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ABSTRACT
Suppose you’re certain that a claim—say, ‘Frida is tall’—does not have a determinate truth value. What attitude should you take towards it? This is the question of the cognitive role of indeterminacy. This paper presents a puzzle for theories of cognitive role. Many of these theories vindicate a seemingly plausible principle: if you are fully certain that A, you are rationally required to be fully certain that A is determinate. Call this principle ‘Certainty’. We show that Certainty, in combination with some minimal side premises, entails a very implausible claim: whenever you’re certain that it’s indeterminate whether A, it is rationally required that you reject A. This is a surprising result, which requires abandoning at least some intuitive views about indeterminacy and cognitive role.
Disclosure Statement
No potential conflict of interest was reported by the authors.
Notes
1 For explicit attempts to answer this question, see Field [Citation2000, Citation2003, Citation2008], Schiffer [Citation2003], Dorr [Citation2003], Smith [Citation2008], Williams [Citation2014a, Citation2014b], and Bacon [Citation2018].
2 For attempts to talk about vague desire and rational belief, desire and decision in the context of vagueness, see, for example, Edgington [Citation1997] and Williams [Citation2016].
3 On our reading of him, the leading defender is Field [Citation2003]. As discussed below, there are some delicate exegetical issues.
4 Discussion of the indeterminate future dates back to Aristotle. For a clear recent articulation of the view on which future contingents are indeterminate, see Barnes and Cameron [Citation2009].
5 See Smith [Citation2008] for a discussion of the degrees of truth approach to indeterminacy, and Smith [Citation2009] for an explicit articulation and defence of this anti-rejectionist position. Truth-value-gap semantics for indeterminacy, by contrast, form a neat theoretical package with indeterminacy, but, in conversation, truth-value-gap theorists whom we have asked about the issue typically bridle at the suggestion that they should embrace rejectionism. Rejectionism is not popular even among those whom one might expect to be its friends.
6 The literature on triviality results was started by Lewis [Citation1976]. See Hajek and Hall [Citation1994] for an overview of early triviality results. For more recent results, see Bradley [Citation2000, Citation2007]; see also Charlow [Citation2016], Russell and Hawthorne [Citation2016], and Goldstein [Citation2019] for extensions of triviality results beyond conditionals.
7 In Hartry Field’s work on the cognitive role question for indeterminacy, he moves between point valued and interval valued formalisms. In the point valued formalism, his view is that a rational agent certain that A is indeterminate should have credence 0 in A. In the interval valued setting, his thesis is that a rational agent certain that A is indeterminate should take interval-valued attitude [0,1] toward A. Starting from the point-valued representation, it is natural to read him as a rejectionist, recommending the adoption of a state of minimal confidence to known-indeterminate claims (an interpretation that’s enforced, we submit, by his discussion of norms of logic and the liar paradox). Redescribed in the interval valued formalism, however, the corresponding ‘strength of belief’ will be [0,1]. This is perfectly consistent with rejectionism, so long as one adopts a reading of interval-valued strengths of belief where all intervals [0,x] are states of minimal confidence. There is a rival (and more common) reading of interval valued formalism on which [0,1] represents an agnostic state, strictly more confident than [0,0] but less confident than [1,1] and incomparable to, e.g., [0.5,0.5] (more generally, [a,b]<[c,d] iff b<c). If one forced this reading onto Field’s theory, then he would not count as a rejectionist by our lights, and the argument to follow, if sound, would be a reductio of the position, rather than an argument for it.
8 An alternative that works equally well for our purposes is to understand conditional degree of belief in terms of supposition: Cr(B|A) denotes an agent’s degree of belief in B, on the supposition that A. This alternative might allay some worries raised by the update-based construal of conditional probabilities. (For example, one might worry that update involves learning a proposition with certainty only in rare occasions.) Thanks to an anonymous referee here.
9 In fact, via this route we get a stronger principle than rejectionism—i.e. a principle with a weaker antecedent:In our discussion, we stick with rejectionism because it seems to be the philosophically more interesting principle.
10 We can also argue for it, given a few more principles: (a) rational degree of belief doesn’t drop over logical consequence; (b) determinacy is factive: det A ⊨ A.
11 An anonymous referee raises a worry: what if we say that it’s irrational to believe that the future is indeterminate? In that case also, the rejectionist can grant that subjects have rational positive credence towards propositions about the future (since that credence merely tracks uncertainty, rather than indeterminacy). We grant the referee’s point, although we hasten to point out that the view that they describe is very strong.
12 For an argument for an exclusionary answer to the cognitive role question on this sort of basis, see Williams [Citation2014b].
13 A referee suggests that we might be being overly charitable to rejectionism, and that it is not even a prima facie option for the cognitive role of (an instance of) indeterminacy. If the referee is right, that would only intensify the puzzle that this paper is articulating, but we do not endorse anything so strong. The formal articulation of rejectionism described elsewhere by Williams [Citation2016] should help interested readers to see that rejectionism could at least be made coherent, regardless of whether or not it is plausible.
14 Though see the discussion of degrees of determinacy and conditional probability in Williams [Citation2016] for a precedent.
15 Proof. Assume that Cr(B|A) = 1. Then, via ratio, Cr(A ∧ B) = Cr(A). Via conjunction, Cr(B) ≥ Cr(A ∧ B); by replacing Cr(A ∧ B) with Cr(A) in the inequality, it follows that Cr(B) ≥ Cr(A). Incidentally, notice that ratio assumes that multiplication is well-defined on degrees of belief. So, in order to claim that bound follows from ratio, we need more substantial assumptions about degrees of belief than the ones that we have taken up in section 2.
16 A clarification: here, and in what follows, we assume that borderline cases are cases of indeterminacy. (Some theorists, notably epistemicists, do not subscribe to this use of ‘borderline’, since they take all borderline sentences to have determinate truth values.)
17 Thanks to Jason Turner for this kind of case.
18 Here, we’re understanding ≳ as follows: x ≳ y ⇐⇒ x ≥ y ∨ x ≈ y.
19 Higher order weak determinacy is little-explored, but is of obvious relevance here. For example, on Williamson’s fixed-width margin of error models for higher order vagueness [Citation1992, Citation1994], nothing is higher-order weakly determinate at all orders. This could form the basis for an independent objection to weak certainty. However, our initial investigations show that there are natural variants of these models that avoid this feature. An objection from this quarter would have to dig into the plausibility of the various detailed modelling assumptions that are in play.
20 This response was first put to us by Jason Turner in comments on this paper at the 2019 APA. Compare Lewis [Citation1986].
21 If credences are real numbers, we have x ≈ y is true iff |x − y| ≤ ϵ, and, analogously, x ≈≈ y iff |x − y| ≤ 2ϵ, x ≈≈≈ y iff |x − y| ≤ 3ϵ, etc. x >≈ y ≔ x ≥ y ∨ x ≈≈ y.
22 Someone might have the concern that a belief in A being within ϵ of 1 doesn’t guarantee that our belief in det A is within ϵ of 1. But this interlocutor may endorse a suitably hedged variant of the principle—that the consequent follows if A meets some tighter bound, within some δ of 1, where δ < ϵ. Writing ≃ for this tighter approximation, we can combine approximate and hedged versions of our argument via the following premises:
There are contexts where this variant of our argument—strengthened in several dimensions—is required.
23 For discussion of this material, we are indebted to Mike Caie and Branden Fitelson.
24 More precisely: Caie introduces the notion of a determinacy fixed-point, defined as follows: A is a determinacy fixed-point just in case det A = A. Caie shows that certainty is equivalent to the claim that evidential propositions are determinacy fixed-points.
25 The first point is due to Richard Bradley (see, e.g., Bradley [Citation2000, Citation2007]). For examples of triviality arguments applied to epistemic modals, see, e.g., Russell and Hawthorne [Citation2016] and Goldstein [Citation2019].
26 Thanks to Thomas Brouwer, Mike Caie, Branden Fitelson, and Jason Turner for extensive help and discussion. Thanks also to audiences at the Josh Parsons memorial conference at St Andrews, the Pacific APA 2019, the Australian National University, and Glasgow University.