Commentary Aristotle Posterior Analytics
Lecture 36
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 A 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 reached.
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]).
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 (84b38) 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.