A nonhybrid logic for crossworld predication

Abstract

Some sentences of natural languages, such as John might have been taller than Mary is, ascribe relations to objects each of which is associated with a possible world. In this example, Mary is associated with the actual world whereas John is associated with a possible world that can be distinct from the actual one. This phenomenon — the phenomenon of crossworld predication — cannot be reflected by means of standard modal logic. Some nonstandard logics accommodating crossworld predication have been proposed; the most expressive among them is Kocurek’s hybrid logic. All of them make use of nonstandard operators such as the operators of hybrid logic. In this paper, I propose a nonhybrid first-order modal logic for crossworld predication, CWPL, that demonstrates significant expressive power (though is less expressive than Kocurek’s hybrid logic) and has that advantage over other proposals that it makes no use of any nonstandard operators. Semantically, CWPL is based on crossworld interpretation of predicates that assigns extensions to each n-ary predicate with respect to n-tuples of possible worlds rather than single possible worlds. To be able to employ crossworld interpretation of predicates when evaluating formulae, I suggest relativizing truth values to partial functions from variables to possible worlds. In the paper, I describe CWPL’s syntax and semantics, provide a translation of CWPL into a two-sorted first-order logic, and compare it with Kocurek’s hybrid logic in terms of expressive power.