Counterpart theory as a semantics for modal logic

Abstract

Counterpart theory is supposed to provide a semantics for intensional languages. In counterpart theory de re modality is represented in terms of an object's counterparts in other worlds. Unlike our standard possible worlds semantics, an object and its counterparts are not identified with one another. This is because the counterpart relation is only a relation of similarity. I argue that the counterpart relation has not been shown to be a relation of similarity and that an analysis of the adequacy of counterpart theory as a semantical theory should rely solely on the consequences arising from the non-transitivity and non-symmetry of the counterpart relation. Furthermore, I argue that this requires formal analysis. I show that basic modal principles, such as K and L(α ˆ β)  Lβ, fail in counterpart theory. I also show that supposing an individual to have more than one counterpart in a given world results in the invalidity in our modal language of either universal instantiation or L(α ˇ ~α). The solution which I postulate is that every object has exactly one counterpart in every world, or in terms of a semantics involving an accessibility relation, every object has exactly one counterpart in every world to which it has access. I investigate the problem of proving such a modal logic to be complete. I show logics which I call KC+ and TC+ to be complete. The difference between the revised counterpart theoretic semantics and standard semantics for modal logics continues to lie in the non-transitivity and non-symmetry of the counterpart relation. If we want to identify an object with its counterparts, as in our standard semantics, the result is an indexical account of an individual. Relative to a world, we may regard an individual as the sum of its counterparts.