Vagueness and Quantification

This paper deals with the question of what it is for a quantifier expression to be vague. First it draws a distinction between two senses in which quantifier expressions may be said to be vague, and provides an account of the distinction which rests on independently grounded assumptions. Then it suggests that, if some further assumptions are granted, the difference between the two senses considered can be represented at the formal level. Finally, it outlines some implications of the account provided which bear on three debated issues concerning quantification.

1. Westerståhl [29], Stanley and Szabo [27]. For simplicity we will not consider pragmatic accounts of domain restriction, that is, accounts on which the determination of domains is left to pragmatic factors which determine the communicated content as distinct from what is literally said, such as that outlined in Bach [1].

2. Peters and Westerståhl define quantifiers this way in [21], pp. 62-64. Note that in Definitions 1-3 no index is attached to A and B to show that they depend on D, but such effect could easily be obtained with some minor adjustment. For example, the notation adopted in Lappin [16] makes A and B systematically depend on D.

3. Supervaluationism is consistent with (VP) both in its standard version outlined in Fine [10] and in non-standard versions such as that provided in McGee and McLaughlin [19]. Epistemicism is consistent with (VP) at least in the version advocated in Williamson [31]. Other views consistent with (VP) are those suggested in Braun and Sider [8] and in Iacona [12], which qualify as neither supervaluationist nor epistemicist. Finally, (VP) is consistent with some views according to which vagueness is in rebus, as in Barnes [2] and in Barnes and Williams [4] and [3].

4. Definition 4 is in line with the suggestion in Barwise and Cooper [5], p. 163, and the account in Westerståhl [30], pp. 405-406. In the latter work, two readings of ‘most’ are considered. But if Definition 4 is adopted there seems to be no reason to do that.

5. The examples (7) and (8) are drawn from Peters and Westerståhl [21], pp. 213.

6. The hypothesis that ‘most’, ‘few’ and ‘many’ can be treated along the way suggested is adopted in Barwise and Cooper [5] and in Westerståhl [30]. Instead, Keenan and Stavi [15] and Lappin [16] provide differents accounts of ‘few’ and ‘many’.

7. Iacona [14] provides an argument against the uniqueness assumption.

8. Sainsbury [23] suggests a criterion of adequate formalization that rests on the idea that formalization must preserve what is said, pp. 161-162.

9. Brun [9], p. 27, and Baumgartner and Lampert [6], p. 104, provide some considerations in support of the claim that the logical form of a set of sentences is expressed by an adequate formalization of the set.

10. Barwise and Cooper provide a proof of the first order undefinability of more than half of in [5], pp. 213-214. Peters and Westerståhl, in [21], pp. 466-468, spell out a proof method that extends to other proportional quantifiers.

11. A direct proof of the first order expressibility of ‘more than half of’ is provided in Iacona [13]. The theorems presented in this section provide a generalization of that result.

12. Here it is assumed that contextual restrictions are formally represented in the way suggested in Section 4. But note that one would get the same result even if one adopted a formal representation in which a separate predicate letter expresses the restricting condition, because in that case (12) and (13) would be replaced by two formulas ∀x(R x⊃(P x⊃Q x)) and ∀x(S x⊃(P x⊃Q x)) which differ in the first predicate letter.

13. Note that the converse entailment clearly does not hold. For it may be the case that the sentences containing a quantifier expression e (as it is used on a given occasion) admit non-minimal variation in formal representation even if e does not exhibit quantifier indeterminacy. This is shown by the case of ‘more than half of’, which does not exhibit quantifier indeterminacy even though (3) may be represented as (17) or (18).

14. Barwise and Cooper [5], p. 162.

15. Peters and Westerståhl [21], pp. 334-335, Bonnay and Westerståhl [7], Section 8.

16. Note that, given the restriction mentioned in Section 1, ‘sentence’ refers to simple quantified sentences such as (1)-(6).

17. Instead, there is a straightforward connection between logicality so understood and first order definability. Iacona [13] proves that every logical quantifier expression is first order definable.

18. Moss [20], Section 8.2, provides a complete axiomatization of a class of inferences involving sentences containing either ‘most’ or ‘some’. The explanation suggested here seems to hold at least for that class.

19. Williamson [32], pp. 424-427 and 452-460. Glanzberg [11] argues against Williamson that, for every every domain purporting to contain everything, there are in fact things falling outside the domain.

20. Peters and Westerståhl, among others, assume that domains are sets, see p. 48. In Section 5 the same assumption is adopted for the sake of argument.

21. Lewis [17], p. 213, Sider [24], pp. 128-129, Sider [25], pp. 137-142.

22. Lopez De Sa [22] and Sider [26] elaborate and defend the argument. Liebesman and Eklund [18] and Torza [28] argue against it.

23. Note, however, that it might be unclear whether ‘all’ is used unrestrictedly, in which case a similar kind of indeterminacy would arise. Note also that, just like ‘all’ may involve a restriction, the same goes for the general term ‘thing’ as it occurs in ‘all things’. As it is made clear in Lopez de Sa [22], pp. 405-406, (UP) is compatible with recognizing that there might be restricted uses of ‘thing’ that are vague. For in that case, quantifying over every thing in that sense is not the same thing as quantifying over absolutely everything.

24. I presented the material for this paper in talks at the University of Milan (spring 2014), at the University of Barcelona (spring 2015), and at the University of L’Aquila (fall 2014). The paper has benefited enormously from the questions, objections, and suggestions I have received on those occasions. Special thanks go to Dan López De Sa, Sven Rosenkranz and Elia Zardini. I also owe much to two anonymous referees for their sharp and accurate comments.

Authors and Affiliations

1. Center for Logic, Language, and Cognition, Department of Philosophy and Education, University of Turin, Via S. Ottavio 20, 10124, Torino, Italy

   Andrea Iacona

Corresponding author

Correspondence to Andrea Iacona.

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