Epistemicism and modality

Abstract

What kind of semantics should someone who accepts the epistemicist theory of vagueness defended in Timothy Williamson’s Vagueness (1994) give a definiteness operator? To impose some interesting constraints on acceptable answers to this question, I will assume that the object language also contains a metaphysical necessity operator and a metaphysical actuality operator. I will suggest that the answer is to be found by working within a three-dimensional model theory. I will provide sketches of two ways of extracting an epistemicist semantics from that model theory, one of which I will find to be more plausible than the other.

Keywords:

• Vagueness
• modality
• epistemicism
• Timothy Williamson

Acknowledgments

I would like to thank Catharine Diehl, Cian Dorr, Kit Fine, Peter Fritz, Jeremy Goodman, John Hawthorne, Jon Litland, Mark McCullagh, Ofra Magidor, Beau Madison Mount, Jeff Sanford Russell, Gabriel Uzquiano, Sara Kasin Vikesdal, Tim Williamson, and audiences at the University of Oxford and at the 2015 Williamson on Modality workshop in Montreal for helpful discussions and for comments on earlier drafts of this paper.

Notes

1. Hawthorne (2006).

2. That is to say, a notion of truth that satisfies each instance of the schema ‘“S” is true iff S,’ where instances are obtained by replacing ‘S’ with a suitable sentence. A sentence is suitable just in case inserting it into the schema does not result in any trouble, such as, but not limited to, inconsistency (e.g. sentences that purport to ascribe truth to themselves are also not suitable). The relevant notion of suitability is notoriously difficult to make precise.

3. Williamson (2003, 710).

4. An equivalent definition of definiteness in terms of borderline cases is perhaps more intuitive (it is definite that S iff S and it is not borderline whether S), but in this paper I take definiteness as primitive, as this is more natural for logical and semantic investigations, in which ‘definitely’ is standardly treated like ‘necessarily.’

5. For example, Shapiro (2007) has this to say about epistemicism: ‘Here I do not muster a sustained argument against that view, and it is not polite to stare’ (7). This remark is directed at both Williamson (1994) and Sorensen’s (1988) earlier development of an epistemic theory of vagueness.

6. See Hawthorne (2006), Kearns and Magidor (2008), Caie (2012), and Magidor (forthcoming).

7. See Williamson (1994, Section 8) and (2003, 710).

8. The content of A, as of any indexical, will vary with context. In this paper, the logical constants are given a syncategorematic treatment — the usual practice — but they could also be assigned characters in obvious ways.

9. A different approach is suggested by a passage in Kaplan’s ‘Demonstratives’: in general we should take the proposition expressed by ϕ in w to be the proposition that would be expressed by ϕ if it were used in w. (According to Remark 1 of Section XIX of Demonstratives, the ‘Content of a sentence in a context is, roughly, the proposition the sentence would express if it were uttered in that context’ [Kaplan 1977, 546].) One problem with this suggestion is that, on just about anyone’s conception of worlds, it is a non-contingent matter which sentences are used in which worlds (and more generally, what is so in a given world), wherefore a sentence that is not used in a world could not have been used in that world. (Kaplan’s contexts, I should note, are not simply worlds: they also include agents, times, and locations. Nevertheless, the same worry applies: given that it is impossible for an utterance to occur in w unless it does occur in w, it is also impossible for an utterance that doesn’t occur in w to occur in 〈w, a, t, l〉 if this means, as it is naturally interpreted, that the utterance occurs in w, and is, in w, produced by agent a at time t and location l.)

10. One rarely encounters a compositional treatment of variable-binding: see Yli-Vakkuri (2013).

11. As in, e.g. Kaplan (1977).

12. The assumption that there is an intended model is something of an idealization, inter alia, for the simple reason that there may be too many entities being theorized about (e.g. points of evaluation) to form a set — a well-known general limitation of set-theoretic semantics, which I will set to one side.

13. See Williamson (1994, Section 5) and (1999).

14. Caie (2012, 365).

15. Ibid.

16. See note 23.

17. See Davies and Humberstone (1980).

18. Kaplan’s contexts, however, are not simply worlds: see note 9.

19. The example is inspired by Williamson (1994, 253–254). I use it because it is difficult to come up with other kinds of uncontroversial examples of borderline identity statements involving proper names. Many of the standard examples are arguably definite cases where the source of the intuition of vagueness is in the vagueness of, say, ‘is located at’ or ‘is part of.’

20. Thanks to Peter Fritz and Jeremy Goodman for discussion here.

21. Cresswell (1990) takes just this approach. There are no expressions in his quantified modal–temporal language whose interpretation is fixed in all of his models: consequently, ‘there will be no wff true in all interpretations, and thus no intensional logic.’ Yet he says of this language that ‘among its possible interpretations is one which comes closest to reflecting the meanings of particular words in a particular natural language, for convenience English’ (7).

22. Caie’s (2012) main counterexample to otherworldly semantics exploits this flaw, and some of the arguments in Kearns and Magidor (2008) could be reconstructed as doing so, although the latter are not directly concerned with epistemicist interpretations of .

23. Caie and Hawthorne both argue that Williamson’s epistemicism has problems accounting for the definiteness of certain disquotational sentences. Both arguments, however, rely on premises that seem to me less secure than the ones I use.

Caie (2012, Section 5) does not discuss T-sentences but what I call (below) R-sentences: sentences of the form ‘“N” refers to N.’ The questionable assumption in Caie’s argument is that (schematically), for some name ‘N’ there is a close world in which ‘N’’s referent is different from its actual referent but in which ‘refers’ expresses the same content as it actually does. (An analogous assumption about sentences and the truth predicate could be used to argue for the non-definiteness of some healthy T-sentences.) But Hawthorne’s (2006) ‘domestic stability’ solution seems to involve rejecting this assumption.

Hawthorne’s (2006, Section 13) argument concerns T-sentences and involves what seems to me a questionable move from the plausible claim (given Semantic Plasticity) that the truth predicate expresses a content different from its actual content in some close worlds to the further claim that there is a close world where the content the truth predicate has in that world determines an extension different from the extension determined in that world by the actual content of the truth predicate (197). There may be a good argument for why an epistemicist committed to Semantic Plasticity would have to accept the latter claim, but Hawthorne does not offer one.

24. Suppose that (a) |ϕ|_@,w ≠ |ϕ|_w,w, (b) @Rw, and that (c) ϕ is suitable. Note that |ϕ|_w,w = 1 iff ϕ ∈ ⟦T⟧(@, w) by (1) and (c)

iff ⟦^⌜ϕ^⌝⟧(@, w) ∈ ⟦T⟧(@, w) by (ix)

iff |T(^⌜ϕ^⌝)|_@,w = 1 by (viii),

so |ϕ|_w,w = |T(^⌜ϕ^⌝)|_@,w. Then, by (a), |ϕ|_@,w ≠ |T(^⌜ϕ^⌝)|_@,w, so |T(^⌜ϕ^⌝) ≡ ϕ|_@,w = 0. By (b) and either (A₁) or (A₂), |(T(^⌜ϕ^⌝) ≡ ϕ)|_w,w = 0. Note that this argument requires the assumption that R is symmetric. There is, I think, an equally plausible argument for the existence of healthy T-sentences that fail to be definite in some close worlds that does not require the assumption that R is symmetric: we can simply assume that, in the case at hand, both wR@ and @Rw. I take it to be fairly clear that there will be such cases if one or another of the actualistic semantics is correct, but I won’t supply the philosophical argument for this for lack of space.

25. |Ref(⌜τ⌝, τ)|_@,w = 1 iff 〈⟦⌜τ⌝⟧(@, w), ⟦τ⟧(@, w)〉 ∈ ⟦Ref⟧(@, w) by (viii)

iff 〈τ, ⟦τ⟧(@, w)〉 ∈ ⟦Ref⟧(@, w) by (ix)

iff ⟦τ⟧(w, w) = ⟦τ⟧(@, w) by (3)

so

|Ref(⌜τ⌝, τ)|_@,w = 0 iff ⟦τ⟧(w, w) ≠ ⟦τ⟧(@, w),

so, in particular,

(3*) If ⟦τ⟧(w, w) ≠ ⟦τ⟧(@, w) and wR@ then |Ref(⌜τ⌝, τ)|_w,w = 0.

26. It is not entirely clear to me that Caie’s two options are the most natural ones. In fact, his taxonomy of truth conditions for ϕ at proper points omits the one that seems to me to most closely follow the letter of Williamson’s account of ignorance of borderline matters. Consider, for example:

What distinguishes vagueness as a source of inexactness is that the margin for error principles to which it gives rise advert to small differences in meaning, not to small differences in the objects under discussion (Williamson 1994, 231).

Suppose I am on the ‘thin’ side of the boundary but only just. […] The sentence ‘TW is thin’ is true, but could easily have been false without any change in my physical measurements or those of the relevant comparison class (Williamson 1994, 231).

The most straightforward way of extrapolating a truth condition for ϕ at a proper point from remarks like these would seem to be this: ϕ is true at 〈w, w〉 iff, for all v such that wRv and the truth value of the proposition ϕ expresses in w is the same in w and in v, ϕ is true at 〈v, v〉. This is consistent with two truth definition clauses:

() ϕ iff, for all u such that vRu and ϕ |ϕ| , |ϕ| .

() ϕ iff, for all u such that wRu and |ϕ| |ϕ| , |ϕ| .

() and () share the logical shortcomings, respectively, of (O₁) and (O₂), and these are accompanied by similar semantic shortcomings. Unlike (O₁) and (O₂), however, () and () both entail that a borderline sentence expresses, in some close world, a proposition different from the one it actually expresses. This is another reason why () and () seem to me to be closer matches, within the 2D framework, to the letter of Williamson (1994) than (O₁) or (O₂).

27. See Williamson (2003) for discussion.

28. See Williamson (1994, Section 9.3).

29. Thanks to Peter Fritz for this evocative label.

30. I am inclined to think that we should not require R to be symmetric, but the assumption that R is symmetric helps simplify some arguments in this paper.

31. In addition to note 30, there are some delicate issues in this area, such as the status of the principles ϕ ϕ and □ ϕ ϕ, both of which the model theory validates, which I discuss in my work with Litland. A model theory in which is interpreted by a relation on W × W × W rather than on W offers a more flexible framework for logical and semantic theorizing about L_MV. For maximal generality we can also add to the models a second accessibility relation to interpret □. My work with Litland explores model theories with both features.

32. For this idea to have any hope of working, we must not be too permissive about what to count as a fact ‘about character.’ For example, it is, in some intuitive sense, a fact about the character of ‘Tim is thin’ that the character of ‘Tim is thin’ is its actual character. Furthermore, it is a known fact. But then it cannot be a fact about character in the intended sense, because if it were, then no world in which ‘Tim is thin’ has a character different from its actual character will be R-related to the actual world; in any such world the known fact that the character of ‘Tim is thin’ is its actual character fails to obtain.

33. This principle is plausible on it own, but it also becomes a theorem under the metasemantic interpretation of the 3D model theory, if we augment the model theory by assigning metasemantic characters to the logical constants in any of the obvious ways. Then all of the constants except for A will have constant characters in every metasemantic context of every model. (We would also have to add functors and an identity predicate to the language to handle the Meter Sentence.)

34. See Remark 11 of Section XIX of Demonstratives (Kaplan 1977, 551) for a classic articulation of the consensus view.

35. In Hawthorne’s original example, we introduce an indexical singular term, ‘Actuality,’ and a non-indexical singular term, ‘Worldy,’ to designate the actual world. The identity ‘Worldy = Actuality’ is then known but, given natural assumptions about how the 3D model theory would have to deal with singular terms and identity, is not definite according to the metasemantic interpretation.

36. See note 8.

37. It is much less clear that I am in a position to know this much about the supervenience of content facts on ‘metaphysical groundfloor’ facts, for any natural interpretation of that phrase. For example, it is difficult to see how I could be in a position to know anything at all about how the content facts supervene on the microphysical facts, unless the relevant notion of ‘being in a position to know’ packs in the idealization I have access to Chalmers’ ‘cosmoscope’ or a similar device (see Chalmers 2012, 114–115).

38. Of course, models for epistemic logic do not usually come with a designated actual epistemic possibility, nor with contexts that determine epistemic possibilities (because they usually do not come with contexts at all), but this is just an artifact of the model theory. A semantics must recognize, for each parameter shifted by an operator, a function that assigns to each context the value of that parameter that is ‘realized’ or ‘present’ (etc.) in that context.

39. It would be natural to strengthen this condition by adding the conjunct: ‘and w matches actuality with respect to the truth value of the proposition ϕ actually expresses,’ because possibilities in which one is in error because the facts that actually ϕ concerns (e.g. whether Tim is thin) are different than they actually are irrelevant to vagueness-induced ignorance (see Williamson 1994, 231). However, even the weaker condition does not hold.

40. Specifically, we must assume that the logic of validates each instance of the K axiom schema (ϕ ϕ , and that ϕ is valid whenever ϕ is a truth-functional tautology — call this latter principle Tautological Definitization. Because ♯ is true and χ is not, by K, ¬(♯  χ) is true. Furthermore, because χ is borderline, ¬¬χ is true, and we can show that ¬¬(♯  χ) is true as follows. First we assume for a reductio:

(i) ¬(♯  χ).

(ii) is valid by K.

(ii) (¬(♯  χ)  ¬χ)  (¬(♯  χ)  ¬χ),.

and (iii) is valid by Tautological Definitization.

(iii) (¬(♯  χ)  ¬χ).

Because χ is borderline,

(iv) ¬¬χ.

(i), (ii), and (iii) imply ¬χ, which contradicts (iv). We get:

¬¬(♯  χ) ∧ ¬(♯  χ),

which is definitionally equivalent to (♯  χ).

41. This argument does not essentially depend on the assumption that ♯ expresses all of the use facts; any kind of fact (e.g. microphysical) on which content facts supervene will do. Nor does the argument require the assumption that ♯ specifies all of the facts of the relevant kind; it is enough that ♯ specifies enough of them for it to be the case that, necessarily, if the proposition actually expressed by ♯ is true, then χ expresses p_@. Thanks to John Hawthorne for discussion here.

42. While I find this argument to be pretty decisive, I should note that at least one philosopher sympathetic to epistemicism rejects two of its assumptions. Magidor (forthcoming) rejects one of the ‘very minimal assumptions about the logic of definiteness’ mentioned in the main text, namely the K axiom for (see note 40), and Kearns and Magidor (2012) reject the supervenience of content facts on use facts (as well as on microphysical facts, and apparently on any facts other than the content facts themselves), so they would reject (**).

43. Williamson himself appears to posit such ignorance. One relevant passage is this:

Since the content of the concept depends on the overall pattern, you have no way of making your use of a concept on a particular occasion sensitive to the overall pattern. Even if you did know all the details of the pattern (which you could not), you would still be ignorant of the manner on which they determined the content of the concept (1994, 231–232).

See also Williamson (1994, Section 7.4).

44. Williamson (1994, 230).