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Absolute Generality and Semantic Pessimism

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Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language

Part of the book series: Synthese Library ((SYLI,volume 373))

Abstract

Semantic pessimism has sometimes been used to argue in favour of absolutism about quantifiers, the view, to a first approximation, that quantifiers in natural or artificial languages sometimes range over a domain comprising absolutely everything. Williamson argues that, by her lights, the relativist who opposes this view cannot state the semantics she wishes to attach to quantifiers in a suitable metalanguage. This chapter argues that this claim is sensitive to both the version of relativism in question and the sort of semantic theory in play. Restrictionist and expansionist variants of relativism should be distinguished. While restrictionists face the difficulties Williamson presses in stating the truth-conditions she wishes to ascribe to quantified sentences in the familiar quasi-homophonic style associated with Tarski and Davidson, the expansionist does not. In fact, not only does the expansionist fare no worse than the absolutist with respect to semantic optimism, for certain styles of semantic theory, she fares better. In the case of the extensional semantics of so called ‘generalised quantifiers’, famously applied to natural language by Barwise and Cooper, it is argued that expansionists enjoy optimism and absolutists face a significant measure of pessimism.

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Notes

  1. 1.

    See, for instance, Williamson [48, sec. II]. Kent Bach [1] provides an opposing view; see Stanley and Szabó [44] for discussion.

  2. 2.

    Lewis [21], McGee [28] and Williamson [48, §V] press the concern that relativism cannot be coherently stated. Glanzberg [14], Fine [12] and Lavine [19] reply on behalf of the relativist.

  3. 3.

    See, for instance, Cartwright [7, p. 7], Williamson [48, p. 425], and Rayo and Uzquiano [39, p. 6].

  4. 4.

    Compare Boolos [5, pp. 223–4].

  5. 5.

    To avoid such commitment, the absolutist needs to maintain, contrary to Quine [34, 35], that plural reference to and quantification over objects in English is not disguised singular reference and quantification. We shall make this assumption throughout. See Boolos [3, 4] for an influential case in favour of treating plural quantification in plural terms.

  6. 6.

    See, for instance, Russell [41].

  7. 7.

    PFO is presented in Linnebo [24].

  8. 8.

    Parsons appeals to what he calls the ‘systematic ambiguity’ of certain sentences to achieve such generality [31, 32]. See also Glanzberg [14]. Lavine [19] develops a relativist-friendly account of schematic generality. A different approach employed by Fine [12] is to introduce suitable modal operators, allowing us to recapture absolute generality by embedding our quantifiers within them. The resulting view falls somewhere in between absolutism and relativism, as traditionally conceived, and we set it aside here.

  9. 9.

    See, for instance, Russell [40], Dummett [9, chs. 15–6; 10, ch. 24] and, more recently, Glanzberg [14, 15], Fine [12], Hellman [16], Shapiro and Wright [43]. Boolos [5], Cartwright [7] and Williamson [48] respond on behalf of the absolutist.

  10. 10.

    See Hellman [16] and Parsons [32].

  11. 11.

    See Williamson [48, secs. VI–VIII]. Glanzberg [14], Fine [12], Parsons [32] and Lavine [19] respond on behalf of the relativist.

  12. 12.

    Compare Williamson [48, p. 449] and Linnebo [23, p. 150].

  13. 13.

    We adopt the logician’s convention of omitting quotes from expression from formal languages.

  14. 14.

    Assignments may be treated in the standard way as (set-)functions from variables to objects. As usual, the assignment σ[v∕a] agrees with σ on every variable other than v and maps v to a.

  15. 15.

    The analogues of elementary set-theoretic operations like intersection and union are readily accommodated within the No Domain Theory of Domains. \(D^{\star } \cap D^{c}\) may be treated as a plural term denoting the things that are both one of the members of D ⋆ and one of the members of D c.

  16. 16.

    Fine makes this distinction in the context of defending his third-parameter version of relativism, procedural postulationism, mentioned in n. 23, but we should separate the two. The distinction between restriction and expansion has wider significance in the debate about absolute generality.

  17. 17.

    Although Fine does not gloss the distinction in these terms, it seems to provide a natural regimentation of the view he elucidates. Our primary concern, however, remains semantic pessimism rather than Fine exegesis.

  18. 18.

    A non-circular elucidation follows in Sect. 16.4.1.

  19. 19.

    We define subdomain (also ‘subuniverse’, etc.) as the obvious analogue of subset: D is a subdomain of D′ (in symbols: D ⊆ D′) if every member of D is a member of D′. A subdomain D of D′ is said to be a proper subdomain of D′ (D ⊂ D′) if, moreover, D′ is not a subdomain of D.

  20. 20.

    More extreme versions of relativism are also possible, according to which both barriers are imposed: domains are always restricted and universes are always expandable.

  21. 21.

    We assume here that Glanzberg does not go in for the more extreme version of relativism mentioned in n. 20. Note that there is a sense in which domains are expanded on this account—shifts in context lead us to wider domains—and Glanzberg applies the label ‘expansionism’ to his view [15, n. 5]. Terminological issues aside, however, what matters in the context of semantic pessimism is that these domains are always proper subdomains of the universe of the entire language.

  22. 22.

    We borrow Evans and Altham’s [11] famous example of reference shift.

  23. 23.

    Fine [12, sec. 2.6] outlines what seems to be a third option according to which shifts in universe result from a shift in ‘ontology’ distinguished as a third parameter, distinct from both shifts in the circumstances and shifts in semantic content.

  24. 24.

    See also the example attributed to Peter Ludlow by Stanley and Williamson [45].

  25. 25.

    Williamson [48, p. 419] presents a similar example; compare the example presented by Stanley and Szabó [44, p. 243].

  26. 26.

    Williamson generalises his argument to sortal versions of restrictionism [48, sec. VIII].

  27. 27.

    Barwise and Cooper often use the label ‘model-theoretic semantics’. We deviate from their terminology to avoid blurring the distinction between model theory and semantics.

  28. 28.

    See also Lewis [20, p. 40].

  29. 29.

    See, for instance, Frege [13].

  30. 30.

    Here most is taken to have its weakest sense; so interpreted, MOST(η)(θ) says roughly that more than half of the satisfiers of η satisfy θ.

  31. 31.

    The extensional approach may be naturally generalised to intensional languages. Since issues pertaining to intensionality do not concern us here, we continue to simplify by focusing on extensional semantics.

  32. 32.

    Compare Williamson [48, p. 449]. See Peters and Westerståhl [33, sec. 4.5] for an overview of conservativity.

  33. 33.

    Plural or higher-order resources have often been called on in order to formulate absolutist-friendly formulations of model theory. See, for instance, McGee [27, 29], Rayo [38], Williamson [48]. Linnebo and Rayo [26] extend such accounts into the transfinite.

  34. 34.

    Note first that we may encode a pair of pluralities \(\langle xx,yy\rangle\) as, for instance, the plurality comprising the pair \(\langle 1,x\rangle\) for each member x of xx and the pair \(\langle 2,y\rangle\) for each member y of yy and nothing else. (Compare Linnebo and Rayo [26, app. B.2].) A determiner-extension may then be encoded as a superplurality zzz of such pairs. Each plurality-predicate-extension xx occurring as the left co-ordinate of a plurality-pair in zzz is mapped to the superplurality-quantifier-extension yyy, comprising those pluralities yy such that \(\langle xx,yy\rangle\) is a member of the superplurality-determiner-extension zzz.

  35. 35.

    See also, for instance, Keenan [17], Keenan and Stavi [18], Westerståhl [47], and Peters and Westerståhl [33].

  36. 36.

    See Rayo [36, sec. 4] and Linnebo and Rayo [26, app. B] for further details. Compare Shapiro [42, thm 5.3].

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    J. P. Studd

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    Alessandro Torza

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Studd, J.P. (2015). Absolute Generality and Semantic Pessimism. In: Torza, A. (eds) Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language. Synthese Library, vol 373. Springer, Cham. https://doi.org/10.1007/978-3-319-18362-6_16

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