Commentary Aristotle Posterior Analytics
Lecture 43
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 principles 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.