Abstract
This paper examines a theory of vague language often taken to support supervaluationist logic, and argues that the theory supports subvaluationism equally well, which is to say not well at all. Instead, it’s shown that the theory naturally gives rise to truth-functional theories of vague language. Two such theories are presented and evaluated; the theory based on the logic LP is preferred. A potential objection stemming from higher-order vagueness is examined and responded to. Finally, the import of an LP-based theory for the sorites argument itself is examined.
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Notes
- 1.
Although there are many different ways of presenting a supervaluational system, I’ll ignore these distinctions here; my remarks should be general enough to apply to them all, or at least all that adopt the so-called ‘global’ account of consequence. (For discussion, see Varzi 2007.) Similarly for subvaluational systems.
- 2.
- 3.
- 4.
NB: S’valuationists differ in the extent to which they take vagueness to be like ambiguity, but they all take it to be like ambiguity in at least this minimal sense. Smith (2008) draws a helpful distinction between supervaluationism and what Smith calls ‘plurivaluationism’. Although both of these views have both travelled under the name ‘supervaluationism’, they are distinct. Supervaluationism makes use of non-bivalent semantic machinery (for example, the machinery in Fine 1975), while plurivaluationism makes do with purely classical models, insisting merely that more than one of these models is the intended one. My discussion of supervaluationism here is restricted to the view Smith calls supervaluationism.
- 5.
At least in most normal contexts. Vague predicates seem particularly context-sensitive, although they are not the only predicates that have been claimed to be (see e.g., Recanati 2004; Wilson and Sperber 2002). For the purposes of this paper, I’ll assume a single fixed (non-wacky) context; these are theories about what happens within that context. Some philosophers (e.g., Raffman 1994) have held that taking proper account of context is itself sufficient to dissolve the problems around vagueness. I disagree, but won’t address the issue here.
- 6.
There are many ways to build s’valuational models. In particular, one might not want to have to fully precisify the language in order to assign truth-values to just a few sentences. Nonetheless, the approach to be presented here will display the logical behaviour of s’valuational approaches, and it’s pretty simple to boot. So we can get the picture from this simple approach.
- 7.
And from propositional variables directly to classical truth-values, if one wants bare propositional variables in the language. Vague propositional variables can be accommodated in this way as well as precise ones.
- 8.
Note that this is a multiple-conclusion consequence relation. One can recover a single-conclusion consequence relation from this if one is so inclined, but for present purposes the symmetrical treatment will be more revealing. See e.g., Restall (2005) for details, or Hyde (1997) for application to s’valuations. See also Keefe (2000) for arguments against using multiple-conclusion consequence, and Hyde (2010) for response.
- 9.
Explosive sentences are sentences from which one can derive any conclusions at all, just as logical truths are sentences that can be derived from any premises at all. It’s a bit sticky calling them ‘logical falsehoods’, as may be tempting, since some sentences (in SB at least) can be false without failing to be true. And I want to shy away from ‘contradiction’ here too, since I understand by that a sentence of the form A ∧ ¬A, and such a sentence will be explosive here but not in the eventual target system.
- 10.
Here I prove only the LTR directions, but both directions indeed hold; see Hyde (1997) for details.
- 11.
This, essentially, is Tappenden’s ‘objection from upper-case letters’ (Tappenden 1993). With multiple conclusions, there’s no need for upper-case letters; the point can be made in any typeface you like.
- 12.
For example, consider the sentence ‘There is a last noonish second’. It is true for the supervaluationist, but there is no second x such that ‘x is the last noonish second’ is true for the supervaluationist.
- 13.
Pace Fine (1975), which, in a footnote, proposes SP as a logic for ambiguous language. As noted above, this would make it impossible to explain how one resolves a contradiction by finding an ambiguity—a very bad result.
- 14.
Lewis (1982) argues for LP, in particular, as a logic of ambiguity, and mentions vagueness as one sort of ambiguity.
- 15.
See Ripley (2011) for evidence that ordinary speakers agree with such claims as (64c) and (64d).
- 16.
See e.g., Beall (2008) for a discussion of revenge.
- 17.
- 18.
Again, assume a context where this is true.
- 19.
For simplicity, we look at only one predicate: N for ‘noonish’. This set is then a set of sets of precisifications for ‘noonish’. Let {x–y} be the set of times between x and y inclusive.
- 20.
Actually I don’t see that anything does.
- 21.
For proof, see Priest (1984).
- 22.
Proof and details can be found in Priest (1984). Note as well that the result can be iterated past ω into the transfinite; I don’t think that’ll be necessary here, since every new level is created to address the vagueness of some finite predicate.
- 23.
Some have accepted the conclusion or rejected the first premise (e.g., Unger 1979), but to take such a position seriously is to remove much of the sense of ‘noonish’. And to take it seriously for every vague predicate would make it very hard indeed to talk truly at all. There are other more radical approaches, too: we might reject transitivity of entailment (as in Zardini 2008 or Cobreros et al. 2012), or universal instantiation (as in Kamp 1981). The LP-based solution offered here keeps to the more conservative side of the street.
- 24.
This is because modus ponens on ⊃ is equivalent to disjunctive syllogism, which anyone who takes contradictions seriously ought to reject. See Priest (1979) for discussion.
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Acknowledgements
Research partially supported by the French government, Agence Nationale de la Recherche, grant ANR-07-JCJC-0070, program “Cognitive Origins of Vagueness”, and by the Spanish government, grant “Borderlineness and Tolerance” ref. FFI2010-16984, MICINN. Many thanks as well to Jc Beall, Rachael Briggs, Mark Colyvan, Dominic Hyde, Joshua Knobe, William Lycan, Ram Neta, Graham Priest, Greg Restall, Keith Simmons, Mandy Simons, and Zach Weber, as well as audiences at the Fourth World Congress of Paraconsistency, the University of Queensland, and Carnegie Mellon University for valuable discussion, insight, and support.
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Ripley, D. (2013). Sorting out the Sorites. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_18
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