Future Contingents, Supervaluationism, and Relative Truth

i i “Libro” — 2019/2/23 — 8:24 — page 39 — #39 i i FUTURE CONTINGENTS, SUPERVALUATIONISM, AND RELATIVE TRUTH ROBERTO CIUNI ∗ CARLO PROIETTI † [email protected] [email protected] Department FISPPA, Section of Philosophy, Department of Philosophy, University of Padova University of Lund Abstract: The problem of future contingents is one of the most ancient and debated puzzles in Western philosophy, and Supervaluationism is, today, one of the most prominent solutions to the problem. Recently, John MacFarlane has carried a well-known criticism to Supervaluationism (and all other standard approaches to future contingents) and put forward a new solution of the problem of future contingents, which is known as Double Time Reference Theory (DTRT). Here, we compare DTRT with Supervaluationist semantics, and we show that the success of MacFarlane’s criticism crucially depends on the expressivity of the language adopted. Once a reasonable expressive power is granted, however, MacFarlane’s criticism no longer applies. Keywords: Future contingents, Supervaluationism, relative truth, truth-attribution, assertion, MacFarlane. 1 Introduction A future contingent is a statement about some future state of affairs (or fact) that is neither impossible nor inevitable. The problem of future contingents is: ‘If the present state of the world is not sufficient to determine all subsequent facts (as indeterminists purport), how are we to attribute a truth-value to a future contingent?’ The question is pressing, since indeterminism enjoys today a great popularity. One possible reply, which is usually traced back to Aristotle, is that future contingents are neither true nor false. In today’s philosophical logic, this is the main tenet of Supervaluationism, a view that takes a statement about the future to be true (false) if and only if it is satisfied (dissatisfied) relative to every history passing through the moment of evaluation. MacFarlane (2003) criticizes Supervaluationism and the other standard approaches to future contingents. In particular, he argues that Supervaluationism would not constitute a good ground for a theory of the assertion of future contingents, since it could not keep together three features that such a theory should Research for this paper was carried while Roberto Ciuni was a Piscopia Fellow with the Marie Curie Cofund DYTEBEL project at the Department FISPPA, University of Padova (2016-2018). † Work by Carlo Proietti was supported by the Riksbankens Jubileumsfond (P16-0596:1). ∗ i i i i i i “Libro” — 2019/2/23 — 8:24 — page 40 — #40 i i 40 ROBERTO CIUNI AND CARLO PROIETTI satisfy (see Section 4). In order to fix this problem MacFarlane (2003) formulates the nowadays famous Double Time Reference Theory (DTRT), also known as relativistic postsemantics. Also, MacFarlane (2008) implies that Supervaluationism cannot define a suitable actuality operator in its semantics (Section 6). After providing some background and introducing Supervaluationism (Sections 2 and 3), in this paper we show that the criticisms by MacFarlane have a punch just if a limited expressive power is taken into account (Sections 5 and 6). However, if a greater (and reasonable) expressive power is granted, then Supervaluationism can be the ground for a relativist theory of assertion-truth and satisfy the desiderata from Section 4, and it can express the situations where an actuality operator proves crucial. 2 Future contingents and indeterminism The sentence (?) Tomorrow there will be a sea-battle is a paradigmatic example of a future contingent: it expresses (the futurity of) a fact that it is not logically necessary or physically determined. By contrast, “Tomorrow I will either run from Marathon to Athens, or fail to do so” does not qualify as a future contingent, since the instance of Excluded Middle it expresses is inevitable.1 Indeterminism on time is the view that ‘[a]t a given moment [. . . ] in the world there are a variety of ways in which affairs might carry on’ (Belnap and Green, 1994, p. 365).2 Today, the majority of indeterminist theories come with a branching-time representation of time: each moment is preceded by a linearly order sequence of earlier moments, but it may be followed by many incomparable later moments; also, maximal chains of moments (histories) represent complete possible developments of the present state of the world.3 It is generally held that truth-attribution to future contingents poses a problem to indeterminism: if at each moment we have a number of alternative real possibilities open, then which among them is relevant to evaluate (?) or its assertion? 1 At least, if we are using Classical Logic, which is the standard choice in temporal logic. Notice that this kind of indeterminism wishes to capture an objective feature of the world, and must not be confused with epistemic indeterminism, which just states our systematic ignorance of the future – see (Belnap and Green, 1994, p. 369). 3 We refer the reader to (Belnap and Green, 1994; Belnap et al., 2001; Prior, 1967; Thomason, 2002) for this. 2 i i i i i i “Libro” — 2019/2/23 — 8:24 — page 41 — #41 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 41 We have possible futures where, tomorrow, a sea-battle occurs, and possible continuations where no sea-battle occurs. It is not clear which one should decide for the truth of (?). 2.1 The Indeterminacy Intuition One possible reply available to the indeterminist is: (assertions of) future contingents are neither true nor false, since the conflicting future possibilities are on the same plan and there is, then, no way to break their symmetry. MacFarlane (2003) calls this the ‘Indeterminacy Intuition’. The intuition is usually traced back to Aristotle. According to the so-called standard interpretation, in Chapter IX of De Interpretatione Aristotle purports that if a sentence is true now, then it is so by necessity. If we join this with bivalence,4 then we must conclude that either (?) holds by necessity, or its negation does. Along these lines, Aristotle’s apparent way out is to deny that bivalence applies to future contingents.5 Remark 1 Chapter IX of De Interpretatione received a number of different interpretations. According to the non-standard interpretation – argued for by Anscombe (1956), Hintikka (1973), Rescher (1963) and Sainati (2011), among others – the Aristotelian solution does not entail failure of bivalence. It just entails that the necessity of future contingent statements holds only sensu composito, i.e., necessarily (there will be a sea battle or not), but not sensu diviso, i.e., necessarily there will be a sea battle or necessarily there will not be. We refer the reader to (Sainati, 2011) for this. However, it is the standard interpretation above to be relevant to our story.6 Aristotle’s standard interpretation has inspired a formal approach to future contingents that is known as Supervaluationism (Thomason, 1970).7 This approach restricts bivalence to formulas where no future tense operator occurs, and 4 That is, the principle that every sentence is either true or false. ‘[S]ome things happen as chance has it, and of the affirmation of negation neither is true rather than the other’ (19a7). Also: ‘what holds for things that are does not hold for things that are not but may possibly be or not be’ (19a39). By contrast, Aristotle insists that bivalence holds for statements about the present and the past (18a28). 6 We refer the reader to the second chapter of (Mariani, 2018) for a critical discussion of an array of interpretations of Chapter IX of De Interpretatione. 7 Supervaluationism is a semantical tool that is used to interpret a number of phenomena beside future contingents, including vagueness – see for instance (Fine, 1975). Here, when talking about ‘Supervaluationism’, we will always refer to the application of a supervaluationist semantics to temporal reasoning. 5 i i i i i i “Libro” — 2019/2/23 — 8:24 — page 42 — #42 i i 42 ROBERTO CIUNI AND CARLO PROIETTI yet is able to keep Excluded Middle and all classical laws in their full generality (see Section 3).8 In particular, if we read F φ as ‘It will be the case that φ ’, Supervaluationism makes the inference F φ _ ¬F φ |= F φ , ¬F φ invalid: one cannot validly infer either disjunct from a disjunction. By contrast, Supervaluationism makes the formula F φ _ ¬F φ valid – and the same goes for F(φ _ ¬φ ). This fact is universally taken as a desirable feature of Supervaluationism. Indeed, the problem of future contingents is motivated by sentences whose truth would depend on future facts, which are in turn not inevitable.9 Instances of logical laws (e.g., Excluded Middle), by contrast, are inevitable, since they hold no matter how the world turns to be. Thus, we are not urged to consider them neither true nor false.10 2.2 The Determinacy Intuition and the retrogradation of truth The indeterminist is not forced to drop bivalence, and indeed some indeterminist solutions to the problem of future contingents retain the principle, thus purporting that future contingents have a definite truth-value (Prior, 1967). MacFarlane (2003) calls this the ‘Determinacy Intuition’. The appeal of this intuition is usually justified with the semantical principle of the retrogradation of truth. To put it with MacFarlane (2003, p. 321): after all, once the sea battle has happened (or not), it seems quite strange to deny that the assertion [of our sentence (?)] was true (or false). More in general, the principle states that if it is true now that φ , then it is true that in the past it would have been the case that φ , or equivalently: if φ is true relative to m (and some other possible 8 Together with a host of other proposals, Supervaluationism may also be used to provide a formal reconstruction of some of the arguments discussed in Chapter IX of De Interpretatione. In turn, this proves crucial in assessing the different interpretations of Aristotle’s view on future contingents. For an example of this use, see the first and second chapters of (Mariani, 2018). 9 The standard interpretation of Aristotle’s approach has been very influential among philosophers. Beside the Supervaluationism by Thomason (1970), also the three-valued approach to future contingents by Łukasiewicz has been inspired by the standard interpretation. The approach follows Aristotle in dropping bivalence, but ends up invalidating all logical laws. Finally, notice that Prior (1967) lists Thomas Aquinas and Peter de Rivo among the proponents of the Aristotelian approach. 10 Notice that, according to the so-called standard interpretation of De Interpretatione, Chapter IX, Aristotle’s solution indeed wished to retain Excluded Middle. i i i i i i “Libro” — 2019/2/23 — 8:24 — page 43 — #43 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 43 relevant parameter), then PF φ and the stronger HF φ are also true relative to m (and the same possible relevant parameter). In a compositional semantics, the principle implies that, if φ is true at m, then F φ is true at every earlier moment m0 (again, relative to the same additional parameters). This fact grounds the Determinacy Intuition. As is clear from the above quote, the retrogradation of truth seems to hold good not only for sentences, but also for assertions. As for the former, if there is a sea-battle today, then it is true that (?) was the case yesterday. As for the latter, if Jake asserted (?) yesterday, and a sea-battle is raging today, then Sally is justified in telling him ‘Your assertion yesterday turned out to be true’ (MacFarlane, 2003, pp. 324–325). Solutions that retain bivalence. A host of (indeterminist) solutions to the problem of future contingents have insisted on retaining bivalence. In particular, the Actualist solutions stemming from William of Ockham keep bivalence while maintaining that a sentence can be true without being necessarily true. These hinge either on Prior’s Ockhamist semantics (Prior, 1967), or on the so-called Thin red line (TRL) semantics.11 We will not discuss these solutions here, but we will introduce Prior’s Ockhamist semantics in the next section, since it is presupposed by Supervaluationism. 3 Branching-time logic and Supervaluationism Indeterminism relies on a specific view of time: each moment leaves many possible futures open, but has only one past course of events. In contemporary philosophical logic, this view has found a rigorous formal definition in the so-called branching-time structures, or trees: Definition 1 (Trees) A tree T is a pair hM, <i where: • M is a set {m, m0 , m00 , . . . } of moments. • < is an earlier/later order on M satisfying backward linearity: 8m, m0 , m00 2 M : m0  m and m00  m ) m0  m00 or m00  m0 where m  m0 is short for ‘m < m0 or m0 = m’. Backward-linearity secures that there is just one past course of the events. Linearity does not hold ‘forward’, 11 For TRL semantics, we refer the reader to (Belnap and Green, 1994; Braüner et al., 1998, 2000; MacFarlane, 2003; Malpass, 2012; Øhrstrøm, 1983, 1984). i i i i i i “Libro” — 2019/2/23 — 8:24 — page 44 — #44 i i 44 ROBERTO CIUNI AND CARLO PROIETTI though: given m, m0 , m00 2 M, we can have m < m0 , m < m00 , and yet m0 6 m00 and m00 6 m0 . This qualifies m0 and m00 as part of two mutually exclusive possible developments of the world. This last notion is given a precise formal rendering by the notion of a history: Definition 2 (Histories) For every tree T , a history is a maximal -chain h of moments in M. That is, where HT is the set of histories defined on tree T , the following holds: m0 2 / h ) h [ {m0 } 2 / HT (maximality) 8h 2 HT : m 6 m0 and m0 6 m ) m 2 / h or m0 2 / h (chain) We call Hm = {h 2 HT | m 2 h} the set of the histories passing through m. Trees provide the indeterminist with an intuitive tool to interpret the standard temporal language L for branching time: Definition 3 Given a nonempty set P of atomic formulas, the language LP of branching-time temporal logic is defined by the following BNF (Backus-Naur Form): φ ::= p | ¬φ | φ _ ψ | φ ^ ψ | φ ! ψ | Pφ | F φ | ⇤φ where p 2 P and the connectives receives their standard interpretations. Pφ reads ‘It was the case that φ ’, with H φ = ¬P¬φ being its dual and reading ‘It was always the case that φ ’. F φ reads ‘It will be the case that φ ’, with Gφ = ¬F¬φ being its dual and reading ‘It will always be the case that φ ’. Finally, ⇤φ reads ‘It is inevitably the case that φ ’, with ⌃φ = ¬⇤¬φ being its dual and reading ‘It is possibly the case that φ ’. We omit reference to P when possible. A host of different semantics have been devised in order to interpret L on trees and similar structures. Here, we are interested in Prior’s Ockhamist branchingtime semantics, since Supervaluationism presupposes its satisfaction relation. Given a tree T , we define a model (T , v), where v : P ! 2M is a valuation function assigning to each variable p 2 P a set of moments – ideally, the set of moments where p is true. Definition 4 (Ockhamist satisfaction relation) The satisfaction relation |=ock between models, moment-history pairs and formulas in L is defined recursively as i i i i i i “Libro” — 2019/2/23 — 8:24 — page 45 — #45 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 45 follows:12 (T , v), (m, h) |=ock p , m 2 v(p) (T , v), (m, h) |=ock ¬φ , (T , v), (m, h) 6|=ock φ (T , v), (m, h) |=ock φ _ ψ , (T , v), (m, h) |=ock φ or (T , v), (m, h) |=ock ψ (T , v), (m, h) |=ock Pφ , 9m0 2 h : m0 < m and (T , v), (m0 , h) |=ock φ (T , v), (m, h) |=ock F φ , 9m0 2 h : m < m0 and (T , v), (m0 , h) |=ock φ (T , v), (m, h) |=ock ⇤φ , 8h0 2 Hm : (T , v), (m, h0 ) |=ock φ Ockhamist-satisfaction at a model and Ockhamist-validity on trees are standardly defined. Figure 1 illustrates a tree where “There will be a sea-battle” (F p) is true at m relative to h, but false at the same moment relative to h0 . h0 h p r m0 B B ⇥ B⇥ J ⇥ J h00 B B ⇥ B⇥ h000 ⇥ Jr m Notice that Ockhamist semantics retains bivalence in its full generality.13 In particular, we have: F φ _ ¬F φ |=ock F φ , ¬F φ Beside, it is easy to check that Ockhamist semantics makes Excluded Middle (and hence F φ _ ¬F φ ) valid. Also, 6|=ock F φ ! ⇤F φ (the tree in Figure 1 provides a countermodel): Ockhamist semantics is designed as to make F φ true without necessarily making it settled, thus turning down one tenet that is apparently presupposed by Chapter IX of Aristotle’s De Interpretatione.14 12 We omit the definition for Boolean constructions ^ and !, which can be defined via _ and ¬. It is clear from Definition 4 that if (T , v), (m, h) 6|=ock φ , then (T , v), (m, h) |=ock ¬φ . 14 However, notice that the inference from truth to necessity holds if no future operator is involved. Indeed, the clause for atomic formulas and Boolean construction, and backward linearity imply that |= φ ! ⇤φ , if φ contains no occurrence of F. 13 i i i i i i “Libro” — 2019/2/23 — 8:24 — page 46 — #46 i i 46 ROBERTO CIUNI AND CARLO PROIETTI 3.1 Supervaluationist semantics Central to Supervaluationism is a notion of truth, which is known as supertruth: in order for a formula φ to be supertrue, φ must be satisfied relative to all the relevant parameters of evaluation. In the application to future contingents, the relevant parameters are the histories passing through the moment of evaluation. Notice that Supervaluationism usually does not consider the settledness operator ⇤ relevant, and it is based on the language L , which is the fragment of L where no formula of the form ⇤φ occurs. The following defines supervaluationist semantics: Definition 5 (Supertruth) For every model (T , v), the satisfaction relation |=sup (supertruth) between models, moments and formulas in L is defined as follows: (T , v), m |=sup φ , 8h 2 Hm : (T , v), (m, h) |=ock φ In a nutshell, a formula φ is supertrue at m iff it is Ockhamist-true at m relative to all the histories passing through that moment. By contrast, φ is superfalse (at m) if 8h 2 Hm : (T , v), (m, h) 6|= φ . This in turn equates with (T , v), m |=sup ¬φ : a formula is superfalse (at m) iff its negation is supertrue (at m). We are particularly interested in two special cases of Definition 5: (T , v), m |=sup Pφ , 8h 2 Hm : (T , v), (m, h) |=ock Pφ (T , v), m |=sup F φ , 8h 2 Hm : (T , v), (m, h) |=ock F φ We believe it is clear why supertruth does not satisfy bivalence: Definition 5 of course allows for having both 9h 2 Hm : (T , v), (m, h) 6|= φ and 9h0 2 Hm : (T , v), (m, h0 ) |= φ , which means that φ is neither supertrue nor superfalse (at m). This very consideration suffices to realize that, in Figure 1, F p is neither supertrue nor superfalse. However, notice that, if φ contains no occurrence of operator F, then supertruth collapses into Ockhamist truth. Thus, “there is a seabattle” is supertrue at m under the same conditions at which it is Ockhamist-true at (m, h), for an arbitrary history h 2 Hm . A more general fact is worth stressing: supervaluationist semantics is non-compositional, as is clear from Definition 5. 4 Future contingents, assertion, and relative truth MacFarlane (2003) criticizes the standard formal approaches to future contingents, and in particular TRL semantics and Supervaluationism. This criticism i i i i i i “Libro” — 2019/2/23 — 8:24 — page 47 — #47 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 47 is articulated, but its main rationale is entwined with MacFarlane’s focus on assertion15 – in particular, on the need of designing a satisfactory theory of truthattribution for assertions of future contingents. However, MacFarlane (2008) also confronts Supervaluationism on its ability to provide a good theory of truthattribution to future contingent sentences and propositions as well. We consider MacFarlane’s criticisms in turn. In particular, MacFarlane (2003) claims that Supervaluationism does not provide a good account of truth-attribution to assertions of future contingents, since: 1. It is unable to accommodate both the Indeterminacy Intuition and the Determinacy Intuition. 2. It cannot account for the retrogradation of the truth of assertions. 3. It cannot account for the fact that the truth of an assertion is relative not only to a moment of assertion, but also to a moment of assessment. 4.1 The Absoluteness Thesis of assertion-truth MacFarlane (2003) notices that the orthodox view on assertions is that they do not change their truth-values in time. According to this insight, if the assertion of (?) – relative to moment m – is true at m, then that assertion is true at any time.16 By contrast, if the assertion of (?) – relative to m – is neither true nor false, then that assertion is neither true nor false at any time. MacFarlane (2003) calls this the Absoluteness Thesis of assertion-truth, which we can sum up as follows: (AT) The truth of an assertion depends just on the context of assertion. In particular, according to AT, ‘[t]he truth-value of an utterance is independent of the context from which the utterance is being assessed’ (MacFarlane, 2003, p. 322). The notion of a context of utterance (or context of assertion, as we say in this paper) is basic in the philosophy of language, and it can be traced back to (Kaplan, 1989) on indexicals. The context of assertion is the set of those parameters that are relevant to evaluate the sentence expressed by the assertion. When indexicals are at stake, like in ‘I am here’, these parameters will typically include a world, a time, a location, and a speaker. If a basic temporal language like L (or L ) is at stake, the parameters will include a moment (of assertion), and maybe a 15 MacFarlane (2003, 2008) talks about ‘utterances’. In this paper, we will refer to ‘assertions’ rather than ‘utterances’. We believe that this is justified, since MacFarlane (2003, 2008) only deals with assertive utterances and, as a consequence, his points equally apply to assertion. Also, assertion seems to be the real center of attention of his works. 16 The same goes is the assertion is false at m, of course. i i i i i i “Libro” — 2019/2/23 — 8:24 — page 48 — #48 i i 48 ROBERTO CIUNI AND CARLO PROIETTI history (‘the history of assertion’). As for this option, MacFarlane (2003) seems to follow Belnap and Green (1994) in thinking that the choice of one history would be either arbitrary (in the case of Ockhamist semantics), or incompatible with indeterminism (in the case of TRL semantics) – see especially (MacFarlane, 2003, pp. 325–326). Here, we accept this view by Belnap and Green (1994) and MacFarlane (2003). As a consequence, from now on we will take moments as the only relevant parameters to attribute truth to assertions of sentences from L or L . One point is worth stressing: in general, a given set of parameters that are relevant according to a semantic theory can fail to provide a context of assertion – see the example of the assertion of ‘I am here’ from (MacFarlane, 2003, p. 329) for this. However, since moments are the only relevant parameters when assertions of sentences from L and L are at stake (at least for the sake of the argument by MacFarlane), this possibility does not apply: the dynamic of the example from (MacFarlane, 2003, p. 329) suffices to understand that moments alone never fail in providing a context of assertion. 4.2 The Absoluteness Thesis and the assertion of future contingents AT has a crucial import on the assertion of future contingents: if we endorse the thesis, then the Indeterminacy Intuition and the Determinacy Intuition turn incompatible: either my assertion of (?) made at m is neither true nor false at any time, or it is true (or false) at any time. In particular, if Supervaluationism gets coupled with AT, then it cannot accommodate the Determinacy Intuition along the Indeterminacy Intuition, and it cannot account for the retrogradation of truth of the assertion of, say, (?).17 To MacFarlane, inability to accommodate both the Indeterminacy and the Determinacy intuitions would be fatal to Supervaluationism as for any formal theory of the assertion of future contingents. Indeed, (MacFarlane, 2003, pp. 325–326) believes that (1) the Indeterminacy Intuition is indispensable to any indeterminist theory worth its name, and yet (2) the retrogradation of truth should not be dropped. Since retrogradation seems to imply that an assertion of a future contingent like (?) has a definite truth-value at least at some moment, there is only one way to have both features: ‘reject the absoluteness assumption [and] relativize the truth of utterances to a context of assessment and the truth of sentences to both a context of utterance and a context of assessment’ (MacFarlane, 2003, p. 322). 17 For take Figure 1, and suppose m is the moment when Jake asserts (?), and m0 is the moment where Sally assesses Jake’s assertion (remember that p is “there is a sea-battle”). Since the assertion of (?) is neither supertrue nor superfalse at m, AT holds Sally wrong in saying, at m0 , that what Jake asserted turned out to be true. i i i i i i “Libro” — 2019/2/23 — 8:24 — page 49 — #49 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 4.3 49 The double time reference theory of assertion-truth According to MacFarlane (2003), the passage from an AT-based theory of (the assertion of) future contingents to a relativist theory can be accomplished via two simple moves: 1. Endorse a theory that relativizes the truth of an assertion to a moment of assessment and the truth of a sentence to both a moment of evaluation and a moment of assessment. 2. Come up with a new theory of truth-attribution to future contingents. The rationale behind 1 is clear from the quote closing the previous subsection. As for 2, MacFarlane (2003) seems to believe that standard theories of future contingents are designed under the presupposition that AT is true, or at least the influence of AT is so strong that the theories have been built, maybe unintentionally, in order to go along with AT.18 Thus, the right way to ensure freedom from the wrong presupposition would be to design a theory that is made to fit a relativist theory of assertion-truth (at least relative to future contingents). MacFarlane’s proposal to fulfill 1 is a double time reference theory of assertiontruth: (DTRT) The truth of an assertion varies both with features of the context of assertion and with features of the context of assessment. The context of assessment is, in case our language is like L or L , a moment at which we are considering (maybe retrospectively) whether a given assertion is true or not. A natural option is that this is some moment m0 identical or later than the moment m of assertion (notice: we can have many different moments of assessment, once a moment of assertion is fixed).19 Thus, in Figure 1, m could be the moment of the assertion of (?), while m0 could be one possible moment of assessment of such an assertion. The following informal definition turns DTRT into a theory of truth-attribution, namely the so-called double time reference (DTR)postsemantics (MacFarlane, 2003, p. 331):20 18 MacFarlane’s belief is not groundless: no proponent of any standard theory discusses how the Indeterminacy and Determinacy intuitions could go together, and they tend to support just one of them. See (MacFarlane, 2003, pp. 321–322). 19 However, this needs not be the case: in principle, we could even be assessing a possible assertion that ‘is made’ in some moment that is later than, or incomparable with, the moment at which we assess the assertion. 20 A postsemantics is a procedure that outputs a context of assertion (and possibly, a context of assessment) and a truth-relation from a satisfaction relation and the relevant parameters that are selected by an initial semantics. The notion has been first introduced by MacFarlane (2003). i i i i i i “Libro” — 2019/2/23 — 8:24 — page 50 — #50 i i 50 ROBERTO CIUNI AND CARLO PROIETTI Double time reference postsemantics: φ is true [false] at a context of assertion u and context of assessment a iff φ is true [false] at every point (m, h) such that (1) m = the moment of u, h passes through m and (3) if the moment of a > m through the moment of a as well. It is easy to check that the following definition conforms with the informal one given by MacFarlane: Definition 6 (Double Time Reference Postsemantics) For every model (T , v), the satisfaction relation |=DT R (double time truth) between models, pairs of moments and formulas in L is defined as follows: ( 8h 2 Hm0 , (T , v), (m, h) |= φ if m < m0 0 DT R φ , (T , v), (m, m ) |= 8h 2 Hm , (T , v), (m, h) |= φ if m 6< m0 That is, if the context m0 of assessment is in the future of the context of assertion, then ‘[w]e evaluate φ with respect to the moment of utterance and all of the histories passing through both it and the moment of assessment’21 (MacFarlane, 2003, p. 331); otherwise, ‘we just look at the [histories] overlapping at the context of use’ (MacFarlane, 2008, p. 91). We believe it is clear that the postsemantics introduced by MacFarlane provides a relativist theory of the assertion of future contingents – thus fulfilling point 2 from the list of desiderata above. DTR-postsemantics is able to accommodate both the Indeterminacy and the Determinacy intuitions. For take Figure 1 and suppose the assertion of (?) is made at m. By Definition 6, the assertion is neither true nor false at (m, m) – that is, if assessed at the moment of assertion itself – since 9h 2 Hm , (T , v), (m, h) 6|=ock F p and 9h 2 Hm , (T , v), (m, h) |=ock F p. By the same definition, however, the assertion is true at (m, m0 ), that is, if the moment of assessment is m0 . Indeed, since (T , v), (m0 , h) |= p for any h 2 Hm0 , we have that 8h 2 Hm0 , (T , v), (m, h) |= F p. Thus, the assertion has a definite truth-value relative to m0 (as the Determinacy Intuition dictates), but lacks any value relative to m (as the Indeterminacy Intuition dictates). Also, DTR-postsemantics can account for the retrogradation of the truth of assertions: Sally can correctly state to Jake ‘Your assertion yesterday turned out to be true’, since at m0 it is true that in the past p would have been the case: Figure 1 and Definition 6 imply (T , v), (m0 , m0 ) |=DT R PF p. 21 It is easy to check that, if m  m0 , then Hm0 = Hm \ Hm0 . i i i i i i “Libro” — 2019/2/23 — 8:24 — page 51 — #51 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 51 5 Assessing the criticism to Supervaluationism We come back to MacFarlane’s criticism to Supervaluationism, and we show that its success crucially depends on the expressive power of the temporal language: if a greater (and at the same time reasonable) expressive power is allowed for, then claims 1–3 (Section 4) can be resisted. Notice that, here, we will agree with the relativist tenets by MacFarlane. Our point of departure with respect to MacFarlane is exactly on whether Supervaluationism can give us a relativist theory of the assertions of future contingents. Contrary to MacFarlane, we believe that, once a reasonable expressive power is granted, Supervaluationism can do that. 5.1 Supervaluationism and relative truth In order to appraise MacFarlane’s criticisms (1)–(3), we must first understand how Supervaluationism could define the notion of assertion-truth – a crucial enterprise, which is neglected by MacFarlane (2003). This requires, first, to come up with a working formal definition of an assertion. We just need the minimal conceptual insight required to our purpose. Contrary to a sentence, an assertion is something that occurs at a given moment. Also, the crucial difference between the assertion of (?) made at m (by any speaker whoever) and the assertion of (?) made at a distinct m0 (by the same speaker) is not in the sentence F p that is asserted in either occurrence, but in the fact that m and m0 are not the same moment. For the sake of simplicity, we abstract from speakers here (suppose we are adopting a kind of single-agent perspective). The following working definition then proves adequate: Definition 7 (Assertions) For every tree T = (M, <), formula φ 2 L , and moment m 2 M, the assertion of φ at m is the pair (φ , m). If the language we take into account is L (or L ), then MacFarlane is right in claiming that the Supervaluationist cannot to provide clauses for assertion-truth that conform to a relativist tenet. However, L and L are limited with respect to our linguistic practices. The majority of our statements about the future presuppose some sort of metric over time – think for instance of ‘I will be there in five minutes’, or again, ‘tomorrow there will be a sea-battle’, and the two languages cannot express this kind of tense statements. In order for us to express them in a formal language, we must expand L or L with the metric operators Fk and i i i i i i “Libro” — 2019/2/23 — 8:24 — page 52 — #52 i i 52 ROBERTO CIUNI AND CARLO PROIETTI Pk , defined by Prior (1967), and endow our semantics with a notion of distance d(m, m0 ) between moments m and m0 .22 Definition 8 (Ockhamist satisfaction for metric operators) The following defines the satisfaction clauses for metric operators (T , v), (m, h) |=ock Pk φ , 9m0 2 h : m0 < m, d(m, m0 ) = k and (T , v), (m0 , h) |=ock φ (T , v), (m, h) |=ock Fk φ , 9m0 2 h : m < m0 , d(m, m0 ) = k and (T , v), (m0 , h) |=ock φ Once the above operators are available, we can define: Definition 9 (Supervaluationist Assertion-truth) For every model (T , v): ( (T , v), m0 |=sup Pk φ if m < m0 , for k = d(m, m0 ) 0 (φ , m) is true at m in (T , v) , if m 6< m0 (T , v), m |=sup φ That is, if the moment m0 of assessment is k steps later than m, then an assertion (φ , m) is true at m0 if it is supertrue at m0 that Pk φ . If the moment m0 of assessment is m itself, then (φ , m) is true at m0 if the asserted sentence is supertrue at m. The same if m0 is earlier than m, or the two are incomparable. We believe it is clear why this is a reasonable choice, given the indeterminist tenet. Notice that Definition 9 connects assertion-truth with supertruth, which is the notion of truth that Supervaluationism proposes for sentences. Also, it is straigthforward to check that Definition 9 is equivalent with Definition 6. 5.2 Supervaluationism and the two Intuitions Once we endow our language with the reasonable expressive power we claimed for above, Supervaluationism can account for the fact that the truth of an assertion is relative not only to a moment of assertion, but also a moment of assessment, contrary to claim (3) from Section 4. One crucial point is that m0 |= Pk φ implies that it is supertrue at m0 that in the past it was Ockhamist-true that φ , but it does not imply that, in the past, it 22 Our notion of distance is basically the one from definition 1.3 from the first chapter of (Mariani, 2018), with one important difference: we drop condition 4 and assume that d(m, m0 ) is undefined if m 6 m0 and m0 6 m. i i i i i i “Libro” — 2019/2/23 — 8:24 — page 53 — #53 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 53 is supertrue that φ . This fact, which is ensured by Definition 5, is also crucial in securing that Supervaluationism secures retrogradation of the truth of sentences, while maintaining that future contingents are neither true nor false. Given Definition 9, we can check in our model of Figure 1 that (F p, m) is neither true nor false at m, while the very same assertion is true at m0 . Therefore, a suitably expressive version of Supervaluationism can account for the retrogradation of the truth of assertions of future contingents, contrary to claim (2) from Section 4. We believe it is clear at this point that, if given a reasonable expressive power, Supervaluationism can accommodate both the Indeterminacy and the Determinacy Intuitions for assertions, contrary to claim (1) from Section 4. Indeed, Definition 9 gives Supervaluationism all it takes to hold that the assertion of a future contingent has no definite truth-value at the moment of assertion (thus giving the Indeterminacy Intuition is due), and yet it can turn out true (false) at some later moment of assessment (thus giving the Determinacy Intuition is due). In sum, the criticism to Supervaluationism by MacFarlane (2003) turns out to be justified only due to the limited expressivity of L. One striking consequence of our rejection of the criticism is that, contrary to what MacFarlane (2003) implies, Supervaluationism proves a viable ground for a relativist theory of assertion-truth. 6 Actuality operator MacFarlane (2008) shifts focus from assertion-truth, with which he has been mainly concerned in (MacFarlane, 2003), to proposition-truth. The reason is that propositions would be the main bearers of truth, and so assertion-truth would be somehow derivative on proposition-truth (MacFarlane, 2008, p. 16). MacFarlane states that, once this shift is made and retrospective assessments are thought of as involving propositions, they do not pose a problem for supervaluationist semantics (MacFarlane, 2008, p. 18). MacFarlane seems to follow the kind of reasoning we have also been following. In particular, he seems to accept that Supervaluationism coherently goes along with the view that ‘What I said yesterday [say, F p] was true’ holds good (at m0 ) if and only if (m0 , h) |= PF p for every h 2 Hm0 . In case p holds at m0 , this is indeed the case for the supervaluationist. So, the supervaluationist holds that F p lacks a truth-value at the moment of assertion, and yet we may later on state that it was true, in a retrospective assessment. In the end, MacFarlance admits that Supervaluationism provides a basic theory of proposition-truth that makes room for both the indeterminacy and the determinacy intuition. i i i i i i “Libro” — 2019/2/23 — 8:24 — page 54 — #54 i i 54 ROBERTO CIUNI AND CARLO PROIETTI Actuality kicks in MacFarlane, however, maintains that Supervaluationism does no longer provide a suitable notion of proposition-truth if we enrich the standard tense language with an actuality operator @. In particular, MacFarlane considers the statement, asserted by John (say): Tomorrow there will actually be a sea-battle (@F p) At the moment of assertion (m), this statement lacks a truth-value for the supervaluationist, exactly as ‘Tomorrow there will be a sea-battle’. Of course, when witnessing a sea-battle the day after (m0 ), we would still like to say ‘What John stated yesterday was true’. According to MacFarlane, the double-time theorist would guarantee this, while ‘according to the supervaluationist, it should be correct for me to say (now) that [What John stated yesterday] was false’ (MacFarlane, 2008, p. 23). The actuality operator To obtain this result MacFarlane (2008) adopts a specific interpretation of the actuality operator @, called Actually1 by Belnap et al. (2001, p. 246). Actually1 has a specific interpretation in the Ockhamist semantics based on moment/history pairs: Truth of @φ at (m,h) for DTR: Given a context of assertion m and a context of assessment m0 , @φ is satisfied at (m, h) iff φ is satisfied at (m, h0 ) for all h0 2 Hm0 . Moreover, MacFarlane claims that, since the Supervaluationist does not keep track of the distinction between the context of assertion and the context of assessment, he is forced to accept the following clause for @: Truth of @φ at (m,h) for Supervaluationist semantics: @φ is satisfied at (m, h) iff φ is satisfied at (m, h0 ) for all h0 2 Hm . In other words, for a Supervaluationist the actuality operator should work as a standard ⇤. Given these definitions we can show how MacFarlane’s claim is justified via our model in Figure 1. Let us consider the formula P@F p at m0 , in order to assess the sentence ‘what I said yesterday [there will actually be a sea battle tomorrow (@F p)] is true’. For both DTRT and Supervaluationism P@F p is (super-)true if it is ockhamist-true for all h 2 Hm0 . For the Supervaluationist this is true if i i i i i i “Libro” — 2019/2/23 — 8:24 — page 55 — #55 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 55 @F p is true at all (m, h), but this is not the case. Indeed, the operator @ forces to quantify over all h0 2 Hm , and F p is false for some such histories. By contrast, for the DTR-theorist quantification at (m, h) is restricted over all h0 2 Hm0 and, since p is true at (m0 , h0 ) for all such histories, F p is true at m for all the h0 at stake. As a consequence, the Supervaluationist and the DTR-theorist give two different truth-values to the same formula. Two considerations are in order here. First, it is not clear that, given the DTR reading of @, the formula P@F p expresses the proper meaning of “what I said yesterday [there will actually be a sea battle tomorrow] is true”. Indeed, when I utter the sentence “there will actually be a sea battle tomorrow” at m I usually refer to the context of m, not to that of a later m0 . Second, and more important, MacFarlane can claim that Actually1 is not definable by the Supervaluationist only because the language and semantics are not able to “keep track” of the initial context of assessment m0 where quantification is made. Again, this is due to the limited expressivity of the language adopted rather than to an intrinsic semantic feature. Indeed, both DTRT and Supervaluationism are based on ockhamist satisfaction in order to evaluate truth of sentences. However, if the language is expressive enough to reproduce the semantic distinctions of the example above, then nothing is lost. The language of hybrid modal logic provides such tool by introducing new operators such as the “downarrow binder” # m., one for each m, and the corresponding “satisfaction operators” @m (Braüner, 2011, pp. 5–7). In short these operators work as pointers, whose function is to say “remember the point m” (# m.) and “go back to m” (@m ). Now, the formula # m0 .P # m.@m0 ⇤@m F p has the very same (ockhamist) truth conditions as P@F p in the DRT reading. Notice that, in our example the formula is Ockhamist-true at (m0 , h) for an arbitrary h 2 Hm0 since p is satisfied at m0 . Hence, it is supertrue at m0 . Therefore Supervaluationism, when endowed with a suitable expressive power, can express an actuality operator that does the same job as the @ in DTR. 7 Conclusion In this paper, we have discussed (and resisted) the criticisms that MacFarlane (2003, 2008) makes against Supervaluationism, which maintains that future contingents are neither true nor false (Sections 2 and 3). The criticism from MacFarlane (2003) holds that Supervaluationism is unable to accommodate both the Indeterminacy Intuition and the Determinacy Intuition on the truth-value of (assertions of) future contingents (Section 4). We show (Section 5) that the success of i i i i i i “Libro” — 2019/2/23 — 8:24 — page 56 — #56 i i 56 ROBERTO CIUNI AND CARLO PROIETTI this criticism crucially depends on the expressive power of the temporal language at hand and, once a reasonable expressive power is granted, Supervaluationism menages to accommodate both intuitions. At this condition, Supervaluationism may prove a ground for a relativist theory of assertion-truth, contrary to MacFarlane’s claim. MacFarlane (2008) holds that Supervaluationism cannot define a suitable actuality operator. Again, we show (Section 6) that the success of the criticism depends on the expressive power one wishes to admit. In particular, once the reference-fixing expressive power of hybrid logic is admitted, Supervaluationism may well define a mechanism that perfectly matches that of the actuality operator. Thus, Supervaluationism has no crucial problem with actuality operators, contrary to what MacFarlane (2008) claims. Our interest in the problem of future contingents has its origins in the classes and papers by Mauro Mariani, our supervisor, on Chapter IX of Aristotle’s De Interpretatione. Lectures by Carlo Marletti in the philosophy of language have been, for both of us, an intriguing and fascinating introduction to the discipline. We are glad for the opportunity to contribute a paper combining the interests and expertise that we owe them. References Anscombe, G. (1956). Aristotle on the sea battle. De Interpretatione 9. Mind, 65:1–15. Belnap, N. and Green, M. (1994). Indeterminism and the Thin Red Line. In Tomberlin, E., editor, Philosophical Perspectives, volume 8, pages 365–388. Ridgeview Publishing Company, Atascadero, CA. Belnap, N., Perloff, M., and Xu, M. (2001). Facing the Future. Agents and Choice in our Indeterminist World. Oxford University Press, Oxford. Braüner, T. (2011). Hybrid Logic and its Proof Theory. Springer, Berlin. Braüner, T., Hasle, P., and Øhrstrøm, P. (1998). Ockhamistic logics and true futures of counterfactual moments. International Symposium on Temporal Representation and Reasoning (TIME-98), Kathib, L. and Morris, R., editors, pages 32–139. Braüner, T., Hasle, P., and Øhrstrøm, P. (2000). Determinism and the origins of temporal logic. In Advances in Temporal Logic, volume 12, pages 185–206. Springer, Berlin. Fine, K. (1975). Truth, vagueness and logic. Synthese, 30:265–300. Hintikka, J. (1973). Time and Necessity. Studies in Aristotle’s Theory of Modality. Oxford University Press, Oxford. Kaplan, D. (1989). Demonstratives. In Almog, J., Perry, J., and Wettstein, H., editors, Themes from Kaplan, pages 481–563. Oxford University Press, Oxford. MacFarlane, J. (2003). 53:321–336. Future contingents and relative truth. The Philosophical Quarterly, i i i i i i “Libro” — 2019/2/23 — 8:24 — page 57 — #57 i i FUTURE CONTINGENTS , RELATIVE TRUTH , AND SUPERVALUATIONISM 57 MacFarlane, J. (2008). Truth in the garden of the forking paths. 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