1 2 Quantifier Variance Dissolved 3 4Q1 SUKI FINN AND OTÁVIO BUENO 5 6 7 Abstract 8 Quantifier variance faces a number of difficulties. In this paper we first formulate the 9 view as holding that the meanings of the quantifiers may vary, and that languages 10 using different quantifiers may be charitably translated into each other. We then 11 object to the view on the basis of four claims: (i) quantifiers cannot vary their 12 meaning extensionally by changing the domain of quantification; (ii) quantifiers cannot vary their meaning intensionally without collapsing into logical pluralism; 13 (iii) quantifier variance is not an ontological doctrine; (iv) quantifier variance is 14 not compatible with charitable translation and as such is internally inconsistent. In 15 light of these troubles, we recommend the dissolution of quantifier variance and 16 suggest that the view be laid to rest. 17 18 19 20 1. Formulating Quantifier Variance 21 22 In contemporary metaontological discussions, quantifier variance is 23 the view according to which there is no unique best language to de- 1 24 scribe the world. Two equivalent descriptions of the world may 25 differ for a variety of pragmatic purposes, but none is privileged as 26 providing the correct account of reality. The view, crucially, involves 27 the recognition that there is variability in the use of the quantifica- 28 tional apparatus of a language, and it is this variation in the quantifi- 29 cational apparatus that provides the multitude of equally correct 30 languages to describe the world. Furthermore, these languages can 31 be charitably translated into one another so that speakers of each 32 may understand those of the other and see how their statements are 33 true in virtue of this translation. The purpose of this view is to 34 deflate ontological disputes into merely verbal ones, where, upon 35 charitable translation, the proponents of each side of an ontological 36 dispute can be seen to have some common ground of agreement. 37 Such ontological debates thus become deflated upon recognition 38 that each disputant speaks truthfully in their own language, using 39 40 41 1 E. Hirsch, Quantifier Variance and Realism: Essays on Metaontology 42 (Oxford: Oxford University Press, 2011). 43 doi:10.1017/S135824611800005X © The Royal Institute of Philosophy and the contributors 2018 Royal Institute of Philosophy Supplement 82 2018 1 Suki Finn and Otávio Bueno 44 their own quantificational apparatus, and that none are truer than 45 another. 46 The main advocate of the quantifier variance position is Eli Hirsch, 47 who gives the following outline of the view: 48 The quantificational apparatus in our language and thought – 49 such expressions as ‘thing’, ‘object’, ‘something’, ‘(there) 50 exists’–has a certain variability or plasticity. There is no neces- 51 sity to use these expressions in one way rather than various other 52 ways, for the world can be correctly described using a variety of 53 concepts of ‘the existence of something’.2 54 55 Hirsch is here clearly influenced by the view of conceptual relativity, 56 advanced by Hilary Putnam, who similarly claims that: 57 The logical primitives themselves, and in particular the notion of 58 object and existence, have a multitude of different uses rather 59 than one absolute ‘meaning’.3 60 61 As we can see from these quotes and their discussion of existence, the 62 quantifier-variance theorist is most interested in providing a variance 63 in the meaning of the existential quantifier ∃ (which we prefer to call 64 the particular quantifier, for reasons that will become clear by the end 65 of this paper). Ordinarily, ∃ is taken to mean ‘some things in the 66 domain’, which is contrasted with the universal quantifier ∀ that is 67 taken to mean ‘all things in the domain’. ∃ has this meaning by 68 ranging over the domain of quantification and when it is applied to 69 an open formula, the resulting sentence is true as long as the open 70 formula is satisfied by something in the domain. We argue that ∃ 71 always has this role, as it invariably has the function of ranging over 72 the domain and signaling that some, rather than none, of its 73 members satisfy the relevant formula. Yet the quantifier-variance 74 theorist requires ∃ to have multiple meanings. But to have a variation 75 in the meaning of ∃ forces the quantifier not only to mean ‘some’, but 76 to have other candidate meanings. This raises the issue of how the 77 meaning of a quantifier can differ, and what the other meanings 78 could be. And it is this issue that we tackle, arguing that one cannot 79 make sense of variation in quantificational apparatus in the way that 80 the quantifier-variance theorist demands. 81 82 83 2 Eli Hirsch, ‘Quantifier Variance and Realism’, Philosophical Issues 12 84 (2002), 51. 85 3 Hilary Putnam, ‘Truth and Convention: On Davidson’s Refutation of 86 Conceptual Relativism’, Dialectica 41 (1987), 71. 2 Quantifier Variance Dissolved 87 2. Domain Variation 88 89 The first attempt to understand quantifier variance is by taking the 90 meaning of quantifiers to be identified by their domains, in other 91 words, by extension. If this were the case, then a difference in 92 members of the domain would result in a difference in the meaning 93 of the quantifier ranging over that domain. This would thus establish 94 quantifier variance via domain variance. A difference in the domain 95 can arise in many ways, including being the result of shifts in the 96 domain’s members (call this ‘domain shifting’), or emerging from re- 97 strictions in the domain (call this ‘domain restriction’). We argue that 98 neither approach is sufficient to change the meaning of the quanti- 99 fiers and how they operate. 100 101 102 2.1. Domain Shifting 103 104 A domain can vary its members by shifting the sort of thing that is 105 being quantified over. For example, we could quantify specifically 106 only over Ps or only over Qs, such that the domain had only Ps as 107 members or only Qs as members. The question is, however, 108 whether this domain shifting, between Ps and Qs, is sufficient for 109 quantifier variance: would we have a different meaning of the quan- 110 tifier for each of these domains, namely, those constrained by ∃P and 111 ∃Q? It seems not, since regardless of what the quantifier ranges over, 112 whether its domain includes objects of kind P or Q, the way the quan- 113 tifier operates is constant and means the same for each domain. 114 In each domain, our quantifier still operates in the same way: by in- 115 dicating that something in that domain satisfies the relevant formula. 116 As such, ∃ will still mean ‘some thing in the domain’, regardless of 117 what sorts of thing it quantifies over. This should be unsurprising, 118 since similarly to other logical operators and connectives, no variation 119 in meaning occurs with changes in what is in the domain. The 120 meaning of the conjunction symbol does not differ relative to what 121 is being conjoined nor does the meaning of the disjunction differ de- 122 pending on what is being disjointed. Likewise with quantifiers, their 123 meaning does not differ relative to what they quantify over. The only 124 difference that emerges in quantifiers as a result of domain shifts is in 125 their range or scope. 126 However, having domains with different members is not enough to 127 secure quantifier variance, since the type of quantification over these 128 various domains is still the same. Those quantifiers play the same 129 roles as they do with different domains, and abide by the same 3 Suki Finn and Otávio Bueno 130 rules as they would with different domains. Regardless of what the 131 quantifier ranges over, the way the quantifier functions remains con- 132 stant. Therefore, domain shifting is insufficient for quantifier vari- 133 ance, given that it only marks a difference in quantificational scope, 134 whilst in each domain the quantifiers still operate in the same way: 135 the universal quantifier ranges over all objects of the domain, the par- 136 ticular over some. 137 138 139 2.2. Domain Restriction 140 141 Alternatively, instead of varying the domain by shifting the type of 142 thing quantified over, such as P or Q, the variation can be achieved 143 by restricting the domain of quantification. A restricted domain pro- 144 vides a way of getting a variation in the domain’s members, and thus 145 alters the scope or range of quantification by taking a subset of a larger 146 domain. For example, a broad (perhaps maximal) domain of Z things 147 can be restricted to a certain type of thing P, and then these things can 148 be restricted again to the Z things that are both P and R (assuming 149 that the extensions of P and R, although overlapping, are not the 150 same). This process yields progressively more constrained domains 151 with each restriction, in light of the increased number of such restric- 152 tions, and assuming that at each stage the extensions of the overlap- 153 ping domains are not the same. For each restricted case, a different 154 quantifier is used: for the broad (perhaps maximal) domain of Z 155 things, one would use the quantifier ∃Z; for the restricted domain 156 of Z things that are also P, one would use the quantifier ∃ZP, and 157 for the even further restricted domain of Z things that are both P 158 and R, one would use the quantifier ∃ZPR.