Timothy Smiley

What is a syllogism

Lukasiewicz rejected the traditional treatment of syllogisms as arguments and claimed that the authentic Aristotelian syllogism is a conditional whose antecedent is the conjunction of the premisses and whose consequent is the conclusion.1 For Aristotle, however, a syllogism is essentially something with a deductive structure as well as premisses and conclusions. Consider, for example, his distinction between ostensive and per impossibile syllogisms. This is entirely a matter of how their conclusions are derived and not at all a matter of what conclusions are derivable (An. Pr. 45a26, 62b38). Aristotle writes as if he is marking a genuine distinction between two classes of syllogisms, but his way of going about it would be senseless if a syllogism were uniquely determined, as a Lukasiewiczian conditional is, by its premisses and conclusions. Moreover, everything suggests that Aristotle is concerned here with the distinction between direct and indirect patterns of deduction, as exemplified in the contrast between the ostensive argument ‘P, Q, so R’ and the per impossibile one, ‘P, suppose not R, then not Q, so R’.2 Eukasiewicz may be equally ready to distinguish between different patterns of deduction, but for him this will as yet have nothing to do with syllogism: syllogisms for him have no more intrinsic connection with deduction than any other conditionals, and in particular Aristotelian ‘demonstrations’ are mere conditionals and not, ironically, proofs of anything. Thus if Eukasiewicz’s treatment is to accommodate Aristotle’s distinction, he must both show that the distinction makes sense when interpreted as applying to conditionals and that this sense is such as to establish some connection between ostensive and per impossibile conditionals and ostensive and per impossibile deduction. He attempts neither task, and perhaps it is sufficient to give a bare indication of the difficulties he would have to overcome. If we interpret the distinction as applying to conditionals, then, as was said at the beginning, we must not think that we are dividing conditionals as such into two classes; the grounds for calling a conditional ostensive or per impossibile must be sought outside

A Modest Logic of Plurals

We present a plural logic that is as expressively strong as it can be without sacrificing axiomatisability, axiomatise it, and use it to chart the expressive limits set by axiomatisability. To the standard apparatus of quantification using singular variables our object-language adds plural variables, a predicate expressing inclusion (is/are/is one of/are among), and a plural definite description operator. Axiomatisability demands that plural variables only occur free, but they have a surprisingly important role. Plural description is not eliminable in favour of quantification; on the contrary, quantification is definable in terms of it. Predicates and functors (function signs) can take plural as well as singular terms as arguments, and both many-valued and single-valued functions are expressible. The system accommodates collective as well as distributive predicates, and the condition for a predicate to be distributive is definable within it; similarly for functors. An essential part of the project is to demonstrate the soundness and completeness of the calculus with respect to a semantics that does without set-theoretic domains and in which the use of set-theoretic extensions of predicates and functors is replaced by the sui generis relations and functions for which the extensions were at best artificial surrogates. Our metalanguage is designed to solve the difficulties involved in talking plurally about individuals and about the semantic values of plural items.