Relational Modality

Saul Kripke’s thesis that ordinary proper names are rigid designators is supported by widely shared intuitions about the occurrence of names in ordinary modal contexts. By those intuitions names are scopeless with respect to the modal expressions. That is, sentences in a pair like (a) Aristotle might have been fond of dogs (b) Concerning Aristotle, it is true that he might have been fond of dogs will have the same truth value. The same does not in general hold for definite descriptions. Since the scope property is accounted for by means of the intensions of the names and the descriptions, the conclusion is that names do not in general have the same intension as any normal, identifying description. However, the difference in scope behavior between names and description can be accounted for alternatively by appeal to the semantics of the modal expressions. On the account we suggest, dubbed ‘relational modality’, simple singular terms, like proper names, contribute to modal contexts simply by their actual world reference, not by their intension. It turns out to be fully equivalent with the rigidity account when it comes to truth of modal and non-modal sentence (with respect to the actual world), and hence supports the same basic intuitions. It also turns out to be model theoretically equivalent with rigidity semantics with respect to logical consequence, within the class of reflexive models with non-empty domains. Here we give the truth definition for relational modality models, and prove the equivalence results.