Logics for Rigidity

Kripke is famous for holding that when the identity sign is flanked by proper names or natural kind terms, then the result is necessary if true. His conclusion is supported by the idea that proper names and natural kinds terms are rigid designators. To explore the cogency of Kripke’s position, this paper takes on two interlocking projects in the formulation of the semantics for quantified modal logic (QML). Project 1. Define ‘rigidity’ in a way that is faithful to Kripke’s intentions, and for which evidence for the rigidity of proper names and natural kind terms is forthcoming. Project 2. Given a candidate definition of rigidity produced in Project 1, find a well-motivated semantics for QML that validates the claim that ‘t is identical to s’ is necessary if true when terms t and s are rigid in that sense. It is surprisingly difficult to succeed at both projects. Three different definitions for ‘rigidity’ are explored here along with a variety of corresponding QMLs. The few QMLs that survive Project 2 are very different from ones Kripke developed in his seminal paper “Semantic Considerations in Modal Logic”. Furthermore, there are serious costs whichever solution we attempt.