2.4 Non-modal Syllogistic

Aristotle's non-modal syllogistic (Prior Analytics A 1–7) is the pinnacle of his logic. Aristotle defines a syllogism as ‘an argument (logos) in which, certain things having been laid down, something different from what has been laid down follows of necessity because these things are so’. This definition appears to require (i) that a syllogism consists of at least two premises and a conclusion, (ii) that the conclusion follows of necessity from the premises (so that all syllogisms are valid arguments), and (iii) that the conclusion differs from the premises. Aristotle's syllogistic covers only a small part of all arguments that satisfy these conditions.
Aristotle restricts and regiments the types of categorical sentence that may feature in a syllogism. The admissible truth-bearers are now defined as each containing two different terms (horoi) conjoined by the copula, of which one (the predicate term) is said of the other (the subject term) either affirmatively or negatively. Aristotle never comes clear on the question whether terms are things (e.g., non-empty classes) or linguistic expressions for these things. Only universal and particular sentences are discussed. Singular sentences seem to be excluded and indefinite sentences are mostly ignored. At An. Pr. A 7 Aristotle mentions that by substituting an indefinite premise for a particular, one obtains a syllogism of the same kind.
Another innovation in the syllogistic is Aristotle's use of letters in place of terms. The letters may originally have served simply as abbreviations for terms (e.g. An. Post. A 13); but in the syllogistic they seem mostly to have the function either of schematic term letters or of term variables with universal quantifiers assumed but not stated. Where he uses letters, Aristotle tends to express the four types of categorical sentences in the following way (with common later abbreviations in parentheses):

اقتباس :

‘A holds of (lit., belongs to) every B’	(AaB)
‘A holds of no B’	(AeB)
‘A holds of some B’	(AiB)
‘A does not hold of some B’	(AoB)

Instead of ‘holds’ he also uses ‘is predicated’.
All basic syllogisms consist of three categorical sentences, in which the two premises share exactly one term, called the middle term, and the conclusion contains the other two terms, sometimes called the extremes. Based on the position of the middle term, Aristotle classified all possible premise combinations into three figures (schêmata): the first figure has the middle term (B) as subject in the first premise and predicated in the second; the second figure has it predicated in both premises, the third has it as subject in both premises:

اقتباس :

I	II	III
A holds of B	B holds of A	A holds of B
B holds of C	B holds of C	C holds of B

A is also called the major term, C the minor term. Each figure can further be classified according to whether or not both premises are universal. Aristotle went systematically through the fifty-eight possible premise combinations and showed that fourteen have a conclusion following of necessity from them, i.e. are syllogisms. His procedure was this: He assumed that the syllogisms of the first figure are complete and not in need of proof, since they are evident. By contrast, the syllogisms of the second and third figures are incomplete and in need of proof. He proves them by reducing them to syllogisms of the first figure and thereby ‘completing’ them. For this he makes use of three methods:

اقتباس :: (i) conversion (antistrophê): a categorical sentence is converted by interchanging its terms. Aristotle recognizes and establishes three conversion rules: ‘from AeB infer BeA’; ‘fromAiB infer BiA’ and ‘from AaB infer BiA’. All but two second and third figure syllogisms can be proved by premise conversion.

اقتباس :: (ii) reductio ad impossibile (apagôgê): the remaining two are proved by reduction to the impossible, where the contradictory of an assumed conclusion together with one of the premises is used to deduce by a first figure syllogism a conclusion that is incompatible with the other premise. Using the semantic relations between opposites established earlier the assumed conclusion is thus established.

اقتباس :: (iii) exposition or setting-out (ekthesis): this method, which Aristotle uses in addition to (i) and (ii), involves choosing or ‘setting out’ some additional term, say D, that falls in the non-empty intersection delimited by two premises, say AxB and AxC, and using D to justify the inference from the premises to a particular conclusion, BxC. It is debated whether ‘D’ represents a singular or a general term and whether exposition constitutes proof.

For each of the thirty-four premise combinations that allow no conclusion Aristotle proves by counterexample that they allow no conclusion. As his overall result, he acknowledges four first figure syllogisms (later named Barbara, Celarent, Darii, Ferio), four second figure syllogisms (Camestres, Cesare, Festino, Baroco) and six third figure syllogisms (Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison); these were later called the modes or moods of the figures. (The names are mnemonics: e.g. each vowel, or the first three in cases where the name has more than three, indicates in order whether the first and second premises and the conclusion were sentences of type a, e, i or o.) Aristotle implicitly recognized that by using the conversion rules on the conclusions we obtain eight further syllogisms (An. Pr. 53a3–14), and that of the premise combinations rejected as non-syllogistic, some (five, in fact) will yield a conclusion in which the minor term is predicated of the major (An. Pr. 29a19–27). Moreover, in the Topics Aristotle accepted the rules ‘from AaB infer AiB’ and ‘from AeB infer AoB’. By using these on the conclusions five further syllogisms could be proved, though Aristotle did not mention this.
Going beyond his basic syllogistic, Aristotle reduced the 3rd and 4th first figure syllogisms to second figure syllogisms, thus de facto reducing all syllogisms to Barbara and Celarent; and later on in the Prior Analytics he invokes a type of cut-rule by which a multi-premise syllogism can be reduced to two or more basic syllogisms. From a modern perspective, Aristotle's system can be understood as a sequent logic in the style of natural deduction and as a fragment of first-order logic. It has been shown to be sound and complete if one interprets the relations expressed by the categorical sentences set-theoretically as a system of non-empty classes as follows: AaB is true if and only if the class A contains the class B. AeB is true if and only if the classes A and B are disjoint. AiB is true if and only if the classes A and B are not disjoint. AoB is true if and only if the class A does not contain the class B. It is generally agreed, though, that Aristotle's syllogistic is a kind of relevance logic rather than classical. The vexing textual question what exactly Aristotle meant by ‘syllogisms’ has received several rival interpretations, including one that they are a certain type of conditional propositional form. Most plausibly, perhaps, Aristotle's complete andincomplete syllogisms taken together should be understood as formally valid premise-conclusion arguments; and his complete and completed syllogisms taken together as (sound) deductions.

2.4 Non-modal Syllogistic

2.4 Non-modal Syllogistic :: تعاليق

2.4 Non-modal Syllogistic