Rigidity without trans-world-identity?
I wonder how rigidity can be characterized without begging the question against a lot of good semantic theories.
Usually, a rigid expression is defined as an expression which has the same extension in all possible worlds (that is, as an expression with a constant intension, or C-intension).This characterization presupposes literal trans-world-identity between extensions, which is bad, since it carries a commitment to precise essences of individuals on the one hand and (presumably abundant) universals as extensions of predicates on the other, thereby ruling out counterpart theories and accounts on which tropes or classes are the extensions of predicates.
So we'd better say that the intension of a rigid expression specifies for every possibility an extension that is C-related to the actual extension, where C may or may not be identity, and may or may not be an equivalence relation. Now we need another constraint on C to distinguish rigidity from non-rigidity.
Consider rigid 'tiger'. Its intension returns for every possibility w the species in w that is species-identical with the actual species tiger. (If there is no such species in w, it returns undefined.) Likewise, the intension of rigid 'water' returns for every possibility w the substance in w that is substance-identical with the actual substance water. The intension of rigid 'Feynman' returns for every possibility w the person in w that is person-identical with the actual person Feynman. And so on. What is common to all these quasi-identities? (They resemble Geach's 'relative identities'. But when A and B are counterparts of each other, they are person-identical and yet arguably not the same person.)