An axiomatization of a logic for crossworld predication

Some sentences of natural languages that cannot be semantically analyzed in terms of standard possible world semantics because they involve a phenomenon that cannot be 'seen' by standard semantics. In the literature, the phenomenon in question is called crossworld predication. This is ascription of relations to objects, each of which is associated with a possible world. An example is John might have been taller than Mary is: this sentence ascribes the relation of being taller to John as he is in a possible world, and Mary as she is in the actual world. The phrase 'x as it is in w' expresses the association of an object x with a possible world w. Semantic analysis of this sort of sentences requires a special sort of interpretation of predicates - crossworld interpretation, i.e. interpretation that assigns extensions to n-ary predicate letters with respect to n-tuples of possible worlds rather than single possible worlds. Thus, if we want to model reasoning in natural languages involving crossworld predication, we need a logic that should be semantically based on crossworld interpretation of predicate letters. In some recent papers, I elaborated such a logic - a crossworld predication logic (CWPL). In CWPL semantics, we are able to employ crossworld interpretation of predicates when evaluating formulae because we evaluate them with respect to partial functions from variables to possible worlds (VP-functions). Thus, crossworld interpretation of predicates and relativization of truth values of formulae to VP-functions are features of CWPL semantics that distinguish it from standard semantics. CWPL is a first-order modal logic with individual constants, equality and lambda-operator. So far, I presented its semantics and a tableau proof theory for it. In the present article, a Hilbert-style proof theory for a simplified version of CWPL is proposed. The simplifications are as follows: the logic in question (CWPLf) is without individual constants, equality and quantifiers. Besides, CWPL1 is based on propositional modal logic D, whereas CWPL is based on K. To establish its completeness, a version of the method of canonical models is elaborated. I am grateful to I.I. Borisova for editorial assistance. I am grateful to I.I. Borisova for editorial assistance. The author declares no conflicts of interests.

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