Demonstrative vs. Dialectical
The demonstrative premiss differs from the dialectical, because the demonstrative premiss is the assertion of one of two contradictory statements, whereas the dialectical premiss depends on the adversary’s choice between two contradictories.
A perfect syllogism needs nothing other than what has been stated to make plain what necessarily follows; a syllogism is imperfect, if it needs either one or more propositions, which are indeed the necessary consequences of the terms set down, but have not been expressly stated as premisses.
That one term should be included in another as in a whole is the same as for the other to be predicated of all of the first.
Every premiss states that something either is (assertoric), or must be (apodeictic), or may be (problematic) the attribute of something else; some premisses are affirmative, others negative; some are universal, others particular, others indefinite.
In universal attribution the terms of the negative premiss are convertible (e.g. if no pleasure is good, then no good will be pleasure); the terms of the affirmative must be convertible, not however, universally, but in part (e.g. if every pleasure,is good, some good must be pleasure); the particular affirmative must convert in part; but the particular negative need not convert, for if some animal is not man, it does not follow that some man is not animal.
The same manner of conversion hold good also in respect of necessary premisses. The universal negative converts universally; each of the affirmatives converts into a particular. But the particular negative does not convert.
Possibility is used in several senses (what is necessary, what is not necessary, and what is potential is possible). Affirmative statements will all convert in a manner similar to those described. But in negative statements the case is different. Whatever is possible, either because B necessarily is A, or because B is not necessarily A, admits of conversion like other negative statements (e.g. if it is possible that no garment is white, it is also admissible for nothing white to be a garment). But if anything is possible because it is the general rule and natural (and it is in this way we define the possible), the negative premisses can no longer be converted like the simple negatives; the universal negative premiss does not convert, and the particular does.
Three Figures of Syllogism
In every syllogism there must be a universal premiss, and that a universal statement is proved only when all the premisses are universal; in every syllogism either both or one of the premisses must be like the conclusion, not only in being affirmative or negative, but also in being necessary, pure, problematic.
No syllogism can establish the attribution of one thing to another, unless some middle term is taken, which is related to each by way of predication. This is possible in three ways: by predicating A of B, and B of C; or B of both; or both of B.
Every demonstration will proceed through three terms and no more.
1. The First Figure
Whenever three terms are so related to one another that the last is contained in the middle as in a whole, and the middle is either contained in, or excluded from, the first as in or from a whole, the extremes must be related by a perfect syllogism. I call that term middle which is itself contained in another and contains another in itself: in position also this comes in the middle. By extremes I mean both that term which is itself contained in another and that in which another is contained.
If A is predicated of all B, and B of all C, A must be predicated of all C; if A is predicated of no B, and B of all C, it is necessary that no C will be A; If A is predicated of all B, but B of no C, there will be no syllogism in respect of A and C; for it is possible that A should belong to either all or none of C.
If one term is related universally, the other in part only, to its subject, there must be a perfect syllogism whenever universality is posited with reference to the major term either affirmatively or negatively, and particularity with reference to the minor term affirmatively. If all B be A and some C be B. it is necessary that some C is A. And if no B is A but some C is B, it is necessary that some C is not A; but whenever the universality is posited in relation to the minor term, or the terms are related in any other way, a syllogism is impossible. If some B is or is not A, and all C is B; Nor when the major premiss is universal, whether affirmative or negative, and the minor premiss is negative and particular/indefinite, can there be a syllogism: e.g. if all B is A and some C is not B, or if not all C is B.
2. The Second Figure
Whenever the same thing belongs to all of one subject, and to none of another, or to all of each subject or to none of either. The middle term, which is predicated of both subjects, stands outside the latter, and is first in position. A syllogism cannot be perfect anyhow in this figure, but it may be valid. All are made perfect by certain supplementary statements, which either are contained in the terms of necessity or are assumed as hypotheses.
If the terms are related universally a syllogism will be possible, whenever the middle belongs to all of one subject and to none of another (it does not matter which has the negative relation), but in no other way. Let M be predicated of no N, but of all O. Since, then, the negative relation is convertible, N will belong to no M: but M was assumed to belong to all O: consequently N will belong to no O.
3. The Third Figure
If one term belongs to all, and another to none, of a third, or if both belong to all, or to none, of it. The middle term, which both the extremes are predicated of, stands outside the extremes, and is last in position. A syllogism cannot be perfect in this figure either, but it may be valid.
In all the figures, if both the terms are affirmative or negative nothing necessary follows at all, but if one is affirmative, the other negative, and if the negative is stated universally, a syllogism always results relating the minor to the major term; All syllogisms can be reduced to the universal syllogisms in the first figure, though not all in the same way; the universal syllogisms are made perfect by converting the negative premiss, each of the particular syllogisms by reductio ad impossible.
When the major premiss is necessary, the conclusion is necessary. is, e.g. if A is taken as necessarily belonging or not belonging to B, but B is taken as simply belonging to C, A will necessarily belong or not belong to C. But if the major premiss is not necessary, but the minor is, the conclusion will not be necessary; In particular syllogisms, if the universal premiss is necessary, then the conclusion will be necessary; but if the particular, the conclusion will not be necessary.
In the second figure, if the negative premiss, which is convertible, is necessary, then the conclusion will be necessary, but if the affirmative, not necessary.
The expression ‘to be possible’ is used in two ways. In one it means to happen generally and fall short of necessity, e.g. man’s turning grey or growing or decaying, or generally what naturally belongs to a thing. In another sense the expression means the indefinite, generally what happens by chance: for none of these inclines by nature in the one way more than in the opposite. That which is possible in each of its two senses is convertible into its opposite.
1. If the major premiss is universal, and the minor particular:
a) whenever the major premiss is problematic, whether affirmative or negative, and the particular is affirmative and assertoric, there will be a perfect syllogism, just as when the terms are universal; b) whenever the major premiss is assertoric, not problematic, and the minor is particular and problematic, in all cases there will be an imperfect syllogism. Only some of them will be proved per impossibile, others by the conversion of the problematic premiss. c) whenever the particular premiss is assertoric and negative, there cannot be a syllogism;
2. If the minor premiss is universal, and the major particular, whether either premiss is negative or affirmative, problematic or assertoric, nohow is a syllogism possible.
Refuting/Converting a Syllogism
To convert a syllogism means to alter the conclusion and make another syllogism to prove that either the extreme cannot belong to the middle or the middle to the last term. For it is necessary, if the conclusion has been changed into its opposite and one of the premisses stands, that the other premiss should be destroyed. For if it should stand, the conclusion also must stand.
In the first figure the syllogisms are formed through the middle and the last figures, and the premiss which concerns the minor extreme is alway refuted through the middle figure, the premiss which concerns the major through the last figure. In the second figure syllogisms proceed through the first and the last figures, and the premiss which concerns the minor extreme is always refuted through the first figure, the premiss which concerns the major extreme through the last. In the third figure the refutation proceeds through the first and the middle figures; the premiss which concerns the major is always refuted through the first figure, the premiss which concerns the minor through the middle figure.
Aristotle’s Criticism of Plato’s Notion’s Recollection
To know is used in three senses: it may mean either to have knowledge of the universal, or to have knowledge proper to the particular, or to exercise such knowledge. Nothing then prevents a man both knowing and being mistaken about the same thing, provided that his knowledge and his error are not contrary.
The argument in the Meno that learning is recollection may be criticized in a similar way. For it never happens that a man starts with a foreknowledge of the particular, but along with the process of being led to see the general principle he receives a knowledge of the particulars, by an act (as it were) of recognition.
Nemo: The universal principle cannot be demonstrated from the particular, but only the particular from the universal. A man never receives universal principle when encountering the particular, but the particular is demonstrated from the universal as in the Meno. If the man never receives the universal principle from which the particular is demonstrated, he must possess the foreknowledge of the universal. Recollection is not refuted.