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The Application of Constraint Semantics to the Language of Subjective Uncertainty

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Abstract

This paper develops a compositional, type-driven constraint semantic theory for a fragment of the language of subjective uncertainty. In the particular application explored here, the interpretation function of constraint semantics yields not propositions but constraints on credal states as the semantic values of declarative sentences. Constraints are richer than propositions in that constraints can straightforwardly represent assessments of the probability that the world is one way rather than another. The richness of constraints helps us model communicative acts in essentially the same way that we model agents’ credences. Moreover, supplementing familiar truth-conditional theories of epistemic modals with constraint semantics helps capture contrasts between strong necessity and possibility modals, on the one hand, and weak necessity modals, on the other.

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Notes

  1. For similar views, see [65, 53] and [58, 136–141].

  2. Another natural extension is to expand the domain of the semantic interpretation function so that it includes some non-declarative sentences.

  3. For example, epistemic modals are sometimes used to raise possibilities, and we might model this effect by taking \(\mathcal {F}\) to be a proper subalgebra of the power set of W. For discussion see Swanson [65, 77–84] and Yalcin [71] and [82] and [84].

  4. Those who endorse what Richard Foley calls the “Lockean thesis” about belief might replace enriched probability measures with ordinary probability measures (making appropriate tweaks here and there) in giving a constraint semantics for the language of subjective uncertainty. According to the Lockean thesis, “it is epistemically rational for us to believe a proposition just in case it is epistemically rational for us to have sufficiently high degree of confidence in it, sufficiently high to make our attitude towards it one of belief” (Foley [15, 111]).

  5. Timothy Williamson has argued both for a distinction between belief and credence and for the hypothesis that “Outright belief still comes in degrees” (Williamson [80, 99]). It is not clear that the phenomena Williamson adduces in support of the latter hypothesis should be represented in semantics; for simplicity I suppose that they should not.

  6. As Schroeder [58] nicely puts a related thought: “The expressivist strategy is to explain the language in some domain by explaining the thought in that domain” (152). See also Gibbard [24]: “the meaning of normative terms is to be given by saying what judgments normative statements express—what states of mind they express” (84).

  7. For simplicity I write as though we are always extremely precise about credences; this assumption could easily be dropped.

  8. E.g., van Fraassen [17], Levi [44] and [45], Jeffrey [34, 35, 37, 75], and Sturgeon [64].

  9. For similar ways of thinking about semantic value see Swanson [65, 82] and Yalcin [83], and Moss [52]. Although my semantics is in some ways more complicated than Yalcin’s, the complications are necessary to handle the “third grade of [epistemic] modal involvement” (Quine [55]): quantifiers that scope over epistemic modals and other hedges.

  10. To be sure, many interesting constructions are beyond the ambit of this implementation of constraint semantics. For example, consider

    1. (4)

      Few likely winners are tall.

    To handle cases like (4) we would need noun phrases to be of a higher type. (For example, we might make noun phrases type 〈〈s e, a t〉, a t〉 and keep predicates type 〈s e, a t〉; alternatively we might raise both.) Thanks to Daniel Rothschild for questions here.

  11. (8) also has a reading on which its semantic value is the set of admissibles that take the proposition that Betty is tall to an element of [0.5,1] × {0,1}; adding wide-scope negation helps make this reading salient. For example, ‘There isn’t a 50 % chance that Betty is tall’ can be read as saying that there is a less than 50 % chance that Betty is tall. Thanks to Daniel Rothschild for these observations.

  12. On which see (e.g.) Jaynes [32, 42, 79], and van Fraassen [17].

  13. For a nice generalization of van Fraassen’s example, see Seidenfeld [59].

  14. Moreover, it’s possible that our final account of disjunction should be informed by other kinds of non-truth-conditional language as well. See especially the discussions of disjunction in Gibbard [25] and Schroeder [58].

  15. The talk of ‘mixtures’ is meant only as a heuristic: perhaps disjunctive mixtures should be represented using sets of probability spaces, in a way that forgoes anything like ‘mixture.’

  16. Thanks to Daniel Rothschild for pressing this point.

  17. Of course, each quantifier needs to be associated with a set of sets of individual concepts, but the recipe is straightforward: the relevant set for a quantifier phrase ‘QP’ will be {Q | Q is a set of type 〈s, e〉 objects, and for any given world w, the set consisting of the images of w under the members of Q includes QP in w}. For example, for ‘some person’/‘exactly five people’ we take {Q | Q is a set of type 〈s, e〉 objects, and for any given world w, the set consisting of the images of w under the members of Q includes some person/exactly five people in w}. It might seem odd to associate a quantifier phrase like ‘every person’ with a disjunction, but because the conjunctions that make up the disjuncts each target every person, the whole disjunction is equivalent to a conjunction of claims targeting each person.

  18. For discussion of some similar examples, see von Fintel and Gillies [11, 10, 14]; [12]; and [14]; Swanson [65, 40-42]; Dowell [7, 69]; and MacFarlane [48].

  19. Grice [26, 26] is the locus classicus. See Chapman [3, 87–94] for an illuminating discussion of Grice’s historical context. See Swanson [68] and the citations therein for some details on the derivations of such implicatures.

  20. For further discussion see Section 3 of Swanson [69].

  21. Adding an explicit specification of the relevant information—like ‘given what we know about when he left’—can help make the epistemic readings vivid.

  22. See Swanson [65, 56] and von Fintel & Gillies [13] for the evidential feature of epistemic possibility modals. See Huddleston & Geoffrey [31, 186], Copley [6], and Swanson [66, 1203–1205] for the evidential feature of weak necessity modals. Karttunen does write that “… what is true about must is also true of other similar words, such as necessarily…, have to, or is bound to” (Karttunen [38, 13]), but he does not mention weak necessity modals or possibility modals.

  23. For Kratzer’s discussions see especially her [39, 40], and [41]. The view I sketch here differs somewhat from Kratzer’s view. In particular, she treats the ordering source of epistemic modals as representing stereotypicality [41, 644], whereas I treat it as representing premises taken to support belief in the prejacent, where ‘things are proceeding normally’ might be one of those premises.

  24. Without the limit assumption, I favor the view developed in [70].

  25. I don’t have space here for serious discussion of the subtleties of Yalcin’s data involving supposition of epistemic contradictions. But there may be reason to appeal to the credal constraints of strong necessity modals and possibility modals to help explain why the epistemic readings of (i) and (ii) are odd, even though (iii) is fine.

    1. (i)

      # Suppose it is raining and it might not be raining. (Yalcin [81, 234]; Yalcin [82, 984])

    2. (ii)

      # Suppose it is raining and it needn’t be raining.

    3. (iii)

      Suppose it is raining and it shouldn’t be raining. (Then we’d need to recalibrate our instruments.)

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Acknowledgments

Thanks to Dave Baker, David Braun, Aaron Bronfman, Nate Charlow, Janice Dowell, Kit Fine, Danny Fox, Dilip Ninan, Paul Portner, David Sobel, Rich Thomason, Tim Williamson, and Steve Yablo for helpful discussion. Thanks also to audiences at Harvard University; University of California, Berkeley; the Formal Epistemology Festival at University of Konstanz; University of Michigan, Ann Arbor; the Chambers Philosophy Conference at University of Nebraska-Lincoln; and Yale University. Thanks especially to Andy Egan, Kai von Fintel, Thony Gillies, Ned Hall, John MacFarlane, David Manley, Sarah Moss, Mark Richard, Robert van Rooij, Daniel Rothschild, Bob Stalnaker, and Seth Yalcin.

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Swanson, E. The Application of Constraint Semantics to the Language of Subjective Uncertainty. J Philos Logic 45, 121–146 (2016). https://doi.org/10.1007/s10992-015-9367-5

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