Abstract
The standard truth-conditional semantics for substitutional quantification, due to Saul Kripke, does not specify what proposition is expressed by sentences containing the particular substitutional quantifier. In this paper, I propose an alternative semantics for substitutional quantification that does. The key to this semantics is identifying an appropriate propositional function to serve as the content of a bound occurrence of a formula containing a free substitutional variable. I apply this semantics to traditional philosophical reasons for interest in substitutional quantification, namely, theories of truth and ontological commitment.
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Notes
Soames’s point is directly analogous to a well-known observation about rigid designation. To say that a rigid designator designates the same object in every possible world in which that object exists (and nothing else in any world where the object does not exist) is to say that the rigid designator, as we actually use it does this (Kaplan 1989, pp. 493–494). It would be a mistake to conclude that the proper name ‘Tally’ fails to rigidly designate my border collie on the grounds that there are possible worlds in which I use the name to refer to some other dog.
Christopher Hill (1999, pp. 101–102) gives this argument in favor of substitutional quantification in natural language. Joseph Camp (1975) discusses a different kind of example: ‘there are things I have always wanted that don’t exist’. Many of the points I raise at the end of the paper about (2) can be made about Camp’s example. For a different view of examples like Camp’s, see Priest (2005).
It is not strictly required for the example that Tally be the only dog that exists. All that is strictly required is (i) that Tally exists, and (ii) that no existing dog be named ‘Tally’. But I find that the case in which Tally is the only dog that exists and has no name makes it easier to focus on the relevant intuition.
I call this an ‘intuitive’ rule of necessitation to distinguish it from the rule Necessitation stated for formal languages of modal logic.
I have not determined whether all those who lodge this version of the objection have read Van Inwagen, but it is just how I would imagine he would protest. See note 2.
Perhaps the most extreme variation on this objection is Lycan’s (1979) claim that the particular substitutional quantifier is ‘semantically mute’. According to Lycan, we don’t understand sentences like (1) because there is simply nothing that they say, or no proposition that they express. The semantics I present in Sect. 4 of this paper directly answers Lycan’s claim.
I thank an anonymous referee for this observation.
Thanks to Brian Cutter for discussion.
Again, it is not strictly required that Tally be the only dog that existed. See note 4.
For discussion of the notion of logical consequence relative to a context, see Georgi (2014).
The discussion above exposes a tension in our understanding of Sub. Strictly, this argument requires a modified version of Sub along the following lines:
$$\begin{aligned} \ulcorner {\varSigma \text{x}} \ \phi (\text{x}) \urcorner \text{ is } \text{ true } \text{ in } c \text{ if } \text{ and } \text{ only } \text{ if } {\exists \text{t}} (\phi \text{(x/t) } \text{ is } \text{ true } \text{ in } c) \end{aligned}$$(Sub_{c})Yet with this modified version in place, it is unclear what role the truth of (3) relative to a context plays in the evaluation of (1). A related issue is addressed in Objection 4 below.
Thanks to an anonymous referee for helping me to see the issues raised in this aside.
Thanks to Michael Kremer for discussion.
Note that the most plausible way of understanding the reasoning in the first objection is that it fallaciously moves from a true claim based on Sub\(_{2}\) (that if (4) is true at \(w\), then there is some term \(t\) such that \(\ulcorner t\,\text{is}\,\text{a}\,{\text{dog} \urcorner}\) is true at \(w\)) to a false claim apparently based on Sub\(_{1}\) (that (3) is true at \(w\)).
That substitutional quantification is not merely metalinguistic objectual quantification has been emphasized in the literature on substitutional quantification since at least Dunn and Belnap (1968, pp. 184–185). (Though Kripke at least hints that he would accept such an interpretation of substitutional quantification (1976, p. 356).)
According to the rule Sub\(_{2}\), a substitutionally quantified sentence like (1) is necessarily equivalent to the disjunction of all of its substitution instances. See note 21 for more discussion of this observation.
Several of the issues raised in the objections and responses are discussed by Soames in the Appendix of Chapter 3 of Understanding Truth (1999).
Two notes about the rule PS: (i) I adopt a neo-Russellian picture of structured propositions, but that does not mean that I identify propositions with ordered \(n\)-tuples. Rather, \(n\)-tuples stand in for propositions in the statement of the rule. For a selection of current views of structured propositions, see King et al. (2014). (ii) A consequence of PS is that (1) expresses different propositions relative to different substitution classes. But this seems to me to be the right result (or at least not obviously the wrong result), similar to the different proposition expressed by ‘there is no beer’ relative to different domain restrictions in different contexts.
Example: the proposition expressed by ‘\(\forall y \exists z \forall r (r=\) ‘apple’)’ is \({ \left\langle \text{EVERY}, h \right\rangle}\), where for any \(o\), \(h(o)\) is the proposition \(\left\langle \text{SOME,}\,h^{\prime } \right\rangle\), where for any \(o\), \(h^{\prime }(o)\) is the proposition \(\left\langle \text{EVERY},\,h^{\prime \prime } \right\rangle\), where for any \(o\), \(h^{\prime \prime }(o)\) is the proposition that \(o\) = ‘apple’. The propositional function \(h^{\prime \prime }\) is metalinguistic because for any \(o\), \(h^{\prime \prime }(o)\) contains the expression ‘apple’. The propositional function \(h^{\prime }\) is metalinguistic because for any \(o\), \(h^{\prime }(o)\) contains the metalinguistic propositional function \(h^{\prime \prime }\). The propositional function \(h\) is metalinguistic because for any \(o\), \(h(o)\) contains the metalinguistic propositional function \(h^{\prime }\). The proposition \(\left\langle \text{EVERY},\,h \right\rangle\) is metalinguistic because it contains the metalinguistic propositional function \(h\).
An alternative proposal, suggested by an anonymous referee, is to identify the proposition expressed by (1) with the (perhaps infinite, given an infinite substitution class) disjunction of the propositions in the range of \(g^{*}_{S}\). This proposal also satisfies desiderata (i)–(iii). I see at least two disadvantages to this proposal: first, it is questionable whether, on this account, we could understand a sentence like (1), if understanding a sentence requires grasping the proposition that it expresses. We would need some account of what it is to grasp an infinite proposition. Second, on this view the form of the proposition expressed by S is nothing like the form of the sentence that expresses it.
It is this point, I suggest, more than any other, that Van Inwagen (see note 2) fails to understand about substitutional quantification:
I shall assume at the outset that my readers agree with me on one fundamental point: it is not the case that there is something called ‘the existential—or particular—quantifier’ of which philosophers of logic have offered two ‘interpretations’ or ‘readings’ , viz. the objectual or referential interpretation and the substitutional interpretation. It would be better to say that there are two quantifiers, two distinct variable-binding operators: the objectual or referential and the substitutional. (1981, p. 281)
I do not agree. On the view defended in this paper, neither option is correct. There is just one particular quantifier, with one interpretation, but two distinct kinds of variables to be bound.
Thanks to David Braun for suggesting to me the rule PR.
One might worry that the proposition \(\left\langle \text{SOME}_{S},\,g^{\prime \prime } \right\rangle\) shows that our earlier definition of ‘metalinguistic proposition’ is in fact inadequate. For while the proposition satisfies that definition, it may seem intuitively metalinguistic insofar as SOME\(_{S}\) includes some kind of objectual quantification over expressions. But one might also think that substitutional quantification is metalinguistic in this sense.
Note that in order to maintain contact with these traditional debates, I will frame the discussion in terms of two quantifiers ‘\(\exists\)’ and ‘\(\varSigma\)’. This is despite the consequence above that there is really only one particular quantifier.
It may be tempting to respond to this objection to (11) as follows: the content of ‘\(\varSigma\)’ in PS is the property of being a function that maps at least one thing to a true proposition, but there are other neo-Russellian semantic treatments of quantification according to which the content of a quantifier is the property of being a non-empty set (e.g. Soames (1987, p. 73)). Understanding this property does not require grasp of the concept of truth. But this proposal has the unfortunate consequence that (1) is about sets, contrary to our linguistic intuitions.
Alex Orenstein (1984) has also called the ontological neutrality of substitutional quantification into question, on grounds very much like those raised here. One must approach his argument with care, however, as Orenstein is not always careful to distinguish between quantifiers, occurrences of quantifiers, and sentences in which quantifiers appear. In the following passage, he attributes what he calls ‘referential force’ to a whole sentence, but elsewhere he attributes referential force to quantifiers (or uses of quantifiers):
The referential force and ontological import of ‘[(\(\varSigma\)x) x is a Siamese cat]’ would not be due to [Sub] but to the referential aspect of the truth condition for the instances it depends upon. (p. 147)
It seems to me that Orenstein is here concerned about the (primary) occurrence of ‘\(\varSigma\)’ in ‘(\(\varSigma\)x) x is a Siamese cat’. Thus I take Orenstein to be arguing for the same claim as I have stated above.
See also Kripke (1976, p. 333).
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Acknowledgments
I want to thank David Braun, Ben Caplan, Brian Cutter, Michael Glanzberg, Aaron Griffith, Michael Kremer, Scott Soames, two anonymous referees for Philosophical Studies, and the audiences of the 2012 Illinois Philosophical Association, the 2013 Pacific APA, and the 2013 Pittsburgh Area Philosophy Colloquium for comments, suggestions, and feedback.
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Georgi, G. A propositional semantics for substitutional quantification. Philos Stud 172, 1183–1200 (2015). https://doi.org/10.1007/s11098-014-0343-7
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DOI: https://doi.org/10.1007/s11098-014-0343-7