attributes in things.

a11. Now, attributes may be essential for two reasons: either because they are elements in the essential nature of their subjects, or because their subjects are elements in their essential nature. An example of the latter is odd as an attribute of number — though it is number’s attribute, yet number itself is an element in the definition of odd; of the former, multiplicity or the indivisible, which are elements in the definition of number.

a18. In neither kind of attribution can the terms be infinite. They are not infinite where each is related to the term below it as odd is to number, for this would mean the inherence in odd of another attribute of odd in whose nature odd was an essential element: but then number will be an ultimate subject of the whole infinite chain of attributes, and be an element in the definition of each of them. Hence, since an infinity of attributes such as contain their subject in their definition cannot inhere in a single thing, the ascending series is equally finite.

a23. Note, moreover, that all such attributes must so inhere in the ultimate subject—e.g., its attributes in number and number in them—as to be commensurate with the subject and not of wider extent.

a25. Attributes which are essential elements in the nature of their subjects are equally finite: otherwise definition would be impossible. Hence, if all the attributes predicated are essential and these cannot be infinite, the ascending series will terminate, and consequently the descending series too.

a28. If this is so, it follows that the intermediates between any two terms are also always limited in number.

a29. An immediately obvious consequence of this is that demonstrations necessarily involve basic truths, and that the contention of some — referred to at the outset — that all truths are demonstrable is mistaken. For if there are basic truths, (a) not all truths are demonstrable, and (b) an infinite regress is impossible; since if either (a) or (b) were not a fact, it would mean that no interval was immediate and indivisible, but that all intervals were divisible. This is true because a conclusion is demonstrated by the interposition, not the apposition, of a fresh term. If such interposition could continue to infinity there might be an infinite number of terms between any two terms; but this is impossible if both the ascending and descending series of predication terminate; and of this fact, which before was shown dialectically, analytic proof has now been given.

But an analytic process. After showing logically that there is no infinite process upwards or downwards in predicates, the Philosopher now shows the same thing analytically.

And his treatment falls into two parts. In the first he shows the principal proposition. In the second he infers certain corrolaries from the aforesaid, at: it is an evident corollary (84b3) [L. 36].

Concerning the first he does two things. First, he proposes what he intends. Second, he proves his proposition, at: demonstration proves the inherence (84a10).

He says therefore first (84a8) that the fact an infinite process upwards or downwards does not occur in the demonstrative sciences with which we are concerned can be more briefly and quickly manifested analytically than it was logically.

Here we might note that analytic, that is, demonstrative, science which is called judicative, because it resolves to self-evident principles, is a part of logic which even contains dialectics under it. However, it pertains to logic in general to consider predication universally, that is, as containing under it predication which is per se and predication which is not per se. But predication per se is proper to demonstrative science. Therefore, above he proves his proposition logically, because he showed universally in every genus of predication that there is no infinite process. But here he intends to show it analytically, because he proves it only in things which are predicated per se. And this is a more efficient way; furthermore, it suffices for our purpose, because that is the only mode of predication we use in demonstration.

Then, at: demonstration proves the inherence (84a10) he shows his proposition.

About this, he does three things.

First, he proposes which predication analytic, that is, demonstrative, science employs, for it uses per se predication.

Second, he recalls how many modes there are of such predication, at: now, attributes may be essential (84a11; [301]).

Third, he shows that there cannot be an infinite process in any mode of per se predication, at: in neither kind of attribution (84a18; [302]).

He says therefore first (84a10) that demonstration is concerned exclusively with items that are per se in things. For such are its conclusions and from such does it demonstrate, as was established above.

Then, at: now, attributes may be essential (84a11) he lays down two modes of predicating per se. For in the first place those things are predicated per se which are present in their subjects as constituting their essence, namely, when predicates are placed in the definition of a subject. Second, when the subjects themselves are in the essence of the predicate, that is, when the subjects are placed in the definition of the predicate. And he gives examples of each of these ways: for odd is predicated per se of number in the second way, because “number” is placed in the definition of odd. For the odd is a number not divisible by two. Multitude or divisible, however, are predicated of number and are present in its definition; hence these are predicated per se of number in the first way. The other ways, which he mentioned previously, are reduced to these.

Then, at: in neither kind of attribution (84a18) he shows that there must be a stop in each of these modes of per se predication.

In regard to this he does three things.

First, he shows that there must be a stop both upwards and downwards in both of these modes of per se predication.

Second, he concludes that there cannot be an infinitude of middles, at: if this is so, it follows that (84a28; [306]).

Third, he concludes that one cannot proceed to infinity in demonstrations, at: an immediately obvious (84a29; [307]).

Concerning the first he does two things.

First, he shows his proposition in regard to the second way of saying per se, namely, when the subject is placed in the definition of the predicate.

Second, in the first way, when the predicate is placed in the definition of the subject, at: attributes which are essential (84a25; [305]).

In regard to the first he gives two reasons.

The first of these he treats of in the following way. First (84a18) he states ‘his proposition, namely, that in neither of these modes of saying per se does an infinite process occur. Then he proves this of the second mode, as when “odd” is predicated of number. For if one goes further and states that something else should be predicated per se of “odd” according to that mode of saying per se, it follows that “odd” is present in its definition. But “number” is placed in the definition of odd; hence it will follow that “number” is also present in the definition of that third thing which is present per se in “odd.” However, this cannot go on to infinity, so that an infinitude of things would be in the definition of something, as was established above. It remains, therefore, that in such per se predications an upward process to infinity does not occur, that is, on the side of the predicate.

He presents the second reason, at: note, moreover, that all (84a23).

He says that no matter how fair one advances in these per se predications of the second mode, it will be required that all the predicates taken in order be in their first subject, say in number, as predicated of it, because if “odd” is predicated per se of number, it will be required that whatever is predicated of odd be also predicated of number. And it is further required that “number” be in all of them, because if “number” is placed in the definition of odd, it has to be placed in the definition of all those things which are defined by “odd.” And thus it follows that they are mutually in one another. Therefore, they will be convertible and none of wider extent than another: for this is the way proper attributes are related to their subjects. Hence, even though there be an infinitude of per se predicates in this way, it offers nothing to the purpose of one who intends to establish that there is an infinitude of predicates upwards or downwards.

Then, at: attributes which are essential (84a25) he proves his own point in regard to the first mode of saying per se,

and he states that those things which are predicated in essence, that is, as pertaining to the definition of the subject, cannot be infinite; otherwise, definition would be impossible, as we showed above. From this, therefore, he concludes that if all the items predicated in demonstrations are predicated per se, and if there is no infinite upward process in per se predicates, it is necessary that the predicates in demonstrations stop in the upward movement. And from this it also follows that they must stop in the downward movement, because no matter on which side infinity is posited, science and definition are destroyed, as is evident from what has been said above.

Then, at: if this is so, it follows that (84a28) he concludes from the foregoing that if there is a stop upwards and downwards, the middles cannot be infinite. For it has been established above that if the extremes are determinate, there cannot be an infinitude of middles.

Then, at: an immediately obvious consequence (84a29) he further concludes that there is no infinite process in demonstrations.

And he says that if the above statements are true, it is necessary that there be certain first principles of demonstration that are not demonstrated; consequently, there will not be demonstration of everything, as some claim, as was stated in the beginning of this book. Then he goes on to show that his consequence follows. For if it is granted that there are certain principles of demonstrations, it is necessary that they be indemonstrable: for since every demonstration proceeds from things that are prior, as has been established above, then if the principles are demonstrated, it will follow that something would be prior to the principles, and this is contrary to the notion of a principle. And so, if not all things are demonstrable, it will follow that demonstrations do not proceed to infinity. But all these follow from what has been established, namely, from the fact that there is no infinite process in middles, because it is no different to say that any of the foregoing statements is true (namely, that demonstrations proceed to infinity, or that all things are demonstrable, or that there are no principles of demonstrations) than it is to say that no distance is immediate and indivisible, that is, to the position that the two terms of any affirmative or negative proposition belong together only in virtue of a middle. For if any proposition is immediate, it follows that it is indemonstrable; because when something is demonstrated, it is necessary to take a term by interposing, that is, by setting it between the subject and predicate, so that the predicate will be predicated of that term before being predicated of the subject-or removed from it. But the middle in demonstrations is not taken by assuming extraneously; for this would be to assume an extraneous middle and not a proper middle-which occurs in contentious and dialectical syllogisms. Therefore, if demonstrations were to proceed to infinity, it would follow that there is an infinitude of middles between two extremes. But this is impossible if, as has been established above, the predications stop in the upward and downward process. But as we have shown, first logically and then analytically, these predications do stop both upwards and downwards, as explained.

Therefore, in virtue of this conclusion finally induced, he manifests the intent of the entire chapter and why each proposition was introduced.

Certain corollaries from preceding lecture

b3. It is an evident corollary of these conclusions that if the same attribute A inheres in two terms C and D predicable either not at all, or not of all instances, of one another, it does not always belong to them in virtue of a common middle term.

b6. Isosceles and scalene possess the attribute of having their angles equal to two right angles in virtue of a common middle; for they possess it in so far as they are both a certain kind of figure, and not in so far as they differ from one another. But this is not always the case:

b9. for, were it so, if we take B as the common middle in virtue of which A inheres in C and D, clearly B would inhere in C and D through a second common middle, and this in turn would inhere in C and D through a third, so that between two terms an infinity of intermediates would fall — an impossibility. Thus it need not always be in virtue of a common middle term that a single attribute inheres in several subjects, since there must be immediate intervals.

b15. Yet if the attribute to be proved common to two subjects is to be one of their essential attributes, the middle terms involved must be within one subject genus and be derived from the same group of immediate premises; for we have seen that processes of proof cannot pass from one genus to another.

b19. It is also clear that when A inheres in B, this can be demonstrated if there is a middle term. Further, the ‘elements’ of such a conclusion are the premises containing the middle in question, and they are identical in number with the middle terms, seeing that the immediate propositions — or at least such immediate propositions as are universal — are the ‘elements’. If, on the other hand, there is no middle term, demonstration ceases to be possible: we are on the way to the basic truths.

b24. Similarly if A does not inhere in B, this can be demonstrated if there is a middle term or a term prior to B in which A does not inhere: otherwise there is no demonstration and a basic truth is reached. There are, moreover, as many ‘elements’ of the demonstrated conclusion as there are middle terms, since it is propositions containing these middle terms that are the basic premises on which the demonstration rests; and as there are some indemonstrable basic truths asserting that ‘this is that’ or that ‘this inheres in that’, so there are others denying that ‘this is that’ or that ‘this inheres in that’ — in fact some basic truths will affirm and some will deny being.

b32. When we are to prove a conclusion, we must take a primary essential predicate — suppose it C — of the subject B, and then suppose A similarly predicable of C. If we proceed in this manner, no proposition or attribute which falls beyond A is admitted in the proof: the interval is constantly condensed until subject and predicate become indivisible, that is, one. We have our unit when the premise becomes immediate, since the immediate premise alone is a single premise in the unqualified sense of ‘single’.

b38. And as in other spheres the basic element is simple but not identical in all — in a system of weight it is the mina, in music the quarter-tone, and so on — so in syllogism the unit is an immediate premise, and in the knowledge that demonstration gives it is an intuition.

a2. In syllogisms, then, which prove the inherence of an attribute, nothing falls outside the major term.

a3. In the case of negative syllogisms on the other hand, (1) in the first figure nothing falls outside the major term whose inherence is in question; e.g., to prove through a middle C that A does not inhere in B the premises required are, all B is C, no C is A. Then if it has to be proved that no C is A, a middle must be found between and C; and this procedure will never vary.

a7. (2) If we have to show that E is not D by means of the premises, all D is C; no E, or not all E, is C; then the middle will never fall beyond E, and E is the subject of which D is to be denied in the conclusion.

a10. (3) In the third figure the middle will never fall beyond the limits of the subject and the attribute denied of it.

It is an evident corollary. After showing that a process to infinity does not occur in demonstrations, the Philosopher here adduces certain corollaries from what has been established.

In regard to this he does two things.

First, he shows that it is necessary to accept certain first propositions.

Second, how those first things are to be used in demonstrations, at: when we are to prove a conclusion (84a32; [316]).

Concerning the first he does two things.

First, he shows that it is necessary to arrive at a first, when one thing is predicated of several.

Second, when one thing is predicated of one, at: it is also clear that when A inheres in B (84b19; [314]).

In regard to the first he does four things.

First, he states the intended proposition.

Second, he manifests the proposition, at: isosceles and scalene (84b6; [310]).

Third, he proves it, at: for, were it so, if we take B (84b9; [311]).

Fourth, he excludes an objection, at: yet if the attribute to be proved (84b15; [313]).

He says therefore first (84b3) that having demonstrated the aforesaid, namely, that there is no process to infinity in predications and demonstrations, it is clear that if something is predicated of two things, say A of C and of D, such that one of them is not predicated of the other, that is, either not at all, as animal is predicated of man and ox, but neither of them is predicated of the other in any way; or not in all cases, as animal is predicated of man and male, neither of which is universally predicated of the other: when the terms, I repeat, are thus related, it is clear that there is no need for that predicate, which is predicated of both, to be in them in virtue of something common to both and this always in virtue of something else, and so on to infinity.

Then, at: isosceles and scalene (84b6) he cites an example to clarify what he is proposing. For there are two species of triangle, one of which is scalene (none of whose three sides is equal to any other) and the other is isosceles (having two sides that are equal). But neither of these species is predicated of the other, and yet this proper attribute of 'having three angles equal to two right angles' is present in both. Furthermore, this attribute is present in them in virtue of something common, namely, that each is a certain figure, namely, a triangle. However, this process does not continue forever so that something would always belong to it in virtue of something else and so on to infinity, so that “having three” would belong to triangle in virtue of something else again, and so on to infinity.

Then, at: for, were it so, if we take B (84b9) he proves his proposition.

Let the case be that B is predicated of C and of D in virtue of this common feature A. It will then be clear that B will be in C and in D in virtue of that common feature which is A. Now, if A in turn is in them in virtue of something common, which again is in them in virtue of some other common item, there will be an infinite process in the middles. It follows, therefore, that between the two extremes C and B there falls an infinitude of middle terms. But this is impossible. Therefore, if a same thing is in several things, it is not necessary that it be in them always in virtue of something else ad infinitum, because it is necessary to reach certain immediate distances, that is, certain immediate predications, which he calls “distances,” as explained above.

As can be seen from this proof of Aristotle’s, it is not his understanding that it is not always true that when some item is predicated of several, which are not in turn predicated of one another, that item is not in the several in virtue of something common. For this is true in everything predicated as a proper attribute: for if it is in several, it is required that it be in them in virtue of something common which might even be nameless, as we explained above when we treated concerning the universal. But there is no infinite process in that which is common, as this reason introduced by the Philosopher clearly proves. However, if something be taken which is in several as a genus in its species, it will not always be necessary to find something prior in virtue of which it is in them. For example, “living” is in man and in ass in virtue of something prior, namely, in virtue of “animal,” but it is not in animal and plant in virtue of something prior, because these are the first species of “living,” that is, of animate, body.

Then, at: yet if the attribute to be proved (84b15) he excludes an objection.

For someone could say that it is always taken in virtue of something common, in the sense that something from another genus might be found common to them: for example, if we say that “to be self-movent” is in man and in ass in virtue of this common feature which is “animal” and also in virtue of some other common feature such as “having quantity” or having color or other things of this sort that can be taken ad infinitum.

In order to exclude this he says that it is required of the middle terms that they be taken from the same genus and from the same atoms, that is, indivisibles. (By 'atoms' he means those extreme terms between which the middle must be taken, if that common item which is taken as a middle term is to be numbered among things which are predicated per se). Why the middle terms must be taken from the same genus he shows from the fact that, as stated above, a demonstration does not cross from one genus into another.

Then, at: it is also clear that when A (84b19) he shows that it is necessary to arrive at a first among the predicables in which one thing is predicated of one.

First, in affirmatives. Second, in negatives, at: similarly if A does not inhere in B (84b24; [315]).

He says therefore first (84b19) that it is clear that when A is predicated of B, if there is a middle between them, we can use that middle to demonstrate that A is in B: and these are the principles of this kind of conclusion. And whichever things be taken as middles are the principles of the mediate conclusions which are concluded through them. For the elements or principles of demonstrations are none but immediate propositions: and I mean either all such propositions or universal ones.

Now this can be understood in two ways.

In one way, so that universal proposition is taken as set off against singular. For it is not in virtue of a middle that the most special species is predicated of a singular. Hence this proposition, “Socrates is a man,” is immediate, although it is not a principle of demonstration, because demonstrations are not concerned with singulars, since there is no science of such. Consequently, not every immediate proposition, but only one that is universal, is a principle of demonstration.

In another way it can be understood in the sense that of all the propositions of any science, the universal propositions are the common propositions, such as “Every whole is greater than its part.” Hence these are absolutely the principles of demonstrations and self-evident to all. But the proposition, “Man is an animal,” or “The isosceles is a triangle,” is not a principle of demonstration throughout the science but only for some particular demonstrations; hence such propositions are not self-evident to all.

Consequently, if there is a middle for a given proposition, one will demonstrate through a middle until something immediate is readied.

But if there is no middle for a given proposition, it cannot be demonstrated. But this is the way to find the chief principles of demonstrations, namely, to proceed by analysis from mediates to immediates.

Then, at: similarly if A does not inhere in B (84b24) he shows that it is necessary to arrive at a first in negatives.

He says that if A is denied of B, if there is a middle to be taken from which A is removed prior to being removed from B, then this proposition, “B is not A,” will be demonstrable. But if there is no such middle to be taken, that proposition will not be demonstrable but will be a principle of demonstration. And there are as many elements, that is, principles of demonstrations as there are terms at which a stop is reached in such a way that there is no further middle to find. For propositions formed of such terms are principles of demonstrations. Thus, if C be predicated immediately of B, and A be immediately removed from B or immediately predicated of it, B is the final term reached among the middles to be taken; hence each proposition will be immediate and a principle of demonstration. Furthermore, it is clear from the foregoing that just as there are certain indemonstrable affirmative principles in which one thing is predicated of one thing by signifying that this is essentially that (as when a genus is predicated of a proximate species) or that this is in that (as when a proper attribute is predicated of its proper and immediate subject), so there are indemonstrable principles in negatives, namely, by denying that something is an essential predicate or a proper attribute. From this it is evident that there are certain principles of demonstration for demonstrating an affirmative conclusion (which must be concluded from all affirmative propositions), and there are certain principles of demonstration for proving a negative conclusion (to infer which requires that something negative be taken).

Then, at: when we are to prove a conclusion (84b32) he shows how first propositions are to be used in demonstrating.

First, in affirmative demonstrations.

Second, in negatives, at: in the case of negative syllogisms (85a3; [320]).

In regard to the first he does three things.

First, he shows how first and immediate propositions should be taken in demonstrations.

Second, how such propositions are related to demonstrations, at: and as in other spheres (84b38; [318]).

Third, he summarizes, at: in syllogisms, then (85a2; [319]).

He says therefore first (84b32) that when one is required to demonstrate an affirmative conclusion, for example, “Every B is A,” it is first necessary to take something which is first predicated of B before A is, and of which A is also predicated, say C. And, if there is something of which A is predicated before A is predicated of C, and something ahead of that, always proceeding in this way, then no proposition and no term signifying a being outside of A will be taken in the demonstration, because it will be necessary that A be predicated of it per se in such a way that it is contained under it and is not extraneous to it; but it will always be necessary to condense [crowd together] the middles. (Here he is speaking according to an example of men who appear crowded together when they are sitting on a bench and there is no room for anyone else to sit between any two who are seated; in like fashion, the middles in demonstrations are condensed when no middle falls between the terms taken). And this is his meaning when he says, the middle is condensed until the spaces become indivisible, that is, the distances between two terms are such that they cannot be divided into several such distances, but there is only one space. And this occurs when a proposition is immediate. For it is then that a proposition is one, not only actually but potentially, when it is immediate. For if it is mediate, then although it is actually one (because one thing is predicated of one thing), it is in fact several in potency, because when the middle is taken, two propositions are formed. In the same way, a line which is actually one, inasmuch as it is continuous, is several in potency, inasmuch as it is divisible at an intermediate point. Hence he says that an immediate proposition is one as a simple indivisible.

Then, at: and as in other spheres (8038) he shows the relation between an immediate proposition and a demonstration. Here we should note that, as it is stated in Metaphysics X, in every genus there must be one first thing which is the most simple in that genus and is the measure of all the things in that genus. And because a measure is homogeneous to the thing measured, such first indivisibles will vary according to the diversity of genera. Hence these are not the same in all genera: but in regard to weights one indivisible, the ounce or mina, is taken as the minimum weight (even though it is not the absolute minimum, because every weight is further divisible into smaller weights, but it is taken as a minimum by supposition). Again, in melodies there is taken as the one principle a tone, which consists in a proportion of an octave and a half or a diesis, which is the difference between a tone and a semi-tone. Similarly, in diverse genera there are diverse indivisible principles. Now, the principles of a syllogism are propositions; hence it is required that the most simple proposition, which is immediate, be the unit which is the measure of syllogisms. But demonstration has, over and above a syllogism, the added feature that it causes science. Now, “understanding” and science are related as the indivisible unit is related to the many. For science is effected by going from principles to conclusions, whereas “understanding” is the absolute and simple acceptance of a self-evident principle. Hence “understanding” corresponds to the immediate proposition, and science to a conclusion, which is a mediate proposition. Consequently, the indivisible unit of a demonstration regarded as a syllogism is the immediate proposition. But on the part of the science which it causes, the unit is understanding.

Then, at: in syllogisms, then (85a2) he sums up and concludes what was established above, namely, that the middle in affirmative syllogisms does not fall outside the extremes.

Then, at: in the case of negative syllogisms (85a3) he shows how to use immediate propositions in negative syllogisms.

First, in the first figure.

Second, in the second, at: if we have to show that E is not D (85a7; [321]).

Third, in the third, at: in the third figure the middle (85a10; [322]).

He says therefore first (85a3) that in negative syllogisms in the first figure, none of the middles taken while proceeding to immediates will fall outside the genus of the terms of an affirmative proposition; for example, if it is to be demonstrated that “No B is A,” and the middle taken is C, we have the following syllogism: “No C is A; but every B is C: therefore, No B is A.” Now, if it should be necessary to prove that “No C is A,” one will be required to take a middle of C and A, that is, one that will be predicated of C and consequently of B; thus it will belong to the genus of the terms of the affirmative proposition. And so it will always turn out that the middles taken will not fall outside the affirmative proposition, although they will fall outside the genus of the negative predicate, that is, outside the genus of A.

Then, at: if we have to show that E is not D (85a7) he shows how it is in the second figure.

He says that if one is required to demonstrate in the second figure that “No E is D,” by taking C as middle, so as to form the syllogism, “Every D is C; but No E is C or Some E is not C: therefore, No E is D or Not every E is D,” the middle term taken will never fall outside of E. For if it should be necessary to demonstrate that “no E is C,” one will have to find a middle between E and C, because in the second figure it will always be necessary to prove a negative, since an affirmative cannot be concluded in this figure. Hence, just as in the first figure the middles are always taken on the side of the affirmative proposition, so in the second figure the middles will always be taken on the side of the negative proposition.

Then, at: in the third figure the middle (85a10) he shows how it is in the third figure.

He says that in the third figure the middles which are taken will not be outside the predicate which is denied, nor outside the subject of which it is denied. The reason for this is that the middle is the subject in both propositions, whether affirmative or negative: hence if it is necessary to take yet another middle, it will again have to act as the subject of both, whether affirming or denying. And so the middles which are taken will never be taken outside the predicate which is denied, or outside the subject of which it is denied.

a12. Since demonstrations may be either commensurately universal or particular, and either affirmative or negative; the question arises, which form is the better? And the same question may be put in regard to so-called ‘direct’ demonstration and reductio ad impossibile.

a17. Let us first examine the commensurately universal and the particular forms, and when we have cleared up this problem proceed to discuss ‘direct’ demonstration and reductio ad impossibile.

a20. The following considerations might lead some minds to prefer particular demonstration. (1) The superior demonstration is the demonstration which gives us greater knowledge (for this is the ideal of demonstration), and we have greater knowledge of a particular individual when we know it in itself than when we know it through something else; e.g., we know Coriscus the musician better when we know that Coriscus is musical than when we know only that man is musical, and a like argument holds in all other cases. But commensurately universal demonstration, instead of proving that the subject itself actually is x, proves only that something else is x—e.g., in attempting to prove that isosceles is x, it proves not that isosceles but only that triangle is x—whereas particular demonstration proves that the subject itself is x. The demonstration, then, that a subject, as such, possesses an attribute is superior. If this is so, and if the particular rather than the commensurately universal forms demonstrates, particular demonstration is superior.

a31. (2) The universal has not a separate being over against groups of singulars. Demonstration nevertheless creates the opinion that its function is conditioned by something like this—some separate entity belonging to the real world; that, for instance, of triangle or of figure or number, over against particular triangles, figures, and numbers. But demonstration which touches the real and will not mislead is superior to that which moves among unrealities and is delusory. Now, commensurately universal demonstration is of the latter kind: if we engage in it we find ourselves reasoning after a fashion well illustrated by the argument that the proportionate is what answers to the definition of some entity which is neither line, number, solid, nor plane, but a proportionate apart from all these. Since, then, such a proof is characteristically commensurate and universal, and less touches reality than does particular demonstration, and creates a false opinion, it will follow that commensurate and universal is inferior to particular demonstration.

b4. We may retort thus. (1) The first argument applies no more to commensurate and universal than to particular demonstration. If equality to two right angles is attributable to its subject not qua isosceles but qua triangle, he who knows that isosceles possesses that attribute knows the subject as qua itself possessing the attribute, to a less degree than he who knows that triangle has that attribute. To sum up the whole matter: if a subject is proved to possess qua triangle an attribute which it does not in fact possess qua triangle, that is not demonstration: but if it does possess it qua triangle the rule applies that the greater knowledge is his who knows the subject as possessing its attribute qua that in virtue of which it actually does possess it. Since, then, triangle is the wider term, and there is one identical definition of triangle—that is, the term is not equivocal—and since equality to two right angles belongs to all triangles, it is isosceles qua triangle and not triangle qua isosceles which has its angles so related. It follows that he who knows a connexion universally has greater knowledge of it as it in fact is than he who knows the particular; and the inference is that commensurate and universal is superior to particular demonstration.

b15. (2) If there is a single identical definition that is, if the commensurate universal is unequivocal — then the universal will possess being not less but more than some of the particulars, inasmuch as it is universals which comprise the imperishable, particulars that tend to perish.

b18. (3) Because the universal has a single meaning, we are not therefore compelled to suppose that in these examples it has being as a substance apart from its particulars — any more than we need make a similar supposition in the other cases of unequivocal universal predication, viz. where the predicate signifies not substance but quality, essential relatedness, or action. If such a supposition is entertained, the blame rests not with the demonstration but with the hearer.

Since demonstrations may be. After determining about the demonstrative syllogism, the Philosopher now treats of the comparison of demonstrations one to another.

And because science is caused by demonstration, his treatment is divided into two parts.

In the first he treats of the comparison of demonstrations.

In the second of the comparison of sciences, at: the science which is knowledge (87a31; [356]).

In regard to the first he does three things.

First, he raises a doubt concerning the comparison of demonstrations.

Second, he lays down the order of procedure, at: let us first examine (85a17; [325]).

Third, he deals with the doubts raised, at: the following considerations (85a20; [326]).

He says therefore first (85a12) that demonstration is divided in three ways: for in one way it is divided into universal and particular; in another way into categorical and privative, that is, affirmative and negative; in a third way into that which demonstrates ostensively and that which leads to the impossible. In each division, therefore, the question arises as to which is the stronger.

Then, at: let us first examine the commensurately (85a17) he shows what order should be followed.

He says that the comparison of universal to particular demonstration should be treated first. And when this has been done, we shall speak of demonstrations which demonstrate something affirmatively and of those which demonstrate to the impossible, namely, whether the affirmative is stronger and whether the demonstration to the impossible is stronger.

Then, at: the following considerations (85a20) he deals with the problems he has proposed.

First, of the comparison of the particular with the universal.

Second, of the comparison of the affirmative with the negative, at: that affirmative demonstration (86a32; [340]).

Third, of the comparison of the ostensive with that which leads to the impossible, at: since affirmative demonstration (87a1; [349]).

Concerning the first he does three things.

First, he proposes reasons to show that the particular demonstration is more powerful than the universal.

Second, he solves them, at: we may retort thus (85b4; [329]).

Third, he gives reasons to the contrary, at: demonstration is syllogism (85b22; [332]).

In regard to the first he sets forth three reasons, remarking that in virtue of these reasons to be set down forthwith, it will perhaps seem to some that a particular demonstration is of more value than a universal one.

The first reason (85a20) is this: That demonstration is stronger through which we know best in a scientific way. (And he proves it on the ground that the strength of a demonstration consists in knowing in a scientific way. For the most that a thing can do is called its strength: for the strength of a man able to carry 100 pounds is not that he can carry ten but that he can carry 100, which is the limit of his power, as it is stated in On the Heavens I. Now, the most that a demonstration can do is to cause scientific knowledge; consequently, that is the strength of a demonstration. But a thing is more perfect to the extent that it attains the strength appropriate to it, as is clear from Physics VII. Hence this proposition is quite evident, namely, that the better a demonstration causes scientific knowledge, the stronger it is). (He assumes that we know a thing better when we know it according to itself than when we know it according to something else: thus we know more in regard to Coriscus when we know that Coriscus is a musician than when we merely know that some man is a musician). And this proposition is true, absolutely speaking, because that which is per se is always prior to and the cause of that which is through something else, as it is stated in Physics VIII. From these facts the conclusion is gathered that a demonstration which makes one know something according to itself is stronger than one which makes one know something according to something else. But a universal demonstration demonstrates and makes one know something not according to itself, but according to something else, namely, according to the universal, as that a triangle with two equal sides, that is, an isosceles, has three, not because it is isosceles but because it is a triangle. A particular demonstration, on the other hand, demonstrates about a particular thing according to itself. Hence according to this it follows that a particular demonstration is stronger than a universal.

Then, at: the universal has not a separate being (85a31) he gives a second reason.

It is this. The universal is not something apart from singulars, as was proved in Metaphysics VII. But a universal demonstration leads one to think, from the very manner of its demonstration, that the universal is something, that is, a certain nature in the realm of beings: for example, when it demonstrates something of triangle apart from particular triangles, and of figure apart from particular figures, and of number apart from particular numbers. To these two propositions he adds two others: to the first one, which stated that the universal is not something apart from the singulars, he adds this proposition, namely, that a demonstration concerned with being is stronger than one concerned with non-being. But to the second proposition, which stated that a universal demonstration leads one to think that a universal is something existing as a real nature, he adds another proposition, namely, that a demonstration which does not cause error is stronger than one which leads to error. Then he shows that one is led into error on account of a universal demonstration, because one who proceeds according to a universal demonstration demonstrates concerning some universal as of some analogue, that is, as of something common which is referred to many things proportionally, as though there existed something common which is neither a line nor a number nor a solid, that is, a body, nor a plane, that is, a surface, but something apart from these, that is, a universal quantity; or something owing to them, that is, something which must be posited, if they are to have the formality of quantity. Thus, therefore, in virtue of two middles, equivalent as it were to two arguments, he concludes to one conclusion, saying that if the universal demonstration is as described, that is, is less an entity than the particular, and more likely to create a false opinion than the particular, it follows from these two middles that the universal ranks lower than the particular.

Then, at: we may retort thus (85b4) he solves these reasons in order.

First, he solves the first one, saying that the first, that is, the ground on which the first reason rests, is no different in the universal than in the particular, because in both cases we find something which is according to itself and something according to something else. And he shows that something which is according to itself is found in the universal. For “to have three angles equal to two right angles” does not belong to isosceles according to itself, that is, precisely as isosceles, but according as it is a triangle. Consequently, one who knows that a certain triangle, namely, the isosceles, “has three,” has less knowledge of that which is per se than if he knew that a triangle “has three.” And it must be admitted universally that if there be any characteristic which does not belong to triangle as triangle, but that characteristic is nevertheless demonstrated of it, the demonstration will not be true. But if it is in it precisely as it is a triangle, then by knowing it in a universal way of triangle precisely as of triangle, he has a more perfect knowledge. From these facts, therefore, he concludes a conditional statement in whose antecedent three things are placed: one is that “triangle” is in more things than isosceles is; the second is that “triangle” is predicated of isosceles and of those others according to the same formality and not equivocally; the third is that “having three angles equal to two right angles” is present in every triangle. With these three suppositions, the consequent is that “the having of three” does not belong to triangle precisely as it is isosceles, but vice versa.

Now the first two were put in the antecedent on the ground that if “triangle” were not the wider term or if it were predicated equivocally of its several inferiors, it would not be compared to “isosceles” as universal to particular. But he added the third, because if “having three” did not belong to every triangle, it would not belong to isosceles precisely as triangle, but in virtue of being a certain triangle; just as the characteristic of “having three” does not belong to every figure precisely as figure, but because it is a certain figure which is a triangle.

From these statements, therefore, he concludes to the opposite of that which the objection presupposed, namely, he concludes that one who knows in the universal knows the thing per se and as such in a better way than one who knows in the particular. And from this he further concludes his chief proposition, namely, that universal demonstration is stronger than particular.

Then, at: if there is a single identical definition (85b15) he answers the second reason.

He says that if the universal is predicated of several according to one formality and not equivocally, the universal, so far as its formality is concerned, that is, so far as science and demonstration are concerned, will not be less of an entity than the particular, but more. For the incorruptible is more a being than the corruptible; but the formality of a universal is incorruptible, whereas particulars are corruptible, in that they are subject to corruption as to their individual principles, although not as to the formality of the species, which is common to all and conserved through generation. And so in regard to that which pertains to their formality, the universals are beings to a higher degree than particulars. Nevertheless, in regard to natural subsistence the particulars, which are called the first and chief substances, have more being.

Then, at: because the universal has (85b18) he answers the third reason.

He says that although in propositions or demonstrations that are universal, something which is one according to itself, say “triangle,” is signified, nevertheless there is no need for anyone to suppose on this account that “triangle” is some one thing apart from the many, any more than there is need in the case of things which do not signify substance but some genus of accident (when we signify them absolutely, as when we say “whiteness” or “fatherhood”), to suppose that such things exist apart from the substance. For the intellect is able to understand one of the things which are joined in reality without actually thinking of some other one; yet the intellect is not false. Thus, if something white is musical, I am able to think of the white and attribute something to it and demonstrate something of it, say, that it disperses the vision, without adverting at all to musical. However, if one were to understand that the white one is not musical, then the intellect would be false. And so when we say or understand that whiteness is a color, no mention being made of the subject, we are saying something true. But it would be false, were we to say that the whiteness which is a color is not in a subject. In like fashion, when we say that every man is an animal, we are speaking truly, even though no particular man is mentioned. But it would be false were we to say that man is “an animal existing apart from particular men.” Now, if this is so, then demonstration is not the cause of the false opinion according to which someone supposes that the universal is some thing outside the singulars, but it is rather the hearer who understands incorrectly. Hence this does not detract at all from universal demonstration.

Universal demonstration is stronger than particular demonstration

b22. (4) Demonstration is syllogism that proves the cause, that is, the reasoned fact, and it is rather the commensurate universal than the particular which is causative (as may be shown thus: that which possesses an attribute through its own essential nature is itself the cause of the inherence, and the commensurate universal is primary; hence the commensurate universal is the cause). Consequently commensurately universal demonstration is superior as more especially proving the cause, that is the reasoned fact.

b28. (5) Our search for the reason ceases, and we think that we know, when the coming to be or existence of the fact before us is not due to the coming to be or existence of some other fact, for the last step of a search thus conducted is eo ipso the end and limit of the problem. Thus: ‘Why did he come?’ ‘To get the money — wherewith to pay a debt — that he might thereby do what was right.’ When in this regress we can no longer find an efficient or final cause, we regard the last step of it as the end of the coming — or being or coming to be — and we regard ourselves as then only having full knowledge of the reason why he came. If, then, all causes and reasons are alike in this respect, and if this is the means to full knowledge in the case of final causes such as we have exemplified, it follows that in the case of the other causes also full knowledge is attained when an attribute no longer inheres because of something else. Thus, when we learn that exterior angles are equal to four right angles because they are the exterior angles of an isosceles, there still remains the question ‘Why has isosceles this attribute?’ and its answer ‘Because it is a triangle, and a triangle has it because a triangle is a rectilinear figure.’ If rectilinear figure possesses the property for no further reason, at this point we have full knowledge — but at this point our knowledge has become commensurately universal, and so we conclude that commensurately universal demonstration is superior.

a3. (6) The more demonstration becomes particular the more it sinks into an indeterminate manifold, while universal demonstration tends to the simple and determinate. But objects so far as they are an indeterminate manifold are unintelligible, so far as they are determinate, intelligible: they are therefore intelligible rather in so far as they are universal than in so far as they are particular. From this it follows that universals are more demonstrable: but since relative and correlative increase concomitantly, of the more demonstrable there will be fuller demonstration. Hence the commensurate and universal form, being more truly demonstration, is the superior.

a11. (7) Demonstration which teaches two things is preferable to demonstration which teaches only one. He who possesses commensurately universal demonstration knows the particular as well, but he who possesses particular demonstration does not know the universal. So that this is an additional reason for preferring commensurately universal demonstration. And there is yet this further argument:

a14. (8) Proof becomes more and more proof of the commensurate universal as its middle term approaches nearer to the basic truth, and nothing is so near as the immediate premise which is itself the basic truth. If, then, proof from the basic truth is more accurate than proof not so derived, demonstration which depends more closely on it is more accurate than demonstration which is less closely dependent. But commensurately universal demonstration is characterized by this closer dependence, and is therefore superior. Thus, if A had to be proved to inhere in D, and the middles were B and C, B being the higher term would render the demonstration which it mediated the more universal.

a21. Some of these arguments, however, are dialectical. The clearest indication of the precedence of commensurately universal demonstration is as follows: if of two propositions, a prior and a posterior, we have a grasp of the prior, we have a kind of knowledge — a potential grasp — of the posterior as well. For example, if one knows that the angles of all triangles are equal to two right angles, one knows in a sense—potentially—that the isosceles’ angles also are equal to two right angles, even if one does not know that the isosceles is a triangle; but to grasp this posterior proposition is by no means to know the commensurate universal either potentially or actually.

a28. Moreover, commensurately universal demonstration is through and through intelligible; particular demonstration issues in sense-perception.

The preceding arguments constitute our defense of the superiority of commensurately universal to particular demonstration.

Demonstration is syllogism. After answering the arguments which favored the false side, the Philosopher now introduces arguments for the true side, namely, to show that universal demonstration is the more powerful.

In regard to this he sets down seven reasons, adding them to the previous solutions from which the proposition can also be concluded, as was clear from the above.

The first reason (85b22), is this: Demonstration is a syllogism showing the cause and 'propter quid': (for this is the way scientific knowing takes place, as stated above.) But the universal does this better than the particular. For as was shown in the first solution, something is per se in the universal more than in the particular. But that in which something is per se is the cause of this something: (for the subject is the cause of the proper attribute which is in it per se). However, the first thing in which the proper attribute is present is the universal, as is clear from what has been stated above. Hence it is plain that, properly speaking, the cause is that which is universal. From this he concludes the proposition, namely, that universal demonstration is more valuable as better declaring the cause and 'propter quid'.

Then, at: our search for the reason ceases (85b28) he gives the second reason.

This reason is taken from the final causes. Here it should be pointed out that something is the end of another thing both in regard to becoming and in regard to being: in regard to becoming, as generation is for the sake of form; in regard to being, as a house is for the sake of habitation.

He says, therefore, that we search for the 'reason why' something is done or 'why' something is, until we reach the point where there is nothing further to assign than this point reached, as the reason why that comes to be or is, whose 'reason why' is sought. And when we find this, then we think that we know the 'propter quid': the reason being that that which is ultimate in this way, that is, leaving nothing further to be sought, is truly the end and terminus which is being sought when we seek the 'propter quid'. And he gives this example: if we should ask why someone went out, and the answer is given, “to get money,” and this in order to pay a debt, and “this for this other reason,” namely, so that he is not guilty of injustice; and we continue in this way until there is nothing further for the sake of which as for an end—as when we arrive at the ultimate end, which is happiness—we will say that it was for this, as for an end, that he went. And it is the same in all other things that are or are done for the sake of an end: when we arrive at it, we will know best the reason why he went. Thus, if things are so related in the other causes as they are in the final cause, namely, that we know fully when the last one has been reached, then we will know best in the others, when we shall have arrived at the fact that “this is in this” no longer because of something else. And this happens when we shall have reached the universal. To elucidate this he offers the following example: If we should ask concerning this particular triangle, why its exterior angles are equal to four right angles, the answer will be that this happens to this triangle because it is isosceles; and it is so for the isosceles, because it is a triangle; and it is so for a triangle, because it is such and such a rectilinear figure. If no further step can be taken, then we know in the best way. But this happens when the universal has been reached. Therefore, the universal demonstration is stronger than the particular.

Then, at: the more demonstration becomes particular (86a3) he gives the third reason.

He says that the more one proceeds toward particulars, the nearer one gets to what is infinite; because, as it is stated in Physics III, the infinite is appropriate to matter which is the principle of individuation. On the other hand, the more one proceeds toward the universal, the nearer he gets to what is simple and to the end itself; because the universal reason is taken on the part of the form, ic is simple and which has the character of an end insofar as it terminates the infinitude of matter. Now, it is obvious that infinite things as such are not scientifically knowable; rather to the extent that they are finite, to that extent are they knowable; because the principle of knowing a thing is not the matter but the form. Therefore, it is obvious that universals are scientifically more knowable than particulars. Consequently, they are also more demonstrable, because a demonstration is a syllogism that makes one know scientifically. But a demonstration of the more demonstrable is more powerful; for things which are described one in terms of the other grow apace. [???] But demonstration is described in terms of the demonstrable. Consequently, since universals are more demonstrable, universal demonstration will be the more powerful demonstration.

Then, at: demonstration which teaches two things (86a11) he gives the fourth reason and it is this:

Since the end of demonstration is scientific knowledge, the more things a demonstration enables one to know, the more powerful it is. And this is what he states, namely, that a demonstration, according to which a man knows one thing and something additional, is more preferable to one according to which a man knows only the one. But a man who has knowledge of the universal knows also the particular, so long as he knows that the particular is contained under the universal. Thus, one who knows that every mule is sterile, knows that this animal, which he recognizes to be a mule, is sterile. But one who knows the particular does not on that account know the universal. For if I know that this mule is sterile, I do not on that account know that every mule is sterile. It remains, therefore, that universal demonstration, through which the universal is known, is stronger than particular, through which the particular is known.

Then, at: proof becomes more and more proof (86a14) he gives the fifth reason, and it is this:

The nearer to the first principle the middle of a demonstration is, the more powerful is the demonstration. He proves this on the ground that if a demonstration which proceeds from an immediate principle is more certain than one which does not proceed from an immediate, but from a mediate, principle, then it is necessary that to the extent that a demonstration proceeds from a middle nearer an immediate principle, the more powerful it is. But a universal demonstration proceeds from a middle nearer the principle which is an immediate proposition. And he exemplifies this with terms. For if one is required to demonstrate A, which is the most universal, of B, which is the most particular, say “substance” of man, and B and C, say 'living' and 'animal' be taken, such that B is more general than C, as 'living' than 'animal', it is obvious that B, which is the more universal, will be immediate to A; consequently, more will be known through it than through C, which is less universal. Hence it remains that universal demonstration is more powerful than particular.

But he remarks that some of the above reasons are “logical,” because, namely, they proceed from common principles which are not proper to demonstration; especially the third and fourth, which take as their middle something which is common to all knowledge. However, the other three, namely, the first, second and fifth, seem to be more analytic, proceeding as they do from principles proper to demonstration.

Then, at: some of these arguments (86a21) he gives the sixth reason.

He says that the primacy of universal demonstration over particular is most evident from the very propositions from which the two demonstrations proceed. For the universal demonstration proceeds from universal propositions, whereas the particular demonstration proceeds from a particular proposition. Now, the comparison between universal and particular propositions is such that one who has knowledge of the former, that is, of the universal, somehow knows the latter, namely, in potency. For the particulars are potentially in the universal, as the parts are potentially in a whole. Thus, if one knows that every triangle has three angles equal to two right angles, he already knows it potentially of isosceles. But one who knows something in a particular way does not on that account know it universally either potentially or actually. For the universal proposition is neither potentially nor actually contained in the particular. Therefore, if that demonstration is more powerful which proceeds from stronger propositions, it follows that universal demonstration is more powerful.

It should be noted that this reason does not differ from the fourth one given above, except that in the fourth one the comparison was made between the conclusions which are known through demonstration, whereas here the comparison is between the propositions from which the demonstration proceeds.

Then, at: moreover, commensurately universal (86a28) he gives the seventh reason.

It is this: the universal demonstration is intelligible, that is, is terminated in the intellect, because it finishes in”a universal which is known only by the intellect. On the other. hand, a particular demonstration, although it begins in the intellect is terminated in sense, because it concludes a particular which is directly known by sense, and the reason demonstrating reaches out to the particular through a certain application or reflexion. Now, since intellect is more powerful than sense, it follows that universal demonstration is stronger than particular.

Finally, he concludes that this is clear in virtue of all that has been said above.

Affirmative demonstration is stronger than negative

a32. That affirmative demonstration excels negative may be shown as follows.

(1) We may assume the superiority ceteris paribus of the demonstration which derives from fewer postulates or hypotheses — in short from fewer premises; for, given that all these are equally well known, where they are fewer knowledge will be more speedily acquired, and that is a desideratum. The argument implied in our contention that demonstration from fewer assumptions is superior may be set out in universal form as follows. Assuming that in both cases alike the middle terms are known, and that middles which are prior are better known than such as are posterior, we may suppose two demonstrations of the inherence of A in E, the one proving it through the middles B, C and D, the other through F and G. Then A-D is known to the same degree as A-E (in the second proof), but A-D is better known than and prior to A-E (in the first proof); since A-E is proved through A-D, and the ground is more certain than the conclusion. Hence demonstration by fewer premises is ceteris paribus superior. Now, both affirmative and negative demonstration operate through three terms and two premises, but whereas the former assumes only that something is, the latter assumes both that something is and that something else is not, and thus operating through more kinds of premise is inferior.

b10. (2) It has been proved that no conclusion follows if both premises are negative, but that one must be negative, the other affirmative.

b12. So we are compelled to lay down the following additional rule: as the demonstration expands, the affirmative premises must increase in number, but there cannot be more than one negative premise in each complete proof. Thus, suppose no B is A, and all C is B. Then if both the premises are to be again expanded, a middle must be interposed. Let us interpose D between A and B, and E between B and C. Then clearly E is affirmatively related to B and C, while D is affirmatively related to B but negatively to A; for all B is D, but there must be no D which is A. Thus there proves to be a single negative premise, A-D. In the further prosyllogisms too it is the same, because in the terms of an affirmative syllogism the middle is always related affirmatively to both extremes; in a negative syllogism it must be negatively related only to one of them, and so this negation comes to be a single negative premise, the other premises being affirmative. If, then, that through which a truth is proved is a better known and more certain truth, and if the negative proposition is proved through the affirmative and not vice versa, affirmative demonstration, being prior and better known and more certain, will be superior.

b30. (3) The basic truth of demonstrative syllogism is the universal immediate premise, and the universal premise asserts in affirmative demonstration and in negative denies: and the affirmative proposition is prior to and better known than the negative (since affirmation explains denial and is prior to denial, just as being is prior to not-being). It follows that the basic premise of affirmative demonstration is superior to that of negative demonstration, and the demonstration which uses superior basic premises is superior.

b39. (4) Affirmative demonstration is more of the nature of a basic form of proof, because it is a sine qua non of negative demonstration.

That affirmative demonstration. After showing that universal demonstration is more powerful than particular, the Philosopher here shows that affirmative demonstration is more powerful than negative.

In support of this he presents five reasons.

In the first reason (86a32) he presupposes that, all else being equal, the better demonstration is the one which proceeds from fewer postulates or suppositions or propositions.

How these differ is clear from what has been stated above. For propositions might even be taken to mean those per se known statements which are neither suppositions nor postulates, as stated above. But a supposition is not the same as a postulate: for a supposition is a proposition which is not per se known, but is taken by the learner as opined; a postulate, on the other hand, is a proposition which is not per se known and has not been opined by the learner, nothing being said about whether he has opinions to the contrary or not.

Accordingly, he shows in two ways that, other things being equal, the demonstration which employs fewer of these is better.

First, because if it happens that both propositions from which one proceeds are equally known, it follows that one knows more quickly through fewer propositions than through more, because a discourse through fewer propositions comes to an end sooner than one through more. For it is more desirable or more advantageous that a man, learn more quickly. Hence, it remains that a demonstration which proceeds from fewer propositions, provided they are equally known, is better.

Second, he proves the same proposition universally without the above supposition, namely, that all the propositions used are equally known. To prove this he assumes this supposition, namely, that the middles which are of one order are equally known, but the middles which are prior are better known. For this must be universally true. Therefore, supposing this, let there be one demonstration in which it is demonstrated that A is in E in virtue of three middles, BCD, so that a conclusion is reached from four propositions, namely, “Every B is A,” “Every C is B,” “Every D is C” and “Every E is D.” Then let the other demonstration be one which concludes the same conclusion, namely, that A is in E, through two middles which are Z and H. Under these conditions, from the fact that the order of knowing is proportionate to the order of middles, since the prior known are the better known, as has been said, it is obvious that the proposition, “Every D is A” in the first demonstration must be as well known as “Every E is A” in the second, because two middles are used for obtaining each. However, it is also obvious that in the first demonstration this proposition, “Every D is A,” is prior and better known than the proposition, “Every E is A,” because this latter is demonstrated from something prior in the first demonstration-and from what has been established above, it is apparent that that through which something is demonstrated is more credible and more known than that which is demonstrated through it. What remains, therefore, is that this proposition, “Every E is A,” so far as it is concluded by the second demonstration, is better known than the same proposition so far as it is concluded by the first demonstration which used more middles.

Therefore, a demonstration which proceeds from fewer is better than one which proceeds from more. Therefore, having proved the major proposition, Aristotle assumes that an affirmative proposition proceeds from fewer things than a negative: not indeed from fewer terms or fewer propositions materially, because every demonstration, whether affirmative or negative, demonstrates through three terms and two propositions; but a negative demonstration is said to proceed from more according to the quality of the propositions. For an affirmative demonstration takes only “being,” that is, proceeds only from affirmative propositions, but a negative demonstration takes being and non-being, that is, uses an affirmative and a negative. Therefore, the affirmative is stronger than the negative.

Then, at: it has been proved that no conclusion (86b10) he presents the second reason.

Now this second reason is introduced to support the first, which might seem deficient on the ground that it was not subsumed under the major proposition in the way in which it was proved. Therefore, to forestall any subterfuge he adds this second reason to confirm the first.

For it has been shown in Prior Analytics I that a syllogism cannot be formed out of two negative propositions, but one proposition at least must be affirmative and the other may be negative. This clearly shows that affirmative propositions have greater efficacy for syllogizing than do negatives. Hence it follows that an affirmative demonstration which proceeds from affirmatives only, is more powerful than the negative demonstration which proceeds from a negative and an affirmative.

Then, at: so we are compelled (86b12) he presents the third reason.

He says that as a consequence of the foregoing reasons, we can assert that when a demonstration is augmented, that is, by resolving propositions into their principles, it is necessary that there be several affirmative propositions, but it is impossible for there to be more than one negative. Thus, let us take the following negative demonstration: “No B is A; Every C is B: therefore, No C is A.” Therefore, let this demonstration be augmented as to both propositions, if each is mediate, by taking a middle for each, letting the middle of the major proposition, “No B is A,” be D, and the middle of the minor proposition, “Every C is B,” be E. Now, since this proposition, “Every C is B,” is affirmative and was concluded without benefit of a negative, it is necessary that its middle, namely, E, be affirmative to each extreme. Consequently, there are two affirmative propositions, namely, “Every E is B,” and “Every C is E,” from which “Every C is B” is concluded. But the major proposition is negative, namely, “No B is A.” Now, a negative is not concluded from two negatives; but in the first figure, in which demonstration is best made, according to what has been said, that major must be negative and the minor affirmative. Consequently, this middle, D, must be affirmative to B and negative to A, as in the following demonstration: “No D is A; Every B is D; therefore No B is A.” Hence by augmenting a negative demonstration by resolving the propositions into their principles, there will be four propositions: only one of these is negative, namely, “No D is A,” and the other three will be affirmative, namely, “Every B is D,” “Every E is B,” and “Every C is E.” And the same thing will happen in all other syllogisms, because it is always required that the middle, through which the affirmative propositions are proved, be affirmative to both extremes. But the middle through which the negative proposition is proved must be negative to one of the extremes only. Thus it follows that only one proposition is negative and the others all affirmative. From this it is clear that it is mainly through affirmatives that a negative proposition is demonstrated. Therefore, if that through which something is demonstrated is more known and more credible than that which is demonstrated through it (since a negative proposition is mainly proved by affirmatives and not vice versa), it follows that the affirmative proposition is prior and more known and more credible than the negative. Accordingly, the affirmative demonstration will be more valuable.

Then, at: the basic truth of demonstrative syllogism (86b30) he presents the fourth reason.

He says that the principle of a demonstrative syllogism is an immediate universal proposition, in the sense that the proper principle of an affirmative syllogism is an affirmative proposition and the proper principle of a negative syllogism is a universal negative proposition. But the effect of a nobler principle is itself more noble. Therefore, as an affirmative proposition is to a negative, so is an affirmative demonstration to a negative one. But an affirmative proposition is more powerful than a negative one. (He proves this in two ways: first, because the affirmative is prior and better known, since the negative is proved by the affirmative and not vice versa. Second, because affirmation naturally precedes negation, as being is prior to non-being—for although in one and the same thing which passes from non-being to being, the non-being is prior in the order of time, yet in the order of nature, being is prior and, absolutely speaking, is prior even in time, because non-beings are not brought into existence except by something that is). Therefore, it is clear that affirmative demonstration is mo powerful than negative.

Then, at: affirmative demonstration is more (86b39) he presents the fifth reason and it is this:

That upon which something depends is more principal. But negative demonstration depends on affirmative, because there cannot be a negative demonstration without an affirmative proposition, which is not proved except by an affirmative demonstration. Therefore, affirmative demonstration is more principal than negative.

Negative ostensive demonstration is stronger than demonstration leading to the impossible

a1. Since affirmative demonstration is superior to negative, it is clearly superior also to reductio ad impossibile.

a2. We must first make certain what is the difference between negative demonstration and reductio ad impossibile. Let us suppose that no B is A, and that all C is B: the conclusion necessarily follows that no C is A. If these premises are assumed, therefore, the negative demonstration that no C is A is direct.

a7. Reductio ad impossibile, on the other hand, proceeds as follows. Supposing we are to prove that does not inhere in B, we have to assume that it does inhere, and further that B inheres in C, with the resulting inference that A inheres in C. This we have to suppose a known and admitted impossibility; and we then infer that A cannot inhere in B. Thus if the inherence of B in C is not questioned, A’s inherence in B is impossible.

a13. The order of the terms is the same in both proofs: they differ according to which of the negative propositions is the better known, the one denying A of B or the one denying A of C. When the falsity of the conclusion is the better known, we use reductio ad impossible; when the major premise of the syllogism is the more obvious, we use direct demonstration.

All the same, the proposition denying A of B is, in the order of being, prior to that denying A of C; for premises are prior to the conclusion which follows from them, and ‘no C is A’ is the conclusion, ‘no B is A’ one of its premises.

a19. For the destructive result of reductio ad impossibile is not a proper conclusion, nor are its antecedents proper premises. On the contrary: the constituents of syllogism are premises related to one another as whole to part or part to whole, whereas the premises A-C and A-B are not thus related to one another. Now, the superior demonstration is that which proceeds from better known and prior premises, and while both these forms depend for credence on the not-being of something, yet the source of the one is prior to that of the other. Therefore, negative demonstration will have an unqualified superiority to reductio ad impossibile, and affirmative demonstration, being superior to negative, will consequently be superior also to reductio ad impossibile.

Since affirmative demonstration is superior. After showing that universal demonstration ranks higher than particular, and affirmative higher than negative, the Philosopher here shows, third, that ostensive demonstration is more powerful than that which leads to the impossible.

Concerning this he does three things.

First, he proposes what he intends.

Second, he prefaces certain matters needed for showing the proposition, at: he must first make certain (87a2; [350]).

Third, he proves his proposition, at: all the same, the proposition (87a17; [353]).

He says therefore first (87a1) that since we have shown that affirmative, demonstration is more powerful than negative, from this it further follows that an affirmative ostensive demonstration is more powerful than one which leads to the impossible.

Then, at: we must first make certain (87a2) he lays down certain things that are necessary for showing his proposition.

In regard to this he does three things.

First, he shows what a negative demonstration is.

Second, he shows what a demonstration to the impossible is, at: reductio ad impossibile, on the other hand (87a7; [351]).

Third, he concludes the comparison of the one with the other, at: the order of the terms is the same (87a13; [352]).

He says therefore first (87a2) that in order to manifest the proposition it is necessary to point out the difference between them, that is, between the negative demonstration and the one leading to the impossible. If, therefore, one assumes that A is in no B, and B is in every C, and concludes that A is in no C, it will be a negative demonstration.

Then, at: reductio ad impossibile, on the other hand (87a7) he elucidates what a demonstration leading to the impossible is.

And he says that a demonstration leading to the impossible is as follows: Suppose we are to prove that A is not in B. Let us assume the opposite of what we wish to prove, namely, assume that every B is A; then assume that B is in C, using the proposition, “Every C is B”; from these the conclusion follows that “Every C is A,” which is such that everyone knows and admits that it is impossible. From this we conclude that the first proposition, namely, “Every B is A,” is false. Consequently, it will be necessary either that no B is A or at least that some B is not A.

But it must be understood that “A is not in B” follows, when it is clear that B is in C, because if it is obvious that “Every C is A” is false, but not obvious that “Every C is B” is true, it would not, just in virtue of the inference, be plain that “Every B is A” is false; for the falsity of the conclusion might have proceeded from either premise, as was stated above.

Then, at: the order of the terms (87a13) he concludes the comparison of each of the aforesaid demonstrations.

First, he shows wherein they agree, namely, in having a like ordering of their terms. For just as in the negative demonstration, B is taken as the middle between A and C, so too in that which leads to the impossible. Second, he shows wherein they differ, because it makes a difference which negative proposition in each demonstration is better known, namely, whether it be the proposition, “No B is A,” or the proposition, “No C is A”: for in a demonstration leading to the impossible, the proposition “C is not A” is taken to be better known, since from the fact that A is not in C one shows that A is not in B. Consequently, the proposition, “C is not A,” is taken as better known. But when that which is set down as a premise of the syllogism is taken as better known, there is a demonstrative, that is, an ostensive, negative demonstration.

Then, at: all the same, the proposition (87a17) he shows his proposition in the following way: The proposition, “B is not A,” is naturally prior to the proposition, “C is not A.” And this is proved by the fact that the premises, from which the conclusion is inferred, are naturally prior to the conclusion. But in the order of the syllogism, “C is not A” is set down as the conclusion, whereas “B is not A” is set down as that from which the conclusion is inferred. Therefore, “B is not A” is naturally prior.

Then, at: for the destructive result (87a19) he removes an objection.

For someone could say that even the negative “C is not A” is the one from which one concludes “B is not A” in a demonstration to the impossible.

But he excludes this, saying that by the fact that the conclusion is destroyed and from its destruction something in the premises is destroyed, it does not follow that what was first the conclusion is now a principle and vice versa, absolutely and according to nature, but only in a qualified sense. For the relation of conclusion to principle is such that the principle is destroyed by the conclusion’s being destroyed. But that functions as the principle from which a syllogism proceeds, which is related to the conclusion as a whole to a part; while the conclusion is to the principle, as part to whole. For the subject of a negative conclusion is subsumed under the subject of the first proposition. But the propositions AC and AB are not so related to each other that AC is to AB as whole to part. For BA is not subsumed under CA; rather it is just the opposite. Hence it remains that although from the destruction of CA, one concludes to the destruction of BA, nevertheless CA is naturally the conclusion and BA the principle. Consequently, “B is not A” is naturally better known than “C is not A.”

From this, one argues in the following way: That demonstration is the worthier which proceeds from better known and prior principles. But a negative demonstration proceeds from something better known and prior than does a demonstration leading to the impossible. For each causes one to know something in virtue of a negative proposition: but the negative demonstration proceeds to cause belief from the negative proposition, “B is not A,” which is naturally prior. Demonstration to the impossible, on the other hand, proceeds to cause belief from the negative proposition, “C is not A,” which is naturally posterior. What remains, therefore, is that the negative demonstration is more powerful than one which leads to the impossible. Furthermore, as was shown above, the affirmative is stronger than the negative. Therefore, an affirmative ostensive demonstration is much stronger than one which leads to the impossible.

a31. The science which is knowledge at once of the fact and of the reasoned fact, not of the fact by itself without the reasoned fact, is the more exact and the prior science. A science such as arithmetic, which is not a science of properties qua inhering in a substratum, is more exact than and prior to a science like harmonics, which is a science of properties inhering in a substratum; and similarly a science like arithmetic, which is constituted of fewer basic elements, is more exact than and prior to geometry, which requires additional elements. What I mean by ‘additional elements’ is this: a unit is substance without position, while a point is substance with position; the latter contains an additional element.

a38. A single science is one whose domain is a single genus,

a38. namely, all the subjects constituted out of the primary entities of the genus—that is, the parts of this total subject—and their essential properties.

a41. One science differs from another when their basic truths have neither a common source nor are derived those of the one science from those the other.

b1. This is verified when we reach the indemonstrable premises of a science, for they must be within one genus with its conclusions: and this again is verified if the conclusions proved by means of them fall within one genus — that is, are homogeneous.

b5. One can have several demonstrations of the same connexion not only by taking from the same series of predication middles which are other than the immediately cohering term e.g., by taking C, D, and Z severally to prove A-B — but also by taking a middle from another series. Thus let A be change, D alteration of a property, B feeling pleasure, and I relaxation. We can then without falsehood predicate D of B and A of D, for he who is pleased suffers alteration of a property, and that which alters a property changes. Again, we can predicate A of I without falsehood, and I of B; for to feel pleasure is to relax, and to relax is to change. So the conclusion can be drawn through middles which are different, that is, not in the same series — yet not so that neither of these middles is predicable of the other, for they must both be attributable to some one subject.

A further point worth investigating is how many ways of proving the same conclusion can be obtained by varying the figure.

The science which is knowledge. After comparing demonstrations one with another, the Philosopher here treats of the comparison of science, which is the effect of demonstration.

And his treatment falls into two parts.

In the first he compares science to science.

In the second he compares science to other modes of knowing, at: scientific knowledge and its object (89b30; [395]).

Concerning the first he does two things.

First, he compares science to science as to certitude.

Second, as to unity and multiplicity, at: a single science is one (87a38; [361]).

Concerning the first (87a31) he lays down three modes whereby one science is more certain than another.

Laying down the first mode, he says that that science is prior and more certain than another, which, namely, makes one know the same things both 'quia' and 'propter quid'. However, that science is not more certain which knows only the 'quia' apart from one which knows 'propter quid'. But this is the relation of subalternating science to subalternate, as has been said above, namely, that the subalternate science in isolation knows 'quia' without knowing 'propter quid': thus a surgeon knows that circular wounds are healed more slowly, but he does not know 'why'. But such knowledge pertains to the geometer who considers that characteristic of a circle according to which its parts do not lie close enough to form an angle, the nearness of whose sides makes triangular wounds heal more quickly.

He lays down the second mode when he says that a science which is not concerned with a subject is more certain than one which is. Here 'subject' is taken to mean sensible matter because, as the Philosopher teaches in Physics II, some sciences are purely mathematical, those, namely, which abstract according to reason from sensible matter, as geometry and arithmetic; but other sciences are intermediate, namely, those which apply mathematical principles to sensible matter, as optics applies the principles of geometry to the visual line, and harmony, that is, music, applies the principles of arithmetic to sensible sounds. Hence he says here that arithmetic is both more certain and prior to music: it is prior, because music uses its principles for something non-mathematical; it is more certain, because lack of certitude arises from matter’s changes. Hence the closer one gets to matter, the less certain the science.

He lays down the third mode when he says that a science which arises from fewer things is prior and more certain than one which arises from an addition, that is, than one which results from that addition. And he gives the example that geometry is posterior to and less certain than arithmetic: for the things of geometry are the result of adding to the things which pertain to arithmetic. This is easy to see if one admits the postulates of Plato, according to which Aristotle is proceeding here, using them to prove his point (as he frequently uses the opinions of other philosophers as examples in his logical works in order to explain a point).

Now Plato laid it down that “one” is the substance of each thing, because he did not distinguish between “one” which is converted with being and signifies the substance of a thing, and the “one” which is the principle of number and which arithmetic considers. Accordingly, this “one,” as receiving the added characteristic of occupying a position in a continuum, takes on the guise of a point. Hence he said that “one” is a substance not occupying a position; but a point is a substance that does have a position. Consequently, a point adds something to the notion of “one,” namely, position. And just as all numbers not having position are caused from “one,” so from the point, according to the Platonists, all continuous quantities are caused. For a point in motion makes a line; a line in motion makes a plane, and a plane in motion, makes a body. According to this, then, continuous quantities (which are treated in geometry) are the result of additions made to numbers (which are the concern of arithmetic). Hence the Platonists laid it down that numbers are the forms of magnitudes, saying that the point’s form is “one,” and the line’s form is “two,” because it has two endpoints, while a surface’s form is “three,” owing to the first surface’s being the triangle which is terminated by three angles; but the form of bodies is “four,” on the ground that the first solid figure is the triangular pyramid, which is composed of four (non-plane) solid angles, namely, one at the apex and three at the base.

According to this it is obvious that the comparison of the certitude of sciences is here based on two things: for the first mode is taken according to the cause as it is prior to and more certain than the effect; but the other two are taken according to the form as this is more certain than matter, inasmuch as the form is the principle of knowing the matter. But, as it is stated in Metaphysics VII, matter is twofold: one is sensible, according to which the second mode is taken; the other is intelligible, that is, its continuity, according to which the third mode is taken.

And although this third mode was explained according to Plato’s theory, yet even according to Aristotle’s theory, a point results from some addition to “one”: for a point is an indivisible unity in a continuum, abstracting according to reason from sensible matter; but unity abstracts from both sensible and intelligible matter.

Then, at: a single science is one (87a38) he compares sciences one to another according to unity and diversity.

Concerning this he does two things.

First, he shows that there is unity and diversity among sciences both according to subject and according to principles.

Second, he treats concerning both the subjects and principles, at: there is no knowledge by demonstration (88a5; [372]).

Concerning the first he does two things.

First, he shows what makes for unity and diversity of sciences.

Second, he explains something which is needed for understanding what makes for multiplicity of sciences, at: one can have several demonstrations (87b5; [369]).

Concerning the first he does two things.

First, he shows what makes for the unity of a science.

Second, what makes for diversity of sciences, at: one science differs from another (87a41; [365]).

Concerning the first he does two things.

First, he lays it down that the unity of a science is considered from the unity of its generic subject.

Second, he describes the genus which can be the subject of a science, at: namely, all the subjects (87a38; [363]).

He says therefore first (87a38) that a science is said to be one from the fact that it is concerned with one generic subject.

The reason for this is that the process of science of any given thing is, as it were, a movement of reason. Now, the unity of any motion is judged principally from its terminus, as is clear in Physics V. Consequently, the unity of any science must be judged from its end or terminus. But the end or terminus of a science is the genus concerning which the science treats: because in speculative sciences nothing else is sought except a knowledge of some generic subject; in practical sciences what is intended as the end is the construction of its subject. Thus, in geometry the end intended is knowledge of magnitude, which is the subject of geometry; but in the science of building that which is intended as the end is the construction of a house, which is the subject of this art. Therefore, the unity of each science must be considered in terms of the unity of its subject. But just as the unity of one generic subject is more universal than another, for example, being or substance is more common than mobile being, so one science is more general than another. Thus, metaphysics, which treats of being or substance, is more general in scope than physics, which treats of mobile body.

Then, at: namely, all the subjects (87a38) he describes the marks of those genera concerning which there can be sciences.

He lays down two marks.

The first of these is stated when he says: all the subjects constituted out of primary entities, that is, of those subjects of which there is one genus, there is one science.

To understand this it should be noted that, as has been said, the process of science consists in a certain movement of reason passing from one thing to another. But all movement starts from some principle or beginning and is terminated at something definite; hence in the progress of a science, reason must proceed from certain first principles.

Therefore, if there be anything which does not have prior principles from which reason can proceed, there cannot be science of such things, if we take science to mean the effect of demonstration, as we do here. Thus, speculative sciences are not concerned with the very essences of separated substances: for we cannot, through demonstrative sciences, know the essences in them, because the essences of such substances are intelligible of themselves to an intellect proportionate to such intelligibility, and a grasp of the essences of such substances is not obtained by any prior conceptions. The only thing that can be known through speculative sciences about these substances is whether they exist and what they are not; anything else about them is known in terms of likenesses to lower things. But in that case we are really using subsequent things as though they were prior in order to understand them, because things that are subsequent according to nature are prior and better known to us.

And so it is clear that those things concerning which we have science through what is absolutely prior, are of themselves composed of prior items; but things which are known through subsequent items (but prior in reference to us), even though they be simple in themselves, are nevertheless in our knowledge of them composed of things that are first to us.

He lays down the second mark when he says: the parts of this total subject and their per se properties (87a38).

Here it should be noted that the subject of a science can have two types of parts: first, the parts out of which, as out of first things, it is composed, that is, the very principles of the subject; and second, the subjective parts. And although what is stated here can be applied to either of these types of parts, yet it seems to truer of the first type of parts. For in every science there are the principles of its subject, and these must be considered before all else: for example, in natural science the first consideration is about matter and form, and in grammar about the alphabet. But in every science there is also something ultimate, at which the study of that science terminates, namely, that the properties of the subject be manifested. But each of these, namely, the first parts and the properties, can be attributed to something either per se or not per se. For the per se principles and properties of a triangle are not the per se principles and properties of an isosceles triangle precisely as isosceles, but precisely as triangle. Neither are they the per se principles and properties of brass or white, even though the triangle happens to be of brass or be white. Hence, if there were a science which manifests the properties of triangle from the principles of the triangle, the subject of that science would not be isosceles or white or brass, but triangle—whose per se subjective parts would be isosceles equilateral and scalene.

Then, at: one science differs from another (87a41) he shows the reason for sciences being diverse.

First, he lays down this reason.

Second, he explains it, at: this is verified (87b1; [368]).

It should be noted in regard to the first point that although he took the reason for the oneness of a science from the oneness of its generic subject, he does not take the reason for their diversity from the diversity of their subject, but from the diversity of principles. For he says (87a41) that one science is distinct from another when their principles are diverse, in the sense that the principles of the two sciences do not proceed from any prior principles, nor the principles of the one science from those of the other; because if both proceed from the same principles, or if one proceeds from those of another, they would not be diverse sciences.

To understand this it should be noted that a material diversity of objects does not diversity habits, but only a formal diversity. Therefore, since something scientifically knowable is the proper object of a science, the sciences will not be diversified according to a material diversity of, their scientifically knowable objects, but according to their formal diversity. Now, just as the formality of visible is taken from light, through which color is seen, so the formal aspect of a scientifically knowable object is taken according to the principles from which something is scientifically known. Therefore, no matter how diverse certain scientifically knowable objects may be in their nature, so long as they are known through the same principles, they pertain to one science, because they will not differ precisely as scientifically knowable. For they are scientifically knowable in virtue of their own principles. This is made clear by an example, namely, that human voices differ a great deal according to their nature from the sounds of inanimate bodies; but because the consonance of human voices and of the sounds of inanimate bodies is considered according to the same principles, the science of music, which considers both, is one science. On the other hand, if there are things which have the same nature but are considered according to diverse principles, it is obvious that they pertain to diverse sciences. Thus, the mathematical body is never really distinct from a natural body; yet because the mathematical body is known through the principles of quantity, but a natural body through the principles of motion, the science of geometry and the science of nature are not the same. It is clear, therefore, that for sciences to be diverse it is enough that the principles be diverse, this diversity of principles being accompanied by a diversity of scientifically knowable objects. But in order to have one science absolutely, both are required, namely, unity of subject and unity of principles. (That is why above he made mention of unity of subject when he said, “whose domain is one genus,” but made mention of principles when he said, “all the subjects constituted out of the primary entities of the genus.”)

However, it should be further noted that second principles derive their force from the first principles. Hence for diversity of sciences, diversity of first principles is required. But this will not be verified if the principles of diverse things flow from the same principles, as the principles of triangle and square are derived from the principles of figure, or the principles of one are derived from the principles of another, as the principles of isosceles depend on the principles of triangle. Yet it should not be supposed that for the unity of a science it is enough that there be unity of first principles absolutely, but unity of first principles in some scientifically knowable genus. Now, the genera of the scientifically knowable are distinguished according to the diverse modes of knowing: thus things that are defined with matter are known in one mode, and those defined without matter are known in another. Hence the natural body is one genus of the scientifically knowable, and mathematical body is another genus. Hence there are diverse first principles for each of these genera and, consequently, diverse sciences. Furthermore, each of these genera is distinguished into diverse species according to diverse modes and aspects of intelligibility.

Then, at: this is verified (87b1) he explains the reason he laid down.

He says that a sign of the fact that sciences are diversified according to their principles is obtained when one resolves a science into principles that are indemonstrable, for they must be of the same genus as the things demonstrated; because, as stated above, one does not demonstrate by proceeding from an alien genus. Now, the reason why the reaching of indemonstrable principles of one genus is taken as a sign is that the facts demonstrated by them are in the same genus and are co-generic, that is, connatural, or generically proximate to them: for these have the same principles. And so it is clear that the unity of a scientifically knowable genus, precisely as it is the scientifically knowable genus from which was taken the oneness of the science, and the unity of the principles, according to which was taken th diversity among the sciences, mutually correspond.

Then, at: one can have several demonstrations (87b5) he shows how one conclusion can be demonstrated through several principles.

And this can happen in two ways.

In one way, when a number of middles are laid down in the same coordination, and one of these middles is used in one demonstration and another one in another, both leading to the same conclusion. This, of course, requires that the middles be non-continuous. Thus, if the two extremes, A and B, are 'to have three' and 'isosceles' respectively, but the middles, C and D are coordinate, as 'triangle' and 'this type of figure', A could be demonstrated of B with two demonstrations. In the first one, C could be taken as middle, and D in the other, so that in neither demonstration will, there be a middle continuous with the extremes, because in one demonstration a middle continuous with one extreme and not continuous with the other would be taken, and in the other demonstration the converse would be taken.

Or this can happen in a second way, namely, when diverse middles are taken from diverse coordinations; for example, if A, which is the major extreme, were to be altered, and B, which is the minor extreme, were “to be pleased,” and the middles taken were independent, say E would be “to be at rest,” and D “to be in motion.” In this case the same conclusion can be reached through diverse middles that are not of one order. Thus one demonstration would be this: “Whatever is at rest is altered, because it belongs to the same thing to be at rest and to be altered; but whatever is pleased is at rest, because rest in the desired good causes pleasure: therefore, whatever is pleased is altered.” The other demonstration would be: “Whatever is in motion is altered; but whatever is pleased is in motion, because pleasure is a movement of the appetitive faculty [power]: therefore, whatever is pleased is altered.”

Now the statement, “Whatever is pleased is in motion," is Plato’s opinion and is verified in sensible pleasures which involve movement. The other statement, “Whatever is pleased is at rest,” is true according to Aristotle’s opinion, as is clear from Ethics VII and X. And this is verified particularly in intellectual pleasures.

Then he says that just as this has been proved in the first figure, so in the other figures it is easy to see that the same conclusion can be syllogized with diverse middles. Now, the Philosopher mentioned this to show that diverse middles of demonstration sometimes pertain to the same science, as when they are taken from the same coordination, and sometimes to diverse sciences, when they are taken from a different coordination. Thus, astronomy demonstrates that the earth is round, using one middle, namely, the eclipse of the sun and moon; and natural science uses another middle, namely, the motion of heavy objects tending toward the center, as it is stated in Physics II.

Then, when he says: a further point (87b5), he reviews what has been said.

And he says that what has here been shown in the first figure can also be easily considered in the other figures, since the same conclusion follows from the diverse syllogisms.

Science is not concerned with things caused by fortune or with things learned through sense-perception

a5. There is no knowledge by demonstration of chance conjunctions; for chance conjunctions exist neither by necessity nor as general connexions but comprise what comes to be as something distinct from these. Now, demonstration is concerned only with one or other of these two; for all reasoning proceeds from necessary or general premises, the conclusion being necessary if the premises are necessary and general if the premises are general. Consequently, if chance conjunctions are neither general nor necessary, they are not demonstrable.

b28. Scientific knowledge is not possible through the act of perception. Even if perception as a faculty is of ‘the such’ and not merely of a ‘this somewhat’, yet one must at any rate actually perceive a ‘this somewhat’, and at a definite present place and time: but that which is commensurately universal and true in all cases one cannot perceive, since it is not ‘this’ and it is not ‘now’; if it were, it would not be commensurately universal — the term we apply to what is always and everywhere. Seeing, therefore, that demonstrations are commensurately universal and universals imperceptible, we clearly cannot obtain scientific knowledge by the act of perception:

b34. nay, it is obvious that even if it were possible to perceive that a triangle has its angles equal to two right angles, we should still be looking for a demonstration—we should not (as some say) possess knowledge of it; for perception must be of a particular, whereas scientific knowledge involves the recognition of the commensurate universal. So if we were on the moon, and saw the earth shutting out the sun’s light, we should not know the cause of the eclipse: we should perceive the present fact of the eclipse, but not the reasoned fact at all, since the act of perception is not of the commensurate universal. I do not, of course, deny that by watching the frequent recurrence of this event we might, after tracking the commensurate universal, possess a demonstration, for the commensurate universal is elicited from the several groups of singulars.

The commensurate universal is precious because it makes clear the cause; so that in the case of facts like these which have a cause other than themselves universal knowledge is more precious than sense-perceptions and than intuition. (As regards primary truths there is of course a different account to be given.)

Hence it is clear that knowledge of things demonstrable cannot be acquired by perception, unless the term perception is applied to the possession of scientific knowledge through demonstration.

a11. Nevertheless certain points do arise with regard to connexions to be proved which are referred for their explanation to a failure in sense-perception: there are cases when an act of vision would terminate our inquiry, not because in seeing we should be knowing, but because we should have elicited the universal from seeing; if, for example, we saw the pores in the glass and the light passing through, the reason of the kindling would be clear to us because we should at the same time see it in each instance and intuit that it must be so in all instances.

There is no knowledge by demonstration. After assigning the basis of unity and of diversity of sciences on the part of the generic subject and on the part of the principles, the Philosopher here continues with each.

First, with the subjects about which there is science.

Second, with the principles, at: all syllogisms cannot have (89a18; [382]).

In regard to the first it should be noted that above he laid down two marks of the genus which is the subject of the science: one of these is that it be composed of first principles, and the other is that its parts and properties be per se. Now, one of these conditions is lacking in things that happen by fortune, because they do not come about per se, but per accidens and beyond one’s intention, as is proved in Physics II. The other mark is lacking in things that are known through sense, which are first in our knowledge.

Therefore, he shows first that science is not concerned with things which come about by fortune.

Second, that it is not of things that are known through sense, at: scientific knowledge is not possible (87b28; [375]).

He says therefore first (87b19) that demonstrative science cannot be concerned with things that occur through fortune, that is, things that come about per accidens. He proves this in the following way: Every demonstrative syllogism proceeds either from necessary propositions or from propositions which are true for the most part. Now, from necessary propositions a necessary conclusion follows, as has been proved above; in like manner, from propositions which are true for the most part, a conclusion follows which is true for the most part, or perhaps one which is necessary, in the sense that the necessary can follow from the contingent, as the true can follow from the false. However, it never happens that from propositions which are true for the most part, a conclusion will follow which is true in few cases; because that would mean that in certain cases the propositions would be true and the conclusion false, which is impossible, as has been shown. It is necessary, therefore, that the conclusion of a demonstrative syllogism either be necessary or be true in the majority of cases. But that which is by fortune is neither necessary nor is it true in the majority of cases, but it occurs rarely, as is proved in Physics II. Therefore, demonstrative science cannot be concerned with that which is by fortune.

It should be noted, however, that there happens to be demonstration of things which occur, as it were, for the most part, insofar as there is in them something of necessity. But the necessary, as it is stated in Physics II, is not the same in natural things (which are true for the most part and fail to be true in a few cases) as in the disciplines, that is, in mathematical things, which are always true. For in the disciplines there is a priori necessity, whereas in natural science there is a posteriori (which nevertheless is prior according to nature), namely, from the end and form. Hence Aristotle teaches there that to show a 'propter quid', such as if this has to be, say that if an olive is to be generated, it is necessary that this, namely, the olive seed, preexist, but not that an olive is generated of necessity from a given olive seed, because generation can be hindered by some defect. Hence if a demonstration is formed from that which is prior in generation, it does not conclude with necessity, unless perhaps we take as necessary the fact that an olive seed is frequently generative of an olive, because it does this according to a property of its nature, unless it is impeded.

Then, at: scientific knowledge is not possible (87b28) he shows that science is not concerned with things that are known according to sense.

About this he does two things.

First, he shows that science does not consist in sensing.

Second, how sense is ordained to science, at: nevertheless certain points do arise (89a11; [381]).

Concerning the first he does two things.

First, he shows that science is not through sense.

Second, he ranks science above sense, at: the commensurate universal (88a5; [379]).

Concerning the first he does two things.

First, he shows the truth.

Second, he shows the errors of certain ones, at: nay, it is obvious (88b34; [378]).

He says therefore first (88a5) that just as science is not of things which are by fortune, so neither does it consist in knowledge which is through sense. And he proves this in the following way: It is obvious that sense knows the qualities of a thing and not its substance. For the per se object of sense is not the substance and the essence, but some sensible quality, such as hot, cold, white, black, and so on. But qualities of this sort affect singular substances existing in a definite place and time: hence it is necessary that this which is sensed be a this something, that is, a singular substance, and that it be somewhere now, that is, in a definite time and place. From this it is clear that that which is universal cannot fall under sense. For that which is universal is not restricted to the here and now, because then it would not be universal. For we call that a universal which is always and everywhere.

However, this is not to be understood according to the way of an affirmation, as though it pertained to the very notion of a universal or of that which is universal, namely, that it be always and everywhere. For if to exist always and everywhere pertained to the very notion of that which is universal, say to the notion of man or animal, it would then be required that each individual man and animal be always and everywhere, because the notion of man and of animal is found in each of its singulars. Or if this were required by the very notion of the universal, as it is required of a genus that it contain species under it, it would follow that nothing would be universal, if it were not found everywhere and always. According to this, “olive” would not be universal, because it is not found in all lands.

Hence, the statement under consideration must be understood after the manner of a negation or abstraction, that is, that the universal abstracts from every definite time and place. Hence of itself, just as it is found in each thing in one place or time, so it is apt to be found in all.

Thus, therefore, it is clear that the universal does not fall under sense. Therefore, because demonstrations are mainly universal, as has been shown above, it is obvious that science acquired by demonstration does not consist in knowledge obtained through sense.

Then, at: nay, it is obvious (88b34) he excludes the errors of those who believed that science consists in sense-perception.

And this notion seems to belong to those who did not believe that understanding differs from sense and, consequently, that there is no knowledge except sense-knowledge, as it is stated in On the Soul III and Metaphysics IV.

So, to exclude this he says that even if we could perceive through sense that a triangle has three angles equal to two right angles, it would still be necessary to learn this by demonstration if one is to have science; for we do not scientifically know through sense, because sense bears on singulars, whereas science consists in knowing the universal, as has been shown. But because he had given the example of something that cannot be perceived through sense, he makes his point clearer with an example taken from things that can be sensed, namely, from an eclipse of the moon which is caused by the opposition of the earth, which is interposed between the sun and moon, so that the brightness of the sun cannot reach the moon because of the earth’s shadow; and the moon is eclipsed when it enters the shadow. Let us suppose, then, that someone were on the moon and with his senses could perceive the shadow caused by the interposition of the earth. This person, then, would sensibly perceive that the moon is darkened by the shadow of the earth, but this would not mean that he would therefore know totally the cause of the eclipse. For the per se cause of an eclipse is that which universally causes an eclipse. But the universal is not known through sense; rather, we receive universal knowledge from many individually observed cases in which the same thing is found to happen. And so it is through the universal cause that we demonstrate something in the universal way that constitutes science.

Then, at: the commensurate universal (88b34) he shows that science is nobler than sense.

For it is clear that knowledge through the cause is more noble; but the per se cause is the universal cause, as has been said. Therefore, knowledge through the universal cause, such as science is, is more honorable. But because it is impossible to apprehend such a universal cause through sense, it follows that science, which shows the universal cause, is not only more honorable than any sense knowledge, but more honorable than any other intellectual knowledge, provided it concern things which have a cause: for to know something through the universal cause is more noble than any other way of knowing that which has a cause, without knowing the cause. However, the case is otherwise in regard to first things, which do not have a cause. For these are understood in virtue of themselves; and such knowledge of these things is more certain than any science, because it is from such knowledge that science acquires its certitude.

Finally, he concludes to his chief point, namely, that it is impossible through sense to know something demonstrable, unless perhaps one takes “sense” in an equivocal meaning, calling demonstrative science “sense” on the ground that demonstrative science concerns the determinately one just as sense does: on which account sure calculations are called science.

Then, at: nevertheless certain points (89a11) he shows how sense is ordained to science.

For some problematic doubts are caused by the shortcomings of sense. For there would be no need to investigate certain matters, if we were to see them; not because science consists in seeing, but because from things seen by way of experiment the universal is obtained, concerning which there is science. For example, if we should see a porous pane of glass and observed how the light passes through the openings in the glass, we would have 'the scientific answer' as to why the glass is transparent. (He uses this example because it is the opinion of those who supposed that light is a body and that certain bodies are transparent because of openings called pores. But because they cannot be perceived by sight, since they are so small, we wonder why glass is transparent.) A similar example can be given of certain things that have a latent sensible cause. Finally, because he had said that science of such things is not obtained by seeing, he shows that this is true. For in seeing we know isolated singulars; but to know scientifically we must know them all at once in the universal, if we are to know that such and such is the fact in all cases. For we see each individual glass singly, but we receive science that it is thus for all glass.

Principles of all syllogisms are not the same

a18. All syllogisms cannot have the same basic truths. This may be shown first of all by the following dialectical considerations. (1) Some syllogisms are true and some false:

a20. for though a true inference is possible from false premises, yet this occurs once only — I mean if A for instance, is truly predicable of C, but B, the middle, is false, both A-B and B-C being false; nevertheless, if middles are taken to prove these premises, they will be false because every conclusion which is a falsehood has false premises, while true conclusions have true premises, and false and true differ in kind.

a26. Then again, (2) falsehoods are not all derived from a single identical set of principles: there are falsehoods which are the contraries of one another and cannot coexist, e.g., ‘justice is injustice’, and ‘justice is cowardice’; ‘man is horse’, and ‘man is ox’; ‘the equal is greater’, and ‘the equal is less.’ From established principles we may argue the case as follows, confining ourselves therefore to true conclusions.

a31. Not even all these are inferred from the same basic truths; many of them in fact have basic truths which differ generically and are not transferable; units, for instance, which are without position, cannot take the place of points, which have position. The transferred terms could only fit in as middle terms or as major or minor terms, or else have some of the other terms between them, others outside them.

a37. Nor can any of the common axioms — such, I mean, as the law of excluded middle — serve as premises for the proof of all conclusions. For the kinds of being are different, and some attributes attach to quanta and some to qualia only; and proof is achieved by means of the common axioms taken in conjunction with these several kinds and their attributes.

b4. Again, it is not true that the basic truths are much fewer than the conclusions, for the basic truths are the premises, and the premises are formed by the apposition of a fresh extreme term or the interposition of a fresh middle. Moreover, the number of conclusions is indefinite, though the number of middle terms is finite;

b8. and lastly some of the basic truths are necessary, others variable. Looking at it in this way we see that, since the number of conclusions is indefinite, the basic truths cannot be identical or limited in number.

b10. If, on the other hand, identity is used in another sense, and it is said, e.g., ‘these and no other are the fundamental truths of geometry, these the fundamentals of calculation, these again of medicine’; would the statement mean anything except that the sciences have basic truths? To call them identical because they are self-identical is absurd, since everything can be identified with everything in that sense of identity.

b15. Nor again can the contention that all conclusions have the same basic truths mean that from the mass of all possible premises any conclusion may be drawn. That would be exceedingly naive, for it is not the case in the clearly evident mathematical sciences, nor is it possible in analysis, since it is the immediate premises which are the basic truths, and a fresh conclusion is only formed by the addition of a new immediate premise:

b20. but if it be admitted that it is these primary immediate premises which are basic truths, each subject-genus will provide one basic truth.

b22. If, however, it is not argued that from the mass of all possible premises any conclusion may be proved, nor yet admitted that basic truths differ so as to be generically different for each science, it remains to consider the possibility that, while the basic truths of all knowledge are within one genus, special premises are required to prove special conclusions.

b25. But that this cannot be the case has been shown by our proof that the basic truths of things generically different themselves differ generically. For fundamental truths are of two kinds, those which are premises of demonstration and the subject-genus; and though the former are common, the latter — number, for instance, and magnitude — are peculiar.

All syllogisms cannot have the same basic truths. After his treatment concerning that about which there is science, the Philosopher continues here with the principles of sciences and shows that the principles of all syllogisms are not the same.

First, he shows this logically, that is, through reasons common to all syllogisms.

Second, he shows it analytically, that is, through reasons that are peculiar to demonstration, at: if, on the other hand, identity (89b10; [390]).

Concerning the first he does three things.

First, he proves his proposition by showing the difference between false syllogisms and true ones.

Second, by the difference between some false syllogisms and other false ones, at: then again, falsehoods (89a26; [385]).

Third, by the difference between some true syllogisms and other true ones, at: not even all these are inferred (89a31; [386]).

Concerning the first he does two things.

First, he proves his proposition.

Second, he excludes an objection, at: for though a true inference (89a20; [384]).

He says therefore first (89a18) that if we first speculate this matter in a logical way, it is clear that the principles of all syllogisms cannot be the same, inasmuch as some syllogisms are false, that is, conclude false statements, and some are true, that is, conclude to the truth. Now, the principles of false and of true syllogisms are diverse: for the principles of true syllogisms are true, but the principles of false syllogisms are false. Therefore, the principles of all syllogisms are not the same.

Then (89a20) he excludes an objection, at: for though a true inference.

For someone might say that there are false principles even in true syllogisms, because it is possible to syllogize the true from the false.

But he excludes this, saying that although it happens that the true is syllogized from the false, this occurs only once in the first syllogism, in which the true is concluded from the false. But if one is forced to adduce other syllogisms to support the propositions in the premises, then those syllogisms would have to proceed from what is false, because from what is true the false cannot be concluded. Consequently, it is only in the first syllogism that the true is concluded from the false. Then he gives an example of this: Let the proposition, “Every C is A,” be true, and take for each of these extremes a false middle, B, so that A is not in B nor B in C. But if other middles be taken to prove these propositions, all the propositions of the false syllogism will be false, because every false conclusion is concluded from the false, but a true conclusion can be concluded from the true. Hence, when the propositions from which the true is concluded are true, one need not reach anything false. Thus, therefore, since some propositions are true and some false, it follows that the principles of true syllogisms are not the same as the principles of false syllogisms.

Then (89a26) he shows that not even the principles of false syllogisms are the same, at: then again, falsehoods.

For it occurs that some false conclusions are contrary to one another and incompatible with other false conclusions. Thus, the conclusion, “Justice is injustice,” is incompatible with the conclusion, “Justice is fear,” since both are false. For just as fear differs generically from justice, so from injustice. In like fashion, the two conclusions, “Man is a horse,” and “Man is a cow,” are contrary and incompatible. Again, these two propositions are incompatible, “Something equal is greater,” and “Something equal is less.” For it is necessary to arrive at such conclusions from principles, from which, when they are laid down, other things follow. Therefore, since they are contrary and incompatible, so too were the principles from which they were concluded.

Then (89a31) he shows that not even in the case of true syllogisms are the principles the same, at: not even all these.

And he gives four reasons.

The first is based on the differences among proper principles.

Hence he says that not even all these, that is, true conclusions, are inferred from the same basic truths. For the principles of diverse genera are themselves diverse: thus the pripciples of magnitudes are points, and of numbers unities; and these are not exactly the same, for numbers do not have position, but points do. Furthermore, if all principles of syllogisms were in agreement, they would have to meet at some same middle either by ascending upwards to the major term or by descending downwards toward the minor term: because in a syllogism it is necessary that the terms be assumed either within or without.

They are assumed within, when a multiplicity of syllogisms is used to prove the propositions presented. For then it is necessary to take middles that are between the predicates of the propositions and their subjects. Thus, if we form the syllogism, “Every B is A, Every C is B, therefore, Every C is A”; if it is necessary to prove “Every B is A,” we must take a middle between B and A, say D. Again, if it is necessary to prove the minor, we must take another middle between C and B, say E. In this process the terms assumed are always within.

However, they are assumed from without, when the major term is taken as middle by ascending, or the minor by descending: thus, if A is concluded of C through B, and Z is then concluded of B through A, and so on; similarly, by descending, if we conclude B of F through C.

Therefore, in syllogisms sharing principles in common, it is necessary either that the middle of one syllogism be taken above the propositions of another syllogism or that the extremes of one be taken above or below the extremes of the other syllogism.

But this cannot occur in things whose principles are diverse, because points cannot be taken as middles or extremes in syllogisms in which something about number is concluded, nor unities in syllogisms in which something about magnitudes is concluded. What remains, therefore, is that the principles of all syllogisms cannot be the same.

The second reason is presented at: nor can any of the common axioms (89a37), and it is based on common principles.

He says that there cannot be certain common principles from which alone are syllogized all conclusions, as this common principle, “Of each thing there is affirmation or negation,” which is universally true in every genus. Nevertheless, it is impossible that all things be syllogized exclusively from such common principles, because the genera of beings are diverse. Thus, the principles which pertain only to quantities are diverse from those which pertain exclusively to qualities. Such principles must be co-assumed with common principles, if one is to reach a conclusion in each matter. For example, if one wishes in quantities to syllogize from the aforesaid common principle, it is necessary to admit that since it is false that a point is a line, it must be true that a point is not a line; in like manner, in qualities, it is necessary to co-assume something peculiar to quality. Hence what remains is that it is impossible that the principles of all syllogisms be the same.

Then, at: again, it is not true (89b4), he gives the third reason which is based on a comparison of premises with conclusions.

And he says that the principles are not much fewer than the conclusions. They are as a matter of fact fewer, because although two principles are needed to infer one conclusion, that is, two propositions are required, because one conclusion is not concluded immediately except from two premises, nevertheless you can use one proposition to infer a number of conclusions, insofar as many things can be taken under the subject or under the predicate. However, the principles are not much fewer than the conclusions, because most of the facts which are co-assumed with the principles to obtain other conclusions are themselves conclusions. (Here propositions are being called principles). But propositions are formed of terms either added or interposed, that is, propositions in syllogisms are multiplied either by assuming terms extrinsically (either above the major term and below the minor, as has been explained above), or by accepting terms which are in the middle. To this must be added the fact that conclusions are infinite. For anything can be concluded of anything else either affirmatively or negatively. And lest this seem to conflict with the earlier statement that predications do not proceed to infinity, he adds that the terms are finite: and this is why it was stated above that a stop must be made in predications. Nevertheless, an infinitude of conclusions can be derived from a finite number of terms according to diverse combinations, if we take “conclusions” in a general sense, as including those that are per se and those that are per accidens. For we are now speaking of syllogisms in general. Therefore, if the conclusions are infinite and the principles are not much fewer than the conclusions, it follows that principles of syllogisms are also infinite. Therefore, the principles of all syllogisms are not the same.

Then he gives the fourth reason and it is based on the difference between the necessary and the contingent, at: and lastly some of the basic truths (89b8).

And he says that some of the principles which we use in the syllogism are contingent and some necessary, as is clear from Prior Analytics I, where he taught how to syllogize from necessary and from contingent premises. However, the necessary and the contingent are not the same. Therefore, the principles of all syllogisms are not the same. What he concludes from these last two reasons is that in view of what has been established, and because conclusions are infinite, it is impossible that the principles of all syllogisms either be the same or even be finite.

Then he shows the same thing analytically, namely, through reasons proper to the principles by which sciences demonstrate, at: if, on the other hand (89b10).

And he gives three reasons.

Concerning the first of these he says that if someone does not say that the principles of all syllogisms are the same, but says something somewhat different, namely, that some are principles of geometry and some of logic—which are called the principles of syllogisms or reasonings—and some are principles of medicine, and so on for the principles of all sciences, so that in this sense the principles of all demonstrations are the same, this does not support his claim that principles are the same: because from the facts that are given, nothing follows except that each science has its own principles. The contention that the principles of one science are the same as those of another-which would be required if the principles of all scientific syllogisms were the same-is impossible and ridiculous, because according to this it would follow that everything in the sciences would be the same and hence that all the sciences would be the same. For things that are the same as a same thing are the same. But the principles of each science are somehow the same as the conclusions, because they belong to the same genus: for one may not demonstrate by passing from one genus into another, as has been shown above. Therefore, if the principles are the same, it will follow that everything in the sciences would be the same.

Then, at: nor again can (89b15) he gives the second reason and it is this: If a person who contends that the principles of all are the same, intended to say that anything is demonstrated from anything, then he is saying something foolish, because this is possible neither in evident mathematicals nor in analysis. (By 'evident mathematicals', that is, evident considerations or disciplines he means those cases in which a conclusion is drawn immediately from evident premises; but by 'analysis' he means those cases where the assumed propositions are not evident but must be analyzed into others which are more evident). And that this is impossible he proves on the ground that in each of these two cases the principles of the demonstrative syllogism are immediate propositions, which are either immediately assumed in evident mathematicals or doctrines, or are reached by analysis. But we see that a different conclusion is demonstrated, when a different immediate proposition is co-assumed. Consequently, it cannot be that anything is demonstrated from just anything.

Then he excludes a certain objection, at: but if it be admitted (89b20).

For someone could say that there are two genera of immediate propositions: for some are first immediate propositions and some secondary, taking the order of immediate propositions according to the order of the terms. For those immediate propositions which consist of first and common terms, as 'being' and 'non-being', 'equal' and 'unequal', 'whole' and 'part', are first and immediate propositions, as “It does not occur that the same thing both is and is not,” and “Two things which are equal to the same thing are equal,” and so on. But the immediate propositions that are concerned with later and less common terms are secondary in relation to the first, as that “a triangle is a figure,” or that “man is an animal.” Therefore, someone might say that secondary propositions are co-assumed for demonstrating diverse conclusions, but that the first immediate propositions are the same in all demonstrations.

To exclude this he says that if anyone should assert that those first immediate propositions are the principles from which all things are demonstrated, he ought to consider that in each genus there must yet be one principle or one immediate proposition which is first in that genus, even though it is not absolutely first; and that from one which is absolutely first, together with a co-assumed principle proper to a particular genus, one should demonstrate in that genus. Consequently, not all things can be demonstrated from common principles alone, but it is necessary to accept proper principles which are diverse for diverse genera.

Then, having excluded the stupid interpretation of the position against which he is disputing, he concludes to his point, at: if, however, it is not argued (89b22).

And he says that if it is not admitted that anything is demonstrated from just anything—as we must, in view of the foregoing—it follows, that one conclusion is not derived from some principle in such a way that a different conclusion can be derived from it; otherwise just anything is demonstrated from just anything. Hence it is necessary that the principles of diverse sciences be diverse. If it is required that the principles of each science be of the same genus as everything demonstrated from them, it will be required that from these principles be demonstrated these conclusions, and “those from those,” that is, from diverse principles, demonstration being made in diverse sciences which treat of diverse genera.

Then he presents the third reason, at: but that this cannot (89b25).

He says that it is clear from another angle that this does not occur, namely, that the principles of all the sciences are the same; because it has been shown above that the principles of diverse genera are themselves generically diverse. Hence since diverse sciences are concerned with diverse genera, it follows that the principles of diverse sciences are diverse. But because the common principles which all sciences use are in some way the same, he distinguishes among the principles and says that principles are twofold: some of the first principles from which one demonstrates are as the first dignities; for example, “Being and non-being are not the same.” Again, there are other principles, namely, those with which the sciences are concerned, namely, the subjects of the sciences, because we use the definitions of the subjects as principles in demonstrations. Therefore, the members of the first group of principles from which we demonstrate are common to all the sciences; but the principles with which the sciences are concerned are proper to each science, as number to arithmetic, and magnitude to geometry. But the common principles must be applied to these proper principles, if there is to be demonstration. And because one does not demonstrate only from common principles, it cannot be said that the principles of all demonstrative syllogisms are the same, which is what he intended to prove.

b30. Scientific knowledge and its object differ from opinion and the object of opinion

b31. in that scientific knowledge is commensurately universal and proceeds by necessary connexions, and that which is necessary cannot be otherwise.

b33. So though there are things which are true and real and yet can be otherwise, scientific knowledge clearly does not concern them: if it did, things which can be otherwise would be incapable of being otherwise. Nor are they any concern of rational intuition — by rational intuition I mean an originative source of scientific knowledge — nor of indemonstrable knowledge, which is the grasping of the immediate premise. Since then rational intuition, science, and opinion, and what is revealed by these terms, are the only things that can be ‘true’, it follows that it is opinion that is concerned with that which may be true or false, and can be otherwise: opinion in fact is the grasp of a premise which is immediate but not necessary.

a5. This view also fits the observed facts, for opinion is unstable, and so is the kind of being we have described as its object.

a7. Besides, when a man thinks a truth incapable of being otherwise he always thinks that he knows it, never that he opines it. He thinks that he opines when he thinks that a connexion, though actually so, may quite easily be otherwise; for he believes that such is the proper object of opinion, while the necessary is the object of knowledge.

a11. In what sense, then, can the same thing be the object of both opinion and knowledge? And if any one chooses to maintain that all that he knows he can also opine, why should not opinion be knowledge? For he that knows and he that opines will follow the same train of thought through the same middle terms until the immediate premises are reached; because it is possible to opine not only the fact but also the reasoned fact, and the reason is the middle term;

a16. so that, since the former knows, he that opines also has knowledge. The truth perhaps is that if a man grasp truths that cannot be other than they are, in the way in which he grasps the definitions through which demonstrations take place, he will have not opinion but knowledge: if on the other hand he apprehends these attributes as inhering in their subjects, but not in virtue of the subjects’ substance and essential nature possesses opinion and not genuine knowledge; and his opinion, if obtained through immediate premises, will be both of the fact and of the reasoned fact; if not so obtained, of the fact alone.

a23. The object of opinion and knowledge is not quite identical; it is only in a sense identical, just as the object of true and false opinion is in a sense identical. The sense in which some maintain that true and false opinion can have the same object leads them to embrace many strange doctrines, particularly the doctrine that what a man opines falsely he does not opine at all. There are really many senses of ‘identical’, and in one sense the object of true and false opinion can be the same, in another it cannot. Thus, to have a true opinion that the diagonal is commensurate with the side would be absurd: but because the diagonal with which they are both concerned is the same, the two opinions have objects so far the same: on the other hand, as regards their essential definable nature these objects differ. The identity of the objects of knowledge and opinion is similar. Knowledge is the apprehension of, e.g., the attribute ‘animal’ as incapable of being otherwise, opinion the apprehension of ‘animal’ as capable of being otherwise — e.g., the apprehension that animal is an element in the essential nature of man is knowledge; the apprehension of animal as predicable of man but not as an element in man’s essential nature is opinion: man is the subject in both judgements, but the mode of inherence differs. This also shows that one cannot opine and know the same thing simultaneously; for then one would apprehend the same thing as both capable and incapable of being otherwise — an impossibility. Knowledge and opinion of the same thing can co-exist in two different people in the sense we have explained, but not simultaneously in the same person. That would involve a man’s simultaneously apprehending, e.g., (1) that man is essentially animal — that is, cannot be other than animal — and (2) that man is not essentially animal, that is, we may assume, may be other than animal.

Further consideration of modes of thinking and their distribution under the heads of discursive thought, intuition, science, art, practical wisdom, and metaphysical thinking, belongs rather partly to natural science, partly to moral philosophy.

b6. Quick wit is a faculty of hitting upon the middle term instantaneously. It would be exemplified by a man who saw that the moon has her bright side always turned towards the sun, and quickly grasped the cause of this, namely that she borrows her light from him; or observed somebody in conversation with a man of wealth and divined that he was borrowing money, or that the friendship of these people sprang from a common enmity. In all these instances he has seen the major and minor terms and then grasped the causes, the middle terms. Let A represent ‘bright side turned sunward’, B ‘lighted from the sun’, C the moon. Then B, ‘lighted from the sun’ is predicable of C, the moon, and A, ‘having her bright side towards the source of her light’, is predicable of B. So A is predicable of C through B.

Scientific knowledge and its object. After comparing sciences with one another from the viewpoints of certitude, of unity and of diversity, the Philosopher here compares science to everything else that pertains to knowledge.

And his treatment falls into two parts.

In the first he treats of the comparison of science to opinion, which bears on the true and false.

In the second, of the comparison of science to those habits of knowledge which are concerned exclusively with the true, at: further consideration of modes (90a23; [405]).

The first part is divided into two parts.

In the first he determines the truth.

In the second he excludes a doubt, at: in what sense, then (90a11; [402]).

Concerning the first he does three things.

First, he lays down the difference between science and opinion.

Second, he shows what pertains to science, at: in that scientific knowledge (89b31; [396]).

Third, what pertains to opinion, at: so though there are things (89b33; [397]).

He says therefore first (89b30) that science differs from opinion; in like manner the scientifically knowable, which is the object of science, differs from the opinable, which is the object of opinion.

Then he shows what pertains to science, setting down two things which pertain to it, at: in that scientific knowledge (89b31).

One of these is that it is of the universal, for science is not concerned with singulars which fall under sense. (He explained this above). The other is that science is obtained in virtue of necessary things. And he explains that the necessary is that which cannot be otherwise. This, too, was explained above, namely, that demonstration proceeds from necessary things.

Then he shows what pertains to opinion, namely, that it is concerned with things that could be otherwise, either universally or particularly, at: so though there are things (89b33).

This he proves in three ways.

First, by way of a division, saying that in addition to necessary truths, which cannot be otherwise, there are non-necessary truths, which could be otherwise. Now, from what has been said above, it is clear that science is not concerned with the latter, because it would follow that contingent things would be such as could not be otherwise. For science is concerned with such things, as has been said. Similarly, it cannot be said that understanding is concemed with them. “Understanding” is taken here not as a faculty [power] of the soul, but as being the principle of science, that is, as the habit of first principles, from which demonstration proceeds in causing science. Hence in explaining what that understanding is which is the principle of science he adds, “Nor of indemonstrable knowledge” (88b37), that is, not of facts that could be otherwise. As if to say: understanding is nothing more than a science of the indemonstrable. For just as science implies certainty of knowledge acquired through demonstration, so understanding implies certitude of knowledge without demonstration; not because of a shortcoming in some demonstration, but because that about which this certainty is had, is indemonstrable and per se known. Therefore, to explain this he adds that demonstrative science is nothing else than the certain intuition of an immediate proposition. But that understanding is science of the indemonstrable is clear from the fact that he says that it is "the principle of science". For since science is concerned with the necessary, and the necessary is not concluded except from the necessary, as has been proved above, it is required that understanding, which is the principle of science, not be concerned with contingent things.

Having shown, therefore, that neither science nor understanding is concerned with contingent things, he sets down a division and says that understanding and science and opinion happen to be true as that does which is stated by them, that is, which is verbally stated by understanding and science and true opinion. For the true is found both in the intellect’s act of composing and dividing and in the outward expression of this act insofar as it signifies the inward truth of the opinion or science or understanding. Therefore, if every truth belongs either to understanding or science or opinion, and there are some contingent truths, and they belong neither to science nor understanding, what is left is that opinion is concerned with them, whether they be actually true or actually false, provided that they could be other than they are.

In order to elucidate what opinion is, he adds that opinion is the acceptance, that is, the grasping, of a proposition that is immediate and not necessary. And this can be understood in two ways: in one way, so that the immediate proposition in itself is indeed necessary, but it is accepted by opinion as non-necessary; in another way, so that it is in itself contingent. For an immediate proposition is one that cannot be proved through a middle, whether it be a necessary proposition or not. For it has been shown above that there is no process to infinity in predications, neither on the part of the middles nor on the part of the extremes. And this was shown not only analytically in demonstrations, but also logically in general as to syllogisms. Therefore, if there be a contingent proposition which is mediate, it must be reduced to others that are immediate. However, it is not reduced to necessary immediate propositions, because necessary things are not the proper principles of contingent things, nor can contingent things be concluded from necessary things. Hence what remains is that some contingent, propositions are immediate and some mediate. Thus, “The man does not run,” is mediate, for it can be proved through this middle, “The man is not in motion,” which is also contingent, albeit immediate. The acceptance of such immediate contingent propositions is opinion. Yet this does not mean that the accepting of a mediate contingent proposition is not opinion. For opinion is related to contingent things, as science and understanding to necessary things.

Second, he proves the same point by means of that which is commonly observed, at: this view also fits (90a5).

He says that what has been said, mainly, that opinion is concerned with the contingent is confessed, that is, in agreement with things that are observed. For opinion seems to denote something weak and uncertain and to be of such a nature that weakness and uncertainty are in it.

Third, he proves the same point by experiment, at: besides, when a man (90a7).

For no one, when he thinks that it is impossible for a thing to be otherwise, supposes that he is forming an opinion; rather he considers himself to know scientifically. But when he thinks that something is so and that there is nothing to prevent its being otherwise, then it is that he considers himself to have opinions. This would indicate that opinion concerns such things, that is, contingent things, but science necessary things.

Then he raises a doubt concerning the aforesaid, at: in what sense, then (90a11).

First, he raises the doubts.

Second, he solves them, at: so that, since the former (90a16; [403]).

Concerning the first (90a11), he presents two doubts.

One of these is concerned with the opinable and the scientifically knowable. If opinion is concerned with the contingent and science with the necessary, and the contingent is not the same as the necessary, a doubt remains as to how a man can opine and scientifically know the same thing.

The second doubt concerns science and opinion, namely, why opinion is not science; supposing, of course, that one claims that there can be opinion about every known thing. For in regard to everything known a man can opine that it could be otherwise (except perhaps in the case of first principles per se known, whose contraries cannot be conceived, and concerning which there is not science but understanding). But in regard to all mediate truths of which there is demonstration and science, someone can suppose that it is possible that they could be otherwise. Consequently, they can fall under opinion. For opinion is not concerned exclusively with things contingent in their very nature, because then a man could not have opinion about everything he knows. Rather, opinion is concerned with that which is accepted as possible to be otherwise, whether it is that or not.) Therefore, on this supposition, it remains that science and opinion are the same, because both the scientific knower and the man of opinion acquire science and opinion through middles until they reach things which are immediate, as is clear from the foregoing. Hence if someone proceeds through middles to immediate propositions he has science. Now, this is what the man of opinion does, because just as it is possible to have an opinion 'that' something is so, it is also possible to have an opinion 'why' something is so. But when I say, 'why', a middle is implied. Hence it is clear that opinion can proceed to immediates through a middle, whether opinion be of things that are contingent in their very nature, or of things accepted as contingent.

Then he solves the aforesaid doubts, at: so that, since the former (90a16).

First, he solves the second one which is concerned with the identity of science and opinion.

Second, he solves the first one which is concerned with the identity of the scientifically knowable and the opinable, at: the object of opinion (90a23; [404]).

He says therefore first (90a16) that if someone proceeds through middles to immediates in such a way that the middles are not considered capable of being otherwise, but are considered to behave as definitions which are the middles through which demonstrations proceed, there will not be opinion but science. But if someone proceeds to the immediates through certain true middles which, nevertheless, are either not verified of the things of which they are said per se (as definitions which are predicated essentially or signify the species of a thing)—or he does not take them as being in these things per se, then he will have opinion and will not truly know the 'quia' and the 'propter quid' at once, even if he proceeds as far as the immediates: for then he will be forming an opinion through the immediates and will not know scientifically. But if he does not proceed through immediates but through mediates, then he will not be forming an opinion bearing on the 'propter quid' but only the 'quia'. For even scientific knowledge, when it is not propter quid and immediate, is not 'propter quid' science but 'quia'.

Then he solves the first doubt which concerns the identity of the scientifically known and the object of opinion, at: the object of opinion (90a23).

Here it should be noted first of all that it is not unfitting that what is scientifically known by one person be a matter of opinion for another, because what one takes as impossible to be otherwise than stated and as scientifically known, another might take as possible to be otherwise and as a matter for opinion. But to say that the same man has opinion and scientific knowledge of the same thing at the same time, is not entirely true, but only in a way. For just as there can be true and false opinion about the same thing somehow but not absolutely, so also in regard to opinion and scientific knowledge. For if someone were to say that there could be a completely true and completely false opinion about the same thing, as some do say (for example, those who said that whatever seems to be true to a person is true, as it is stated in Metaphysics IV), then a person who would say such a thing would be forced to admit certain absurdities and the many other things (pointed out in Metaphysics IV which follow from stating that the same thing is true and false.) In particular this absurdity would follow, namely, that no opinion is false and that what he opined to be false he did not opine. Hence it cannot be said that there is true and false opinion of the same thing absolutely.

But because “same” is said in many ways, it happens that somehow there is true and false opinion about a same thing, and somehow not. For if “a 'same' thing about which there are opinions” means the same real extramental subject, then there can be true and false opinion about a same thing. But if 'same' refers to the proposition that is opined, then it is impossible. For example, to opine that the diagonal is commensurable with the side of a square is to opine with a false opinion; and it is absurd to say that this statement can be opined as a true opinion. But if we take diagonal as the enunciable subject, then there can be true and false opinion of a same thing. For in regard to a diagonal, one opines truly that it is really incommensurable, and another opines falsely that it is commensurable. Thus, it is clear that although that concerning which there is science and opinion be the same in subject, it is not the same in notion. For the diagonal which is said to be commensurable and not to be commensurable is the same in subject, because the subject of each statement is the same. Nevertheless, it is clear that there is a mental distinction between diagonal insofar as it is taken as commensurable and insofar as it is taken as incommensurable. Consequently, there can be true and false opinion about the same subject, but not of the same subject according to notion.

It is similar in the case of science and opinion. For it is science when someone knows this to be an animal in such a way that it cannot not be an animal; but it is opinion when someone knows it to be an animal in such a way that it might not be an animal. For example, if there is science concerning the fact that man is truly that which is an animal in such a way that it is impossible for it to be otherwise, there will be opinion concerning the fact that man is not truly that which is an animal, and as a result could be otherwise. For it is clear that the subject both known and opined is the same, namely, man; but in notion it is not the same.

Thus, therefore, from the foregoing it is obvious that it is impossible to know scientifically and to opine the same thing at the same time, because a man would at one and the same time have a conjecture that could be otherwise and that could not be otherwise. But this could occur in the case of two men, namely, that one have science about a thing and another have opinion about the same thing, as has been stated. But this does not occur in the same person for the reason given.

Then, at: further consideration (90a23) he compares science to certain habits that are concerned with truth.

First, to those habits which deal with principles and conclusions.

Second, to the habit which is specifically concerned with the middle, at: quick wit is a faculty (90b6; [406]).

He says therefore first (90a23) that in regard to those forms of knowledge that are other than opinion, the question of how they are distinguished into reason and understanding and science and art and prudence and wisdom, pertains somehow to the considerations of first philosophy or even of natural philosophy and somehow to the considerations of moral philosophy which is called ethics.

To understand this it should be noted that in Ethics VI, Aristotle lays down five things which are always concerned with the truth, namely, art, science, wisdom, prudence and understanding, adding two that are concerned with the true and false, namely, suspicion and opinion. Now, the first five are concerned solely with the true, because they imply rectitude of reason. But three of them, namely, wisdom, science and understanding imply reason’s rectitude in regard to necessary truths: science in regard to conclusions, understanding in regard to principles, and wisdom in regard to the highest causes, which are the divine causes. But the other two, namely, art and prudence, imply the rectitude of reason in regard to contingent things: prudence in regard to things to be done, that is, in regard to acts which remain within the one acting, such as to love, hate, choose, and similar acts that pertain to moral activity, of which prudence is the director. Art, however, implies the rectitude of reason in regard to things to be made, that is, things that are done to external matter; for example, cutting and other activities in which art directs. To this list he adds “reason,” which pertains to bringing, principles to their conclusions. Now, to treat of wisdom, namely, what it is and how it works, and to treat of science and understanding and art, pertain somehow to first philosophy. Prudence, on the other hand, pertains to moral consideration; understanding and reason, insofar as they name certain powers, pertain to natural consideration, as is clear in On the Soul.

Then he makes mention of the habit which is specifically concerned with the middle, at: quick wit is a faculty (90b6).

And he says that quick wit is a certain accurate and facile conjecturing of the middle (the reason) why something occurs; and this when one does not have much time to inspect or deliberate, as when a person, seeing that the moon has all its brightness whenever it is turned toward the sun, immediately understands why this is, namely, because it is illumined by the sun. Similarly, in the case of human acts: if a person sees a poor man arguing with a rich man, he knows that the rich man took something from him, and the argument is about its return. Or if he should see persons formerly enemies now made friends, he knows that the reason is because both have a common enemy. Second, he shows in two ways that to know the middle is to know the 'why': first, with the following reason, namely, because such a quick witted person, seeing all the middle causes, knows all the ultimate ones into which they are all finally analyzed, and through which the 'why' is known.

Second, he manifests this with an example presented in a syllogism. Thus, let us suppose the moon to be C, that is, the minor extreme, and “to be bright because of its opposition to the sun,” to be A. the major extreme; but “to be illumined by the sun,” to be B, that is, the middle. For C is B. that is, “The moon has light from the sun,” and A is in B, because “that which has light from the sun is bright when turned toward the sun.” Thus it is proved that A is in C through B. Hence it is clear that quick wit is the faculty of quickly apprehending the middle, and is present in persons as a natural gift or is acquired by training. But he presented various examples of quick wit to show that in all the aforesaid habits, namely, prudence, wisdom, and so on, there can be quick wit.

Each of the four questions which pertain to science is one way or another a question of the middle

b21. The kinds of question we ask are as many as the kinds of things which we know. They are in fact four: (1) whether the connexion of an attribute with a thing is a fact, (2) what is the reason of the connexion, (3) whether a thing exists, (4) What is the nature of the thing.

b26. Thus, when our question concerns a complex of thing and attribute and we ask whether the thing is thus or otherwise qualified — whether, e.g., the sun suffers eclipse or not — then we are asking as to the fact of a connexion. That our inquiry ceases with the discovery that the sun does suffer eclipse is an indication of this; and if we know from the start that the sun suffers eclipse, we do not inquire whether it does so or not.

b28. On the other hand, when we know the fact we ask the reason; as, for example, when we know that the sun is being eclipsed and that an earthquake is in progress, it is the reason of eclipse or earthquake into which we inquire. Where a complex is concerned, then, those are the two questions we ask;

b32. but for some objects of inquiry we have a different kind of question to ask, such as whether there is or is not a centaur or a God. (By ‘is or is not’ I mean ‘is or is not, without further qualification’; as opposed to ‘is or is not [e.g.] white’.) On the other hand, when we have ascertained the thing’s existence, we inquire as to its nature, asking, for instance, ‘what, then, is God?’ or ‘what is man?’. These, then, are the four kinds of question we ask, and it is in the answers to these questions that our knowledge consists.

b38. Now, when we ask whether a connexion is a fact, or whether a thing without qualification is, we are really asking whether the connexion or the thing has a ‘middle’; and when we have ascertained either that the connexion is a fact or that the thing is — that is, ascertained either the partial or the unqualified being of the thing-and are proceeding to ask the reason of the connexion or the nature of the thing, then we are asking what the ‘middle’ is.

a2. (By distinguishing the fact of the connexion and the existence of the thing as respectively the partial and the unqualified being of the thing, I mean that if we ask ‘does the moon suffer eclipse?’, or ‘does the moon wax?’, the question concerns a part of the thing’s being; for what we are asking in such questions is whether a thing is this or that, that is, has or has not this or that attribute: whereas, if we ask whether the moon or night exists, the question concerns the unqualified being of a thing.)

a5. We conclude that in all our inquiries we are asking either whether there is a ‘middle’ or what the ‘middle’ is: for the ‘middle’ here is precisely the cause, and it is the cause that we seek in all our inquiries. Thus, ‘Does the moon suffer eclipse?’ means ‘Is there or is there not a cause producing eclipse of the moon?’, and when we have learnt that there is, our next question is, ‘What, then, is this cause? for the cause through which a thing is — not is this or that, that is, has this or that attribute, but without qualification is — and the cause through which it is — not is without qualification, but is this or that as having some essential attribute or some accident — are both alike the middle’. By that which is without qualification I mean the subject, e.g., moon or earth or sun or triangle; by that which a subject is (in the partial sense) I mean a property, e.g., eclipse, equality or inequality, interposition or non-interposition. For in all these examples it is clear that the nature of the thing and the reason of the fact are identical: the question ‘What is eclipse?’ and its answer ‘The privation of the moon’s light by the interposition of the earth’ are identical with the question ‘What is the reason of eclipse?’ or ‘Why does the moon suffer eclipse?’ and the reply ‘Because of the failure of light through the earth’s shutting it out’. Again, for ‘What is a concord? A commensurate numerical ratio of a high and a low note’, we may substitute ‘What ratio makes a high and a low note concordant? Their relation according to a commensurate numerical ratio.’ ‘Are the high and the low note concordant?’ is equivalent to ‘Is their ratio commensurate?’; and when we find that it is commensurate, we ask ‘What, then, is their ratio?’.

Cases in which the ‘middle’ is sensible show that the object of our inquiry is always the ‘middle’: we inquire, because we have not perceived it, whether there is or is not a ‘middle’ causing, e.g., an eclipse. On the other hand, if we were on the moon we should not be inquiring either as to the fact or the reason, but both fact and reason would be obvious simultaneously. For the act of perception would have enabled us to know the universal too; since, the present fact of an eclipse being evident, perception would then at the same time give us the present fact of the earth’s screening the sun’s light, and from this would arise the universal. Thus, as we maintain, to know a thing’s nature is to know the reason why it is; and this is equally true of things in so far as they are said without qualification to he as opposed to being possessed of some attribute, and in so far as they are said to be possessed of some attribute such as equal to right angles, or greater or less.

It is clear, then, that all questions are a search for a ‘middle’.

The kinds of question. After determining about the demonstrative syllogism in the first book, the Philosopher intends in this [second] book to treat concerning its principles.

But there are two principles of the demonstrative syllogism, namely, its middle and the first indemonstrable propositions. Therefore, this book is divided into two parts.

In the first he determines concerning the knowledge of the middle in demonstrations.

In the second concerning the knowledge of the first propositions, at: as to the basic premises (100b18; [582]). For since it has been established in the first book that every doctrine and every discipline takes its start from preexisting knowledge, and since in demonstrations the knowledge of the conclusion is acquired through some middle and through the first indemonstrable propositions, we are left with the task of investigating how these come to be known.

But the first part is divided into two parts.

In the first he inquires what is a middle in demonstrations.

In the second part he inquires how that middle is made known to us, at: let us now state how essential nature (91a36; [418]).

But because the middle in demonstrations is employed in order to make known something about which there might have been doubt or question, therefore in regard to the first he does two things.

First, he lays down the number of questions.

Second, from these questions he pursues his investigation by showing how the questions pertain to the middle of demonstrations, at: now, when we ask (90b38; [412]).

Regarding the first he does three things.

First, he enumerates the questions.

Second, he explains complex questions, at: thus, when our question (90b26; [409]).

Third, simple questions, at: but for some objects (90b32; [411]).

He says therefore first (90b21) that the number of questions is equal to the number of things that are scientifically known. The reason for this is that science is knowledge acquired through demonstration. But things which we previously did not know are those of which we must seek knowledge by demonstration: for it is in regard to things which we do not know that we form questions. Hence it follows that the things we inquire about are equal in number to the things we know through science. But there are four things that we ask, namely, 'quia' [that is, is it a fact that], 'propter quid' [that is, why, or what is the cause or reason], 'si est' [if it is, that is, whether it is], 'quid est' [what is it]. To these four can be reduced whatever is scientifically inquirable or knowable. However in Topics I he divides questions or problems into four kinds in a different way, but all of them are included under one of the questions listed here, namely, the one called 'quia'. For in the Topics he is concerned only with questions to be disputed dialectically.

Then, at: thus, when our question (90b26) he clarifies the questions he laid down; and first of all the complex ones.

To understand this it should be noted that science bears only on the true, and the true is not signified except by an enunciation; therefore, only the enunciation can be scientifically knowable and so inquirable. But, as it is stated in On Interpretation II, the enunciation is formed in two ways: in one way from a name and a verb without an appositive, as when it is stated that man is; in another way when some third item is set adjacent, as when it is stated that man is white. Therefore, the questions we form can be reduced either to the first type of enunciation so that we get, as it were, a simple question; or to the second type, and then the question will be, as it were, complex or put in number, because, namely, the question concerns the putting together of two items.

According to this latter type a twofold question can be formed.

One of them is whether this is true which is stated. This question he expounds first, saying that when we ask concerning some thing whether that thing is this or that, so that in effect we are somehow putting it in number, namely, by taking two things one of which is predicate and the other subject, as when we ask if the sun is failing because of an eclipse or not, and is man an animal or not, then we are said to ask 'quia': not in the sense that the word 'quia' functions as a question mark, but because we are asking in order to find out 'quia' [that is, that] it is so. An indication of this is that when we have discovered it through demonstration, we cease our questioning; and if we had known it at the very beginning, we would not have asked whether it is so. But inquiry does not cease until that is obtained which was asked. And so, since the question in which we ask whether this is this ceases once we have certified that it is so, it is clear what a question of this kind asks.

Then he clarifies the next question which also puts in number, ‘I saying that when we know that it is so, we ask propter quid [that is, why] it is so, at: on the other hand, when we know (90b28).

For example, when we know 'that' the sun is failing through an eclipse and that the earth is moved during an earthquake, we ask 'why' the sun is failing or why the earth is being moved. Therefore, we ask it in this way, namely, by putting in number.

Then he clarifies the other two questions which do not put in number but are simple, at: but for some objects (90b32).

And he says that we ask certain things in a manner different from the aforesaid questions, namely, by not putting in number; as when we ask whether or not there be centaurs, for in this case the question we ask concerning the centaur is simply whether it exists and not whether the centaur be this—say white—or not. And just as when we knew that this is this, we then asked why, so once we know of something simply that it is, we ask what it is, for example, "what is God" or "what is man".

These, then, and so many are the things we ask; and when we have found the answer, we are said to know scientifically.

Then, at: now, when we ask (90b38) he shows how the aforesaid questions are related to the middle.

Concerning this he does three things.

First, he states what he intends.

Second, he explains what he had said, at: by distinguishing the fact (91a2; [413]).

Third, he proves his proposition, at: we conclude that in all our inquiries (91a5; [414]).

In regard to the first it should be noted that two of the aforesaid questions put in number and two do not. From the first member of each of these groups, he forms another grouping composed of the question 'that it is' and the question 'if it is'.

And he says that when we ask that this is this, or when we simply ask concerning something, if it is, we are not asking anything else than whether or not a middle is to be found of that which we ask.

This is something not conveyed by the form of the question. For when I ask "whether the sun is eclipsed" or" whether man exists", it is not obvious from the form of the question that I am asking whether there is some middle by which it might be demonstrated that the sun is eclipsed or that man exists; but if the sun is eclipsed or man does exist, the consequence is that some middle can be found to demonstrate the things which are inquired. For no one forms a question concerning immediate things which, although they are true, do not have a middle, since things of this sort, being evident, do not fall under a question. Thus, therefore, one who asks whether this is this or whether this absolutely is, as a consequence is asking whether there is a middle of this sort. For in the question 'if it is' or 'that it is', one is asking whether that which is a middle exists, because that which is the middle is the reason of that concerning which one asks whether this is this or simply whether it is, as will be explained below. Nevertheless the question is not being asked under the aspect of middle.

Now it happens that having found the answer to what is asked by these two questions, one knows either 'that it is' or 'if it is': one of which consists in knowing the existence absolutely; but the other in part, as when we know that man is white, because to be white does not signify the existence of man in his entirety but signifies him to be something. This is why when a man is becoming white, we do not say that he is coming to be absolutely, but in a qualified sense. But when it is asserted that "man is", his existence is signified absolutely, so that when a man comes to be, he is said to become absolutely. Therefore, when we know 'that it is' and ask 'why it is', or when we know 'if it is' and ask 'what it is', we are asking what is the middle.

And as in the other cases, so here, this is gathered not from the form of the question but by way of concomitance. For one who seeks the cause why the sun is eclipsed is not seeking it as a middle which demonstrates, but he is seeking that which is a middle, because it is by way of consequence that once he has it, he can demonstrate. And the same applies to the question 'what is it'.

Then, at: by distinguishing the fact (91a2) he manifests what he had said, namely, that 'that it is' and 'if it is' differ as in part and absolutely differ. For when we inquire whether the moon is waning or waxing, it is a question in part, since in a question of this type we are asking if the moon is something, namely, is it waning or has it waxed or not. But when we ask whether the moon exists or whether it is night, the question bears on existence absolutely.

Then he proves his point, namely, that the aforesaid questions pertain to the middle, at: we conclude that in all (91a5).

First, he proves it with a reason.

Second, with a sign, at: cases in which the ‘middle’ (91a5; [416]).

He concludes therefore first (91a5) from the explanation given above that in all the aforesaid questions one is either asking whether there is a middle, namely, in the question 'that it is' and in the question 'if it is', or what the middle is, namely, in the question 'why' and in the question 'what is it'. And he proves that the question 'why' inquires what the middle is. For it is obvious that a cause is the middle in a demonstration which enables one to know scientifically, because to know scientifically is to know the cause of a thing. But it is precisely the cause that is being sought in all the above questions. That this is so he manifests first in regard to the question 'that'. For when it is asked whether the moon is waning, then according to the manner explained above, what is being asked is 'whether' or not something is the cause of this waning. Then he shows this for the question 'why'. For once we know 'that' something is the cause of the moon’s waning, we ask what the cause is; and this is to inquire 'why'. The same applies to the other two questions, as he shows in the following way. For he says that whether we assert not that something is this or that (for example, when I say that "man is white" or is a "grammarian"), but that the substance itself exists absolutely; or do not assert that some thing exists absolutely, but that some thing is something by putting in number (whether that something be a thing predicated per se or a thing predicated per accidens); no matter in which of these ways we take the thing to be, its cause is the middle for demonstrating it.

Then he explains what he means by a substance to exist absolutely when we inquire concerning the moon or the earth or a triangle or any other subject whether it is and then take some middle to demonstrate this. I say that a thing is something when we inquire concerning eclipse in regard to the moon, or inequality in regard to triangle, or whether it is in the middle of the universe or not in regard to the earth.

And he asserts that as far as the present point is concerned it makes no difference which way a thing is taken to be, because in all these cases 'what' it is is the same as 'why'. He manifests this first of all in regard to the waning of the moon. For if one asks what is the eclipse of the moon, the answer is that it is the absence of light in the moon because of the earth’s being set between it and the sun. And this same answer is given when it is inquired why the moon is eclipsed. For we say that the moon is eclipsed because there is a lack of light due to the earth’s opposition. Then he manifests the same idea with another example. For if one asks what is a chord, the answer is given that it is a numerical ratio according to high and low notes. Again, if one asks why a high note and a low note are concordant, the answer is given that it is because the high note and the low note have a numerical ratio. And so the question 'what is it' and the question 'why' reduce to the same thing subjectively, although they differ in formality. Hence because the question 'why' leads us to inquire what the middle is, as has been shown, what is left is that when one asks 'what is it', the middle is likewise inquired.

Then he shows the same thing in regard to the question 'that'. For, as has been said, concordance is a numerical ratio between high and low notes. Men, therefore, one asks whether a high and low note concord, he is inquiring whether there is some numerical ratio of the high and low note; and this is the middle for demonstrating that a high and a low note concord. Consequently in the question 'that', one inquires whether there is a middle. But once we have found that there is a numerical ratio of the high and low note, we then ask what that ratio is: and this is to ask 'what' or 'why'.

Here Aristotle seems to say that the definition of a proper attribute is the middle in demonstration.

However it must be remarked that the definition of the proper attribute cannot be completed without the definition of the subject. For it is obvious that the principles which the definition of the subject contains are the principles of the proper attribute. Hence a demonstration will not reach the first cause unless one takes as the middle of demonstration the definition of the subject. And so the proper attribute must be concluded of the subject by means of the definition of the proper attribute; furthermore, the definition of the proper attribute must be concluded of the subject through the definition of the subject. This is why it was laid down at the very beginning that one must know beforehand the what is it not only of the proper attribute but of the subject also; which would not be required unless the definition of the proper attribute were concluded of the subject through the definition of the subject.

This is clear from the following example: If we wish to demonstrate of triangle that it has three angles equal to two right angles, we first take as middle the fact that it is a figure having an exterior angle equal to its two opposite interior angles-which is, as it were, the definition of a proper attribute. But this in turn must be demonstrated by the definition of the subject, so that we would say: “Every closed figure of three straight lines has an exterior angle equal to its two opposite interior angles; but the triangle is such a figure. Therefore...”

And the same is true if we were to demonstrate that a high note concords with a low note: for we would state the definition of the attribute which in this case consists in their having a numerical ratio; but then to demonstrate this attribute we would have to take the definition of high note and of low note. For a low note is one which is apt to act on the sense for a long time, whereas a high note is one which does so for a short time: but between the long and the short there is a numerical ratio. Therefore, there is a numerical ratio between a high note and a low note. And if the high note and the low note were defined some other way, it would make no difference. For in any case something pertaining to quantity would have to appear in their definition, so that it would be necessary to conclude that there is a numerical ratio between them.

Then, at: cases in which the ‘middle’ (91a5) he manifests his point by means of a sensible sign.

And he says that those cases in which the middle is perceptible by sense clearly show that every question is a question concerning a middle, because, namely, when the middle is known through the senses, no room is left for a question. For we ask one of the aforesaid questions in matters pertaining to sense, when the middle does not appear: thus, we ask whether or not there is an eclipse of the moon because we do not sensibly perceive the middle which is the cause making the moon to be eclipsed. But if we were to situate ourselves in a place above the moon, we would see how the moon became eclipsed by entering the earth’s shadow. Then we would no longer ask if it is or why it is, but both would at once be obvious to us.

But because someone could object that sense-perception bears on singulars—whereas it is universals that are being asked about, just as it is universals that are scientifically known—and consequently, it does not seem that the matter under question can be made known through sense; therefore, as though in answer to this objection he adds that it is precisely because we do sense the particular (namely, that this body of the moon enters this shadow of the earth), that we happen at once to know the universal. For our sense would observe the fact that the light of the sun is now blocked by the earth’s opposition; (and through this it would be clear to us that the moon is now eclipsed.) And because we would conjecture that the eclipse of the moon always occurs in this way, the sense knowledge of the singular would immediately become a universal in our science.

And so from this example he concludes that knowing the quod quidQuod quid refers to any or all of the items that constitute the essential nature of a thing. and the why are the same. For from the fact that we observe the earth situated between the sun and moon, we would know scientifically both what an eclipse of the moon is and why the moon is eclipsed. And one of these, namely, the knowledge what it is, is reduced to the science by which we know that something simply is, and not that something is in some thing. But the 'why' is reduced to our knowledge of things that are in some thing, as when we say that three angles are equal to or greater than or less than two right angles.

Finally, he summarizes and concludes the main point, namely, that it is clear from the foregoing that in all questions there is question of the middle.

a36. Let us now state how essential nature is revealed and in what way it can be reduced to demonstration; what definition is, and what things are definable. And let us first discuss certain difficulties which these questions raise,

b1. beginning what we have to say with a point most intimately connected with our immediately preceding remarks, namely the doubt that might be felt as to whether or not it is possible to know the same thing in the same relation, both by definition and by demonstration.

b3. It might, I mean, be urged that definition is held to concern essential nature and is in every case universal and affirmative; whereas, on the other hand, some conclusions are negative and some are not universal; e.g., all in the second figure are negative, none in the third are universal.

b7. And again, not even all affirmative conclusions in the first figure are definable, e.g., ‘every triangle has its angles equal to two right angles’. An argument proving this difference between demonstration and definition is that to have scientific knowledge of the demonstrable is identical with possessing a demonstration of it: hence if demonstration of such conclusions as these is possible, there clearly cannot also be definition of them. If there could, one might know such a conclusion also in virtue of its definition without possessing the demonstration of it; for there is nothing to stop our having the one without the other.

b14. Induction too will sufficiently convince us of this difference; for never yet by defining anything—essential attribute or accident—did we get knowledge of it.

b16. Again, if to define is to acquire knowledge of a substance, at any rate such attributes are not substances. It is evident, then, that not everything demonstrable can be defined.

b19. What then? Can everything definable be demonstrated, or not? There is one of our previous arguments which covers this too. Of a single thing qua single there is a single scientific knowledge. Hence, since to know the demonstrable scientifically is to possess the demonstration of it, an impossible consequence will follow:—possession of its definition without its demonstration will give knowledge of the demonstrable.

b23. Moreover, the basic premises of demonstrations are definitions, and it has already been shown that these will be found indemonstrable; either the basic premises will be demonstrable and will depend on prior premises, and the regress will be endless; or the primary truths will be indemonstrable definitions.

b28. But if the definable and the demonstrable are not wholly the same, may they yet be partially the same? Or is that impossible, because there can be no demonstration of the definable?

There can be none, because definition is of the essential nature or being of something, and all demonstrations evidently posit and assume the essential nature — mathematical demonstrations, for example, the nature of unity and the odd, and all the other sciences likewise.

b33. Moreover, every demonstration proves a predicate of a subject as attaching or as not attaching to it, but in definition one thing is not predicated of another; we do not, e.g., predicate animal of biped nor biped of animal, nor yet figure of plane — plane not being figure nor figure plane.

b38. Again, to prove essential nature is not the same as to prove the fact of a connexion. Now, definition reveals essential nature, demonstration reveals that a given attribute attaches or does not attach to a given subject; but different things require different demonstrations — unless the one demonstration is related to the other as part to whole. I add this because if all triangles have been proved to possess angles equal to two right angles, then this attribute has been proved to attach to isosceles; for isosceles is a part of which all triangles constitute the whole. But in the case before us the fact and the essential nature are not so related to one another, since the one is not a part of the other.

So it emerges that not all the definable is demonstrable nor all the demonstrable definable; and we may draw the general conclusion that there is no identical object of which it is possible to possess both a definition and a demonstration. It follows obviously that definition and demonstration are neither identical nor contained either within the other: if they were, their objects would be related either as identical or as whole and part.

So much, then, for the first stage of our problem.

Let us now state how essential nature. After showing that every question is in some sense a question of the middle, which is the 'quod quid' and the 'propter quid', the Philosopher now begins to make manifest how the middle becomes known to us.

And it is divided into two parts.

In the first part he shows how the 'quod quid' and 'propter quid' are related to demonstration.

In the second part he shows how the 'quod quid' and 'propter quid' should be investigated, at: so let us now discuss (97a22; [527]).

The first part is divided into two parts.

In the first he manifests how the 'quid est' is related to demonstration.

In the second he manifests how the 'propter quid' which signifies a cause is related to demonstration, at: we think we have (95a20; [490]).

In regard to the first he does two things.

First, he states his proposal.

Second, he pursues his proposal, at: beginning what we have (91b1; [420]).

He says therefore first (91a36) that since every question for whose solution a demonstration is adduced is a question of the middle which is 'quid' and 'propter quid', our first task is to declare how the 'quod quid' is made known to us, namely, is it through demonstration or division or some other way. Furthermore, we must point out how to reduce to the 'quod quid' those items of a thing that are apparent. And because a definition is a statement which signifies the 'quod quid', it is necessary also to know what a definition is and why certain things are definable. Consequently we shall proceed in these matters in the following way.

First, by raising difficulties.

Second, by establishing the truth (94a1; [470]).

Then (91b1) he pursues his proposal in the order stated, at: beginning what we have.

First, he proceeds dialectically by raising difficulties.

Second, by determining the truth, at: we must now start afresh (94a1; [470]).

In regard to the first he does two things.

First, he proceeds by disputing about the definition which signifies 'quod quid'.

Second, about the 'quod quid' which is signified by the definition, at: the next step is to raise (92a12; [431]).

Concerning the first he does three things.

First, he inquires disputatively whether there is definition of all things of which there is demonstration.

Second, whether conversely there is demonstration of all things of which there is definition, at: what then? (91b19; [425]).

Third, whether there can be definition and demonstration of the same thing, at: but if the definable (91b28; [427]).

In regard to the first he does two things.

First, he states his intention.

Second, he pursues his proposal, at: the doubt that might be felt (91b1).

He says therefore first (91b1) that when a number of items are planned for discussion, one should take his start from the item which is most fruitful in settling the “had,” that is, the subsequent, problems.

This would mean that in our case we should begin with the fact that someone might wonder whether the same thing and according to the same aspect could be known through definition and demonstration.

Then, at: it might, I mean (91b3) he proves that there is not definition of all things of which there is demonstration. This he does in four ways.

The first of these is the following: a definition is indicative of the 'quod quid'; but anything which pertains to the 'quod quid' is predicated both affirmatively and universally: therefore, a definition contains or signifies only those things which are predicated affirmatively and universally. But not all syllogisms demonstrate affirmative universal conclusions: in fact some are negative, as, for example, all those in the second figure; and some are particular, as for example, all those in the third figure. Therefore, there is not definition of all things of which there is demonstration.

Second, at: and again, not even (91b7), he shows the same thing when he says that there cannot even be definition of all things which are concluded through affirmative syllogisms-which occurs only in the first figure, as when it is demonstratively syllogized that every triangle has three angles equal to two right angles. Now, the reason for this statement, namely, that there cannot be definition of all things thus syllogized, is that to know something demonstratively is nothing else than to have a demonstration. From this it follows that if the science of all these things is had only through demonstration, there is not definition of them. For things which have a definition are made known through their definition. Otherwise it would follow that a person who does not have a demonstration of these things would have scientific knowledge of them, for the simple reason that nothing hinders a person who knows the definition from not having at the same time the demonstration-although the definition is a principle of the demonstration. For not everyone who knows the principles knows how to deduce the conclusion by demonstrating.

Third, at: induction too will sufficiently (91b14; [423]), he shows the same thing through an induction from which a conviction can arise sufficient for admitting the above-mentioned conclusion. He says, therefore, that demonstration is concerned with things which are per se in something, as is clear from what has been established in the first book. But a definition never gives anyone a knowledge of those items which are per se in a thing, much less of those which are in it per accidens-not that there cannot be definitions of accidents which are in a thing per se or per accidens, as is stated in Metaphysics VII; but because no one has ever given the definition of that which is to be in a thing per se or per accidens, which the syllogism concludes.

Fourth, at: again, if to define (90b16), he shows the same thing with a reason, namely, that the definition makes the substance known:

first of all, because substance is defined in a primary way, whereas accident is defined secondarily by means of a definition which adds something alien, as is stated in Metaphysics VII;

second, because an accident is not defined save insofar as it is signified after the manner of a substance by employing a noun. But the things upon which demonstrations bear are not substances nor things signified after the manner of substances, but things signified after the manner of accidents, namely, after the manner of something inhering in some thing. Hence he concludes that it is not possible that there be a definition of everything of which there is demonstration.

Then, at: what then? (91b19) he inquires whether conversely there is demonstration of everything of which there is a definition.

And he shows that there is not for two reasons,

the first of which has already been touched upon above. For it seems that of one thing, precisely as it is one, there is one science, that is, one way of knowing it. Hence if that which is demonstrable is truly known scientifically in virtue of the fact that a demonstration is had of it, then something impossible follows, if it can also be scientifically known through a definition: for the one having the definition would be knowing something demonstrable without having the demonstration-which seems absurd. And this reason was the second of the four previously presented.

Then, at: moreover, the basic premises (91b23; [426]), he gives the second reason. For definitions are principles of demonstrations, as was established in Book I; but the principles are not demonstrable—otherwise it would follow that there are principles of principles and that demonstrations would proceed to infinity, which is impossible, as we have shown in the first book. Hence it follows that definitions are not demonstrable, being as they are first principles in demonstrations. Consequently, there is not demonstration of all things of which there is definition.

Then, at: but if the definable (91b28) he inquires whether it is possible for there to be definition and demonstration of the same thing in some cases even though not of all.

And with three reasons he shows that there is not.

The first of these reasons is that a definition is manifestive of the 'quod quid' and of the substance, that is, of the essence, of a thing. Demonstrations, on the other hand, do not manifest this but suppose it: thus in mathematical demonstrations it is supposed from arithmetic what unity is and what odd is; and the same applies in other demonstrations. Therefore, there is not demonstration and definition of the same thing.

Then, at: moreover, every demonstration (91b33) he presents the second reason and it is this:

In the conclusion of a demonstration something is predicated of something either affirmatively or negatively; but in a definition nothing is predicated of anything: thus in the definition that man is a two-footed animal, neither 'animal' is predicated of two-footed nor 'two-footed' of animal. Likewise in the definition that a circle or a triangle is a plane figure, neither plane is predicated of figure nor figure of plane. For if the parts of a definition were to be joined to one another, the resulting predication would have to be understood in a manner which suits a definition, namely, in quod quid. But this we do not observe. For the genus is not predicated quod quid of the difference, nor the difference of the genus. Therefore, there is not definition and demonstration of a same thing.

Then, at: again, to prove essential (91b38) he gives the third reason and says that it is one thing to manifest the 'quod quid' and another the 'quia', as is clear from the difference between the questions listed above; but the definition shows the 'quid' of something, whereas a demonstration shows either affirmatively or negatively that something is or is not so of something. But we see that for different things different demonstrations are required, unless those two different things are related as whole and part: for in that case there would be one and the same demonstration concerning both. Thus, the demonstration that a triangle has three angles equal to two right angles applies also to isosceles, which is related to triangle as part to whole. But that is not the case in these two things, namely, in the 'quia' and in the 'quid'; for neither is a part of the other.

Therefore, it has been established that there is not demonstration of everything of which there is definition, nor conversely. Furthermore, it has been possible from this to conclude that these are not of the same thing, and that definition and demonstration are neither the same nor is one in the other as a subjective part in its whole—otherwise it would be necessary that even those things of which they are be related after the manner of whole and part in such a way, namely, that everything definable would be demonstrable, or vice versa—which was disproved above.

Finally, he summarizes and concludes that so far we have proceeded by opposing.

a12. The next step is to raise the question whether syllogism—that is, demonstration—of the definable nature is possible or, as our recent argument assumed, impossible.

a14. We might argue it impossible on the following grounds: — (a) syllogism proves an attribute of a subject through the middle term; on the other hand (b) its definable nature is both ‘peculiar’ to a subject and predicated of it as belonging to its essence. But in that case (1) the subject, its definition, and the middle term connecting them must be reciprocally predicable of one another;

a18. for if A is to C, obviously A is ‘peculiar’ to B and B to C — in fact all three terms are ‘peculiar’ to one another:

a19. and further (2) if A inheres in the essence of all B and B is predicated universally of all C as belonging to C’s essence, A also must be predicated of C as belonging to its essence.

a22. If one does not take this relation as thus duplicated — if, that is, A is predicated as being of the essence of B, but B is not of the essence of the subjects of which it is predicated — A will not necessarily be predicated of C as belonging to its essence. So both premises will predicate essence, and consequently B also will be predicated of C as its essence.

a25. Since, therefore, both premises do predicate essence—that is, definable form—C’s definable form will appear in the middle term before the conclusion is drawn.

a28. We may generalize by supposing that it is possible to prove the essential nature of man. Let C be man, A man’s essential nature — two-footed animal, or aught else it may be. Then, if we are to syllogize, A must be predicated of all B. But this premise will be mediated by a fresh definition, which consequently will also be the essential nature of man. Therefore, the argument assumes what it has to prove, since B too is the essential nature of man.

a32. It is, however, the case in which there are only the two premises — that is, in which the premises are primary and immediate — which we ought to investigate, because it best illustrates the point under discussion.

a35. Thus they who prove the essential nature of soul or man or anything else through reciprocating terms beg the question. It would be begging the question, for example, to contend that the soul is that which causes its own life, and that what causes its own life is a self-moving number; for one would have to postulate that the soul is a self-moving number in the sense of being identical with it.

b1. For if A is predicable as a mere consequent of B and B of C, A will not on that account be the definable form of C: A will merely be what it was true to say of C. Even if A is predicated of all B inasmuch as B is identical with a species of A, still it will not follow: being an animal is predicated of being a man — since it is true that in all instances to be human is to be animal, just as it is also true that every man is an animal — but not as identical with being man. We conclude, then, that unless one takes both the premises as predicating essence, one cannot infer that A is the definable form and essence of C: but if one does so take them, in assuming B one will have assumed, before drawing the conclusion, what the definable form of C is; so that there has been no inference, for one has begged the question.

The next step is to raise the question. After inquiring disputatively whether the definition signifying 'quod quid' can be demonstrated, the Philosopher continues to inquire disputatively and asks whether the 'quod quid' itself, which is signified by a definition, can be proved demonstratively.

First, he states what he intends.

Second, he pursues his proposal, at: we might argue it impossible (92a14; [433]).

First, therefore (92a12), he raises the question whether there can be a syllogism or demonstration of that which is the 'quod quid', namely, in such a way as to conclude that this is the quod quid of this; or can this not be done, as the immediately preceding reason supposed?

Now it was necessary to undertake this disputation on the heels of the previous one, because in a definition one must be sure not only that what it signifies is the quod quid, but also that it be formulated in such a. way as to be suitable for manifesting the quod quid, that is, in such a way as to be composed of prior and better known items and possess the other characteristics which mark a definition. It is significant that he says, "whether there be syllogism or demonstration", for some of the reasons which follow conclude that there is not demonstration of the 'quod quid', and others that there is no syllogism at all of it.

Then, at: we might argue it impossible (92a14) he proceeds disputatively to show that there is not syllogism, or demonstration of the quod quid.

First, he excludes certain special ways by which it might seem possible to demonstrate the 'quod quid'.

Second, he presents common reasons for this, at: to put it another way (93b3; [460]).

Concerning the first he does three things.

First, he shows that 'quod quid' cannot be demonstrated by using convertible terms.

Second, that it cannot be demonstrated by division, at: nor, as was said in my formal logic (92b12; [443]).

Third, that it cannot be demonstrated by accepting that which is required for 'quod quid', at: can we nevertheless (93a6; [451]).

In regard to the first he does three things.

First, he prefaces certain notions required for showing the proposal.

Second, he induces a reason, at: for if A is to C (92a18; [435]).

Third, he discloses the inconvenience which follows, at: it is, however, the case (92a32; [440]).

With respect to the first (92a12) he presupposes two things:

the first of these pertains to the syllogism, namely, that every syllogism proves something about something by means of some middle, as is clear from the above.

The other pertains to what is to be proved by the syllogism, namely, the 'quod quid', for which two conditions are required.

The first of these is that the 'quod quid' be proper, for each thing has its own proper essence or quiddity.

But because not everything which is proper to something pertains to its essence, a second condition is required, namely, that it be predicated in quid.

Furthermore, a third condition must follow on these two, namely, that the 'quod quid' be convertible with that of which it is.

Then, at: for if A is to C (92a18) he lays down a reason in support of his proposal.

Regarding this he does three things.

First, he states what sort of syllogism would be required for concluding a 'quod quid', if this were possible.

Second, he concludes to the inconvenience that follows from this, at: since, therefore, both premises (92a25; [438]).

Third, he uses terms to exemplify this, at: we may generalize (92a28; [439]).

Concerning the first he does three things.

First, he shows what is required for a syllogism concluding the 'quod quid' so far as it is something proper.

Second, what is required so far as it is predicated in quid, at: and further if A (92a19; [436]).

Third, he shows that without these characteristics such a syllogism would not be possible, at: if one does not take (92a22; [437]).

He says therefore first (92a18) that if A which is to be proved of C as its 'quod quid 'is proper to C (which is required of the 'quod quid', as has been said), it will be necessary that the first, namely, A, be proper to the middle which is B; for if A exceeded [that is, were more universal than] B, which is predicated universally of C, it would follow that A exceeds C even more. In like manner, it is clear that B will have to be proper to C: for if B were to exceed C, it would follow that A, which is predicated universally of B, exceeds C; consequently, it would not be proper to it, as was supposed. What remains, therefore, is that if there is to be a syllogism concluding 'quod quid', the terms will have to be so related that all are mutually convertible.

Then, at: and further if A (92a19) he shows what the syllogism in question must have, if the condition is to be fulfilled that what it concludes be predicated in quod quid.

And he says that such a syllogism must be so formulated that the major extreme, which is A, be predicated in quod quid of the middle, which is B; in like manner, B will have to be predicated in quod quid of the minor extreme, which is C. In this way it can be concluded that A is predicated of C in quod quid.

Then, at: if one does not take (92a22) he shows that the aforesaid way of syllogizing is required.

And he asserts that if one does not so take the terms by thus duplicating, that is, observing the two conditions laid down above, that is, taking the 'quod quid' on the side of both, it does not follow of necessity that A would be predicated of C in quod quid. Indeed, if the aforesaid conditions are observed on the part of one only, it is not sufficient for the purpose intended. For even if A were predicated of B in quod quid, it would not on that account follow that it must be predicated in quod quid of everything of which B might happen to be predicated in some random way. Consequently, it follows that the 'quod quid' must be taken on the side of both in such a way, namely, that not only must A be the 'quod quid' of B, but also that B must be the 'quod quid' of C, as being predicated convertibly and in quod quid.

Then, at: since, therefore, both premises (92a25) he leads to the inconvenience.

If, as has been indicated, not only is there found on the side of both something which is predicated in quod quid-as a genus predicated of a species-but it is further found on the side of both that this something is the 'quod quid erat esse'-which a definition signifies-it follows that the 'quod quid erat esse' was already in the middle term, that is, that the middle term is the 'quod quid erat esse' of the minor extreme. Consequently, we would have been supposing the very thing we had set out to prove, namely, the quiddity of C.Quod quid erat esse refers to the essential nature of a being.

Then, at: we may generalize by supposing (92a28) he clarifies what he has just said, using terms.

Thus, if we wish to show what man is, let C, that is, the minor extreme, be 'man', and A, the major extreme, be the 'quod quid' of man, say 'two-footed animal', or something similar to this. Therefore, if this is what we intend to prove by the syllogism, it is necessary to find the definition of some middle, namely, B, such that A will be predicated of every B. Furthermore, to this middle will pertain some other middle definition which, namely, was the definition of the minor extreme. Hence it will follow that this middle will also be the 'quod quid' of man. Consequently, one who syllogizes in this way would be supposing what he was required to prove, namely, that B is the 'quod quid erat esse' of man.

Then, at: it is, however, the case (92a32) he shows how this inconvenience follows from the premises.

To this end he does three things.

First, he indicates the method by which this may be fittingly done.

And he says that what has been said should be considered in two propositions which are first and which have terms that belong to each other immediately. For it would be possible to prove our point either with several propositions (which would lead to several syllogisms), or with only two propositions, but taking them as mediate. But because in the final analysis we must always arrive at two immediate propositions, then for the sake of brevity and dispatch let us assume such propositions from the very start. Thus we shall be able to manifest our point quite easily.

Second, at: thus they who prove (92a35), he states what he intends.

He says, concluding from the foregoing, that those who would demonstrate through convertible terms the 'quod quid' of some thing (say, what is a soul, or what is man, or anything else of this sort), necessarily commit the fault of begging the question. And he cites as an example Plato’s definition of soul. For since the soul is alive and is the body’s cause of being alive, it follows that it differs from the body in the fact that the body lives in virtue of some cause other than itself, but the soul lives in virtue of itself. But Plato posited that number is the substance of all things: for he failed to distinguish between one, which is converted with being and refers to the substance of that of which it is predicated, and one which is the principle of number. As a consequence, it followed that the soul as to its substance is a number, just as any other thing containing many things in itself. Likewise Plato posited that to live is one of the ways of being moved. For the living are distinguished from the non-living on two scores, namely, sense and movement, as it is stated in On the Soul I. Furthermore, he asserted that sensing or knowing is one way of being moved. Thus, therefore, he said that the soul is a self-moving number in addition to saying that the soul is its own cause of being alive. Consequently, if anyone desired to prove what a soul is, namely, that it is something which is its own cause of being alive, and assumed as middle that the soul is a number which moves itself, it would be necessary to beg this, namely, that the soul is a self-moving number, such that this is identical with the soul as being its 'quod quid'. Otherwise, it would not follow, if something is the 'quod quia' of self-moving number, that it is the 'quod quid' of the soul itself.

Third, at: for if A is predicable (92b1), he proves his proposal, namely, that such a proof involves begging the question.

And he says that it does not follow from the fact that A follows on B and B on C, that the major extreme, A, is the 'quod quid' of the minor extreme, C; what follows is simply that A is in C. And if it is further granted that A is the 'quod quid' of something and is predicated universally of B, it does not yet follow that A is the 'quod quid' of C: for it is plain that this, that is, the 'quod quid' of animal is universally predicated of this which is man’s 'quod quid'. For just as it is true that animal is universally predicated of man, so it is true that the definition of animal is universally predicated of the definition of man; but that does not mean that they are entirely one and the same. And so it is clear that unless one takes the terms in such a way that the first is entirely one and the same with the middle, and the middle with the last, it will not be possible to syllogize that A, which is the first, is the 'quod quid' of C—which is the last—and its essence. But if the terms are taken in that way, it follows that prior to its being concluded, the 'quod quid' of C, namely, B, was assumed in the premises. From this it follows that there is not a demonstration but a begging or assuming of the question.

b12. Nor, as was said in my formal logic, is the method of division a process of inference at all, since at no point does the characterization of the subject follow necessarily from the premising of certain other facts: division demonstrates as little as does induction. For in a genuine demonstration the conclusion must not be put as a question nor depend on a concession, but must follow necessarily from its premises, even if the respondent deny it. The definer asks ‘Is man animal or inanimate?’ and then assumes — he has not inferred — that man is animal. Next, when presented with an exhaustive division of animal into terrestrial and aquatic, he assumes that man is terrestrial. Moreover, that man is the complete formula, terrestrial-animal, does not follow necessarily from the premises: this too is an assumption, and equally an assumption whether the division comprises many differentiae or few. (Indeed as this method of division is used by those who proceed by it, even truths that can be inferred actually fail to appear as such.)

b25. For why should not the whole of this formula be true of man, and yet not exhibit his essential nature or definable form?

b26. Again, what guarantee is there against an unessential addition, or against the omission of the final or of an intermediate determinant of the substantial being?

b28. The champion of division might here urge that though these lapses do occur, yet we can solve that difficulty if all the attributes we assume are constituents of the definable form, and if, postulating the genus, we produce by division the requisite uninterrupted sequence of terms, and omit nothing; and that indeed we cannot fail to fulfill these conditions if what is to be divided falls whole into the division at each stage, and none of it is omitted; and that this — the dividendum — must without further question be (ultimately) incapable of fresh specific division.

b33. Nevertheless, we reply, division does not involve inference; if it gives knowledge, it gives it in another way. Nor is there any absurdity in this: induction, perhaps, is not demonstration any more than is division, et it does make evident some truth. Yet to state a definition reached by division is not to state a conclusion: as, when conclusions are drawn without their appropriate middles, the alleged necessity by which the inference follows from the premises is open to a question as to the reason for it, so definitions reached by division invite the same question. Thus to the question ‘What is the essential nature of man?’ the divider replies ‘Animal, mortal, footed, biped, wingless’; and when at each step he is asked ‘Why?’, he will say, and, as he thinks, proves by division, that all animal is mortal or immortal: but such a formula taken in its entirety is not definition; so that even if division does demonstrate its formula, definition at any rate does not turn out to be a conclusion of inference.

Nor, as was said in my formal logic. After showing that 'quod quid' cannot be demonstrated with convertible terms, the Philosopher now shows that it cannot be demonstrated by the method of division.

And in regard to this he does two things.

First, he shows what he proposes.

Second, he excludes a certain solution, at: the champion of division (92b28; [449]).

Concerning the first he does two things.

First, he shows what he proposes by using a reason common to all things which are syllogized.

Second, he shows it with reasons proper to that which is 'quod quid', at: for why should not (92b25; [447]).

He says therefore first (92b12) that just as the quod quid cannot be demonstrated by convertible terms, so neither by the method of division through which, as a matter of fact, nothing is proved syllogistically, as was established in Resolutions Touching Figures, that is, in Prior Analytics I. For just as in the Posterior Analytics resolution to first principles is taught, so in the Prior Analytics resolution is made to certain first simple items which pertain to the arrangement of the syllogism in mood and figure.

That something cannot be syllogized by the method of division he proves from the fact that in this method, given the premises, the conclusion does not follow of necessity, which it should, considering the nature of a syllogism. Rather, the same thing happens in the method of division as happens in the method of induction. For one who induces through singulars to the universal does not demonstrate or syllogize from necessity. For when something is proved syllogistically, it is not necessary to make further inquiry concerning the conclusion or to ask that the conclusion be conceded; what is necessary is that the conclusion be true, if the premises laid down are true.

But, as he shows with certain examples, this does not happen in the method of division. For the method of division consists in assuming something common which is divided into at least two members, so that, one being removed, the other is concluded. For example, if beings are so divided that on the one hand is animal and on the other lifeless things, then having established that man is not a lifeless thing, it is concluded that he is animal. But this conclusion does not follow, unless the hearer grants that man is either an animal or a lifeless thing.

It might be remarked here that he quite fittingly compared division to induction. For in both cases one is required to suppose that he has listed all the things contained under some general heading; otherwise, the person inducing could not conclude a universal from the singulars he assumed, nor could the person dividing conclude to one member just because the others have been eliminated. Thus it is obvious that one cannot in virtue of the fact that Socrates and Plato and Cicero run, induce of necessity the conclusion that every man runs, unless his audience concedes that nothing more is contained under man than the ones listed. In like manner, one who divides cannot, in virtue of having proved that this colored object is neither white nor grey, conclude of necessity that it is black, unless his audience grants him that nothing else is contained under colored object than the things mentioned in the division.

And because a person investigating what man is must assume not only the genus which is animal but also the difference, he goes on to say in his example that if every animal is either of the land or of the water and it is established that because man is not a water animal, he is this whole which is 'land animal', that statement does not follow of necessity from the premises: rather it is further required that he suppose his audience to have granted him something, namely, that animal is sufficiently divided into those of the land and those of the water.

And because in some cases several divisions are used in obtaining the 'quod quid' of a thing, the fore, after laying down two divisions in his example, he adds that it makes no difference whether few or many are used. For the formality is the same in all.

And thus he concludes once more that those who proceed by the method of division, even in matters that could be syllogized, do not use a syllogized proof.

Then, at: for why should not (92b25) he induces two reasons proper to the 'quod quid'.

The first of these is that not everything which is truly predicated of something is predicated in 'quod quid' or signifies its essence. Therefore, even if it be conceded that one has sufficiently proved by the method of division that this whole, namely, 'land animal', is truly predicated of man, neverthless it does not follow that it is predicated of it in 'quod quid' or that it shows the 'quod quid erat esse', that is, that it demonstrates the essence of the thing.

Then, at: again, what guarantee (92b26) he gives the second reason. For the essence of a thing is declared through certain definite items which may neither be added to or subtracted from. But there is nothing to hinder one who proceeds by the method of division from adding something over and above the items which suffice for showing the 'quod quid', or from subtracting some of the things which are necessary for this, or from going beyond and exceeding the essence of the thing, as when it is more common than the thing itself—which happens when the ultimate differences which contract common things are removed. Hence the 'quod quid' is not sufficiently proved by division. And this is what he concludes, namely, that in the method of division the above-mentioned conditions are not satisfied, that is, the conditions that what is concluded be predicated in 'quod quid' and that it neither exceed nor be exceeded.

Then, at: the champion of division (92b28) he excludes a certain solution.

First, he proposes it.

Second, he excludes it, at: nevertheless, we reply (92b33; [450]).

He says therefore first (92b28) that one might solve our objections by saying that when a division is made, one could be taking all the things that are predicated in quod quid, so that as a consequence of the division he accomplishes what was primarily intended, namely, that he obtains a definition signifying the 'quod quid' and leaves out nothing which is required for defining. And if he does these two things, namely, if all the items he assumes in the division are predicated in quod quid and all such things are included in the division so that nothing is left out, then it is necessary that what is obtained be the 'quod quid'. And the reason for this necessity is that in taking everything which is predicated in quod quid and leaving nothing out, that which is found must be something individual, that is, the individual notion of such a thing, so that no further division is required for it to be appropriate to this thing.

Then, at: nevertheless, we reply (92b33) he excludes this solution.

He says that even though under the aforesaid conditions something individual is necessarily obtained, as explained above, the method is nevertheless not syllogistic; although it might make one know the 'quod quid' in some other way. And this is not unbecoming, namely, that something be manifested in a way other than by a syllogism: for one who uses induction does not prove anything syllogistically, but yet he does manifest something. That one who reaches a definition by way of division does not achieve a syllogism he shows by something similar. For if a conclusion is induced from a major proposition, the second proposition being omitted, and the person concluding declares that this must follow from the premises, the hearer could ask why it is necessary—which is something that does not happen in a syllogistic proof. Hence such a method of arguing is not syllogistic. Similarly, in terms of division no syllogism is achieved, because the question why always remains. Thus, if someone wishing to disclose what man is were to assert by the method of division that man is a two-footed mortal animal, or one that has two feet but no wings, then as he adds one item to another in his division, he could be asked in regard to each one, 'why' is it necessary. For one who sets out to manifest a 'quod quid' by division will not only assert but also prove—in keeping with what he thinks—that everything which exists is mortal or immortal. And although it be granted that through this division he might be demonstrating his proposition, nevertheless it is not necessary that the notion so concluded be a definition; for perhaps the items out of which such a notion is formed are not predicated in quod quid or exceed the substance of the thing defined. But even though such a notion might happen to be a definition, it is nevertheless not proved by the syllogism to be a definition, as is clear from what has been established above.

a6. Can we nevertheless actually demonstrate what a thing essentially and substantially is, but hypothetically, that is, by premising (1) that its definable form is constituted by the ‘peculiar’ attributes of its essential nature; (2) that such and such are the only attributes of its essential nature, and that the complete synthesis of them is peculiar to the thing; and thus — since in this synthesis consists the being of the thing — obtaining our conclusion?

a9. Or is the truth that, since proof must be through the middle term, the definable form is once more assumed in this minor premise too?

a12. Further, just as in syllogizing we do not premise what syllogistic inference is (since the premises from which we conclude must be related as whole and part), so the definable form must not fall within the syllogism but remain outside the premises posited. It is only against a doubt as to its having been a syllogistic inference at all that we have to defend our argument as conforming to the definition of syllogism. It is only when someone doubts whether the conclusion proved is the definable form that we have to defend it as conforming to the definition of definable form which we assumed. Hence syllogistic inference must be possible even without the express statement of what syllogism is or what definable form is.

a20. The following type of hypothetical proof also begs the question. If evil is definable as the divisible, and the definition of a thing’s contrary — if it has one the contrary of the thing’s definition; then, if good is the contrary of evil and the indivisible of the divisible, we conclude that to be good is essentially to be indivisible.

a23. The question is begged because definable form is assumed as a premise,

a24. and as a premise which is to prove definable form. ‘But not the same definable form’, you may object. That I admit, for in demonstrations also we premise that ‘this’ is predicable of ‘that’; but in this premise the term we assert of the minor is neither the major itself nor a term identical in definition, or convertible, with the major.

a28. Again, both proof by division and the syllogism just described are open to the question why man should be animal-biped-terrestrial and not merely animal and terrestrial, since what they premise does not ensure that the predicates shall constitute a genuine unity and not merely belong to a single subject as do musical and grammatical when predicated of the same man.

a34. How then by definition shall we prove substance or essential nature? We cannot show it as a fresh fact necessarily following from the assumption of premises admitted to be facts — the method of demonstration: we may not proceed as by induction to establish a universal on the evidence of groups of particulars which offer no exception, because induction proves not what the essential nature of a thing is but that it has or has not some attribute. Therefore, since presumably one cannot prove essential nature by an appeal to sense perception or by pointing with the finger, what other method remains?

Can we nevertheless actually demonstrate. After showing that the 'quod quid' cannot be demonstrated either by convertible terms or by the method of division, the Philosopher here shows that it cannot be demonstrated by taking that which is required for a 'quod quid'.

In regard to this he does two things.

First, he shows what he proposes.

Second, he concludes from all of the foregoing that there is no way in which 'quod quid' maybe demonstrated, at: how, then, by definition (93a34; [459]).

Concerning the first he does two things.

First, he employs reasons proper to what he proposes.

Second, he employs a reason common to what is now being said and to what was said above, at: again, both proof (93a28; [458]).

Concerning the first he does two things.

First, he shows that the 'quod quid' cannot be demonstrated by taking that which pertains to the notion of its 'quod quid'.

Second, he shows that the 'quod quid' of this thing cannot be demonstrated by taking the 'quod quid' of some other thing, at: the following type (93a20; [455]).

In regard to the first he does two things.

First, he raises the question.

Second, he argues for his proposal, at: or is the truth (93a9; [453]).

He asks therefore first (93a6) whether one might demonstrate what something is according to its substance by supposing that the 'quod quid' of a thing can be obtained if one satisfies the conditions which are proper to the 'quod quid': (for example, if one were to prove that two-footed land animal is the 'quod quid' of man, having taken as middle the fact that this notion is converted with man and is composed of genus and difference); which are the only conditions required for a 'quod quid'. Furthermore, this whole, which has been stated, is proper to the 'quod quid' because it is the being or reality of it, that is, of that which is 'quod quid'. It is as though he were saying that being a convertible notion composed of a genus and difference is exactly the same as being the 'quod quid'.

Then, at: or is the truth (93a9) he lodges two objections against the above question, showing that the 'quod quid' cannot be demonstrated in the way it suggests.

The first of these is that just as the previously mentioned ways of demonstrating failed because they assumed the very thing in question, so too in the present instance. For it, too, assumes the 'quod quid erat esse', as for example when it is assumed that every convertible notion composed of genus and difference signifies the 'quod quid est'; consequently, the proof is not acceptable. For one must never take as middle the very thing the demonstration is intended to prove; rather it is necessary to demonstrate it through some other middle.

Then, at: further, just as in syllogizing (93a12) he gives the second reason, which is based on a likeness to a syllogism.

For when a person syllogizes, he is not required to use the definition of syllogism in order to syllogize; because the items from which a syllogism proceeds are such that each item is always either a whole proposition, that is, a universal or major proposition, or a part, that is, a particular or minor proposition subsumed under the major one. Consequently, the definition of a syllogism is not one of the things from which a syllogism proceeds. In like manner if someone wishes to syllogize the 'quod quid erat esse' of something, he is not required to mention what a 'quod quid erat esse' is; but he must have this in reserve in his mind over and above the items which are put in the definition or in the syllogism.

For these notions of a syllogism and of a definition function in defining and syllogizing as rules of art which an artisan must respect in his works. Now, an artisan who makes a knife does not do so by establishing the rules according to which he works, but in the light of the rules he has in his mind he examines whether the knife was made well. So, too, a person who syllogizes does not mention the notion of a syllogism as he syllogizes, but in view of the notion of a syllogism he examines the syllogism he made, in order to see whether it is good.

Hence if a person should wonder, once a syllogism has been formed, whether it syllogized or not, the one who formed the syllogism can settle this doubt by showing that it does syllogize, on the ground that a syllogism is such and such a thing. So, too, a person who intends to syllogize the 'quod quid erat esse' should have reserved in his mind the notion of what a 'quod quid erat esse' is, so that if someone should say that a 'quod quid erat esse' has not been syllogized, he can answer that it has, because such and such is put down as the 'quod quid erat esse'. Thus, therefore, it is clear that a person who syllogizes a 'quod quid erat esse' should not state what a syllogism is or what a 'quod quid erat esse' is.

Then, at: the following type of hypothetical (93a20) he shows that the 'quod quid' of one thing cannot be demonstrated from the 'quod quid' of another thing.

Concerning this he does two things.

First, he proposes what he intends.

Second, he proves what he proposes, at: the question is begged (93a23; [456]).

He says therefore first (93a20) that the 'quod quid' is not proved, if someone wished to prove it by supposing the 'quod quid' of some other thing; for example, if someone were to proceed in such a way as to assume that the being or reality of divisible and of evil is the same, that is, that division is the 'quod quid' of evil, and then argued in the following manner: “In all things which have a contrary the 'quod quid' of one contrary is a contrary; but the contrary of good is evil, and of divisible the contrary is indivisible: therefore, it follows that indivisible is the 'quod quid' of the good.” And these examples are taken according to Plato’s opinion that the notion of the one and of the good are the same. For we notice that everything seeks unity as its own good. But one is the same as indivisible; and so in virtue of the opposition it follows that evil is the same as divisible. For everything shuns division of the self, because through division it tends to what is shattered and imperfect.

Then, at: the question is begged (93a23) he proves what he has proposed, namely, that the 'quod quid' cannot be demonstrated:

and this by giving two reasons.

The first of these is that even according to this method one demonstrates by assuming the 'quod quid erat esse', and thus he assumes what he should have proved.

Then, at: and as a premise (93a24) he gives the second reason, saying that not only is it improper to take the 'quod quid erat esse' when demonstrating, but this impropriety is compounded when one takes a 'quod quid erat esse' to demonstrate a 'quod quid erat esse': because even in demonstrations in which this is proved of this, say a proper attribute of a subject, a 'quod quid erat esse' is taken as middle; nevertheless the one so taken is not the very thing which was to be concluded or something which has the same notion and is converted. But good and indivisible, and evil and divisible, have the same notion; and each pair is converted, because one being posited, the other is posited, and vice versa.

Then, at: again, both proof by division (93a28) he presents a common reason against anyone who demonstrates from supposition and division.

And he says that the same opposition is directed against both, namely, against one who tries to demonstrate the 'quod quid' by division and against one who proceeds by laying down the 'quod quid' in the syllogism. For clearly a definition signifies something which is one: hence to signify this oneness the items laid down in the definition should be arranged without a conjunction; thus one should say that man is a two-footed land animal, but not that man is an animal and two-footed. Hence if anyone desires to prove the 'quod quid', he must prove that one thing is formed from the things assumed. But it is not required, according to the above-cited methods of division and supposition, that one predicate be formed from these items which are taken for defining: since there might be several, for example, if one should state that man is grammatical and musical. It seems therefore that according to the ways mentioned above, the 'quod quid' is not proved.

Then, at: how, then, by definition (93a34) he concludes from the foregoing that there is no way to prove the 'quod quid'.

He says: If, then, the 'quod quid' cannot be proved either by convertible terms or division or opposition, how will the definer be able to demonstrate the substance of a thing or the 'quod quid'? For it is now clear from the foregoing that he does not prove it according to the method whereby, in virtue of items that are per se known, one manifests that something else must follow through things which are said-which is required for a demonstration. However, besides these three modes there is a fourth, namely, through induction. But it turns out that the 'quod quid' cannot be proved by manifest singulars, namely, in such a way that something is predicated of all and is not anything that might be otherwise: because one who thus proceeds by induction will not demonstrate the quod quid but will demonstrate that something is or is not; for example, that every man is an animal, or that no animal is a stone. But no other method is available for demonstrating the 'quod quid', save perhaps the method of demonstrating something to the senses, as when something is pointed out with the finger.

But it is obvious that this method cannot be invoked in the present instance, because the 'quod quid' is not an object of sense but of intellect, as it is stated in On the Soul III. It remains, therefore, that there is no way in which the 'quod quid' can be demonstrated.

b3. To put it another way: how shall we by definition prove essential nature? He who knows what human—or any other—nature is, must know also that man exists; for no one knows the nature of what does not exist—one can know the meaning of the phrase or name ‘goat-stag’ but not what the essential nature of a goat-stag is.

b7. But further, if definition can prove what is the essential nature of a thing, can it also prove that it exists? And how will it prove them both by the same process, since definition exhibits one single thing and demonstration another single thing, and what human nature is and the fact that man exists are not the same thing?

b11. Then too we hold that it is by demonstration that the being of everything must be proved — unless indeed to be were its essence; and, since being is not a genus, it is not the essence of anything. Hence the being of anything as fact is matter for demonstration; and this is the actual procedure of the sciences, for the geometer assumes the meaning of the word triangle, but that it is possessed of some attribute he proves. What is it, then, that we shall prove in defining essential nature? Triangle? In that case a man will know by definition what a thing’s nature is without knowing whether it exists. But that is impossible.

b18. Moreover it is clear, if we consider the methods of defining actually in use, that definition does not prove that the thing defined exists: since even if there does actually exist something which is equidistant from a centre, yet why should the thing named in the definition exist? Why, in other words, should this be the formula defining circle? One might equally well call it the definition of mountain copper. For definitions do not carry a further guarantee that the thing defined can exist or that it is what they claim to define: one can always ask why.

b26. Since, therefore, to define is to prove either a thing’s essential nature or the meaning of its name, we may conclude that definition, if it in no sense proves essential nature, is a set of words signifying precisely what a name signifies.

b28. But that were a strange consequence; for (1) both what is not substance and what does not exist at all would be definable, since even non-existents can be signified by a name:

b30. (2) all sets of words or sentences would be definitions, since any kind of sentence could be given a name; so that we should all be talking in definitions, and even the Iliad would be a definition:

b32. (3) no demonstration can prove that any particular name means any particular thing: neither, therefore, do definitions, in addition to revealing the meaning of a name, also reveal that the name has this meaning.

b35. It appears then from these considerations that neither definition and syllogism nor their objects are identical, and further that definition neither demonstrates nor proves anything, and that knowledge of essential nature is not to be obtained either by definition or by demonstration.

To put it another way. After considering each of the ways in which something might be demonstrated and showing that none can demonstrate the 'quod quid', the Philosopher now shows what he proposes through common reasons.

Concerning this he does three things.

First, he lays down something which is necessary for proving what he proposes.

Second, he shows what he proposes, at: but further, if definition (93b7; [462]).

Third, he sums up what has been stated, at: it appears then from these (93b35; [469]).

He says therefore first (93b3) that there does not seem to be any possible way for a person to demonstrate that some 'quod quid' is man: and this because whoever knows a 'quod quid' to be of man or of any other thing is required to know that the thing exists. For since there is no quiddity or essence of a non-being, it is impossible to know the 'quod quid' of something which is not; but one might know the meaning of the name or have a notion composed of several names. Thus a person is capable of knowing the meaning of the word 'tragelaphus' or 'goat-stag, because it signifies an animal composed of goat and stag, but it is impossible to know the 'quod quid' of a goat-stag, because there is no such thing given in nature.

Then, at: but further, if definition (93b7) from what has been established he goes on to prove what he proposed.

In regard to this he does two things.

First, he shows that the 'quod quid' cannot be shown by demonstration.

Second, that it cannot be shown by definition, at: since, therefore, to define (93b26; [465]).

In regard to the first he presents three reasons.

The first of these (93b7) is the following: just as a definition is brought forward to manifest something which is one in the sense that from the parts of the definition something is formed which is one per se and not one per accidens, so a demonstration which employs a definition as its middle must demonstrate something which is one; for the conclusion must be proportionate to the middle. Consequently, it is clear that one cannot in virtue of one and the same definition demonstrate things that are diverse. But the quod quid of man is one thing and his being or existence is another—(for it is only in the First Principle of being, Who is being essentially, that to be and quiddity are one and the same; in all other things, which are beings by participation, the to be has to be other than the quiddity.) Therefore, it is not possible to demonstrate both the quid and the quia with the same demonstration.

Then, at: then too we hold (93b11) he presents the second reason and it is this: According to a common opinion of wise men, it is necessary that all that is, the whole demonstrated by a demonstration, be the 'quia est', unless perchance someone would want to say that the 'quia est' itself is the substance of a thing. However, this is impossible. For the to be is not the substance or essence of anything existing in a genus; otherwise it would be required that 'being' be a genus, because a genus is something predicated in quod quia. But, as it is proved in Metaphysics III, being is not a genus. For this reason God Who is His own essence is not in a genus. Furthermore, if the 'quia est' were the substance of any thing, then in the same breath that one shows the 'quia est' he would be showing the 'quid est'; consequently, the whole which a demonstrator manifests would not be the 'quia est'. But this is false. Therefore, it is clear that demonstration manifests only the 'quia est', for it demonstrates some enunciation which signifies that something is or is not. This is apparent also in the way a science proceeds. For geometry assumes what the name 'triangle' signifies, and demonstrates quod sit, as when it demonstrates that an equilateral triangle has been formed on a given straight line. Therefore, if one demonstrated merely what a triangle is apart from the method of demonstrating which the sciences employ, he would not be demonstrating this whole which is that a triangle exists, but only what I call triangle. For just as, owing to the fact that to be is not the substance of a thing, one who demonstrates to be demonstrates that and nothing more, so if someone demonstrates the 'quid est', that is all he demonstrates. It would follow, therefore, that someone who knew the quid est in virtue of a definition would not know an est—which is impossible in view of what has been established above.

Then, at: moreover it is clear (93b18) he presents the third reason. Now, this reason uses ordinary definitions as examples to manifest the same fact that was concluded in the previous reason, namely, that one who demonstrates the 'quid est' is not demonstrating the 'quia est'. Wherefore he says that it is clear not only in the light of the foregoing but also in view of the modes of terms, that is, of definitions, which are now in use, that those who define do not manifest the 'quia est'. For example, one who defines a circle as something all the lines from whose center to the circumference are equal, is still left with the question why there must be such a thing as he has defined, such as why it is required to posit that there is a circle which is defined in the way mentioned. For it is acceptable to give a like description of a brass mountain by saying, for example, that it is a brass body which is lofty and extensive; but the task would still remain to establish whether there is such a thing in nature. And this is because terms, that is, defining notions, do not state that their counterpart either exists or is capable of existing; rather, whenever such a notion is formulated, it is legitimate to ask why such a thing should exist. Clearly then, it is impossible to demonstrate quid est and quia est simultaneously.

Then, at: since, therefore, to define (93b26), by leading to something absurd, he shows that the 'quod quid' cannot be shown by demonstration.

First, therefore, he shows what would follow from this.

Second, he shows that what does follow is unacceptable, at: but that were a strange consequence (93b28; [466]).

He says therefore (93b26) that since a definer might show either the quid est or merely what a name signifies, it follows that a definition does not necessarily manifest the 'quod quid', which belongs to definition in the proper sense; otherwise, it will follow that a definition signifying 'quod quid is' nothing more than a statement signifying in the same way that a name does: for the only thing that a definition adds to such a notion is that it signifies the essence of some thing. Hence, if there is no thing whose essence the definition signifies, it will be no different from a statement explaining the signification of some name.

Then, at: but that were a strange consequence (93b28) he shows that it is unacceptable for a definition to be nothing more than a statement explaining the signification of a name.

This is for three reasons.

The first of these is that it happens that even things which are neither substances nor beings in a universal sense can be signified by a name. Now, a name can be explained by some interpretation. Consequently, if a definition were nothing more than a statement interpreting a name, it would follow that a definition could be given of non-substances and of things that do not exist at all. But this is clearly false, for as it is shown in Metaphysics VII, definition bears principally on substance and on other things insofar as they are related to substance.

Then, at: all sets of words (93b30) he gives the second reason and it is this: To any notion, that is, to any statement signifying something, it is possible to apply a corresponding name which is explained by that notion. Therefore, if a definition is nothing more than a notion explaining a name, it would follow that all notions were definitions. As a consequence, it would follow that when we dispute or converse with anyone, the disputations or our discussions are definitions; similarly, it would follow that the Iliad, that is, Homer’s poem about the Trojan War, is a definition.

Then, at: no demonstration can prove (93b32) he gives the third reason and it is this: No science demonstrates that a given name signifies a given thing. For names are arbitrary signs; hence they must be used according to the will of the one inventing them. But it is obvious that definitions are given in every science. Therefore, it is obvious that definitions do not signify this, that is, a mere interpretation of a name.

Then, at: it appears then (93b35) he sums up what has been so far discussed disputatively.

And he says that from the foregoing it is seen to follow that definition and syllogism are neither the same nor concerned with the same; and furthermore that a definition demonstrates nothing, since it is not concerned with the same thing as a demonstration. In like manner, it seems to have been shown that it is not possible to know quod quid either by a definition or a demonstration, because a definition merely shows the what and a demonstration the 'quia'. But to know the 'quod quid' a knowledge of 'quia' est is required, as has been shown.

a1. We must now start afresh and consider which of these conclusions are sound and which are not, and what is the nature of definition, and whether essential nature is in any sense demonstrable and definable or in none.

a4. Now, to know its essential nature is, as we said, the same as to know the cause of a thing’s existence, and the proof of this depends on the fact that a thing must have a cause. Moreover, this cause is either identical with the essential nature of the thing or distinct from it; and if its cause is distinct from it, the essential nature of the thing is either demonstrable or indemonstrable. Consequently, if the cause is distinct from the thing’s essential nature and demonstration is possible, the cause must be the middle term, and, the conclusion proved being universal and affirmative, the proof is in the first figure. So the method just examined of proving it through another essential nature would be one way of proving essential nature, because a conclusion containing essential nature must be inferred through a middle which is an essential nature just as a ‘peculiar’ property must be inferred through a middle which is a ‘peculiar’ property; so that of the two definable natures of a single thing this method will prove one and not the other. Now, it was said before that this method could not amount to demonstration of essential nature — it is actually a dialectical proof of it —

a16. so let us begin again and explain by what method it can be demonstrated. When we are aware of a fact we seek its reason, and though sometimes the fact and the reason dawn on us simultaneously, yet we cannot apprehend the reason a moment sooner than the fact; and clearly in just the same way we cannot apprehend a thing’s definable form without apprehending that it exists, since while we are ignorant whether it exists we cannot know its essential nature.

a21. Moreover we are aware whether a thing exists or not sometimes through apprehending an element in its character, and sometimes accidentally, as, for example, when we are aware of thunder as a noise in the clouds, of eclipse as a privation of light, or of man as some species of animal, or of the soul as a self-moving thing.

a24. As often as we have accidental knowledge that the thing exists, we must be in a wholly negative state as regards awareness of its essential nature; for we have not got genuine knowledge even of its existence, and to search for a thing’s essential nature when we are unaware that it exists is to search for nothing. On the other hand, whenever we apprehend an element in the thing’s character there is less difficulty. Thus it follows that the degree of our knowledge of a thing’s essential nature is determined by the sense in which we are aware that it exists.

a28. Let us then take the following as our first instance of being aware of an element in the essential nature. Let A be eclipse, C the moon, B the earth’s acting as a screen. Now, to ask whether the moon is eclipsed or not is to ask whether or not B has occurred. But that is precisely the same as asking whether A has a defining condition; and if this condition actually exists, we assert that A also actually exists. Or again we may ask which side of a contradiction the defining condition necessitates: does it make the angles of a triangle equal or not equal to two right angles? When we have found the answer, if the premises are immediate, we know fact and reason together; if they are not immediate, we know the fact without the reason, as in the following example: let C be the moon, A eclipse, B the fact that the moon fails to produce shadows though she is full and though no visible body intervenes between us and her. Then if B, failure to produce shadows in spite of the absence of an intervening body, is attributable A to C, and eclipse, is attributable to B, it is clear that the moon is eclipsed, but the reason why is not yet clear, and we know that eclipse exists, but we do not know what its essential nature is. But when it is clear that A is attributable to C and we proceed to ask the reason of this fact, we are inquiring what is the nature of B: is it the earth’s acting as a screen, or the moon’s rotation or her extinction? But B is the definition of the other term, viz. in these examples, of the major term A; for eclipse is constituted by the earth acting as a screen. Thus, (1) ‘What is thunder?’ ‘The quenching of fire in cloud’, and (2) ‘Why does it thunder?’ ‘Because fire is quenched in the cloud’, are equivalent. Let C be cloud, A thunder, B the quenching of fire. Then B is attributable to C, cloud, since fire is quenched in it; and A, noise, is attributable to B; and B is assuredly the definition of the major term A. If there be a further mediating cause of B, it will be one of the remaining partial definitions of A.

b15. We have stated then how essential nature is discovered and becomes known, and we see that, while there is no syllogism — that is, no demonstrative syllogism — of essential nature, yet it is through syllogism, viz. demonstrative syllogism, that essential nature is exhibited. So we conclude that neither can the essential nature of anything which has a cause distinct from itself be known without demonstration, nor can it be demonstrated; and this is what we contended in our preliminary discussions.

We must now start afresh. After inquiring disputatively how the definition and the 'quod quid' are known, the Philosopher here determines the truth.

First, he states what his intention is.

Second, he carries out what he proposes, at: now, to know its essential (94a4; [471]).

Third, he summarizes, at: our discussion has therefore made plain (95a13; [489]).

He says therefore first (94a1) that after the disputative process we must once more consider, but this time by establishing the truth, how much of the foregoing has been well stated and how much not well stated: and this both in regard to definition, by considering what a definition is, and in regard to 'quod quid', by considering whether it might somehow or other be manifested through a definition or a demonstration or not at all.

Then, at: now, to know its essential (94a4) he carries out what he has proposed.

First, in regard to the 'quod quid'.

Second, in regard to definition which is a statement signifying it, at: since definition is said to be (94b28; [483]).

Concerning the first he sets down two methods of manifesting the 'quod quid'.

First, the method of logical proof.

Second, the method of demonstrative proof, at: so let us begin again (93a16).

Regarding the first he reviews first of all (94a4) that which was established above, namely, that it is the same thing to know quid est and to know the cause involved in the question 'an est', just as it is the same thing to know 'propter quid' and to know the cause involved in the question 'quia' est. The reason for this, that is, for the fact that it is the same thing to know the 'quid est' and the cause involved in the 'si est', is that there must be some cause of the fact that a thing exists, (for something is referred to as caused by reason of the fact that it has a cause of its existing.) Now, the cause of existing is either the same, that is, the same as the essence of a thing, or not the same: (the same, indeed, as matter and form, which are parts of the essence; not the same, as agent and end, which two causes are somehow the cause of the form and matter, for the agent acts for an end and unites the form to the matter). And if we consider the cause which is other than the essence of a thing, sometimes it is such a cause that a demonstration can be formed in virtue of it, and sometimes not. For the effect does not follow of necessity from every efficient cause; but from the supposition of the end it follows that what is required for the end also exists, as is proved in Physics II.

So let us suppose that an effect exists which has as its cause not only the essence of the thing but also another cause such that something can be demonstrated through it; for example, we might say that if a man has attained happiness, it is necessary that virtue preexist. Now, let us assume that the essence of virtue is 'a habit which operates according to right reason'. Therefore, it can be demonstrated that there is a habit which operates according to right reason, if there is a habit conducive to happiness. Let that other cause, therefore, which is demonstrative, be taken as the middle, and let a syllogism be formed in the first figure (which must be done, because the 'quod quid' must be universally and affirmatively predicated of the thing of which it is the quod quid): the syllogism then will be: "every habit conducive to happiness is a habit operating according to right reason; but virtue is such a habit: therefore, etc."

He concludes, therefore, that this method which is now being examined is one method of showing the 'quod quid' through something else which is a cause. And this method is evidently suitable, because, as was established above, it is necessary that the middle taken for proving 'quod quid' be the 'quid est'; and similarly that the middle taken for proving properties be something proper. But there is the further consideration that since the 'quid est' is the cause of the very being of the thing, then according to the diverse causes of one thing, there are various ways of assigning the 'quod quid' of the same thing. For example, the 'quod quid' of house can be formulated in terms of its material cause, so that we might say that it is something composed of wood and stones; and also in terms of the final cause, namely, that it is an artifact constructed for habitation. Thus, therefore, it will occur that since there are several 'quod quid’s' of the same thing, one of them will be demonstrated and another not demonstrated but supposed. Hence it does not follow that there is a begging of the question, because one 'quod quid' is supposed and the other proved. Nor is this a method proving the 'quod quid' demonstratively, but of syllogizing it logically, because by this method it is not sufficiently proved that what is concluded is the 'quod quid' of the thing of which it is concluded, but merely that it is in it.

Then, at: so let us begin again (94a16) he shows how by demonstration the 'quod quid' can be achieved.

Regarding this he does two things.

First, he shows how the 'quod quid' can be manifested in certain instances through demonstration.

Second, he shows that it is not so in all cases, at: now, while some things (94b22; [479]).

Concerning the first he does three things.

First, he lays down certain things needed for showing what he proposes.

Second, he manifests what he has proposed, at: let us then take the following (94a28; [477]).

Third, he summarizes, at: we have stated then how essential (94b15; [478]).

With respect to the first (94a16) he lays down three things.

The first of these is a comparison between the 'quod quid' and the 'propter quid'.

He says, therefore, that in order to show how it occurs that the 'quod quid' can be achieved through demonstration, we must start once more from the beginning. The first thing to be considered is the fact that a person may be in either of two states as regards knowledge of the 'propter quid'. For sometimes we have the 'quia' in our knowledge and are still seeking the 'propter quid'; and sometimes both are manifest to us at one time. A third case is impossible, namely, that one know the 'propter quid' of something before the 'quia'. And the same holds for the 'quod quid erat esse': sometimes we know a thing to be without perfectly knowing the quid sit; sometimes we know both at once. Again, a third case is impossible, namely, to know quid est without knowing if it is.

Then, at: moreover, we are aware (94a21) he lays down the second preliminary remark.

He says that there are two ways in which we might know something to be without perfectly knowing the quid est: in one way, insofar as we might know one of its accidents, as for example, when by the speed of its motion we judge that something is a hare; in another way, by knowing something of its essence. (This of course is possible in the case of a composite substance, as when we comprehend something to be a man in virtue of the fact that it is rational, without yet knowing the other items which complete the essence of man. However, in simple substances this does not occur, because one cannot know anything concerning the substance of a simple thing without knowing the entire essence, as is clear from Metaphysics IX.) At any rate, a person who knows something to be must know it through something of that thing, namely, something outside the essence of the thing or something pertaining to its essence. And he clarifies this with the example of knowing thunder to be, because we perceive a sound in the clouds—which of course pertains to the essence of thunder, albeit not the entire essence, because not every sound in the clouds is thunder—or of knowing a defect, that is, an eclipse, of the sun or moon to be, because there is a failure of light-although not every failure of light is an eclipse. And the same applies if someone perceives a man to be, because there is an animal; or a soul to be, because something is moving itself.

Then, at: as often as we have accidental (94a24) he sets forth the third preliminary remark.

He says that those things which we know to exist through some accident belonging to them are such that there is no correlation between this knowledge and knowledge of their 'quid est', because in virtue of such accidents we do not even truly know that they are. We do of course know these accidents to be theirs, but because the accidents are not the things themselves, we do not thereby know that the things themselves exist. Moreover, it is fruitless to seek the quid est if one does not know quia est; but those things which we know to exist through something of their own can be more easily known by us as to what they are. Hence it is plain that as we are in regard to knowing 'that something is', so we are in regard to knowing 'what it is'.

Then, at: let us then take the following (94a28) in the light of the foregoing he manifests what he proposed.

And he says that in regard to a thing which we know exists because we know something of its essence, let us consider the following example in which A, the major extreme, is defect, that is, eclipse; and in which C, the minor extreme, is moon; and interposition of the earth between sun and moon is B, the middle. In this case, therefore, to ask whether the moon is eclipsed or not is the same as to ask whether B exists or not; but to ask whether B exists or not is to ask if there is some reason for that eclipse. For B, that is, the interposition of the earth, is the reason for the moon’s eclipse; and if there is an interposition of the earth we say that the other exists, that is, the eclipse of the moon. Or, similarly, if we should seek the reason for one side of a contradiction, that is, does it lie in having or in not having two right angles. But when we find that what we seek exists, for example, that there is an eclipse, then we know the 'quia' and the 'propter quid' at once, if what is sought is found through an appropriate middle which is the cause; and if not, but it is found through something extrinsic, we will know the 'quia' and not the 'propter quid'. For example, let the moon be C and the eclipse A, and let us take as middle, that is, B, the fact that the moon, although it is full, is now unable to cause a shadow when we put an object between it and us. For the moon, whenever it is not eclipsed, causes a shadow if some body is interposed; yet the fact that it is now unable to cause a shadow is not the cause of the eclipse, but rather its effect. Therefore, if B be predicated of C (that is, that the moon is now unable to produce a shadow of an object which is interposed), and if moreover A is true of that middle (that is, if we assume that whenever this happens, the moon is eclipsed), it will be plain that the moon is eclipsed. But it will not yet be clear 'why' the moon is eclipsed. Similarly, we will know 'that' there is an eclipse but not know what an eclipse is, when it is clear that A is in C that is, that the moon is eclipsed. Now, just as in this example neither the 'propter quid' nor the 'quid' is known, so to inquire 'why' [that is, propter quid] something is, is the same as to inquire [quid], that is, what it is. Thus, if we should ask propter quid, that is, why the moon is eclipsed, namely, is it because the earth is between the sun and moon, or because the moon has turned, that is, because the dark side of the moon has been turned toward us, as some say, or even because the light of the moon has been extinguished in something moist: to ask whether it is due to any of these causes that the moon is eclipsed is nothing more than to ask whether the eclipse of the moon consists in the interposition of the earth, or the turning of the moon, or the extinction of its light. And this middle is the reason of the other extreme: thus, in the foregoing example, it is the reason for A which is the major extreme, because the eclipse of the moon is nothing more than the moon’s light being blocked by the earth. He gives a further example, namely, inquiring what is thunder. According to the opinion of Anaxagoras and Empedocles it is the quenching of fire in a cloud; (although according to his own opinion in Meteorology II, thunder is caused by a dry exhalation in a cold environment beating against a cloud. But he frequently uses the opinions of others in his examples.) So, if according to the above-mentioned opinion, one should ask why it thunders, the answer might be given that it is because fire is being quenched in a cloud. Therefore, let the cloud be C, the minor extreme, and thunder A, the major extreme, and the quenching of fire B, the middle. Then we may syllogize in the following manner: "In C is B, that is, in the cloud is a quenching of fire; but every quenching of fire is a sound: therefore in the cloud is a sound of thunder." And thus it is clear that to arrive at the 'propter quid' through demonstration, we take the 'quid est', because the very middle which shows the 'propter quid' is a statement defining the first term, that is, defining the major extreme. But if we had to take some other middle to demonstrate this, it would be taken from the remaining statements, that is, from the definition of the minor extreme and of other extrinsic causes. For since the subject is the cause of its proper attribute, it is required that the definition of the proper attribute be demonstrated by the definition of the subject. And this is clear in the other example; for since the moon is a body fixed by nature to move in such and such a way, it is necessary that at some time the earth be interposed between it and the sun.

Then, at: we have stated then (94b15) he sums up what has been said.

He says that it has been said how the 'quod quid' is taken and made known, namely, by taking the 'propter quid'. We have also said that there is neither syllogism nor demonstration of the 'quid est' in the sense of properly syllogizing or demonstrating the 'quod quid', although the 'quod quid' is manifested by syllogism and demonstration, insofar as the middle in a 'propter quid' demonstration is the 'quod quid'. Hence, it is clear that a 'quod quid' having another cause cannot be known without a demonstration, but the demonstration is nevertheless not of the 'quod quid', as was proved in the objections. And according to this the above objections are true.

To attain quod quid through demonstration is not possible in all cases: relation of definition to demonstration

b22. Now, while some things have a cause distinct from themselves, others have not. Hence it is evident that there are essential natures which are immediate, that is are basic premises; and of these not only that they are but also what they are must be assumed or revealed in some other way. This too is the actual procedure of the arithmetician, who assumes both the nature and the existence of unit. On the other hand, it is possible (in the manner explained) to exhibit through demonstration the essential nature of things which have a ‘middle’, that is, a cause of their substantial being other than that being itself; but we do not thereby demonstrate it.

b28. Since definition is said to be the statement of a thing’s nature, obviously one kind of definition will be a statement of the meaning of the name, or of an equivalent nominal formula. A definition in this sense tells you, e.g., the meaning of the phrase ‘triangular character’. When we are aware that triangle exists, we inquire the reason why it exists. But it is difficult thus to learn the definition of things the existence of which we do not genuinely know — the cause of this difficulty being, as we said before, that we only know accidentally whether or not the thing exists. Moreover, a statement may be a unity in either of two ways, by conjunction, like the Iliad, or because it exhibits a single predicate as inhering not accidentally in a single subject. That then is one way of defining definition.

b38. Another kind of definition is a formula exhibiting the cause of a thing’s existence.

b39. Thus the former signifies without proving, but the latter will clearly be a quasi-demonstration of essential nature, differing from demonstration in the arrangement of its terms.

a2. For there is a difference between stating why it thunders, and stating what is the essential nature of thunder; since the first statement will be ‘Because fire is quenched in the clouds’, while the statement of what the nature of thunder is will be ‘The noise of fire being quenched in the clouds’. Thus the same statement takes a different form: in one form it is continuous demonstration, in the other definition. Again, thunder can be defined as noise in the clouds, which is the conclusion of the demonstration embodying essential nature.

a9. On the other hand, the definition of immediates is an indemonstrable positing of essential nature. We conclude then that definition is (a) an indemonstrable statement of essential nature, or (b) a syllogism of essential nature differing from demonstration in grammatical form, or (c) the conclusion of a demonstration giving essential nature.

a13. Our discussion has therefore made plain (1) in what sense and of what things the essential nature is demonstrable, and in what sense and of what things it is not; (2) what are the various meanings of the term definition, and in what sense and of what things it proves the essential nature, and in what sense and of what things it does not; (3) what is the relation of definition to demonstration, and how far the same thing is both definable and demonstrable and how far it is not.

Now, while some things have a cause. After showing that in some instances the quod quid is acquired through demonstration, the Philosopher now shows that this is not possible in all.

And in order to show this he repeats (94b22) that of some things there is a cause other than the thing, but of other things not. Therefore, because the 'quod quid' is acquired by a demonstration whose middle is a cause, it is manifest that there are certain things whose 'quod quid' must be taken as an immediate principle, in the sense that one must suppose both the to be and the quid of that thing, or manifest it by some means other than demonstration, say by an effect or by a likeness or some other way.

Here it should be noted that his statement that some things have a cause not distinct from themselves can be understood in three ways.

In one way, it means that it simply and absolutely has not a cause of its being. And this is proper to the First Principle alone, which is the cause of the being and truth of all things. For there is nothing to prevent even things which exist of necessity from having some cause of their necessity, as is stated in Metaphysics V. And therefore, although the Philosopher is speaking in the plural here, his words should not be taken to mean that there are several things which have no cause of their being.

In another way it can be understood as referring to the order of the causes of one thing. For it is clear that in things which have four causes, one cause is somehow the cause of another one. For since matter is for the sake of form and not vice versa, as is proved in Physics II, a definition which is based on the formal cause is the cause of the definition which is taken from the material cause of the same thing. And because the thing produced obtains a form in virtue of the action of the generator, it follows that the agent is somehow the cause of the form, and the definition the cause of the defined. Furthermore, every efficient cause acts for an end; hence, too, the definition taken from the end is somehow the cause of the definition which is taken from the efficient cause. Beyond this it is impossible to go in the genera of causes: hence the dictum that the end is the cause of the causes. Yet in the individual genera of causes one can proceed from the subsequent to the prior; but definitions should be given through proximate causes. And according to this interpretation we find it stated in certain books that definitions which are made in terms of the species have no middle by which they might be demonstrated: but that definitions made in terms of matter can have a middle, in the sense that definitions which are given according to the material cause, can be demonstrated by those which are given according to the formal cause. But a definition which is given according to formal cause cannot be further demonstrated by an intrinsic principle of the thing which belongs in a proper way to the quod quid as entering into the essence of the thing. But if it were to be demonstrated by the efficient and final cause, one would have to say that a higher cause is related as formal to the lower. "However, these words are not present in the Greek version; hence they seem to be a Gloss introduced into the text by a copyist’s error".