Predication Versus Membership in the Distinction Between Logic as Language and Logic as Calculus

Abstract

Two types of framework are distinguished regarding the nature of logic and the logical analysis of natural language. In the first, logic is a calculus subject to varying set-theoretic interpretations over domains of varying cardinality, and in this sense is based on a theory of membership in a set. This type need not restrict its analyses of natural language to extensional discourse only; e.g., Richard Montague's sense-denotation intensional logic, which has been used to provide analyses of intensional discourse, is really a type-theoretical set theory supplemented with a theory of senses. The analyses this type of framework provides are not entirely satisfactory, however, for reasons related to the way that intensional entities are analyzed in terms of membership in a set. The second type of framework, where logical forms are semantic structures in their own right, is based on predication as described in a formal theory of universals. This type of framework: gives a more adequate analysis of natural language and can be developed in a type-free way without generating the logical antinomies. Also, because a set-theoretic semantics provides only an extrinsic characterization of validity for this type of framework, such a semantics cannot be used to show that the laws of logic of this type of framework must be essentially incomplete.