Probabilistic Approaches to Vagueness and Semantic Competency


Wright (Synthese 30:325–365, 1975) holds that the following two theses are jointly incoherent: (T1) Rules determine correct language use. (T2) These rules are discoverable via internal reflection on language use. I argue that incoherence is derivable from (T1) alone and examine two types of probabilistic accounts that model a modification of (T1), one in terms of inexact knowledge, the other in terms of viewing semantic rules as reasons for linguistic actions. Both accommodate tolerance by breaking the link between justified assertion and truth, but incoherence threatens their conception of justified assertion (the ‘relocation problem’). I argue that the rules-as-reasons approach can relocate sharp boundaries to a place where they are not only more tolerable, but to be expected.

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  1. 1.

    The formulation given here was popularised in, for example, van Rooij (2011), Cobreros et al. (2012) and Smith (2008).

  2. 2.

    A fully model theoretic representation could be given. For Models \({\mathcal {M}}=\langle {\mathcal {D}}, [\![\cdot ]\!] \rangle \) with variable assignment g: \({\text {For}}\,{\text {all}}\, d,\,d'\in {\mathcal {D}}_{{\mathcal {M}}},\, {\text {if}}\, [\![P(x) \wedge x\sim _P y ]\!]_{{\mathcal {M}}}^{g[x:= d,y:= d']} = 1,\, {\text {then}}\, [\![P(y)]\!]_{{\mathcal {M}}}^{g[y:= d']}=1\)

  3. 3.

    This may, however, implicitly restrict our discussion to declarative sentences or sentences used with assertoric force, given that, on the governing view, it is not clear that imperatives and interrogatives should receive a truth-conditional interpretation. However, semantics for interrogatives are frequently proposed in terms of sets of propositions/possible answers [a classic reference for which is Roberts (1996)]. Starr (2016) provides a helpful overview of the imperatives literature and provides a proposal for the semantics of imperatives in terms of a dynamic semantics inspired update on interlocutors’ preferences (an update on the order of preference for alternatives).

  4. 4.

    It is not enough for a defender of the governing view to simply claim that one knows the vague truth conditions without providing an analysis of what being a vague truth condition amounts to. In the following, the inexact knowledge approach that will be discussed could be read as one way of filling out this claim. However, as we shall see, this approach makes some revisions to the governing view.

  5. 5.

    The alternative being that we drop the idea that rules determine correct usage. However, one could doubt the utility of rules that determine incorrect/not fully correct usage.

  6. 6.

    There may be more strategies available than the inexact knowledge or rules-as-reasons approaches. However, I will limit my investigations to just these two.

  7. 7.

    On which there is a highly comprehensive commentary and discussion in Égré and Barberousse (2014).

  8. 8.

    Both Sutton (2013, 2015) and Lassiter and Goodman (2015) argue that vagueness can be characterised in terms of metalinguistic uncertainty.

  9. 9.

    However, MacFarlane’s (2009) account involves uncertainty about degrees of truth.

  10. 10.

    Although Égré and Barberousse (2014) say that Borel’s account bears affinity with a graded version of supervaluationism in that degrees are “parasitic on bivalent verdicts” (p. 1055), the sense in which Borel’s probabilities are grounded in use, and/or the judgements of competent speakers lends justification to this association.

  11. 11.

    That Hampton (2007) calls his model the Threshold Model could be confusing in this context. The threshold being referred to is a vaguely defined threshold after which two objects are both members of a class, yet differ with respect to their typicality as members of that class.

  12. 12.

    As pointed out by an anonymous reviewer, this may be a contingent fact about ‘red’. There could be world in which actual borderline cases for vague predicates never arise.

  13. 13.

    Lassiter and Goodman (2015) are also explicitly quietist on the “metaphysical” questions normally posed regarding vagueness in philosophy: “The present account is intended as an answer to the psychological question of how people understand and use scalar adjectives. We do not propose an answer to the metaphysical questions that have occupied much of the discussion of vagueness, involving when a scalar adjective really is applicable to an object, and to what degree. [...] On some plausible assumptions about the nature of meaning, the latter type of question should be illuminated–perhaps even resolved–by an answer to the former. However, we will not attempt this philosophical project here.” One way of understanding the project in this article is as pursuing this philosophical project further.

  14. 14.

    An alternative would be to use the more sophisticated model in Lassiter and Goodman (2015) in which Soft-max choice rules are incorporated.

  15. 15.

    The PLK/PJC distinction is not clearly demarcated by situation versus possible worlds theories. For example, Larsson and Fernández (2014) defend a version of PLK within a situation theoretic formalism, and Hampton (2007) is not formulated in a situation or world theoretic system.

  16. 16.

    Rather than Situation Types, Cooper et al. (2015) use Record Types. For a discussion on the connections between records, record types, situations, and situation types, see Cooper (2005).

  17. 17.

    A possible concern is that situation types are not the right kinds of entities to form a probability space, the more traditional entities being possible worlds. However, as Cooper et al argue, although the literature on probability theory tends to adopt possible words talk, it is far from clear that full-blown possible worlds are intended. Take the above example of a dice roll. It is not clear that when probability theorists describe a world in which the dice lands even, that they are describing a fully fledged possible word, rather than a situation type.

  18. 18.

    As pointed out by an anonymous reviewer, a more sophisticated model could make use of Bayesian confirmation theory. This is an intriguing possibility, which certainly merits future detailed consideration. At least for now, an account based on the simplifying assumption seems sufficient for the limited goals of this paper.

  19. 19.

    This is itself a drastic oversimplification, not least on the role of comparison classes [see, for example, Kennedy (2007)]. Whereas Cooper et al. (2015) focus on nouns, Sutton (2015) gives a semantics for adjectives that incorporates semantic features of adjectives from the linguistics literature. In brief, adjectives are treated as functions on probability distributions, where the distributions they modify are given explicitly by the argument noun, by in implicit comparison class, or from a combination of the two. For example, we have reasonable expectations of heights for basketball players. The use of ‘tall’ with respect to this comparison class encodes a raising of our expectations as to the heights of basket ball players.

  20. 20.

    In practice, however, values for priors can be given a more empirical basis.

  21. 21.

    This does not, however, imply that below the threshold one must always apply the negation of the expression.

  22. 22.

    With the restriction that all resultant values are in the range [0, 1].

  23. 23.

    Giving values in the form \(p(\phi )\) is a simplification for readability. Such probabilities should, on PJC, be conditional on the situation being described.

  24. 24.

    For example, Davies (2014) considers cases in which the belief in rape myths of members of a jury can prevent a rape victim from performing the illocutionary act of denying she had given consent.

  25. 25.

    I use notation from Égré (2011).


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This research was funded by a Strategic Research Fund (Strategischer Forschungsfonds—SFF) from Heinrich-Heine-University and by the DFG Collaborative Research Centre 991, project C09. The original impetus for writing this paper was for the VII Navarra workshop on vagueness. Thanks to the organisers and attendees of this workshop as well as to two anonymous Erkenntnis referees.

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Sutton, P.R. Probabilistic Approaches to Vagueness and Semantic Competency. Erkenn 83, 711–740 (2018).

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