Logic of existence judgements and syllogistic

Abstract

We set out a formal system for logical analyses of existence judgements. Its language contains the constant of existence, atomic formulas are formed by the concatenation of this constant with any finite sequence of general terms (positive and negative). Complex formulas are formed by means of propositional connectives. We formulate a natural semantics for this language. The extension of a positive general term in a model is a subset of the domain, the extension of a negative general term is the complement to the extension of the respective positive term. An atomic formula is valid in a model iff the intersection of the extensions of all terms in it is non-empty. We set out an axiomatization of the set of generally valid formulas. This calculus formulates on the base of classical propositional logic. We notice that categorical judgements can be defined with the aid of existence judgements and raise a question about metatheoretic interrelations among our logic and different syllogistic theories. We prove that the logic of existence judgements is recursively equivalent to the syllogistic with indefinably-placed constant which is the generalization of constant a. We formulate the embedding translations from each of these two systems to another. Also, we prove soundness and completeness theorems for the calculus of existence judgements.