Abstract
Quantifier variance entails that ‘there exists’ has a variety of meanings. Determining what makes all these meanings quantifier meanings is a problem associated with this view (the problem of meaning variance). A reasonable candidate suggested by Hirsch is the set of formal rules governing quantification. However, the collapse argument presents a notorious objection to the viability of the candidate: there cannot be more than one quantifier obeying the same rules up to logical equivalence. It is proposed that a quantifier variantist who intends to retain Hirsch’s proposed solution can admit a many-sorted language in order to block the collapse argument. Within the many-sorted language, a Kaplanian thesis about the rules governing sorted quantifiers saves quantifier variance from meaning variance.
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Notes
Within the relevant literature there is an agreement on the regimentation of existence as existential quantifier but not, e.g. as a predicate. Here I will not question this common-ground and presuppose it throughout the discussion. All that is said here, arguably, can be paraphrased according to the alternative presumptions such as regimenting existence as a predicate.
Following the literature around quantifier variance, I will use ‘meaning’ and ‘concept’ interchangeably.
Later, Hirsch slightly changed his view and characterized QV more broadly: “[QV] says that there is no uniquely best ontological language with which to describe the world” (2011, xii). There, thus, might be no quantifier expression in some of these languages; hence the name quantifier ‘elimination’ (2011, xiv). Although I maintain (and I think Hirsch does) that these two characterizations are interdependent, I will concentrate on Hirsch’s original characterization of QV.
It is worth noting that this problem is not the same as some other problems (or objections) that have been raised against Hirsch's metaontological views. For example, he says that none of these quantifier expressions are metaphysically privileged. What is the exact meaning of being metaphysically superior, inferior or equivalent? Or he says that alternative languages that use these quantifier expressions provide us with possible perspectives and there are many possible perspectives on ‘the existence of objects,’ which all are adequate for describing the same facts. What does it mean to say that each of these languages is adequate for this end and what is the meaning of ‘fact’? And there are many other actual or possible problems with his metaontological views. The literature around Hirsch's metaontology is full of discussions about such questions and other problems; for instance see Dorr 2005, Eklund 2007, 2009, 2011 Hawthorne 2009, Koslicki 2005, Sider 2009. Here, I discuss none of these problems. The only problem that I am concerned with here is the meaning variance problem.
In the introduction to his collected papers on metaontology Hirsch takes an agnostic stance with regard to MV and claims that nothing important hangs on it (2011, xiv). Peculiarly, he urges there again that quantifier expressions in different languages have something in common: formal roles (Ibid).
Turner (2010, 25–27), discussing the collapse argument against the “ways of being” thesis, proposes that ∃1 and ∃2 obey free-logical, but not classical, rules; therefore, he claims that the argument is not valid. McDaniel (2009) characterizes QV as a species of the “ways of being” thesis which he claims to be “that there is more than one fundamental quantifier expression” (314). If McDaniel were right, Turner’s free-logic route, which was originally proposed for the “ways of being” thesis, would automatically apply to QV; hence there would be no need for the next section's many-sorted logic solution. Fortunately, McDaniel’s claim is not well-motivated. Because first, quantifier variantists claim that there is no fundamental (metaphysically privileged) meaning for quantifier expressions, not that there are more than one such meanings; and second, according to the “ways of being” thesis, domains for quantifier expressions are mutually exhaustive of all entities, while, for the quantifier variantist, these domains are supposed to be alternatives. QV is thus far from the “ways of being” thesis. Yet, Turner (2012, 432–424) gives further considerations for the claim that these two theses are orthogonal. Of course, the disparity between QV and the “ways of being” thesis does not entail that the logic of QV is not free. Maybe it is. My aim here is just to argue that many-sorted logic is suited for QV. There might be more than one logic appropriate for a single thesis.
There is a simplification in the text. There must be a further constraint, namely, no variable from one language could be substituted for a term from another language. Unless, the collapse argument would be rehabilitated: from Fx1, by substitution we have Fx2; then the original line of proof follows. This constraint, however, will be automatically fulfilled later when the underlying formalism made explicit in the next section.
The suggestion is not supposed to work in general; namely in cases that some quantifier expressions obey non-classical rules. As Hjortland (2013) argues, it is questionable whether such a core meaning can be found even for less controversial logical constants such as conjunction. My aim here is less tendentious. The account is suggested for a limited range of many-sorted languages whose quantifiers are governed by classical rules of inference. Actually, quantifier variantists' debate with their opponents falls into this limited range.
Here, I merely consider introduction rules. Full treatment is deferred to the next section.
This is much like Beall and Restall’s abstract statement of Tarski thesis for logical consequence. The generalized Tarski Thesis (GTT) states that: “[an] argument is valid if and only if, in every case in which the premises are true, so is the conclusion” (2006, 29). The similarity is that as GTT is not a real logical consequence relation, abstract rules are not real rules, either. Accordingly, by determining x, GTT results in a real logical consequence and by determining i, abstract rules result in real ones. The parallelism goes further: GTT has some explanatory role for real logical consequences' being logical consequences; likewise, abstract rules have an explanatory role in real rules' being so. These similarities show that the current view is as viable as Beall and Restall's Pluralism.
I have said that the proposed solution for the problem of MV is Hirschian. Actually, there are more affinities between the presented idea and Hirsch's overall metaontological views. Here I mention two. First, Hirsch, following Putnam, claims that there is no God's quantifier (2002, 63). Many-sorted languages confirm this idea, since there may be no quantifier that includes all other quantifiers. Second, Hirsch, pace Lewis, states that the compositional nihilist's quantifier is not contextually, but semantically restricted "because of the semantic rules implicit in [the nihilist's] language" (2005, 76). The familiar semantics of many-sorted languages is an apt model for Hirsch's view, since its semantic rules require the quantifiers to be restricted to sorted domains.
These constraints on Lis and L* are not arbitrary in the context of QV. The sorted domains should be allowed to intersect, because the domain of compositional nihilism and universalism intersect. Or we should assume that these alternative languages share their set of predicate symbols, in order to make room for Hirsch's conciliatory translations (Hirsch 2005). The unified language should be the mere aggregate because there is no interaction between sorted languages; those languages are alternative descriptions of reality. Besides, constants, functions and sorted universal quantifiers can be defined by means of the primitive symbols.
The formalism developed here suggests another solution for the problem of MV. The problem was determining what explains why quantifier expressions in different languages are quantifiers. Here is the new choice: the abstract quantifier. The abstract quantifier has all the features of a Kaplanian character shared by the sorted quantifiers. Firstly, it is a function that takes a sort and gives the relevant sorted quantifier. Secondly, the content of a sorted quantifier is divided into two parts: a character and a sort. And finally, the character of sorted quantifiers is sort-independent and remains constant during various languages. So, the character of sorted quantifiers is what can explain why all sorted quantifiers in different languages are quantifiers.
Does this new explanation amount to a Hirschian variantism?. The answer to this question depends on the presupposed meaning theory of logical constants. If our variantist is a proof-theorist about the meanings of logical constants, he may accept that the sorted introduction and elimination rules governing a quantifier exhaust its meaning. If so, he is probably prepared to concede that the meaning of the abstract quantifier would be exhausted by the abstract introduction and elimination rules. Hence the new explanation consists of nothing more than the proposed Hirschian variantists' claim that the rules governing the sorted quantifiers have the same character.
McDaniel (2009) interprets this priority as being a more natural quantifier. Appealing to the minimal conceptual priority is adequate for the current argument to go through, however.
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Acknowledgements
Thanks to Eli Hirsch and Matti Eklund for helpful correspondence, to Mohsen Haeri for constructive suggestions, and to the anonymous referees who provided many fruitful comments on earlier versions of this paper.
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Hosseini, D. Quantifier Variance Without Meaning Variance. Axiomathes 30, 313–325 (2020). https://doi.org/10.1007/s10516-019-09457-8
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DOI: https://doi.org/10.1007/s10516-019-09457-8