Commentary Aristotle Posterior Analytics
Lecture 20
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.
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.