Quantified Quinean \(S5\) (Q1310623)

The modal system \(N5\) developed here is Quinean in the sense that it treats necessity, not as an object language operator on formulas, including open formulas, but rather as a metalinguistic predicate applied to terms referring to syntactical objects. \(N5\), however, responds to Quine's third grade of modal involvement, in which there is quantification into modal contexts, by allowing formulas of the form \(\forall x N[\varphi x]\), where \(N\) is the necessity predicate and \([\forall x]\) is a singular term referring to an object language formula \(\varphi x\) with \(x\) free. Since, however, `\(x\)' as it occurs in `\([\varphi x]\)' is not a variable but a part of a closed term referring to a formula containing a variable, this is not the full-blooded quantifying into modal contexts that Quine despaired of. Rather it amounts to quantifying over syntactical structures. Nevertheless, something of the effect of quantifying in is achieved through a Matesian quasi-substitutional interpretation of the quantifier. Within the framework of \(N5\), necessity is as validity in arbitrary sets of first- order models, which is an analog for the Kripkean interpretation of necessity as truth throughout arbitrary sets of possible worlds. Given that, it is then proved that \(N5\) is formally equivalent to standard quantified \(S5\), and hence that the syntactical predicate approach to necessity is able to do everything the operator can do, while remaining thoroughly extensional. Montague, however, demonstrated that such syntactical interpretations of modality are liable to inconsistency, at least in any language rich enough to contain arithmetic. That inconsistency is avoided in \(N5\) through a strict hierarchical control on the formation of names of formulas.