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At least not false, at most possible: between truth and assertibility of superlative quantifiers

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Abstract

Generalized Quantifier Theory defines superlative quantifiers at most n and at least n as truth-conditionally equivalent to comparative quantifiers fewer than n+1 and more than n \(-\)1. It has been demonstrated, however, that this standard theory cannot account for various linguistic differences between these two types of quantifiers. In this paper I discuss how the distinction between assertibility and truth-conditions can be applied to explain this phenomenon. I draw a parallel between the assertibility of disjunctions and superlative quantifiers, and argue that those assertibility conditions are essentially modal. I use epistemic logic to formalize the assertibility conditions and revisit some of the linguistic puzzles related to superlative quantification.

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Notes

  1. For our purposes only formulas with quantification under the scope of model operators are needed, and the philosophical questions (though interesting) that arise from allowing modalities under the scope of quantifiers are not relevant for this paper.

  2. Since in the current paper the analysis is limited to sentences of form at most n/at least n A are B, here and in all definitions below, \(\phi \) is a formula \((A(x) \wedge B(x))\), where \(A\),\(B\) are unary FO predicates.

  3. $$\begin{aligned}&\exists ^{=n}x\phi (x) \overset{dfn}{\iff } \exists x_1...\exists x_n [ \bigwedge _{i=1}^{n}\phi (x_i)\wedge \bigwedge _{1\le i<j\le n}(x_i\ne x_j) \wedge \forall y (\bigwedge _{i=1}^{n} y\ne x_i \rightarrow \lnot \phi (y))]\end{aligned}$$
    (VI)
    $$\begin{aligned}&\exists ^{>n}x\phi (x) \overset{dfn}{\iff } \exists x_1...\exists x_{n+1} \left[ \bigwedge _{i=1}^{n+1}\phi (x_i)\wedge \bigwedge _{1\le i<j\le n+1}(x_i\ne x_j)\right] \end{aligned}$$
    (VII)
    $$\begin{aligned}&\exists ^{<n}x\phi (x) \overset{dfn}{\iff } \forall x_1...\forall x_{n} \left[ \bigwedge _{1\le i<j\le n}(x_i\ne x_j)\rightarrow \bigvee _{i=1}^{n}\lnot \phi (x_i)\right] \end{aligned}$$
    (VIII)

  4. Note that \(\Diamond _S \exists ^{<n}x\phi (x)\) is not equivalent to \(\bigwedge _{i=o}^{n-1} \Diamond _S \exists ^{=i}x\phi (x)\), where \(\exists ^{=0}x\phi (x)\) means that \(\lnot \exists x\phi (x)\).

  5. Such a requirement would enforce the following formulation of the belief condition: \(\bigwedge _{i=n}^{N} \Diamond _S \exists ^{=i}x\phi (x)\), where \(N\) is the cardinality of the quantifier’s domain.

  6. “Smallest” is used here from the point of view of economy principles governing verification procedure, i.e. the “first” one that results in falsification failure.

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Acknowledgments

For comments on the manuscript I would like to thank Henk Zeevat and Jarmo Kontinen. For linguistic corrections and remarks I am grateful to Kevin Reuter. This research was funded by the Foundation Stiftung Mercator.

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  1. Department of Philosophy, Ruhr University of Bochum, Universitätsstraße 150, 44780, Bochum, Germany

    Maria Spychalska

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Spychalska, M. At least not false, at most possible: between truth and assertibility of superlative quantifiers. Synthese 195, 571–602 (2018). https://doi.org/10.1007/s11229-014-0615-y

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