Theory of (categorical) syllogisms (A4-6 about the three „figures”) Three terms: A, B, C Suppose that relation a holds between B and C and a or e holds between A and B. Then there is a perfect syllogism about A and C, namely: By reference to the semantical definition. Medieval mnemonic names: Barbara, Celarent. Vowels refer to the forms of major, minor and conclusion. B, C: the first resp. second syllogism of the first figure. The third is called Darii, the fourth Ferio. They are called complete and have semantic justification again. But what is a figure?

How to reconstruct Aristotle’s combinatorics? How to characterise the figures (schémata)? Why do we have three (and not four) figures? We have three terms and two premises. One of the terms occurs in both propositions, it is called middle (medius, meson). The two others are called extremes, one of them is the major (meidzon), the other is the minor (elatton). The setting of major and minor is different in the different figures. Both premises express some of the four (a, e, i, o) relation. We are seeking for a proposition of the same form that expresses a relation between the two extremes. Premises are called major resp. minor after the extreme term included. In „this” (later: first) figure the major term is the predicate of the major premise and the minor is subject.

In the syllogisms stated in ch. 4-6 always the major is the predicate and the minor is the subject of the conclusion. If we take it as a general rule, we have four possibilities: The major is predicate and the minor is subject in the respective premise(first figure) The middle is predicated about both extremes (second figure) Both extremes are predicated about the middle (third figure) The major is subject and the minor is predicate of its premise (fourth figure, missing at Aristotle) But he does mention all the syllogisms belonging to the fourth figure as derivative syllogisms of the first figure! E.g. Calemes: Every A is B, No B is C, therefore no C is A. By Celarent (and change of the order of premises), we can conclude to „No A is C”, and by e-conversion, we can get Calemes. In this (traditional) reading A is the major, C is the minor term. A more Aristotelian reading: We have a pair of premises according to the second syllogism of the first figure (i.e., Celarent) with C as the major, A as the minor term and by conversion, we get the conclusion.

Alternative reading: The classification into figures belongs to pairs of premises, irrespective to the conclusion and the order of the premises. Either the middle term is subject in the one and predicate in the other premise, or predicate both times, or subject both times. We should decide which one is the major and which is the minor afterwards. The setting of the major and the minor term refers to some diagram. „this is also middle in position” Aristotle asks first how the major relates to the minor. But we could ask (with him) how the minor relates to the major. In the second and the third figure, we get the answer simply by changing the order of the premises. Or – what is virtually the same – by changing the role of the major and the minor term. E.g. Cesare and Camestres of the second figure. In the first figure, we need some conversion in addition.

In each figure, we have 4*4= 16 pairs of premises. If a pair does not give a syllogism, Aristotle proves this fact by pairs of opposite. examples. I. e., „Every B is A” and „No C is B” gives no conclusion, because it can happen both that every C is an A and that no C is an A – every horse is an animal but no stone is an animal.

Apparent reference to a diagram again. Potential: dunatos

Cesare and Camestres reduced to Celarent by repeated e-conversion and tacit change of order. Not complete because a deduction was needed.

Proof by reduction to impossibility (agein eis to adunaton): easy to reconstruct. Camestres: „Every N is M but no X is M, therefore no X is N” Let us suppose that some X is N (indirect hypothesis). Then some N is X (i-conversion). The second premise and this give that some N is not M (Ferio). Contradiction with the first premise. The other two syllogisms are proved by impossibility (only)

Third method of proof: ekthesis (exemplification, „setting-out”) Third figure, P is the major, R the minor and S the middle. …

A7: theory of reduction Darii is proved by reduction to impossibility, using Camestres. Camestres was reduced to Celarent, therefore no circularity there. (Ferio on a similar way.) Therefore, all syllogisms are reduced to Barbara and Celarent.

Modern reconstructions: Łukasiewicz 1951: Frege-Hilbert style calculus with four (not Aristotelian) axioms. Presupposes propositional logic as backgroud theory Objection: not a theory of consequence but a theory of the four relation within the frames of 0-order logic. The notion of consequence comes from there. Corcoran, 70es (several papers) : system of natural deduction. Starting schemes and transformation rules are the Aristotelian ones (i.e. Barbara, Celarent resp. conversion rules and rule of indirect proof).