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Coming true: a note on truth and actuality

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Abstract

John MacFarlane has recently presented a novel argument in support of truth-relativism. According to this, contextualists fail to accommodate retrospective reassessments of propositional contents, when it comes to languages which are rich enough to express actuality. The aim of this note is twofold. First, it is to argue that the argument can be effectively rejected, since it rests on an inadequate conception of actuality. Second, it is to offer a more plausible account of actuality in branching time, along the line of David Lewis (Noûs 4:175–88, 1970; Postscripts to ‘Anselm and actuality’, Philosophical papers I, Oxford University Press, Oxford, 1983).

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Notes

  1. According to MF (pp. 93–94), one may find the argument anticipated—in a nutshell—in some parts of MF (2003). For the sake of clarity, our discussion will focus on his argument in MF (2008).

  2. The term ‘truth-value’ is used here in a broader sense, not in the strict sense of semantic values, which are functions of the semantic values of immediate components and the syntax. Specifically, for supervaluationist frameworks such as the ones discussed by MF, which are a type of modal logics, the modal statuses of ‘necessary truth’ (or ‘determinate truth’), ‘necessary falsity’ (or ‘determinate falsity’), and ‘contingency’ (or ‘lack of determinate truth and lack of determinate falsity’) are treated as ‘truth-values’.

  3. The term ‘truth-relativism’ may be traced back to Egan et al. (2005). Truth-relativism, the view that a sentence’s truth-value may vary with the assessment context, should be distinguished from content relativism, the view that a sentence’s content may vary with the assessment context. For a statement, and defence, of the latter view, see Cappellen (2008).

  4. In cases of the supposed kind, we have a pair of statements s 1 and s 2 where (i) s 1 predicates truth of a given utterance u while s 2 denies u to be true, and where (ii) neither s 1 nor s 2 involve any fault on the part of the speaker making the respective statement. This definition, of course, leaves it entirely open whether there are any instances of this kind, and if so, whether we should give a semantic account of faultlessness (as ‘truth’).

  5. We rely here on the pretheoretically given basic notions of a ‘moment’ and a ‘history’—which are to be distinguished from the technical senses in which these terms are defined in the formal semantics, as set out in the Appendix (see Appendix 1).

  6. For a survey on this literature, see Øhrstrom and Hasle (2006, Sect. 5).

  7. Two notes are here in order: (i) One may conceive of variants of AV where future contingent utterances are either true or false (as assessed at the utterance moment), but where it is indeterminate (as assessed at the utterance moment) which classical truth-value they have. For a variant of AV in this vein, see Belnap et al. (2001, Sect. 2). MF’s argument as well as our counterargument could be easily reformulated in terms of this variant of AV. For simplicity, we leave the focus here on AV in the introduced sense. (ii) It is rather controversial whether Aristotle himself subscribed to AV. In using the label “Aristotelian View”, we just adopt a common convention.

  8. That FL is essential for the argument was first observed in Heck (2006).

  9. For the same point, see Strawson (1950, p. 130).

  10. MF still disagrees also for this case with Kaplan, as far as the evaluation indices is concerned. Whereas Kaplan includes an index for points of time, to treat tenses as operators, MF can do without an index for points of time, since he favours a quantificational approach to tense; cf. Appendix.

  11. Compare constraint (36) on p. 95.

  12. This type of account was first proposed in Belnap and Green (1994). There is no common ground on the account of actuality for branching-time frameworks, though. For alternatives to the given indexical account, see the discussion in Sect. 4.

  13. For further details on the formalisation of these and other examples, see Appendix 4.

  14. In case we assess utterances that might have been made but that in fact, were not made, the relativist notion of utterance truth is no different from the contextualist one. Since this type of case is not relevant for our considerations, it can be ignored henceforth.

  15. For details on the suggested valuation rules for connectives, quantification and historical modalities, see Appendix 1.

  16. On the varieties of using “what is said” informally, see also Stojanovic (2006).

  17. Speaking in terms of Stalnaker’s distinction, semantic contents in Kaplan’s sense are horizontal contents. Another witness for the thesis that informal uses of ‘what is said’ can be adequately modelled by either a horizontal content or, in Dummett (1991)’s terms, by an ingredient sense, is Stanley (1997, p. 577).

  18. Perry (1997, p. 17) writes “We can say that in at least the vast majority of cases, the common sense concept of “what is said” corresponds to content C [i.o.w., the semantic content].” If this is correct, why should ‘what is said’ in cases of reassessment of the relevant type be bound to refer to something distinct from the semantic content?

  19. Since the literature on this issue is vast, for brevity, the reader is referred here to Devitt (2006, Chap. 7), which offers a critical survey and which argues for distinguishing theories of the syntax/semantics of a language from theories of the language faculty.

  20. It is worth noticing that the same conclusion can be established without having to rely on the—very intuitive, we think, but admittedly disputable—assumption that (4) is neither true nor false if (3) is. The argument could be run assuming a generalisation of MF’s own initial redundancy constraint, viz. that \(\ulcorner P \urcorner\) and \(\ulcorner @P \urcorner\) must be true in exactly the same utterance contexts (see MF, p. 98). The generalisation in question additionally requires that P and @P also be false in the same utterance contexts. It then follows that \(\ulcorner P \urcorner\) and \(\ulcorner @ P \urcorner\) must be untrue and unfalse in the same utterance contexts, which, again, is inconsistent with @ i –C: on indexical accounts of actuality, if \(\ulcorner P \urcorner\) is a future contingent, then \(\ulcorner @ P \urcorner\) must be false as assessed at the utterance moment.

  21. More generally, @ i –C coincides with @ i –R for any sentence P and utterance moment c and assessment moment a if either the semantic content of P or its set-theoretic complement is settled at c relative to a. Consequently, we do not even to assume that the assessment moment and the utterance moment are identical. What is merely required is the assumption that the difference between the utterance and the assessment moment does not make a difference with respect to the historical modal status of P.

  22. It deserves mentioning here that MF has taken this point (p.c.).

  23. Lewis’ own views on how to accommodate the intuitive asymmetry between an ‘open future’ and a ‘closed past’ in terms of a standard possible worlds framework (set out in Lewis 1979) and his criticism of branching-time views of the open future (in Lewis 1986, pp. 199–209) can be put aside, since AV is taken for granted here, for the sake of argument.

  24. Note that in Kaplanian semantics, utterance truth of a sentence amounts to its truth as uttered at the context and as evaluated with respect to the utterance world.

  25. Lewis’ ambiguity thesis has been recently endorsed by Wehmeier, who also mentions the following shifty use of ‘actually’: ‘Under certain circumstances, no-one would believe in aliens, though there would actually be aliens’ (2005, p. 195, n. 6). Kai F. Wehmeier further suggests that ‘there is a contextual feature determining which world will be invoked by this adverb: viz., the mood of the predicate to which it is attached.’ To wit, according to Wehmeier, indexical uses are mandated by predicates in the subjunctive mood, whereas shifty uses are mandated by predicates in the indicative mood. Humberstone (2004, p. 29) briefly mentions Lewis’ ambiguity thesis, and writes of the shifty uses adduced by Lewis and Wehmeier that ‘one would want to make sure that [the shifty uses involve] the logical operator ‘actually’ and not ... the use of this word as a merely rhetorical device – say, for emphasis or contrast. Indeed one would like some justification for the claim that these are different uses of the [same] word’ (2004, p. 59, n. 14). Wehmeier’s conjecture indeed supports the assumption that future contingent statements such as (4), where an indicative mood attaches to the predicate, actuality can be only used shiftily. But notice that Humberstone’s suggestion that the examples of shifty uses of ‘actually’ may involve a different word altogether—not what he calls the logical operator ‘actually’—is also consistent with our main claim in the text below, viz. that not all uses of the English word ‘actually’ are indexical, and that some uses only have pragmatic force: ‘actually’ here does not modify the truth-conditions of the (contents expressed by the) sentences in which it occurs.

  26. Belnap and Green (1994) suggest that the meaning of ‘actually’ is captured by @ i –C alone. Brogaard (2008), by contrast, suggests that ‘actually’ is always used shiftily, when arguing that it is used across the board merely as a pragmatic device, which does not affect the semantic content of utterances in which it is used. Belnap et al. (2001, pp. 246–247) mention, further to an indexical account, as another option a hybrid account that combines both an indexical and a shifty interpretation. According to this, ‘actually’ is used in the shift sense in contexts where histories are considered that are still candidate histories at the utterance moment, and used in the indexical sense otherwise. Leaving aside the problem that the resulting account looks gerrymandered, it is easy to see that it is too restrictive. E.g., suppose we just tossed a coin, with the coin landing heads. We can then truly say “A short while ago, it was still possible that tails would actually be the outcome”. On the hybrid account, however, this statement must be in any event false.

  27. MF (p. 99, n. 20) mentions in passing also the possibility of a shifty account, dismisses this account though with the comment that there is need for a non-shifty account of ‘actually’—thereby, it seems, suggesting that ‘actually’ is not ambiguous.

  28. MF’s original example sentence begins with the temporal adverb ‘today’. We leave it out here since it makes no difference for MF’s point nor for the point we are going to make.

  29. Our suggestions regarding the sense in which the notion of ‘actuality’ is used in certain types of utterances are backed by both by considerations on language use and by considerations on ‘folk-linguistic’ evidence: e.g., in support of an interpretation of ‘actually’ in (8) as a redundant operator, one may submit that a report of the form (8) should be assented to just in case the same obtains for (9), and the same for dissenting behaviour; or, that what is said by an utterance of (8) just comes to the same as what is said by an utterance of (9). Both types of reasoning are defeasible, of course. However, in the absence of undermining or even rebutting evidence, it is fair to say that our counterargument poses a serious challenge to MF’s univocal indexical conception of actuality.

  30. One might object that not too much weight should be put on such sentences, which—one might argue—are rather odd things to say outside a philosophical context. Moreover, one might add, ‘actually’ can be even omitted salva veritate in sentences such as (10) and (11), which, therefore, fail to provide evidence for introducing a non-shifty actuality operator. To this potential concern, we respond that the examples originally introduced at the end of the 1970s for motivating the introduction of an actuality operator in the context of the metaphysical modalities, e.g. (to borrow a more recent example from Wehmeier (2005))

    $$ \hbox{It is possible that all the astronauts who actually flew to the moon didn't fly to the moon.} $$
    (14)

    can be easily adapted to the case of the historical modalities, as in

    $$ \hbox{Before 1969, it was still possible that all the astronauts who actually flew to the moon wouldn't fly to the moon.}$$
    (15)

    Similar sentences, we suggest, effectively call for a non-schifty interpretation of the actuality operator, irrespective of whether ‘actually’ is actually used in English. That is, we agree that ‘actually’ may be omitted salva veritate in (10), (11) and (15), though we maintain that such uses are elliptical, and that a proper semantic (and charitable) interpretation of such uses requires a non-shifty actuality operator, or some logically equivalent device. We finally note that any worry about sentences such as (15) as a motivation for introducing an actuality operator in the context of the historical modalities would transfer, by parity of reasoning, to sentences such as (14) used as a motivation for introducing an actuality operator in the context of the metaphysical modalities. In keeping with Hazen (1976, 1979), Davies and Humberstone (1980), Wehmeier (2005), Fara and Williamson (2005), we take sentences such as (14) and (15), and also (10) and (11), to provide compelling evidence for introducing a non-shifty actuality operator in modal logic. We thank an anonymous referee for raising this potential concern.

  31. The most natural way of shifting historical modal perspectives in English seems to be to use tense in combination with historical modalities: that p is not possible, but it was possible that p at some point in the past. MF’s discussion is simplifying in that it ignores the interaction of actuality with tense—which is a bit surprising, considering that he takes a stance on the semantics of tenses earlier on in his paper. For a tentative thought on how to model an indexical sense that furthermore takes into account tense in accordance with MF’s line, see n. 35 in the Appendix.

  32. In fact, MF (p. 99, n. 20) mentions @ i*–C, without explaining why he dismisses it in favour of @ i –C.

  33. See p. 82, where MF endorses a quantificational approach to tense. For a defence of this approach, e.g., see King (2003).

  34. A chain on a set \(\mathcal{M}\) ordered by a relation < is any subset of \(\mathcal{M}\) such that for all \(m_1, m_2 \in h,\) if m 1 ≠ m 2, then m 1 < m 2 or m 2 < m 1. A chain h on \(\langle \mathcal{M}, < \rangle\) is maximal iff for all chains g on \(\langle \mathcal{M}, < \rangle,\) if \(h \subset g,\) then h = g.

  35. The natural way of extending these valuation rules to tenses, e.g., for past tense, may run as follows: \(\langle m, h, M \rangle \models {\text{past}}\, z (\varphi)[v]\) iff \(\langle m, h, M \rangle \models \varphi[v^*], \) for some v * that differs at most from v in that for any history h *v *(z)(h *) < v(z)(h *), where z is a moment variable.

  36. See Thomason (1970). The details can be left aside, since nothing in MF’s argument hinges on provisos on the logic of the object-language.

  37. For the two-place function H, see Sect. 3.1 in this paper.

  38. In fact, one can strengthen this point and show that for u 1 to be unfalse (i.e., true with respect to some candidate history at the utterance moment), u 0 is to be true. Since this epicycle does not make any difference for our points, we do not pursue this line further here.

  39. If days are for the present purposes identifiable with moments, ‘tomorrow’ should designate the immediate successor to the utterance moment, with respect to each candidate history at the utterance moment. Otherwise, further constraints would need to be introduced to make sure that ‘tomorrow’ can be modelled as an indexical moment constant.

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Acknowledgements

Earlier versions of this article were presented in 2007–2009 at the University of Bergamo, the Institut Jean Nicod (Paris), UNAM (Mexico City), and the Universities of St Andrews and L’Aquila. For comments, many thanks to the audiences, especially to John MacFarlane, and to an anonymous referee of this journal. Richard Dietz gratefully acknowledges financial support from the Formal Epistemology Project, which was funded by the Research Foundation Flanders (FWO) and based at the University of Leuven in 2007–2010. Julien Murzi gratefully acknowledges financial support from the Royal Institute of Philosophy, the University of Sheffield, the Analysis Trust and the Alexander von Humboldt Foundation.

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Appendix

Appendix

MF’s outlines of the truth-relativist framework he has in mind are rather sketchy. This may be cause for concern that his position might be flawed at an even more fundamental level than we suggest and that it cannot be even soundly implemented in formal semantic terms. To dispel such concerns, MF’s proposal is cashed out in more exact terms (Appendix 2), along with the intended contextualist base semantics, i.e., a supervaluationist branching-time semantics (cf. Thomason 1970; Belnap et al. 2001) (Appendix 1). The intended base framework deviates from the standard branching-time frameworks in that predicates are indexed to moments—a natural prerequisite for modelling tenses not (the standard way) as operators but as quantifiers.Footnote 33 Because tense is not essential for MF’s argument, however, we will follow MF in omitting the semantics of tense. Historical modalities are accordingly indexed to moments as well. MF’s way of analysing temporal adverbs such as ‘tomorrow’ or of locutions like ‘the weather tomorrow’ may not be the most elegant one, but to make sure that our reconstruction does not deviate essentially from his suggestions, we will enrich the syntax accordingly—modelling temporal adverbs like ‘tomorrow’ as indexical terms designating moments, and phrases like ‘the weather tomorrow’ as indexical functions that take individuals as values for moments as arguments. With the introduced contextualist base framework in place, the crucial flaw in MF's argument can be pinpointed (Appendix 3). Finally, we give formalisations of the examples that are presented by MF as cases in point for instances of RT—which are in the line of MF’s accounts and only with regard to the indexical valuation rule for ‘actuality’ (Appendix 4).

1.1 1. Supervaluationist branching-time semantics

To set the stage, we start with the syntax, semantics and logic of a language \(\mathcal{L}\) of predicate logic with identity, historical modality and actuality.

1.1.1 Syntax

The lexicon of \(\mathcal{L}\) has the following ingredients:

  • logical connectives: \(\neg\) (negation) and ∧ (conjunction)

  • an existential quantifier: ∃

  • identity: =

  • a countable set of moment variables: \({\hbox{MOM-VAR}_{\mathcal{L}} = \lbrace z_1, z_2, \ldots \rbrace}\)

  • a countable set of moment constants: \({\hbox{MOM-CON}_{\mathcal{L}} = \lbrace g_{1}, g_{2}, \ldots \rbrace}\)

  • a countable set of individual variables: \({\hbox{IND-VAR}_{\mathcal{L}} = \lbrace x_1, x_2, \ldots \rbrace}\)

  • a countable set of individual constants: \({\hbox{IND-CON}_{\mathcal{L}} = \lbrace c_1, c_2, \ldots \rbrace}\)

  • a countable set of individual functor constants: \({\hbox{IND-FUNCT}_{\mathcal{L}} = \lbrace f_1, f_2, \ldots \rbrace}\)

  • a countable set of n-place predicate constants: \({\hbox{PRED}^{n}_{\mathcal{L}} = \lbrace F^{n}_{1}, F^{n}_{2}, \ldots \rbrace}\)

  • an operator of ‘historical necessity’: □

  • an operator of ‘actuality’: @

  • parentheses: ( and )

The set of moment terms of \({\mathcal{L}}, \hbox{MOM-TERM}_{\mathcal{L}},\) is given by \(\hbox{MOM-VAR}_{\mathcal{L}} \cup \hbox{MOM-CON}_{\mathcal{L}}.\) The set of individual functor terms of \({\mathcal{L}},\) \(\hbox{IND-FUNCT-T}_{\mathcal{L}},\) is given by the smallest set that contains f(t), for any \(f \in \hbox{IND-FUNCT}_{\mathcal{L}}\) and any \(t \in \hbox{MOM-TERM}_{\mathcal{L}}.\) The set of individual terms of \({\mathcal{L}}, \hbox{IND-TERM}_{\mathcal{L}},\) is given by \(\hbox{IND-VAR}_{\mathcal{L}} \cup \hbox{IND-CON}_{\mathcal{L}} \cup \hbox{IND-FUNCT-T}_{\mathcal{L}}.\)

The set of atomic formulas of \({\mathcal{L}}, \hbox{ATFORM}_{\mathcal{L}},\) is given by the smallest set such that:

  1. 1.

    If \({F \in \hbox{PRED}_{\mathcal{L}}^0}\) and \(t \in \hbox{MOM-TERM}_{\mathcal{L}},\) then \(F_t \in \hbox{ATFORM}_{\mathcal{L}};\)

  2. 2.

    If \(F \in \hbox{PRED}_{\mathcal{L}}^n,\) for any \(n \geq 1, t \in \hbox{MOM-TERM}_{\mathcal{L}},\) and \(i_1, \ldots, i_n \in \hbox{IND-TERM}_{\mathcal{L}},\) then \(F_t(i_1, \dots, i_n) \in \hbox{ATFORM}_{\mathcal{L}};\)

  3. 3.

    If \(i_1, i_2 \in \hbox{IND-TERM}_{\mathcal{L}},\) then \(i_1 = i_2 \in \hbox{ATFORM}_{\mathcal{L}}.\)

The set of formulas of \(\mathcal{L}, \hbox{FORM}_{\mathcal{L}},\) is the smallest set such that:

  1. 1.

    \(\hbox{ATFORM}_{\mathcal{L}}\subseteq \hbox{FORM}_{\mathcal{L}}\)

  2. 2.

    If \({\varphi \in \hbox{FORM}_{\mathcal{L}},}\) then \({\neg \varphi \in \hbox{FORM}_{\mathcal{L}}}\)

  3. 3.

    If \({\varphi, \psi \in \hbox{FORM}_{\mathcal{L}},}\) then \({(\varphi \wedge \psi) \in \hbox{FORM}_{\mathcal{L}}}\)

  4. 4.

    If \({x \in \hbox{IND-VAR}_{\mathcal{L}}}\) and \({\varphi \in \hbox{FORM}_{\mathcal{L}},}\) then \({\exists x\varphi \in \hbox{FORM}_{\mathcal{L}}}\)

  5. 5.

    If \({\varphi \in \hbox{FORM}_{\mathcal{L}}}\) and \({t \in \hbox{MOM-TERM}_{\mathcal{L}},}\) then \({\square_{t}(\varphi) \in \hbox{FORM}_{\mathcal{L}}}\)

  6. 6.

    If \({\varphi \in \hbox{FORM}_{\mathcal{L}},}\) then \({@(\varphi) \in \hbox{FORM}_{\mathcal{L}}}\)

Other symbols can be defined as follows:

  • disjunction: \((\varphi \vee \psi) := \neg(\neg \varphi \wedge \neg \psi)\)

  • material conditional: \((\varphi \rightarrow \psi) := (\neg \varphi \vee \psi)\)

  • universal quantifier: \(\forall x \varphi := \neg \exists x \neg \varphi\)

  • historical possibility: \(\diamond_{t} \varphi := \neg \square_{t}(\neg \varphi)\)

1.1.2 Semantics

A frame for \(\mathcal{L}\) is a triple \(\langle \mathcal{M}, < , \mathcal{D}\rangle,\) where

  • \(\mathcal{M}\) is a non-empty class (of ‘moments’);

  • < is tree-like ordering relation (of ‘being earlier than’) on \(\mathcal{M},\) i.e., a relation on \(\mathcal{M}\) that is (i) transitive and (ii) backwards-linear (i.e., for all moments m 1, m 2 and \(m_3 \in \mathcal{M},\) if m 1 < m 2 and m 3 < m 2, then m 1 < m 3, or m 3 < m 1, or m 1 = m 3;

  • \(\mathcal{D}\) is a non-empty class (of ‘individuals’).

Given a frame \(\mathcal{F} : = \langle \mathcal{M}, <, \mathcal{D}\rangle, \) the set of ‘histories’ for \(\mathcal{F}\) is given by: \(\lbrace h \vert h\) is a maximal chainFootnote 34 on \( \langle \mathcal{M}, < \rangle \rbrace. \)

A model for \(\mathcal{L}\) is a quadruple \(M := \langle \mathcal{M}, <, \mathcal{D}, i\rangle, \) where \(\mathcal{F} := \langle \mathcal{M}, <, \mathcal{D} \rangle\) is a frame, with \(\mathcal{H}\) being the associated set of histories, and where i is an interpretation for the non-logical constants of \(\mathcal{L}\) at moments \(m \in \mathcal{M}\) in the frame M such that:

  • For every \({g \in \hbox{MOM-CON}_{\mathcal{L}}: i_{\langle m,M \rangle}(g) \in \lbrace k \vert k: \mathcal{H} \rightarrow \mathcal{M} \rbrace.}\)

  • For every \({c \in \hbox{IND-CON}_{\mathcal{L}}: i_{\langle m,M \rangle}(b) \in \lbrace k \vert k :\mathcal{H} \rightarrow \lbrace d \rbrace,}\) where \(d \in \mathcal{D} \rbrace. \)

  • For every \({f \in \hbox{IND-FUNCT}_{\mathcal{L}}: i_{\langle m,M \rangle}(f) \in \lbrace k \vert k : \mathcal{M} \rightarrow \mathcal{D}\rbrace. }\)

  • For every \({F \in \hbox{PRED}_{\mathcal{L}}^{0}: i_{\langle m,M \rangle}(F) \in \lbrace k \vert k : \mathcal{M} \rightarrow \lbrace {0,1} \rbrace \rbrace. }\)

  • For every \({F \in \hbox{PRED}_{\mathcal{L}}^{n}}\) (for any \(n \geq 1): i_{\langle m,M \rangle}(F) \in \lbrace k \vert k : \mathcal{M} \times \mathcal{D}^{n} \rightarrow \lbrace {0, 1} \rbrace \rbrace.\)

Assignment functions in a model M are functions v on \({\hbox{VAR}_{\mathcal{L}}}\) such that: (i) for variables in \({\hbox{IND-VAR}_{\mathcal{L}}, }\) it takes values in \(\lbrace k \vert k : \mathcal{H} \rightarrow \lbrace d \rbrace\), where \(d \in \mathcal{D} \rbrace, \) and (ii) for variables in \({\hbox{MOM-VAR}_{\mathcal{L}},}\) it takes values in \(\lbrace k \vert k : \mathcal{H} \rightarrow \lbrace m \rbrace, \) where \(m \in \mathcal{M} \rbrace.\) To introduce some notation for interpretations of terms and predicates at moments \(m \in \mathcal{M}\) relative to an assignment v in a model \(M = \lbrace \mathcal{M}, <, \mathcal{D}, i\rbrace, \) we say:

  • terms:

    1. 1.

      \(I_{\langle m,v,M \rangle}(c) = i_{\langle m,M \rangle}(c),\) if \({c \in \hbox{MOM-CON}_{\mathcal{L}} \cup \hbox{IND- CON}_{\mathcal{L}}.}\)

    2. 2.

      \(I_{\langle m,v,M \rangle}(c) = i_{\langle m,M \rangle}(f) \circ i_{\langle m,M \rangle}(t),\) if \({c \in \hbox{IND-FUNCT-T}_{\mathcal{L}},}\) being composed of an \({f \in \hbox{IND-FUNCT}_{\mathcal{L}}}\) and a \({t \in \hbox{MOM-TERM}_{\mathcal{L}}. }\)

    3. 3.

      \(I_{\langle m,v,M \rangle}(x) = v(x),\) if \({x \in \hbox{MOM-VAR}_{\mathcal{L}} \cup \hbox{IND-VAR}_{\mathcal{L}}.}\)

  • predicates:

    • \(I_{\langle m,v,M \rangle}(F) = i_{\langle m,M \rangle}(F),\) if \({F \in \hbox{PRED}^n_{\mathcal{L}}.}\)

Let \(\langle \mathcal{M}, <, \mathcal{D}, i \rangle\) be a model for the language \(\mathcal{L}, \) variable ‘h’ range over the associated set of histories, and v be an assignment function. The relation \(\langle m,h,M \rangle \models \varphi[v], \) reading ‘\(\varphi\) as uttered at a moment m is true relative to a history h in a model M relative to assignment v’, is then defined inductively for \(\mathcal{L},\) as follows:

  • atomic formulas:

    1. 1.

      for any \({F \in \hbox{PRED}^0_{\mathcal{L}},}\) any \({t \in \hbox{MOM-TERM}_{\mathcal{L}}:}\)

      $$ {\langle m, h, M \rangle \models F_t[v]\,\hbox{iff} \, I_{\langle m,v,M \rangle}(F)(I_{\langle m,v,M\rangle}(t)(h))= 1.}$$
    2. 2.

      for any \({F \in \hbox{PRED}^n_{\mathcal{L}},}\) for any n ≥ 1, any \({i_1, \ldots, i_n \in \hbox{IND-TERM}_{\mathcal{L}},}\) any \({t \in \hbox{MOM-TERM}_{\mathcal{L}}:}\)

      $$ \langle m, h, M \rangle \models F_t(i_1, \ldots, i_n)[v]\,\hbox{iff}\, I_{\langle m,v,M \rangle}(F)(I_{\langle m,v,M \rangle}(t)(h), I_{\langle m,v,M \rangle}(i_1)(h), \ldots, I_{\langle m,v,M \rangle}(i_n)(h))=1. $$
    3. 3.

      for any \({i_1, i_2 \in \hbox{IND-TERM}_{\mathcal{L}}:}\)

      $$ \langle m, h, M \rangle \models i_1 = i_2[v]\,\hbox{iff}\,I_{\langle m,v,M \rangle}(i_1)(h) = I_{\langle m,v,M \rangle}(i_2)(h). $$
  • formulas:

    1. 1.

      negation:

      $$ \langle m, h, M \rangle \models \neg \varphi[v]\,\hbox{iff}\,\langle m, h, M \rangle \nvDash \varphi[v]. $$
    2. 2.

      conjunction:

      $$ \langle m, h, M \rangle \models (\varphi \wedge \psi)[v]\,\hbox{iff}\,\langle m, h, M \rangle \models \varphi[v]\,\hbox{and}\,\langle m, h, M \rangle \models \psi[v]. $$
    3. 3.

      existential quantification:

      $$ \langle m, h, M \rangle \models \exists x \varphi[v]\,\hbox{iff}\,\langle m, h, M \rangle \models \varphi[v^{\prime}],\,\hbox{for some assignment}\, v^{\prime}\,\hbox{that differs from}\,v\,\hbox{at most with respect to}\,x. $$
    4. 4.

      historical necessity:

      $$ \langle m, h, M \rangle \models \square_t \varphi[v]\,\hbox{iff}\,\langle m, h^{\prime}, M \rangle \models \varphi[v^*],\,\hbox{for some}\,h^{\prime}\,\hbox{such that}\,I_{\langle m,v,M \rangle}(t)(h) \in h^{\prime},\,\hbox{where}\,v^*\,\hbox{differs at most from}\,v\,\hbox{in that}\,v^*(z) = I_{\langle m,v,M \rangle}(t),\hbox{for all}\,z \in \text{MOM-VAR}_{{\mathcal{L}}}. $$
    5. 5.

      actuality indexical –contextualist (@ i –C):Footnote 35

      $$ \langle m, h, M \rangle \models @ \varphi[v]\,\hbox{iff}\,\langle m, h^{\prime}, M \rangle \models \varphi[v^*],\,\hbox{for all}\,h^{\prime}\,\hbox{such that}\,m \in h^{\prime},\,\hbox{where}\,v^*\,\hbox{differs at most from}\,v\,\hbox{in that}\,v^*(z)\,\hbox{takes for any history,}\,m\,\hbox{as value, for all}\,z \in \hbox{MOM-VAR}_{{\mathcal{L}}}. $$

With this in place, the notion \(\langle m,M \rangle \models \varphi, \) reading ‘\(\varphi\) is true (simpliciter) as uttered at m in model M’ is defined in a supervaluationist fashion in terms of the relativised notion \(\langle m,h,M \rangle \models \varphi[v]\) as follows:

  • Utterance Truth–Relativist (UT–R): \(\langle m,M \rangle \models \varphi\) iff \(\langle m,h,M \rangle \models \varphi[v], \) for all h such that \(m \in h, \) for any assignment v such that v(z) takes m for any history, for all \({z \in \hbox{MOM-VAR}_{\mathcal{L}}. }\)

Suffice it to say here that this suggests an associated natural account of logical consequence in terms of preservation of utterance truth (for all moments in all models).Footnote 36

1.2 2. A truth-relativist adaptation

MF’s proposed truth-relativist framework essentially differs in the suggested valuation rule for actuality, and along with this, with the suggested account of utterance truth. Truth-relativist models though are no different from models in the contextualist base framework (as given in Appendix 1). To make room for potential assessment-sensitivity on the level of valuation-rules for formulas, the contextualist two-place notion of an interpretation for non-logical constants is accordingly replaced by a three-place notion of ‘truth at a moment m (of utterance) and a moment a (of assessment) in a model’. That is, starting from a model \(M = \langle \mathcal{M}, <, \mathcal{D}, i \rangle, \) we replace \(i_{\langle m,M\rangle}\) by \(i_{\langle m,a,M\rangle}, \) with ‘a’ ranging over \(\mathcal{M}\)—where the third place notion is defined as a trivial extension of the contextualist counterpart notion, i.e.: for any non-logical constant c in \(\mathcal{L},\) we have for all \(a \in \mathcal{M}, i_{\langle m,a,M\rangle}(c) := i_{\langle m,M \rangle}(c). \) With an associated relativised notion of an interpretation of non-logical constants at a moment m (of utterance) and a moment a (of assessment) relative to an assignment v in a model \(M, I_{\langle m,a,v,M\rangle}, \) in place, we obtain an associated generalisation notion of \(\varphi\) being true at a moment m (of utterance), a history h, and a moment a (of assessment) in model M relative to an assignment \(v',\,\langle m,h,a,M\rangle \models \varphi[v]. \) The only valuation rule that essentially deviates from its contextualist counterpart rule concerns the indexical account of actuality:

  • actuality indexical relativist (@ i R): \(\langle m,h,a,M \rangle \models @(\varphi)[v]\) iff \(\langle m,h^{\prime},a,M \rangle \models \varphi[v],\) for all \(h^{\prime} \in H(m \vert a). \) Footnote 37

UT–R is accordingly revised as follows:

  • Utterance Truth–Relativist (UT–R): \(\langle m,a,M \rangle \models \varphi\) iff \(\langle m,h^{\prime},a,M \rangle \models \varphi[v], \) for all \(h^{\prime} \in H(m \vert a),\) for any assignment v such that v(z) takes the value m for any history, for all \({z \in \hbox{MOM-VAR}_{\mathcal{L}}. }\)

1.3 3. MF’s argument blocked

To cash out the remaining premises in MF’s argument, the relevant instances of WIS can be precisified as follows:

  • What Is Said \(^{\prime} ({\bf WIS}^{\prime}\)): What is said by the utterance of a sentence (in the here relevant fragment of English), p, can be modelled as follows: for some model \(M := \langle \mathcal{M}, <, \mathcal{D}, i \rangle\) of \(\mathcal{L}, \) with the associated set of histories being called \(\mathcal{H}, \) for some \(m \in \mathcal{M}, p\) is identifiable with the set \(\lbrace h \in \mathcal{H} \vert \langle m, h, M \rangle \models \varphi[v],\) for any assignment v such that v(z) takes the value m for any history, for all \({z \in \hbox{MOM-VAR}_{\mathcal{L}} \rbrace. }\)

On MPT, we don’t need to revise the notion of frames nor the syntax of \(\mathcal{L}\) to model truth-predications of propositions. Hence we are free to assume for any frame \(\langle \mathcal{M}, <, \mathcal{D} \rangle, \) with \(\mathcal{H}\) being the associated set of histories, that there is an associated frame \(\langle {\mathcal{M}}, <, {\mathcal{D}}^{*}\rangle,\) where \(\mathcal{D}^{*} := \mathcal{D} \cup P({\mathcal{H}})\) (i.e., the union of \(\mathcal{D}\) and the powerset of \(\mathcal{H}\)). Since \(P(\mathcal{H})\) represents the set of propositions that are definable in terms of any frame with the domain \(\mathcal{M}\) (of moments) and the ordering <, the assumption that propositions are included in the domain of individuals is hence safe. Now, with a frame in place where the domain of individuals includes all relevant propositions, we can precisify MPT as follows:

  • Monadic Propositional Truth \(^{\prime} ({\bf MPT}^{\prime}\)) Let \(\langle \mathcal{M}, <, \mathcal{D}, i\rangle\) be a model of \(\mathcal{L}\) with \(\mathcal{H}\) being the associated set of histories, where \(\mathcal{D}\) includes the associated set of propositions, \(P(\mathcal{H}). \) Then any monadic predicate F of \(\mathcal{L}\) is a predicate of truth for propositions iff its interpretation satisfies the following constraint: For any \(m \in \mathcal{M}, i_{\langle m,M \rangle}(F) \in \lbrace k \vert k : \mathcal{M} \times \mathcal{D} \rightarrow \lbrace {0,1} \rbrace \rbrace, \) where for any pair of a moment \(m \in \mathcal{M}\) and any \(d \in P(\mathcal{H}), k\) takes the value 1 just in case \({m} \in {d.}\)

With these provisos in place, MF’s argument for instances of RT runs as follows: Suppose u 0 is a future contingent utterance of a sentence s. Speaking in terms of a model \(\langle \mathcal{M}, <, \mathcal{D}, i\rangle\) of \(\mathcal{L},\) with \(\mathcal{H}\) being the associated set of histories, this case can be represented by a formula \(\varphi\) of \(\mathcal{L},\) where for some moment, representing the ‘moment of utterance’, \(m_0, \langle m_0,M \rangle \nvDash \varphi\) and \(\langle m_0,M \rangle \nvDash \neg \varphi. \) Let \(u_0^{\prime}\) be another utterance that was made at the same time, where the sentence uttered has the logical form of \(@(\varphi). \) Now consider a later utterance u 1 that predicates truth of what is said in \(u_0^{\prime}. \) Assuming \(\hbox{WIS}^{\prime}, u_{1}\) predicates truth of the semantic content of \(@(\varphi)\) at m 0, that is, the set \(\lbrace h \in \mathcal{H} \vert \langle m_0, h, M \rangle \models @(\varphi)[v],\) for any assignment v such that \(v(z) = \lbrace \mathcal{H} \rightarrow \lbrace m_0 \rbrace \rbrace, \) for all \({z \in \hbox{MOM-VAR}_{\mathcal{L}} \rbrace}\)—call this set p. Starting from \(\hbox{MPT}^{\prime},\) we can represent u 1 in our model by an atomic formula of \(\mathcal{L}, F_t(c), \) such that for some moment \(m^{\prime}\) later than m: (i) F is a monadic predicate of \(\mathcal{L}, \) where (for every assignment v) \(I_{\langle m_1,v,M \rangle}(F)\) satisfies the constraint for truth-predicates of propositions as given in \(\hbox{MPT}^{\prime}. \) (ii) t is a moment term of \(\mathcal{L},\) where (for every assignment v) \(I_{\langle m_1,v,M \rangle}(t) = k : \mathcal{H} \rightarrow \lbrace m_1 \rbrace. \) (iii) c is an individual term of \(\mathcal{L}, \) where (for every assignment v, ) \(I_{\langle m_1,v,M \rangle}(c) = k : \mathcal{H} \rightarrow \lbrace p \rbrace. \) From this, by UT–R and @ i –C, it follows that for u 1 to be true, u 0 is to be true as well: for \(\langle m_1,M \rangle \models F_t(c)\) only if \(\langle m_0,M \rangle \models \varphi. \) Footnote 38 However, since, by UT–R, assessment moments are just idle wheels, if u 0 is true as assessed at a later moment, then so it is assessed at the utterance moment, that is, it should fail to be a future contingent. Contradiction.

On our account, the argument is valid. However, we argued that it fails to be sound, in that it essentially rests on the premise @ i –C. We submit that in the examples discussed, @ s –C is to be adopted instead. Specifically, in the given framework, this comes to:

$$\begin{aligned}\hbox{actuality}_{SHIFTY}{-}\hbox{contextualist}(@_{S}{-}\hbox{C}):\\ \langle m,h,M \rangle \models @ \varphi[v]\iff \langle m,h,M \rangle \models \varphi[v]\end{aligned} $$

On this supposition, though, u 1 can be true without u 0 being true. Precisely, for the outlined framework, it is easy to see that this is the case whenever the semantic content of u 0p, is neither settled to be true nor settled to be false at the moment of u 0 whereas p is settled to be true at the moment of u 1.

1.4 4. MF’s ‘cases in point’

MF gives the following formalisation for (4):

$$ @(\hbox{Sunny}_{\rm tomorrow}), $$

where ‘Sunny’ is zero-place predicate and where ‘tomorrow’ is a term that as used at the utterance moment designates some later moment relative to each candidate history.Footnote 39 By parity of reasoning, he analyses (6) as:

$$ @(\hbox{Sunny}_{\rm tomorrow}) \vee @(\hbox{Cloudy}_{\rm tomorrow}), $$

with ‘Cloudy’ being another zero-place predicate. (8) is analysed as:

$$ \diamond_{\rm today}(\exists x \exists y ((x \neq y \wedge \hbox{Weather(tomorrow)} = x) \wedge @(\hbox{Weather(tomorrow)} = y)), $$

where ‘today’ functions like ‘tomorrow’ in the preceding examples, and where ‘Weather’ is an individual functor term. On our account, the occurrences of ‘@’ are all deletable, since ‘actuality’ is used shiftily in the given examples.

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Dietz, R., Murzi, J. Coming true: a note on truth and actuality. Philos Stud 163, 403–427 (2013). https://doi.org/10.1007/s11098-011-9822-2

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