# Modality and Quantification

# MODALITY AND QUANTIFICATION

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 substitution fails opaque contexts and argues that terms occupying them do not play their normal referring roles. Both '9' and 'the number of planets' refer to nine, so something other than the terms' referents must explain why the truth values of (2) and (3) differ. Presumably, the difference is in the manner of referring to or describing nine. Now note that the standard truth condition for ∃ says that (4) is true if and only if (iff) the open sentence (5) is true of some object:

(4) ∃x(necessarily, x is greater than 7)

(5) Necessarily, x is greater than 7

However, (5) results from putting 'x' for either '9' in (2) or 'the number of planets' in (3), and (2) and (3) were sensitive to the manner in which nine is described. Since 'x' does not describe anything at all, information needed to make sense of (5) being true of an object is now missing. As Quine puts it, what object is (5) true of? Presumably, it is nine, that is, the number of planets. However, the number of planets appears not to satisfy (5), since (3) was false.

Arthur F. Smullyan (1948) was one of the first to respond to Quine's argument. He notes that when 'the number of planets' is translated away according to Bertrand Russell's theory of definite descriptions, (1)–(3) does not constitute a violation of the law of substitution (LS). On the analysis that matches the intuition that (3) is false, it is not possible to derive the translation of (3) from (2) and the translation of (1) in predicate logic, even given LS. If one adopts the position that any purported failure of substitution for an expression is a good reason to treat it as a definite description, then there are no terms in opaque contexts in the first place, and Quine's reasoning does not get off the ground. However, this solution, Quine notes, is limited to those cases where Russell's technique can be plausibly applied.

Alonzo Church (1943) and Rudolf Carnap (1947) propose a different tactic. Presuming that variables of quantification range over concepts rather than objects, Quine's complaint that satisfaction of (5) by an object is unintelligible does not apply. However, Quine finds quantification over concepts ontologically disreputable; and furthermore, citing an alternative treatment of quantification would not rebut an argument concluding the incoherence of quantifying in for quantification over objects, a result damaging enough to QML.

There are a number of different strategies for responding to Quine's objection in the case of quantifying over objects. One popular tactic, exemplified in David Kaplan's "Quantifying In" (1969), involves selecting a privileged class of terms (for Kaplan, the so-called vivid names). Although the truth values of (3) and (2) are sensitive to the ways nine is described, one argues that there is no corresponding indeterminacy in (5) because one of these ways is privileged. Presuming '9' is privileged, (2), and not (3), is used to resolve the status of (5). Since (2) is true, (5) is true of nine, and the fact that that (3) is false is irrelevant.

In note 3 of "Quantifying In" (1969) Kaplan suggests another way to circumvent Quine's objections to (5) without using privileged terms. The idea is (roughly) to revise the truth condition for ∃ so that ∃x(necessarily, x is greater than 7) is T iff some object satisfies the open sentence (6):

(6) x bears the property of being necessarily greater than seven

Since 'x' in (6) lies outside the scope of 'necessarily', substitution holds in this position, and Quine's worries no longer apply. (Something like this tactic is used by Quine himself in "Quantifiers and Propositional Attitudes" [1955] to analyze quantification into belief contexts.)

Kaplan's (1969) strategy is reflected in a solution implicit in the earliest published QML. The system (developed by Ruth Barcan Marcus [1946]) includes the axiom ∀x∀y(x = y → □x = y), which is now known to correspond to the condition that variables are rigid designators, that is, they pick out the same object in every possible world. Under these circumstances (5) is equivalent to (6), and so (6) can be used to make sense of (5).

Kit Fine's "The Problem of De Re Modality" (1989) makes yet another contribution to the problem. Here a formal definition of satisfaction by objects for open sentences like (5) is provided in cases where 'necessarily' indicates logical or analytic necessity.

In "A Backward Look at Quine's Animadversions on Modalities" (1990) Marcus records how the force and variety of such criticisms of Quine's argument led him to a strategic retreat. He conceded that quantifying in is at least coherent, but raised a different objection. Quine perceived early on that attacks on his argument appear to pay a serious price. Appeals to privileged ways of describing things, to rigid designators, or to the cogency of (6) boil down to having to make sense of the idea that some objects bear necessary properties that other objects do not. Quine complains that this amounts to an unacceptable form of essentialism. What sense can it make to assert of an object itself (apart from any way of describing it) that it has necessary properties?

An influential response to this worry appears in the early pages of "Naming and Necessity" (1972), where Saul Kripke undermines Quine's presumption that it only makes sense to attribute necessary properties to an object under a description. Here, the focus shifts from brands of logical or analytical necessity, which were the main concern when Quine first wrote, to metaphysical or physical necessity. Kripke defends the view that objects in themselves do have essential properties. For example, molecules of water are necessarily composed of hydrogen and oxygen, because water just is H_{2}O.

Kripke and others rescued some brands of essentialism from the negative reputation it had when Quine first wrote. However, one need not respond to Quine by arguing for the coherence of a robust essentialism. In "Opacity" (1986) Kaplan argues that the essentialism produced by quantifying in is so weak as to be entirely innocuous. Terence Parsons, in "Essentialism and Quantified Modal Logic" (1969), reports the technical result that sentences of QML that express a controversial essentialism will not be theorems, nor will they be derivable from any collection of premises expressing (nonmodal) facts.

Parsons (1969) and others point out that while quantifying in allows one to assert essentialist claims, this hardly qualifies as a reason for abandoning it. QML should provide an impartial framework for analyzing and evaluating argumentation on all philosophical positions, however misguided. That quantifying in provides resources to express (even the most obnoxious) essentialism is a point in its favor. In any case, Quine's complaint that QML is essentially essentialist amounts to a retraction of the view that quantifying in is (literally) incoherent, for if that were true, quantifying in would not entail essentialism, it would express nothing at all.

It is important to note that Quine's main argument against quantifying in would appear to apply equally well to expressions for propositional attitudes such as "Ralph believes that," for these also create opaque contexts. However, in the case of belief, the situation is different, since charges of essentialism are out of place. In "Intensions Revisited" (1981) Quine explores failings for belief that are analogs to essentialism for necessity.

Despite attacks on Quine's main argument, many still accept the conclusion that quantifying in is incoherent. Graeme Forbes (1996) notes that adherents of this view face a new puzzle, posed by strong intuitions in favor of the intelligibility of English sentences like those represented by ∃x(Ralph believes that x is a spy). So those adherents need an alternative analysis of the logical form of propositional attitude sentences that avoids quantifying in, one Forbes sets out to provide. A tension Quine faces here is that explanations placating intuitions that quantifying in is coherent for belief will provide tools that resolve his worries about necessity.

QML has come a long way in the sixty years since Quine first launched his attack on it. Possible worlds semantics has flourished, bringing a wealth of technical results. For example, soundness and completeness have been proven for a variety of systems that allow quantifying in but reject LS in modal contexts. Theorems are also available on exactly how and where essentialist features arise in QML (e.g., see Fine 1978, 1981). Though work in modal semantics employs ideas that are anathema to Quine (notably the notion of a possible object), it provides tools for better understanding worries about quantifying in. An interest in answering Quine's objections to QML has motivated many of these developments. So, oddly, Quine's legacy has enriched what he hoped to disinherit.

** See also ** Modal Logic.

## Bibliography

### quine's seminal articles on quantifying in

"Notes on Existence and Necessity." *Journal of Philosophy* 40 (1943): 113–149.

"Quantifiers and Propositional Attitudes." In *Reference and Modality*, edited by Leonard Linsky, 101–111. New York: Oxford University Press, 1955.

"Reference and Modality." In *From a Logical Point of View*. New York: Harper and Row, 1963.

"Three Grades of Modal Involvement." In *The Ways of Paradox and Other Essays*. Cambridge, MA: Harvard University Press, 1966.

### some other important works on quantifying in

Barcan, Ruth. "A Functional Calculus of First Order Based on Strict Implication." *Journal of Symbolic Logic* 2 (1946): 1–16.

Carnap, Rudolf. *Meaning and Necessity*. Chicago: University of Chicago Press, 1947.

Church, Alonzo. "Review of Quine." *Journal of Symbolic Logic* 8 (1943): 45–52.

Fine, Kit. "Model Theory for Modal Logic." *Journal of Philosophical Logic* 7 (1978): 125–156, 277–306; 10 (1981): 293–307.

Fine, Kit. "The Problem of De Re Modality." In *Themes from Kaplan*, edited by Joseph Almog, John Perry, and Howard Wettstein. New York: Oxford University Press, New York, 1989.

Forbes, Graeme. "Substitutivity and the Coherence of Quantifying In." *Philosophical Review* 105 (1996): 337–372

Kaplan, David. "Opacity." In *The Philosophy of W. V. Quine*, edited by Lewis Edwin Hahn and Paul Arthur Schilpp, 229–289. LaSalle, IL: Open Court, 1986.

Kaplan, David. "Quantifying In." In *Words and Objections*, edited by Donald Davidson and Jaakko Hintikka. Dordrecht, Netherlands: D. Reidel, 1969.

Kripke, Saul. "Naming and Necessity." In *Semantics of Natural Language*, edited by Donald Davidson and Gilbert Harman, 253–355. Dordrecht, Netherlands: D. Reidel, 1972.

Marcus, Ruth Barcan. "Essentialism in Modal Logic." *Noûs* 1 (1967): 91–96.

Marcus, Ruth Barcan. "A Backward Look at Quine's Animadversions on Modalities." In *Perspectives on Quine*, edited by Robert B. Barrett and Roger F. Gibson. Cambridge, MA: Blackwell, 1990.

Parsons, Terence. "Essentialism and Quantified Modal Logic." *Philosophical Review* 78 (1969): 35–52.

Quine, W. V. O. "Intensions Revisited." In *Theories and Things*. Cambridge, MA: Harvard University Press, 1981.

Smullyan, Arthur F. "Modality and Description." *Journal of Symbolic Logic* 13 (1948): 31–37.

*James W. Garson (2005)*

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