In 'Three grades of modal involvement', W. V. Quine proposes that 'necessary' be construed as a predicate in natural languages, and not as the sentential operator of standard modal logics. 1 As a predicate, 'necessary' will attach to the names of sentences or propositions, rather than attaching directly to sentences or propositions themselves. But David Kaplan and Richard Montague show that a language with a finitely axiomatizable syntax that permits self-reference in attributions of necessity can give rise to semantical paradox. If the necessity predicate may be iterated, and if it satisfies counterparts of the axioms of the modal system T, it is possible to construct a modal version of the Hangman Paradox. 2
Quine argues that if we interpret 'necessary' as a predicate true of sentences, we will be unlikely to want to iterate it. 3 But if natural languages serve as their own metalanguages, as they appear to, only a very narrow view of necessity can justify ruling out such a move altogether. Recent work by Saul A. Kripke and others, though, suggests that there may be a more satisfactory solution to the modal Hangman than an ad hoc prohibition on iteration. They show that it is possible to circumvent the difficulties of the Liar and related paradoxes in languages containing their own truth predicates by interpreting these languages with a truth-value-gap semantics. 4 Under an interpretation with truth-value gaps, paradoxical sentences will be neither true nor false. Extending this technique promises a way around the modal paradox in languages containing their own necessity predicates, s
Up until now, at least one very good reason to refuse to accept Quine's proposalthat we construe 'necessary' as apredicate in natural languages has been the difficulty about the modal Hangman. One very good reason to accept Quine's proposal if truth-value-gap semantics does offer a solution to the paradox is that necessary truths are truths. Other things being equal, an optimally simple semantics for a natural language will treat truth and necessity similarly. If the primary, non-derivative objects of truth are taken to be Philosophical Studies 37 (