Quantified modal logics combine quantifiers (∀ for all, and ∃, for some) with an intensional operator □ (for such expressions as 'necessarily' and 'Ralph believes that'). Quantifying into intensional contexts (or quantifying in, for short) occurs when a quantifier binds an open variable that lies within the scope of □, as in sentences with the form ∃□Fx. Systems of quantified modal logic (QML) routinely include formulas of this kind, but Willard Van Orman Quine (1963) famously argues that quantifying in is incoherent.

Here is a quick summary of his main line of reasoning. Consider (1)–(3), an apparent counterexample to the law of substitution for identity:

(1) 9 equals the number of planets

(2) Necessarily 9 is greater than 7

(3) Necessarily the number of planets is greater than 7

Although (3) is the result of the substituting 'the number of planets' for '9' in (2), and both (1) and (2) are true, (3) is presumably false. Quine calls term positions where...