Intensional logic in extensional language

Abstract

Frege's conception of reference has been interpreted as calling for a completely extensional language, on the grounds that according to it the reference of a complex expression is a function of the references of its parts. To meet this demand and still allow for oblique contexts, one uses Frege's idea that in an oblique context an expression has as reference what is ordinarily its sense.A purely extensional formalized intensional logic was constructed in the 1940s by Alonzo Church: his logic of sense and denotation [2], [3]. The ambiguity of the “direct” and “oblique” reference of an expression is avoided. For each denoting expression A there is an expression A denoting its sense, which then replaces A in “oblique” contexts. A consequence of this approach is that for each natural number n, in a context with n-fold embedding of intensional operators A is paraphrased by the result of iterating the operation à n times, so that A is treated as if it were infinitely ambiguous. Church's underlying logic is a simple theory of types, in fact just an expansion of the extensional typed λ-calculus so as to contain ωth-order predicate logic.In this paper we study a variant of Church's logic that admits a possible-worlds semantics, i.e. a version of Church's Alternative (2) [3, pp. 4–5], [4]. This type of logic has been perspicuously formulated by David Kaplan [11], [12], and we follow his version.In §1 we formulate Kaplan's logic L-Church-O with two alterations: the typed λ-calculus is replaced by combinators, and unlike Kaplan (but like Church) we allow concepts (that is, senses) that are not concepts of anything.