Identity, Haecceity, and the Godzilla Problem

In standard first order predicate logic with identity it is usually taken that $a=a$ is a theorem for any term $a$. It is easily shown that this enables the apparent proof of a theorem stating the existence of any entity whatsoever. This embarrassing result is a motivation for the construction of free logics, but in most orthodox treatments of first order logic with identity it is generally dealt with by being ignored. We investigate the possibility that this problem can be obviated by dropping the rule that $a=a$ is a theorem and requiring instead that it be treated as a global but in principle defeasible assumption about the objects in the domain of discourse. We propose that any logic in which this is done be called "open." We review some motivations in physics, philosophy, and literature for questioning the classical notion of self-identity, and we show that Carnap's "null object" has a natural role to play in any system of predicate logic where self-identity can come into question.