Can truth relativism account for the indeterminacy of future contingents?

Abstract

John MacFarlane has recently argued that his brand of truth relativism provides the best solution to the puzzle of future contingents: assertions about the future that express propositions that are metaphysically neither necessary nor impossible. In this paper, we show that even if we grant all of the metaphysical, semantic and pragmatic assumptions in terms of which MacFarlane sets and aims to solve the puzzle, his truth relativism is not apt to solve the problem of future contingents. We argue that the theory fails to vindicate the intuition that future contingent propositions are neither true nor false, leaving the theory open to a charge of Reductio. We show that these problems cannot be answered while preserving the core tenets of truth relativism.

1 Introduction

In several recent works, John MacFarlane (2003, 2008, 2014) appeals to his distinctive brand of relativism about truth to resolve the puzzle of future contingents: assertions about the future that express propositions that are neither necessary nor impossible, such as ‘There will be a sea battle tomorrow’. The way MacFarlane sets it up, at the heart of the puzzle lies a metaphysical assumption that he explicitly endorses: that the future is objectively open or unsettled.¹

For instance, suppose that on Monday it is objectively unsettled whether there will be a sea battle on Tuesday; in some possible futures there is a sea battle on Tuesday, in others there is peace. Suppose also that on Monday you assert ‘There will be a sea battle tomorrow’, thereby stating the proposition that there is a sea battle on Tuesday.² According to MacFarlane, when we evaluate this proposition from different temporal perspectives, we have clashing intuitions.³ From Monday’s perspective, the proposition strikes us as unsettled or neither true nor false: it is true in some possible futures, but not in others—we have an ‘indeterminacy intuition’. Yet, if we fast forward to the future, and view the previous day’s assertion from the midst of a sea battle, the proposition strikes us as settled or true—we have a ‘determinacy intuition’. These intuitions appear to lead to an incompatibility, since the same proposition seems to be neither true nor false on Monday and true on Tuesday.

As MacFarlane sees it, ‘a satisfactory account of future contingents must give both intuitions their due’ (2003, pp. 321–232). He claims that his brand of truth relativism is best placed to do so. On his view, future contingent propositions are ‘assessment sensitive’ in that whether they are true depends on features of the context of assessment—the context at which a proposition is evaluated as true or false. In the particular case of future contingent propositions, truth value is sensitive to the time at which a proposition is evaluated as true or false. (MacFarlane, 2014, p. 64) For instance, the proposition that there is a sea battle on Tuesday may have a different truth value whether it is assessed on Monday, at the time it is asserted, or whether it is assessed on Tuesday, at the time a sea battle is said to take place.

MacFarlane’s view has generated considerable interest for a variety of reasons. For one thing, the problem of future contingents is a resilient philosophical problem, rooted in Aristotle’s (1984) discussion of the open future in De Interpretatione IX, combining intricate issues in semantics, logic and metaphysics. Any promise of a well worked out solution to this problem, let alone one that is entirely novel, deserves scrutiny. For another, since MacFarlane uses sophisticated tools from cutting-edge philosophy of language to define his relativized truth-predicate, his formulation of relativism is regarded as one of the clearest available.

In this paper, we argue that MacFarlane’s brand of truth relativism⁴ (henceforth: truth relativism, for simplicity) runs into difficulties when it comes to accounting for the indeterminacy intuition regarding future contingents—for instance, that when you say on Monday that there will be a sea battle tomorrow, what you say is neither true nor false. Our argument focuses on a tension between the specific account MacFarlane offers for future contingents and two core commitments of truth relativism, as a general theory of relative truth. Though these commitments will be explained in due course, they can be briefly glossed as follows. The first core commitment is that the assessment sensitivity of any ordinary proposition P gives rise to the assessment sensitivity of the proposition that P is true (see MacFarlane, 2014, p. 93). For instance, if the proposition that there is a sea battle on Tuesday is assessment sensitive, then the proposition that there is a sea battle on Tuesday is true is also assessment sensitive. That is, assessment sensitive discourse is assessment sensitive whether or not it makes use of ordinary notions of truth and falsity. The second core commitment is that the ordinary English truth predicate is monadic—in particular, it is not a relativized truth predicate like the one MacFarlane defines so as to capture the assessment sensitivity of future contingent propositions. For instance, if someone says ‘P is true’, the truth predicate she uses is monadic: what she says is simply that P is true, not that P is relatively-true.

We argue that when combined with MacFarlane’s proposed account of future contingents, these core commitments of truth relativism generate two interrelated difficulties that center around the indeterminacy intuition. First, it implies that this intuition cannot be truly asserted even on Monday, when we are supposedly in its grip. This, we call the ‘Vindication Problem’, which we show can then be used to bring a charge of Reductio against truth relativism.

The paper runs as follows. In section 2 we outline MacFarlane’s solution to the puzzle of future contingents: in  2.1 we explain how he understands the puzzle; In 2.2 we show how truth relativism is meant to address it. In section 3 we show that the indeterminacy intuition cannot be ‘vindicated’, and that a charge of Reductio can be made. In section 4 we consider and reject four potential responses to the objection.

2 Truth relativism and the puzzle of future contingents

2.1 MacFarlane’s puzzle of future contingents

According to MacFarlane, the future is objectively open or unsettled. A natural way of fleshing out this idea, to which MacFarlane appeals, is through so-called ‘Branching Time Theory’ (cf. Belnap & Green, 1994), an eternalist theory of time according to which past, present and future are equally real; and according to which the unfolding of time involves multiple histories that form a rootless tree, with a single trunk representing the settled past history, and multiple branches representing the unsettled future histories. On this view, there is an asymmetry between past and present histories on the one hand and future ones on the other: while there is at any moment a unique past and present history, there are multiple future histories that are ontologically on a par. The future is objectively unsettled because no future history is yet singled out as the future history of the world (see MacFarlane, 2003, 2005).

Now, here is how our puzzle arises. Suppose that on Monday it is objectively unsettled whether a sea battle will occur on Tuesday and that Alice asserts the following contingent, future tensed sentence:

(1) There will be a sea battle tomorrow.

The situation is represented by Fig. 1. Here, w₁ and w₂ are distinct possible worlds that are qualitatively identical in their past and present, but which represent different ways the future might be: at w₁ a sea battle occurs on Tuesday, but not at w₂. We suppose with MacFarlane that for every world, it is determinate how the future is at that world. C₀, C₁ and C₂ are particular contexts—where contexts are sets of parameter values, including worlds, agents, locations and times.

Fig. 1

The Puzzle of Future Contingents

For convenience, let ‘W(C_i)’ denote the set of worlds contained in a context C_i (where ‘C_i’ is a variable for contexts), such that: at C₀, it is unsettled whether there is a sea battle on Tuesday, since both w₁ and w₂ are in W(C₀). At C₁, it is settled that there is a sea battle since only w₁ is in W(C₁); and at C₂ it is settled that there is no sea battle since only w₂ is in W(C₂).

Against this metaphysical background, we are meant to be torn between two intuitions when we evaluate Alice’s assertion of (1) ‘There will be a sea battle tomorrow’. From the concurrent perspective of Monday, what she says strikes us as neither true nor false. After all, the future is open. But from the perspective of Tuesday, it seems that what she said on Monday not only is true, it was true all along. She was right!

Thus, the two intuitions, call them Indeterminacy and Determinacy, conflict:

• Indeterminacy: What Alice said is neither true nor false.⁵

• Determinacy: What Alice said is true (false).⁶

What Alice said cannot both lack a truth value and be true (false).⁷

2.2 Truth relativism applied to the puzzle of future contingents

Nevertheless, MacFarlane offers a way to give both Indeterminacy and Determinacy their due. The theory he puts forward has two key aspects, which must be viewed as a package: the first is the formal apparatus he uses to account for the assessment sensitivity of future contingents; the second has to do with broader core commitments of truth relativism concerning the nature and scope of assessment sensitivity. We explain them in turn.

The first aspect is a formal semantic, postsemantic and pragmatic account. In general, MacFarlane distinguishes between semantics proper, which delivers a content of an expression at a context and an index, and postsemantics, which delivers the truth value of a sentence at a context of utterance and assessment. His pragmatics then specifies the conditions under which assertions may be made or ought to be withdrawn: it is tied to the postsemantics in that these conditions are stated in terms of truth and falsity as defined by the postsemantics.

We now explain how the postsemantics and pragmatics work and explain how they help address the puzzle raised by the combination of Indeterminacy and Determinacy.

Let us assume with MacFarlane that the proposition stated by Alice when she asserts (1) is P₁:

(P₁) There is a sea battle on Tuesday.

Following MacFarlane, we assume that propositions are eternal and should be construed as sets of possible worlds—though we agree with him that nothing of substance hangs on this. It is however interesting to note here that one reason he offers in favour of working within an eternalist framework is that it makes it easier to show that the ordinary, English truth predicate is monadic: ‘For simplicity, we will work with eternalist propositions. This will help illustrate our earlier claim [t]hat assessment sensitivity does not require propositional truth to be relativized to anything besides possible worlds.’ (MacFarlane, 2014: p. 207) The treatment of the ordinary truth predicate as monadic is a core feature of the account to which we return shortly, since it will play a key role in our argument against truth relativism in Sect. 3.

Now consider again P₁ and the scenario described in Fig. 1. The account aims to show that P₁ is evaluated as neither true nor false from the perspective of Monday (at C₀) and as true from the perspective of Tuesday (at C₁).⁸ This is achieved by appeal to an assessment sensitive truth predicate defined in terms of two key notions: a context of utterance, which is a set of parameter values, including worlds, agents, locations, and times, representing the circumstances in which a sentence is uttered; and a context of assessment, which is a set of shiftable parameters from the context of utterance representing the perspective from which an asserted proposition can be assessed (MacFarlane, 2014, p.78). This truth predicate applies to propositions, and is defined for future contingent propositions as follows (MacFarlane, 2014, p. 226):

Relativist Postsemantics for Truth (RPT): A proposition ϕ is true as used at C_i and as assessed at C_j iff for every w ∈ W (C_i, C_j), ϕ is true at w, where W(C_i, C_j) = W(C_j) if W(C_j) $\subset$ W(C_i) and W(C_i) otherwise.

Given that the postsemantics is non-bivalent, falsity is not the dual of truth, but can be defined as truth of the negation, as follows:

Relativist Postsemantics for Falsehood (RPF): A proposition ϕ is false as used at C_i and as assessed at C_j iff for every world w ∈ W (C_i,C_j), ϕ is false at w, where W(C_i, C_j) = W(C_j) if W(C_j) $\subset$W(C_i) and W(C_i) otherwise.⁹

For simplicity, let RP be the conjunction of RPT and RPF. MacFarlane combines RP with a pragmatic account concerning norms of assertion, which are also relied upon to fully explain our intuitions about the accuracy and inaccuracy of assertions of future contingents. Roughly, an assertion is appropriate at a context of utterance and a context of assessment iff it is true relative to both contexts. As he puts it:

Accuracy: An attitude or speech act occurring at C_i is accurate as assessed from a context C_j just in case its content [proposition] is true as used at C_i and assessed from C_j. (MacFarlane, 2014, p.127)

For MacFarlane, inaccuracy is the denial of Accuracy—for an assertion to be inaccurate, it is sufficient that it is not true as used at C_i and assessed from C_j, in which case it should be retracted:

Retraction Rule: An agent in context C_j is required to retract an (unretracted) assertion of P made at C_i if P is not true as used at C_i and assessed at C_j. (MacFarlane, 2014, p.108)

With this in place, here is how both Indeterminacy and Determinacy can be given their due. Recall that Indeterminacy was the intuition that the proposition,

• (P₁) There is a sea battle on Tuesday,

is neither true nor false when assessed from the perspective of C₀. Indeterminacy is given its due as follows. Given RPT, P₁ is not true when used at C₀ and assessed at C₀, since it is not true at w₂ and so is not true at all worlds w ∈ W(C₀, C₀). Given RPF, P₁ is not false when used at C₀ and assessed at C₀, since it is not the case that it is false at all worlds w ∈ W(C₀, C₀). Thus, P₁ is neither true nor false when used and assessed at C₀.

Furthermore, given Accuracy, Alice’s assertion of (1) is inaccurate when assessed at C₀, since P₁, the proposition that Alice expresses in asserting (1), is not true as used at C₀ and assessed at C₀. Given Retraction Rule, Alice ought to retract her assertion of (1) if she is challenged at C₀ while it is still open that there will be no sea battle on Tuesday.

Let us turn now to Determinacy, the intuition that P₁ is true when assessed from the perspective of C₁. This intuition is given its due as follows. Given RPT, P₁ is true when used at C₀ and assessed from C₁ since it is true at every world w ∈ W(C₀, C₁). Given Accuracy, Alice’s assertion is accurate when assessed at C₁, since P₁ is true at C₁, and Retraction Rule does not require her to retract her (unretracted) assertion of (1) if she is challenged at C₁.

Thus, RP seems to give both Indeterminacy and Determinacy their due, where that is reflected at the pragmatic level by Accuracy and Retraction Rule: it entails that what Alice said is neither true nor false when assessed at C₀, yet it is true when assessed at C₁; when assessed at C₀, Alice’s assertion is inaccurate and should be retracted, but not when assessed at C₁.

The second feature of truth relativism considered as a package is as follows. The relativized truth predicates defined in RP just discussed are used to specify the truth conditions of assessment sensitive discourse as well as the pragmatics of such discourse. They are part of a of a broader theory of relative truth, which also includes the two core commitments mentioned in the introduction, and which are therefore crucial to the evaluation of the overall account of future contingents. We now spell them out in detail.

The first core commitment is that, as MacFarlane puts it, ‘if the language can express any assessment sensitive propositions, “true” will also be assessment sensitive, since if P is assessment sensitive, the proposition that P is true must be assessment sensitive too’ (2014, p.93). We call this commitment ‘ASOT’:

Assessment Sensitivity of Ordinary Truth (ASOT): If the proposition that P is assessment sensitive, then the proposition that P is true is assessment sensitive.

The second core commitment is that, as distinct from the relativist truth predicates defined in RP, the ordinary truth-predicates used in English are not defined in terms of the parameters specified in RP. Rather, according to MacFarlane, the English predicate ‘is true’ is monadic. This is captured in (SMIT):

Semantics for monadic ‘is true’ (SMIT): ‘True’ expresses the same property at every context of use—the property of being true. The extension of this property at a circumstance of evaluation e is the set of propositions that are true at e. (MacFarlane, 2014, p.93)

This ordinary English, monadic truth-predicate naturally obeys the Equivalence Schema (MacFarlane, 2014, p.93):

• Equivalence Schema (ES): The proposition that φ is true iff φ.¹⁰

Here is how these core commitments work together. The first clause of SMIT makes it clear that, as MacFarlane puts it, ‘The relativist… can treat the monadic predicate ‘true’ as just another predicate of the object language—the language for which she is giving a semantics.’ The second clause of SMIT leaves open the possibility that the truth values of some propositions vary from one circumstance of evaluation to another. Furthermore, given ASOT, if our discourse about future contingents is assessment sensitive, then so too are the propositions of that discourse which are asserted using the English predicates ‘true’ and ‘false’. Hence if RP provides the correct post-semantics for future contingents in general, it must provide the correct post-semantics for propositions about future contingents stated using the ordinary English truth predicate.¹¹

In what follows, we show that this package—RP, Accuracy, Retraction Rule, ASOT and SMIT – cannot vindicate Indeterminacy and is open to a charge of Reductio.

3 Vindication and Reductio

At first blush, it looks as if RP solves the problem of future contingents, where that is framed against the backdrop of the assumption that the future is objectively open, and is understood as the problem of giving both Indeterminacy and Determinacy their due. However, we now argue that truth relativism, as a package, faces a difficulty that centers around Indeterminacy, in the following way. The ‘Vindication Problem’ arises when we consider someone stating Indeterminacy at the time when we are supposed to be in its grip. As we shall see, RP entails that the proposition, stated in (1), that what Alice said is neither true nor false, is false when used at C₀ and assessed at C₀. While, as we note later, an analogous problem has been raised for similar (post-)semantic theories (specifically, supervaluationism),¹² we argue that in the context of truth relativism framed in terms of RP, Accuracy, Retraction Rule, ASOT and SMIT, it has more unpalatable consequences than have previously been acknowledged. As we show, it can be used to bring a charge of Reductio against it.

We can set up the Vindication Problem as follows. Suppose that Hugo witnesses Alice’s assertion of (1) on Monday. Suppose also that he gives voice to Indeterminacy by asserting the following concurrent assessment of Alice’s assertion:

(2) What Alice said is neither true nor false.

This situation is represented in Fig. 2.¹³

Fig. 2 

The Vindication Problem

Recall that what Alice states with her assertion of (1) at C₀ is the proposition P₁:

• (P₁) There is a sea battle on Tuesday.

What Hugo states with his assertion of (2) at C₀ is the proposition (P₂):

• (P₂) That there is a sea battle on Tuesday is neither true nor false.

By ASOT, if P₁ is assessment sensitive, so is P₂. That is, if RP applies to future contingent propositions such as P₁, it applies equally to propositions about future contingents such as P₂. The problem is that, though RP entails that P₁ is neither true nor false when assessed at C₀, it entails that P₂ is false when assessed at C₀, exactly when we are supposed to be in the grip of Indeterminacy. This is because, given that there is a sea battle at w₁, P₁ is true at w₁. Since P₂ is the proposition that P₁ is neither true nor false, P₂ is false at w₁. And given that there is no sea battle at w₂, P₁ is false at w₂. Since P₂ is the proposition that P₁ is neither true nor false, P₂ is false at w₂.¹⁴ Therefore, P₂ is false at every world w ∈ W (C₀, C₀); when Hugo gives voice to Indeterminacy, what he says is false when assessed from the very perspective at which we are supposed to be in its grip. Indeed, since P₂ is false at both w₁ and w₂, what Hugo says is false at all contexts of assessment C₀, C₁ and C₂. This is our Vindication Problem.

To state the problem more precisely, it will be useful to mark the distinction between the ordinary truth predicates of English that Hugo uses in the way he states Indeterminacy, and those defined in the theory. Thus, let ‘true’ and ‘false’ denote the ordinary truth predicates of English. Let ‘true_RP’ and ‘false_RP’ denote the truth predicates defined by RP. The Vindication Problem can now be stated as follows:

Vindication Problem: Assessment Sensitivity fails to vindicate Indeterminacy because it entails that a proposition stating Indeterminacy using the ordinary English truth predicates is false_RP, even when it is used and assessed at a time at which we take it intuitively to be true.

The Vindication Problem highlights the fact that RP does not capture the intuitive truth status of what is supposed to be one of our core intuitions about future contingents—Indeterminacy as stated by (2). However, one might expect that an adequate semantics or postsemantics for an area of discourse such as future contingents would capture the intuitive truth status of core intuitions about that discourse. In other words, one might expect a pre-theoretic intuition regarding future contingents—one that can be stated using an ordinary English truth predicate—to be captured using the truth predicate of the (post-)semantics for that discourse. We can state this as the following Adequacy Condition regarding the truth status of shared core intuitions about future contingents that we might have:

Adequacy Condition: if P is a proposition about future contingents that is intuitively true/false/neither true nor false, P should be true/false/neither true nor false by the lights of the relevant (post-)semantic theory of future contingents.

MacFarlane makes apparent his commitment to this condition of adequacy when he discusses rival theories and argues that they fail to predict the truth status of core intuitions about future contingents. For instance, he considers the following propositions, which he takes to be intuitively true (MacFarlane, 2014, p.217):

One or the other will happen: It is possible that it will be sunny tomorrow, it is possible that it won’t be, and either it will be or it won’t be.

He argues that rival theories fail to match the intuitive truth status of these propositions, and that his own theory does. Thus, it seems that, for MacFarlane, a general condition of adequacy on a semantic theory of future contingent propositions is in place requiring that the theory should predict the truth statuses of propositions concerning our core intuitions about future contingents, such as One or the other will happen, and of course, Indeterminacy as stated using (2).

Indeed, coming back to Indeterminacy, recall that Hugo is merely giving voice to an intuition that reflects our ‘ordinary thought and talk about the future’ (MacFarlane, 2014: p. 202), one we are meant to all share with respect to Alice’s assertion, using the ordinary, monadic truth predicate of English. If truth relativism is to satisfy Adequacy Condition, given SMIT and ASOT, RP should not only predict the intuitive truth status of P₁, but also the intuitive truth status of P₂: it should imply that P₂, the proposition that P₁ is neither true nor false, is also true_RP: true when assessed at C₀. Yet, not only does it fail to do so, it implies that P₂ is false_RP: false when assessed at C₀.¹⁵

The Vindication Problem wreaks havoc with the pragmatic theory of assertion associated with RP: Alice says on Monday that there will be a sea battle on Tuesday. When used and assessed on Monday, what Alice says is neither true_RP nor false_RP. Yet, if Hugo asserts on Monday that what Alice just said is neither true nor false, thereby challenging her assertion, what he says is false when it is assessed on Monday: it is false_RP. Thus, by Accuracy, Hugo’s assertion is inaccurate;¹⁶ by Retraction Rule, if Hugo’s assertion were to be challenged, he would be obligated to retract it, since the proposition he asserts is false when used and assessed on Monday. More precisely, Alice asserts P₁ at C₀, and P₁ is neither true nor false when assessed at C₀. By Accuracy, her assertion is inaccurate and by Retraction Rule she should retract it, if challenged at C₀. Hugo’s assertion of P₂ at C₀ is such a challenge. Yet, because P₂ is false when assessed at C₀, by Accuracy his assertion is inaccurate, and if it were to be challenged, by Retraction Rule, it would have to be retracted.

How serious are these difficulties? As we noted at the beginning of this section, the kind of issue raised by the Vindication Problem is not entirely unfamiliar. Though MacFarlane does not address this problem in connection with his own theory, he raises a similar issue in connection with supervaluationism applied to future contingents. He offers the following gloss on the problem: any statement to the effect that a future contingent proposition is neither true nor false is ‘ineffable from the “internal” point of view’ (2008, p.97), i.e. no such statement can be truly stated by the lights of the theory. Supervaluationists, as he notes, may simply bite the bullet, and accept that Indeterminacy cannot be truly stated by using (2).¹⁷ Perhaps that is so, but this option is unavailable in the context of truth relativism taken as a package deal, crucially involving SMIT and ASOT, to which supervaluationists need not be committed. To highlight the seriousness of the difficulty, we show that the Vindication Problem can be used to generate a Reductio of truth relativism.

The Reductio charge we raise makes use of one independently plausible semantic assumption, which concerns the relation of the falsity of a proposition to its negation:

• F. The proposition that ϕ is false_RP → ¬ϕ.

F is an extremely natural assumption to make, one that most would be hard-pressed to give up, since it captures a core feature of our intuitive understanding that if something is false then it is not the case. It is hard to imagine a kind of truth predicate to which it would not apply. Moreover, F entirely fits RP, which regards every instance of F as holding at every context of use and context of assessment.

Informally, here is how the charge of Reductio goes. RP implies that P₁ is neither true_RP nor false_RP. This follows straightforwardly from its application to the proposition that Alice states when she asserts (1). Now Consider P₂. This is just one way to state Indeterminacy, which is intuitively true at the time that it is stated, and is stated using the ordinary English truth predicate governed by SMIT. By ASOT, if P₁ is assessement sensitive, so too is P₂. And if P₂ is assessment sensitive, RP applies to it.

However, as our discussion of the Vindication Problem shows, when applied to P₂, RP entails that P₂ is false_RP. Now, consider an application of F to the claim that P₂ is false_RP. Applied to this claim, it follows that not-P₂. But one of starting points was that RP entails P₂. Thus, it appears that RP can be used together with SMIT, ASOT, and F to derive a contradiction.

More formally, the argument goes as follows.

Reductio of truth relativism

(i) RP:

(Ass.)

(ii) RP → P₁ is neither true_RP nor false_RP:

(RP applied to P₁ at C₀, C₀)

(iii) P₂:

(Ass.: Indeterminacy stated by (2))

(iv) P₂ is assessment sensitive:

(Ass.: SMIT, ASOT)

(v) RP → P₂ is false_RP:

((iv), RP applied to P₂ at C₀, C₀)

(vi) P₂ is false_RP:

((i), (v), Modus Ponens)

(vii) ¬P₂:

((vi), F)

(viii) ∴ ¬RP:

((iii), (vii), Reductio ad Absurdum)

Thus, RP is false. This is our Reductio charge, which rests on the Vindication Problem, SMIT, ASOT and F.

In the next section, we consider several avenues of response to this charge of Reductio. Let us first narrow the field of possible responses that we will consider. Rejecting step (i) would be bizarre, since it is the theory to be defended. While we think this theory ultimately should be rejected, we do not consider alternative semantic theories, or modifications to RP itself, as this would broaden the discussion too far to be manageable. One could reject step (ii), but that would be to reject the solution to the problem of future contingents delivered by RP, to reject the whole point of appealing to truth relativism in the first place. One could reject F, and thus block the Reductio at step (vii), but F is highly intuitive, and so difficult to give up. Moreover, giving it up would leave the Vindication Problem untouched, which would only constitute a partial solution to the difficulties we have highlighted. It is clear that one can hardly reject (vi) or (viii) without rejecting Modus Ponens or Reductio ad Absurdum.

So, the most promising responses to the foregoing argument will focus on reinterpreting step (iii) so as to block step (v) of the Reductio. Simply rejecting P₂, so as to block step (iii), is not really an option; this amounts to either rejecting Indeterminacy, one of the intuitions that truth relativism sets itself to account for, or requiring that Indeterminacy cannot be stated using ordinary English truth predicates, which would be a distinctively odd thing to require. It would also mean failing to satisfy Adequacy Condition, the requirement that a (post-)semantics for future contingents predicts the intuitive truth status of a propositions about future contingents. Rather, we will look at replies that suggest that Hugo states a different proposition from P₂ when he asserts (2), thereby blocking the Reductio at step (v). We will look at what these strategies mean for our understanding of step (iv). Thus, according to these replies, even if we could reach a step analogous to (iii), with a different proposition from P₂, the argument could not proceed further and no Reductio occurs. These will be in large part the focus of the next section.

4 Possible responses

We now consider four lines of response to block the charge of Reductio. The first three proceed along the lines just suggested. They effectively reject the idea that the truth predicate used by Hugo in (2) is the ordinary English truth predicate, which MacFarlane takes to satisfy SMIT. The first re-interprets (2) as stating a proposition involving—not truth—but determinate truth, so what Hugo says is really that P₁ is neither determinately true nor determinately false. The second re-interprets (2) as  really involving a dyadic truth predicate, roughly glossed as ‘true at C_i, C_j’. The third re-interprets (2) as really involving a metalinguistic truth predicate of some kind, such as one defined in the spirit of MacFarlane’s RP. We show that none of these responses addresses the difficulties we have raised above. More precisely, we show that if P₂ can be stated at all, then RP is false; and if P₂ cannot be stated, but some proposition can—one stated using a different truth predicate from the ordinary monadic English truth predicate—then we are effectively giving up on ASOT, a core feature of MacFarlane’s truth relativism.

Finally, the fourth response takes a different path, that of severing the link between RP on the one hand and the pragmatic account in terms of Accuracy and Retraction Rule on the other. This response, we argue, is difficult to motivate, and ultimately leaves the difficulties we have raised untouched.

4.1 Determinate truth and determinate falsehood

The first response we consider is one that reformulates Indeterminacy in terms of lack of determinate truth value rather than lack of truth value. Though we informally express the intuition we are meant to share as Hugo does when he asserts (2), one might think that, strictly speaking, the intuition that we share is not that what Alice said is neither true nor false, but that what Alice said is neither determinately true nor determinately false. Thus, a proper statement of Indeterminacy involves a determinacy operator, and if Hugo is giving voice to Indeterminacy, then he must implicitly make use of this operator. This response has the potential benefit of evading the Vindication Problem and blocking step (v) of the Reductio, the step that is highlighted by the Vindication Problem. It does so by requiring that Hugo expresses a different proposition from P₂, thus revising step (iii). Let us consider this response in more detail.

The response first says that the proper way of understanding the indeterminacy intuition is as Indeterminacy* rather than Indeterminacy:

• Indeterminacy*: what Alice said is neither determinately true nor determinately false.

That is, when Hugo asserts (2) on Monday, he in fact gives voice to Indeterminacy*, and thus states the proposition P₃:

• (P₃) That there is a sea battle on Tuesday is neither determinately true nor determinately false.

P₂ is simply not the content of the indeterminacy intuition that Hugo is voicing.

How might we understand ‘is neither determinately true nor determinately false’? The natural way is to understand this phrase as involving supervaluationist, non-bivalent, truth predicates; for instance, these could be defined as follows¹⁸:

• 
  

  Determinate Truth: A proposition ϕ is determinately true at C_i iff for every w ∈ W (C_i), ϕ is true at w.

  
  

  Determinate Falsehood: A proposition ϕ is determinately false at C_i iff for every w ∈ W (C_i), ϕ is false at w.

Up to now, we have assumed that the truth predicate Hugo is using is the English truth predicate. The response currently under consideration denies this assumption, in an effort to avoid the difficulties we have raised. The proposal is that it is not the English truth predicate that is involved in the assertion of (2), but a distinct truth predicate defined by Determinate Truth and Determinate Falsehood. According to this proposal, when Hugo asserts (2), he is really voicing Indeterminacy*, since the predicates ‘is true’ and ‘is false’ as he uses them are defined in terms of Determinate Truth and Determinate Falsehood. Thus, the proposition he expresses is P₃ rather than P₂. Our step (iii) of the Reductio now is step (iii_P3):

(iiiP3) P3

(Assumption: Indeterminacy*)

With all this in place, this response successfully avoids the Vindication Problem and blocks the move in the Reductio at step (v): P₃ is not false_RP. Indeed, given RP, P₃ is true at (C₀, C₀), since it is true at every world w ∈ W (C₀, C₀). While at w₁ there is a sea battle on Tuesday, it is true at w₁ that the proposition that there is a sea battle on Tuesday is neither determinately true nor determinately false when used at C₀ and assessed at C₀. Similarly, while at w₂ there is no sea battle on Tuesday, it is true at w₂ that the proposition that there is a sea battle on Tuesday is neither determinately true nor determinately false when used and assessed at C₀. Thus, the Vindication Problem does not apply if the proposition stated by Hugo is P₃.

As we said, this response blocks the Reductio at step (v), at which RP is applied to P₂ at C₀, C₀. If instead RP is applied to P₃, the corresponding step would be:

• (v_P3) RP → P₃ is false_RP.

Thus, (v_P3) is false. As we have just seen, when RP is applied to P₃, P₃ is true at every context of assessment.

The trouble with this response is that it radically departs from the initial, intuitive way in which the problem of future contingents was set up, and that it sacrifices a core commitment of truth relativism. First, if a response along these lines is to fend off the difficulties we have raised, it must be assumed that there is no ordinary English truth predicate—that does not contain a determinacy operator—for Hugo to use, and thus that P₂ is truly ‘ineffable’; he cannot so much as state it. Otherwise, if it is at all possible for Hugo to state P₂, then defining this alternative truth predicate would leave the original difficulties untouched. Not only is this implausible, but it departs from MacFarlane’s own articulation of Indeterminacy, which is framed using ordinary English truth predicates.

Second, this line of response requires giving up ASOT, according to which if P is assessment sensitive, then P is true is assessment sensitive. After all, P₁ is assessment sensitive, but the proposition P₃, that P₁ is neither determinately true nor determinately false, is not. So, rewriting step (iv) of our Reductio as (iv_P3):

• (iv_P3) P₃ is assessment sensitive

would yield a false proposition.

Yet, ASOT is even more central to the theory, since a commitment to it is taken by MacFarlane to be a defining feature of what makes one a relativist about truth. Thus, to reject ASOT is in effect to reject truth relativism altogether.

Finally, the proposal under consideration radically departs from MacFarlane’s original understanding of the puzzle, which involved Indeterminacy, not Indeterminacy*. Indeterminacy and Determinacy were meant to articulate ordinary intuitions we have about the puzzle of future contingents, ordinary intuitions that we have about the openness of the future and the closedness of the past.¹⁹ An argument would need to be given as to why, besides it being semantically convenient, these intuitions must be framed in terms of Indeterminacy* and Determinacy*, or why it is impossible for one to state proposition P₂ by an assertion of (2).²⁰

4.2 Dyadic truth and falsity

The second response we consider once again rejects step (v) of the Reductio as illicit on the grounds that P₂ is not the proposition that Hugo states when he asserts (2), so in the first place step (iii) has to be revised. Unlike the foregoing response, this is because the truth predicate that he uses is dyadic, with the upshot that the proposition that he asserts contains an explicit relativization of truth value to contexts of utterance and assessment. That is to say, the response under consideration has it that when Hugo asserts (2), he does not use the monadic truth predicate of English, which according to MacFarlane obeys SMIT, but a dyadic truth predicate, which does not. Using this dyadic predicate, he states the following proposition:

• (P₄) That there is a sea battle on Tuesday is not true at (C₀, C₀) and not false at (C₀, C₀).

Once again, it is easy to see how an appeal to a dyadic truth predicate helps to evade the Vindication Problem and block step (v) of the Reductio. First, consider the Vindication Problem. Given RP, P₄ is true at (C₀, C₀), since it is true at every world w ∈ W (C₀, C₀). While at w₁ there is a sea battle on Tuesday, it is true at w₁ that the proposition that there is a sea battle on Tuesday is neither true nor false when used at C₀ and assessed at C₀. Similarly, while at w₂ there is no sea battle on Tuesday, it is true at w₂ that the proposition that there is a sea battle on Tuesday is neither true nor false when used and assessed at C₀. Thus, the Vindication Problem does not arise if the proposition stated by Hugo is P₄.

This response blocks the charge of Reductio at step (iv). By the lights of RP, if Hugo’s assertion of (2) states P₄, then what Hugo says is not only true at (C₀, C₀), it is also true as assessed at C₁, and as assessed at C₂ because it is true at both w₁ and w₂, and hence true at every world w ∈ W (C₀, C₁) and at every world w ∈ W (C₀, C₂). There is no assessment sensitivity to be found there. So step (iv_P4), which is equivalent to step (iv) but concerns P₄,

• (iv_P4) P₄ is assessment sensitive,

would be false. Moreover, the response blocks the charge of Reductio at step (v_P4), which would be the equivalent of step (v) but applied to P₄:

• (v_P4) RP → P₄ is false_RP.

Again, (v_P4) is false because when RP is applied to P₄, P₄ is true at every context of assessment: P₄ is true_RP.

Though this response addresses both the Vindication Problem and the charge of Reductio, it is really a non-starter for anyone sympathetic to truth relativism, because it is incompatible with both SMIT and ASOT. This response is inconsistent with SMIT because, according to SMIT, ‘P is true’ does not express the proposition that P is true at (C_i, C_j); it simply expresses the proposition that P is true. Moreover, once again, for this response to succeed in evading the foregoing difficulties, it must be assumed that there just is no monadic truth predicate for Hugo to use in this context, and hence that he simply cannot state the proposition P₂, which represents an even more radical departure from a core commitment of truth relativism, and is implausible to boot.²¹ Once again, this response is inconsistent with ASOT because while P₁ is assessment sensitive, P₄, effectively the proposition that P₁ is neither true nor false at (C_i, C_j) is not—and, as we have noted, ASOT is a defining feature of relativism about truth. Since the dyadic route is fundamentally at odds with both SMIT and ASOT, both of which are core tenets of MacFarlane’s view, this is not a route that he will wish to take.

4.3 Metalinguistic truth and falsity

Another way to attempt to block the Reductio at step (v) is to go metalinguistic. On this view, the truth predicate that occurs in Hugo’s assertion of (2) is a metalinguistic truth predicate to which RP was never intended to apply. Thus, this attempt follows the previous ones in suggesting that the proposition stated by Hugo when he asserts (2) is not P₂ but another one. Again, as before, the proposal would be to revise step (iii) first.

There are two general strategies one might have here. The first one is inspired by a kind of Tarskian thought (see Tarski, 1944) whereby the object language does not as such contain a truth predicate. When Hugo asserts ‘What Alice said is neither true not false’, he is not stating P₂ but rather P₅:

• (P₅) That there is a sea battle on Tuesday satisfies ‘is neither true nor false’.

The second one has it that when Hugo uses a truth predicate to voice Indeterminacy, he has to be understood as using a theorist’s truth predicate, not simply a truth predicate of English, to which truth relativism is meant to apply. It would be natural in this context to take this theorist to be committed to truth relativism and thus working with the truth predicates defined in RP. On this strategy, when Hugo asserts ‘What Alice said is neither true not false’ he is not stating P₂ but rather P₆:

• (P₆) That there is a sea battle on Tuesday satisfies ‘is neither true_RP nor false_RP’.

Appealing to this second truth predicate avoids the Vindication Problem and blocks the move in the Reductio at step (v) in the same way as appealing to the dyadic truth predicate did. First, consider the Vindication Problem. Given RP, P₆ is true at (C₀, C₀), i.e. it is true_RP, since it is true at every world w ∈ W (C₀, C₀). While at w₁ there is a sea battle on Tuesday, it is true at w₁ that the proposition that there is a sea battle on Tuesday is neither true nor false when used at C₀ and assessed at C₀; i.e. it is neither true_RP nor false_RP. Similarly, while at w₂ there is no sea battle on Tuesday, it is true at w₂ that the proposition that there is a sea battle on Tuesday is neither true nor false when used and assessed at C₀; i.e. it is neither true_RP nor false_RP. Thus, the Vindication Problem does not arise if the proposition stated by Hugo is P₆.

Similarly, the step analogous to (iv) of the Reductio applied to P₆ does not hold:

• (v_P6) RP → P₆ is false_RP

Indeed, (v_P6) is false.

It is easy to see that the core objection which arose in relation to both of the foregoing responses to the difficulties applies here. The present response rejects a core commitment of our truth relativism, namely ASOT, since P₁ is assessment sensitive, but P₆, the proposition that P₁ satisfies ‘is neither true_RP nor false_RP’, is not. As we have pointed out in our discussion of the previous responses, to give up ASOT is to give up truth relativism altogether.

Moreover, as with the previous proposed responses, this one too departs from the suggestion that Indeterminacy is an intuition that we are all meant to share, though the departure is sharper still, since on the present response, it is assumed that when Hugo voices Indeterminacy he must assume the role of the theorist. This assumption is starkly at odds with the thought that Indeterminacy is a core pre-theoretic intuition we all share, including those not well versed in the theory and those who oppose it: it voices a bystander’s reaction to what Alice says in (1). Furthermore, if Indeterminacy is understood to involve the very truth predicates defined by RP, it is hardly an achievement that RP gives Indeterminacy its due.

Now consider the Tarski inspired suggestion that English does not contain a truth predicate at all, but that the words ‘true’ and ‘false’ belong to a metalanguage. On this suggestion, (2) expresses the proposition P₅, which involves these metalinguistic truth predicates. The point of invoking these truth predicates, however they are defined more precisely, is to block the application of RP, on the ground that the metalinguistic truth predicates are not assessment sensitive. Of course, if these truth predicates are not assessment sensitive, then for reasons that parallel those just given above, P₅ is true at every context of use and assessment, and the corresponding line of the Reductio,

• (v_P5) RP → P₅ is false_RP,

is false.

The key complaint here is as above. Though this move might respect the spirit of SMIT, it does so at the cost of the assessment sensitivity of the English language truth-predicate, which is at odds with a core thesis of truth relativism, namely ASOT. This is non-negotiable, since to deny the assessment sensitivity of the truth-predicate Hugo uses is to give up on truth relativism.

Furthermore, the Tarski inspired metalinguistic proposal seems to be a non-starter for reasons that go beyond these theoretical commitments. What underpins this proposal is the wholesale view that the English language does not contain its own truth-predicates, a view that is contentious at best, and will certainly feel like an overreaction in the case of debates about future contingents. It is simply bizarre to think that there is no object language truth-predicate for Hugo to use when he is voicing Indeterminacy—a view that few will find attractive.

4.4 Changing the pragmatics of indeterminacy

The responses we have considered so far have all attempted to evade the Vindication Problem and block the charge of Reductio by postulating an alternative account of the proposition Hugo states when he asserts (2). As we have seen, all of these responses have a common cost: ASOT. We now turn to a response that attempts a very different approach to the difficulties that we have raised, one that starts with pragmatics of assertion.

We have taken Hugo to perform the speech act of assertion when he gives voice to Indeterminacy by uttering (2), and assumed that what he asserts is the proposition P₂. However, one might argue that in giving voice to Indeterminacy, Hugo does not perform the speech act of assertion, but some other speech act. Though there may be many different ways to flesh this response out in detail,²² to fix ideas, we can consider the possible response that what Hugo does when he gives voice to Indeterminacy is not to assert that what Alice said is neither true nor false, but to reject that what Alice said is either true or false. The response involves appealing to the suggestion that there is a sui generis speech act of rejection of a proposition that is not equivalent to the assertion of the negation of that proposition or to the assertion that that proposition is false.²³

Thus, suppose that Hugo merely rejects that that there is a sea battle on Tuesday is either true or false. Then step (iv) of our Reductio charge could be blocked as follows: there just isn’t a proposition P₂ that is asserted by Hugo using (2) that comes out false according to RP. By merely rejecting the proposition that there is a sea battle on Tuesday is either true or false, he is not thereby asserting anything; in particular, he is not asserting P₂. That is to say, there is no Vindication Problem, since this would require P₂ to be asserted. Rather, P₁ is rejected as either true or false.

Now, obviously, to give a full discussion of the matter, we would need to know more about how to understand rejection, how exactly the speech act of rejection is to be distinguished from the speech act of assertion, and how all this might be integrated to the overall architecture of truth relativism.²⁴ But this would lead us too far afield. At any rate, two points can be made here, which suggest that further development of this line of response is not likely to bear fruit. First, the proposal does not seem to be in the spirit of MacFarlane’s own pragmatic account articulated in terms of Accuracy and Retraction Rule. Roughly, Accuracy tells you that an assertion of a proposition is accurate at a context of assessment iff it is true at that context, while Retraction Rule tells you that an assertion of a proposition that is not true at a context of assessment is one that is to be retracted at that context of assessment. The natural way to read these norms is as going along with the standard Fregean view that rejection is the denial of assertion. When one has to retract, one has to retract an assertion because the proposition asserted is not true, and the natural way to understand this is as requiring the assertion of the negation of the proposition—or some speech act that amounts to, or commits one to, such an assertion.

Second, if the rejection of a proposition is not an assertion of the negation of that proposition, then Hugo’s rejecting P₁ would not suffice to trigger Retraction Rule, which states that an agent should retract an assertion of a proposition if that proposition is not true at a context of assessment. If Hugo were merely to reject Alice’s assertion, that would not amount to his stating that what she said was not true, and hence would not meet the requisite for a demand for retraction. But this is odd, because it makes in some sense that demand impossible to formulate.

Finally, the proposal would work only if it implied that it is impossible to assert that P₁ is neither true nor false. After all, if it is so much as possible for Hugo to assert that P₁ is neither true nor false, then it is possible for him to assert P₂, and both the Vindication problem and the Reductio arise once more. So, the proponent of this line of reply would have to argue that Indeterminacy could only be expressed through the speech act of rejection. But why think that? Perhaps it could be argued that it is simply impossible to assert that a proposition is neither true nor false. Yet, such a suggestion is too implausible to merit serious consideration. Or perhaps it could be argued, more plausibly, that there is an available speech act of rejection alongside assertion, but that Hugo in fact performs the speech act of rejection rather than assertion. However, it is difficult to see how one might argue on that basis that it is not possible for Hugo to assert that P₁ is neither true nor false altogether.

5 Concluding remarks

We have leveled related charges at MacFarlane’s articulation of truth relativism: the Vindication Problem and a charge of Reductio. They can only be evaded at a high cost: that of leaving no room for an ordinary, monadic truth predicate, as defined in SMIT, to be used to talk about future contingents, with the effect that ordinary English sentences containing a truth predicate are not themselves assessment sensitive. This cost is too high to be borne: it sacrifices SMIT, the monadicity of the ordinary English truth predicate, and it sacrifices ASOT, the natural assumption that if P is assessment sensitive then so too is P is true. Both are not only highly intuitive, but also core tenets of truth relativism. The responses imply that even if the language contains sentences that express assessment sensitive propositions, and even if one can evaluate those propositions as true or false, when one does so, what one says cannot be assessment sensitive. This is counterintuitive because evaluations of truth and falsity appear to be in the same boat as the propositions they evaluate. Ultimately, these responses cannot accommodate the existence of the truth predicate that we are intuitively using when we speak English.

It is interesting to note that the foregoing argument is akin to a self-refutation charge against global relativism—which of course is as old as relativism itself. The classical charge is roughly that global relativism can in some sense be ‘turned against itself’ (see Kölbel, 2011, p.11). Without going into the vexed issue of how we might understand what self-refutation amounts to more exactly, there is a sense in which the foregoing arguments show how truth relativism can be turned against itself. We have pressed the view specifically on the interaction between truth in the object language, ordinary truth, and truth about truth in the object language. We have found that truth relativism cannot vindicate one of its core implications, that it ultimately implies a contradiction of one of its core implications, and thus is turned against itself. The only plausible responses to these difficulties sacrifice core tenets of truth relativism, and thus turn the theory against itself once more.