Diamonds are forever Things are only impossible until they are not. Jean-­‐Luc Picard ABSTRACT: We defend the thesis that every necessarily true proposition is always true. Since not every proposition that is always true is necessarily true, our thesis is at odds with theories of modality and time, such as those of David Kaplan and Kit Fine, which posit a fundamental symmetry between modal and tense operators. According to such theories, just as it is a contingent matter what is true at a given time, it is likewise a temporary matter what is true at a given possible world; so a proposition that is now true at all worlds, and thus necessarily true, may yet at some past or future time be false in the actual world, and thus not always true. We reconstruct and criticize several lines of argument in favor of such a picture, and argue against it on the grounds that such a picture is inconsistent with certain sorts of contingency in the structure of time. I Nothing impossible has ever happened. If a proposition is necessarily true, then it is always true. Call this principle Perpetuity. Obvious as it sounds, Perpetuity has fallen on hard times. We aim to defend it.1 Three points of clarification. First: Some philosophers are propositional eternalists—they think that every true proposition is always true. On this view, Perpetuity is true but uninteresting, since it is uncontroversial that every 1 Assuming an S5 logic of necessity, Perpetuity is equivalent to the principle that what is possibly true is always possibly true—hence our title. Proof: Suppose that p is possibly true; then it is necessarily possibly true (by the 5 axiom), and is therefore always possibly true (by Perpetuity). In the other direction, suppose that p is necessarily true; then it is possibly necessarily true (by the T axiom), and is therefore always possibly necessarily true (by the titular principle). Since possible necessary truth entails truth (by the B axiom), it follows by the closure of ‘always’ under entailment that p is always true. 1 necessarily true proposition is true. We think that even propositional temporalists—those who deny propositional eternalism—should accept Perpetuity. We will therefore proceed on the assumption that there are some temporarily true propositions and argue that they are all contingently true. Second: Some philosophers are nominalists and think that there aren’t any propositions at all. This view also makes Perpetuity true but uninteresting. But surely nominalists should want to find some way of reframing the debate over Perpetuity that renders it non-­‐trivial by their lights—one needn’t believe in propositions to wonder whether necessity entails eternity. One attractive reframing strategy involves replacing quantification over propositions with quantification into sentence position: the debate about Perpetuity will then be reframed as a debate about whether, for all p: if necessarily p, then always p.2 We think this is also a perfectly intelligible debate. Since in what follows we will be freely intersubstituting ‘Φ’ and ‘the proposition that Φ is true’, it would be a trivial exercise to rewrite our discussion of Perpetuity using quantification into sentence position in place of quantification over propositions.3 Third: Perpetuity is a claim about metaphysical necessity. Given propositional temporalism, there are certainly other notions of necessity which do not entail eternity. For example, epistemic necessity does not entail eternity, since the proposition that there are dogs is known to be true but has not always been true. Or consider the “historical” notion of necessity, according to which truths about the past are automatically necessary. Since there are non-­‐eternal truths about the past—e.g., that it has rained at least once—historical necessity also fails to entail eternity. But how could something that was true yesterday be metaphysically impossible today, or something that is metaphysically necessary today be false 2 See Prior 1971; Williamson 2003, §9; Williamson 2013, §5.9. 3 Even those who harbor no doubts about the existence of propositions sometimes reject the intersubstitutability of ‘Φ’ and ‘the proposition that Φ is true’: for example, some think that if Socrates had not existed, then the proposition that Socrates does not exist would not have existed either, and so would not have been true. We will ignore such worries in what follows, since they do not, so far as we can tell, have any bearing on Perpetuity. 2 tomorrow? We hope you share our judgment that it obviously couldn’t be. If you do, then you might be surprised to learn that this judgment of obviousness is, as we have found in conversation, far from universal. In fact, Perpetuity is inconsistent with most well-­‐developed theories of the interaction of tense and modal operators. Admittedly, many of those who develop combined modal-­‐temporal logics do not specify that they are talking about metaphysical necessity, and indeed some of them are explicitly concerned with something else.4 But two of the most influential treatments of the interaction of modality and tense—David Kaplan’s logic of demonstratives (1989), which is the starting point for much subsequent theorizing about tense and modality in the philosophy of language, and Kit Fine’s modal tense logic (1977), which has been influential in the metaphysics literature—are explicitly concerned with metaphysical necessity.5 While neither of these authors explicitly considers Perpetuity, as we shall see they are committed to its falsity. Moreover, many treatments of tense and modality in the linguistics literature are intended to apply to all available readings of modal adverbs in natural language, and seem to entail that Perpetuity is false no matter how we resolve the context-­‐sensitivity of ‘necessarily’.6, 7 One underlying motivation behind these views is the desire for a symmetric treatment of modality and tense. For consider the converse of Perpetuity: the claim that every proposition that is always true is necessarily true. This claim is not entirely unprecedented: Diodorus Cronus argued that ‘the possible is that which 4 For example, much of the literature on modal-­‐temporal logics (e.g. Thomason 1984) deals with a kind of necessity that applies to all truths if physical determinism is true. This notion is not metaphysical necessity, since physical determinism clearly does not entail the extreme Spinozistic thesis that all truths are metaphysically necessary. 5 Zalta (1988) also very clearly embraces principles inconsistent with Perpetuity. 6 See, e.g., Cresswell 1990; Kaufmann, Condoravdi, and Harizanov 2006; Schlenker 2006; and Rini and Cresswell 2012. More carefully: the views in question seem to entail that if propositional temporalism is true then Perpetuity is false. It is a vexed question whether or not such views take a stand on propositional temporalism (see King 2003; Ninan 2012; Schaffer 2012). 7 We take it that in adopting a metaphysical interpretation of ‘necessarily’ philosophers are exploiting the normal context-­‐sensitivity of the natural-­‐language ‘necessarily’ rather than coining something altogether alien. This does not require thinking that the metaphysical interpretation is widespread outside of philosophy. 3 either is or will be’, which entails (given a standard conception of necessity as dual to possibility) that the necessarily true propositions are exactly those that are true and always will be true, and hence include all the eternally true propositions.8 Of course Diodorus may not have been talking about metaphysical necessity. If he was, he was wrong, since there are metaphysically contingent propositions that are always true, for example the proposition that it sometimes rains. Thus Perpetuity, if true, constitutes an important asymmetry between necessity and eternity.9 The plan for the rest of the paper is as follows. Since all of the aforementioned authors articulate their views in model-­‐theoretic terms, getting a more precise understanding of the view we oppose requires taking a model-­‐
theoretic detour. In §II we present a version of their model theory for languages containing modal and tense operators and show that, according to it, Perpetuity is false (assuming propositional temporalism). In §III–§VI we defend Perpetuity against four arguments; although these arguments have not actually been articulated in the literature, they have some prima facie appeal, they are valid by the lights of the aforementioned model theory, and some of them can be seen as implicit in the informal remarks Fine uses to motivate his view. (We will not appeal to model-­‐theoretic considerations in assessing these arguments, so those who would prefer not to bother with model theory can skip straight to §III.) In §VII we argue directly against the Perpetuity-­‐denying metaphysical vision that emerged in the previous sections, on grounds having to do with contingency in what times there are. In §VIII we criticize a certain fallback position to which opponents of Perpetuity might retreat in the face of the arguments of the previous section. In §IX we consider and rejects the suggestion that the whole debate is merely verbal, and in §X we conclude. 8 For an influential conjecture about Diodorus’s argument see Prior 1955. During the medieval period it became customary to distinguish a notion of “possibility per accidens”, conforming to Diodorus’s definition, from other, less demanding notions of possibility—possibility per se and God’s possibility—for which analogues of Perpetuity seem to have been accepted: see Knuutila 1982. 9 Compare the widely held view that metaphysical necessity implies nomological necessity but not vice versa. 4 II In its simplest form, the aforementioned model theory—henceforth, the symmetric model theory—for languages with modal and tense operators is the result of stitching together the standard model theories for modal logic and for tense logic as follows.10 A model is a sextuple ⟨W,T,α,η,<,⟦∙⟧⟩, where W and T are nonempty sets; α∈W; η∈T; < is a strict total order on T; and ⟦∙⟧ is a function from sentences to subsets of W×T (the set of ordered pairs whose first member is in W and whose second member is in T). ⟦∙⟧ is required to obey the standard rules for truth-­‐
functional connectives and the following rules for the modal operators □ (‘necessarily’) and @ (‘actually’) and tense operators G (‘it will always be the case that’), H (‘it has always been the case that’), and N (‘now’): ⟦□Φ⟧ = {⟨w,t⟩: ⟨w′,t⟩∈⟦Φ⟧ for all w′∈W}11 ⟦@Φ⟧ = {⟨w,t⟩: ⟨α,t⟩∈⟦Φ⟧} ⟦GΦ⟧ = {⟨w,t⟩: ⟨w,t′⟩∈⟦Φ⟧ for all t′∈T such that t<t′} ⟦HΦ⟧ = {⟨w,t⟩: ⟨w,t′⟩∈⟦Φ⟧ for all t′∈T such that t′<t} ⟦NΦ⟧ = {⟨w,t⟩: ⟨w,η⟩∈⟦Φ⟧} We will treat ‘◇Φ’(‘possibly Φ’), ‘FΦ’ (‘it will be the case that Φ’), ‘PΦ’ (‘it has been the case that Φ’), ‘AΦ’ (‘always Φ’) and ‘SΦ’ (‘sometimes Φ’) as metalinguistic abbreviations for ‘¬□¬Φ’, ‘¬G¬Φ’, ‘¬H¬Φ’, ‘Φ∧GΦ∧HΦ’ and ‘Φ∨FΦ∨PΦ’, respectively. A sentence Φ is true in a model ⟨W,T,α,η,<,⟦∙⟧〉 just in case ⟨α,η〉∈⟦Φ⟧. 10 To be precise, the symmetric model theory is the product of the standard model theories for modal logic and for tense logic: see Kurucz 2006. 11 One could generalize the present model theory by enriching the models with a binary accessibility relation on worlds and modifying the clause for □ accordingly. A further generalization, in which models are enriched with a time-­‐relative accessibility relation R(t,w,w′) and ⟦□Φ⟧ is {⟨w,t⟩: ⟨w′,t⟩∈⟦Φ⟧ for all w′ such that R(t,w,w′)}, has been widely discussed in the literature on “branching time” and is a natural way of modeling the notion of historical necessity mentioned in §I. We will ignore such modifications in what follows, since in making it is easier for necessity claims to be true without modifying the truth conditions for eternity claims, they make the model theory even less friendly to Perpetuity. 5 An argument is valid on a class of models just in case every model in the class in which all of the argument’s premises are true is one in which its conclusion is true. Note that the logic of any class of models is one in which every modal operator (□ and @) commutes with every tense operator (G, H and N). It also satisfies the following condition of adequacy on any model theory for a combined language of modality and tense: For any class of models C and formula schema X, if either every modal-­‐operator-­‐free instance of X is valid on C or if every tense-­‐operator-­‐free instance X is valid on C, then all instances of X are valid on C. In this way, the logics of these models extend the logics of their purely modal and purely temporal fragments. It is already clear from these clauses that the symmetric model theory is not friendly to Perpetuity: all it takes for □Φ→AΦ to be false in ⟨W,T,α,η,<,⟦∙⟧⟩ is for ⟦Φ⟧ to include every ordered pair of the form ⟨w,η⟩ while excluding some ordered pair of the form ⟨α,t⟩. To properly address claims like Perpetuity, however, we will need to extend the language and its model theory to handle propositional quantification.12 This is naturally done by modeling propositions as arbitrary subsets of W×T. We extend the language to include countably many variables pi (of the same syntactic type as sentences) and quantifiers to bind these new variables, and extend ⟦∙⟧ to map both closed and open sentences to subsets of W×T, stipulating that ⟦∀piΦ⟧ = {⟨w,t⟩: ⟨w,t⟩∈⟦Φ⟧′ for all ⟦∙⟧′ that agree with ⟦∙⟧ on all atomic sentences other than the variable pi}. Given this apparatus, we can symbolize Perpetuity as ∀p(□p→Ap), assuming a scheme for translating sentences of the formal language into English on which ‘∀pi’ 12 This is because some (e.g. Salmon 1983, Appendix C) accept a version of the symmetric model theory that invalidates the schema □Φ→AΦ but are also propositional eternalists and so accept Perpetuity. One way to get a sense for such views is to think of tense operators as binding time variables in their embedded clauses. □Φ∧¬AΦ then fails to entail ∃p(□p∧¬Ap) for the same reason that the consistent reading of ‘He is happy but not every man is such that he is happy’ fails to entail the absurd conclusion that, for some proposition p, p is true although not every man is such that p is true. 6 is translated as ‘For every proposition pi’ while occurrences of ‘pi’ in sentence position are translated as ‘pi is true’.13 Notice that ∀p(□p→Ap) is true in a model ⟨W,T,α,η,<,⟦∙⟧⟩ just in case T = {η}. This is also the condition for ∀p(p→Ap) (‘Every true proposition is always true’) to be true in a model. So, as advertised, Perpetuity turns out to be logically equivalent to propositional eternalism by the lights of the symmetric model theory.14 Having added propositional quantifiers to our formal language, it is natural to consider also adding devices for explicitly talking about worlds and times. The obvious way to do this is as follows: we enrich our language with countably many “world variables” wi, and “time variables” ti; require the interpretation function ⟦∙⟧ to assign a member of W to each world variable and a member of T to each time variable; and let ∀wi and ∀ti vary these assignments in the obvious way. We can also add new predicates ‘Actualized’ and ‘Present’ that take, respectively, a world variable and a time variable as arguments, subject to the following clauses: ⟦Actualized(wi)⟧ = {⟨⟦wi⟧,t⟩: t∈T} ⟦Present(ti)⟧ = {⟨w,⟦ti⟧⟩: w∈W} Alternatively or additionally, we can add an ‘At’ operator that takes a world or time variable and a sentence as arguments, subject to the clauses: 13 Those who have taken up our invitation to rewrite our discussion using quantification into sentence position in place of quantification over propositions can take such quantification at face value. 14 This argument depends on the fact that propositional quantifiers range over the full power set of W×T. Some will want to reject this feature of the semantics on the grounds that contingency or temporariness in what individuals there are induces a corresponding contingency or temporariness in what propositions there are. But this view about the metaphysics of propositions seems orthogonal to the question whether necessity entails eternity. In any case—with one exception, to be explicitly discussed in §VII—the only principle concerning the existence of propositions we will make essential use of in the arguments that follow is the comprehension schema ∃p(Φ ↔ Φ[ψ/p]) where ψ is a closed sentence, which should be acceptable to those who think that it is contingent or temporary what propositions there are; see Williamson 2013: section 6.6. The model-­‐theoretic implementation of such variability raises many subtle issues: see CITATION DELETED FOR BLIND REVIEW. 7 ⟦At wi Φ⟧ = {⟨w,t⟩: ⟨⟦wi⟧,t⟩∈⟦Φ⟧} ⟦At ti Φ⟧ = {⟨w,t⟩: ⟨w,⟦ti⟧⟩∈⟦Φ⟧} If we have both ‘Present’ and ‘At’ in the language, the above clauses render ‘Present(ti)’ logically equivalent to ‘∀p(p↔At ti p)’, and also render ‘At ti Φ’ logically equivalent to both ‘A(Present(ti)→Φ)’ and ‘S(Present(ti)∧Φ)’; mutatis mutandis for ‘Actualized’ and world variables. But be warned that the last two equivalences will become controversial, and indeed jointly inconsistent, if we modify the model theory to allow for contingency in the composition of the time series, as will be discussed in §VII.15 15 An alternative to explicit quantification over times is to enrich the language with devices of “temporal anaphora” (Kamp 1971, Vlach 1973, Cresswell 1990). We enrich the language with countably many “time-­‐storage” operators ↑i and countably many “then” operators ↓i. We can treat these in the model theory by letting the interpretation function ⟦∙⟧ assign each “then” operator a member of T, with the following compositional clauses: ⟦↓iΦ⟧ = {⟨w,t⟩: ⟨w,⟦↓i⟧⟩∈⟦Φ⟧} ⟦↑iΦ⟧={⟨w,t⟩: ⟨w,t⟩ ∈ ⟦Φ⟧t→ i} where ⟦∙⟧t→i is an interpretation function like ⟦∙⟧ on syntactically simple expressions except that ⟦↓i⟧t→i = t. As Cresswell (1990) points out, this apparatus is in a model-­‐
theoretic sense expressively equivalent to explicit quantification over times, assuming time variables can only occur as arguments of ‘At’ and ‘Present’. Consider the translation functions * and † defined as follows: Present(ti)†= ∀p(p↔ ↓ip) (↓iΦ)*=At ti Φ* †
† (At ti Φ) =↓iΦ
(↑i Φ)* = ∃ti(Present(ti)∧Φ*) †
†
(∀ti Φ) =↑jA↑i↓j Φ (where j is a new index) (* and † commute with all other operators and quantifiers.) For any formula Φ, Φ† contains no time variables and Φ* contains no time-­‐storage or ‘then’ operators. It is easily shown that, whenever Φ is a closed sentence in which each ‘then’ operator is in the scope of the corresponding storage operator, Φ is logically equivalent to any formula obtained from Φ by first relabeling its time variables so that they don’t share any indices with any time-­‐storage or ‘then’ operators and then applying either * or †. All of the above applies mutatis mutandis to quantification over worlds; see Correia 2007. It is worth noting that ontological squeamishness is not the only possible motivation for theorising about temporal and modal anaphora using operators rather than explicit quantification over times and worlds. The standard approach to representing modal and temporal anaphora in natural-­‐languages involves postulating unpronounced free world and time variables in the logical form of every natural language sentence, and thus arguably entails that the propositions expressed in context by natural language sentences are invariable eternal and non-­‐
contingent. While this startling conclusion has been embraced by some (notably 8 (As Fine (1977) has shown, the addition of quantification over times and worlds is expressively redundant relative to the symmetric model theory. In place of such quantification, we can quantify over ‘time-­‐propositions’ (intuitively, propositions equivalent to a particular time’s being present) and ‘world-­‐
propositions’ (intuitively, propositions equivalent to particular world’s being actualized), where these notions are defined as follows: WorldProp(Φ) =df ◇A(Φ ∧∀p(p→□(Φ→p))) TimeProp(Φ) =df S□(Φ∧∀p(p→A(Φ→p))) The model-­‐theoretic upshot of these definitions is that ⟦WorldProp(Φ)⟧ = W×T iff for some w∈W, ⟦Φ⟧ = {⟨w,t⟩:t∈T}, otherwise ⟦WorldProp(Φ)⟧ = ∅; ⟦TimeProp(Φ)⟧ = W×T iff for some t∈T, ⟦Φ⟧ = {⟨w,t⟩:w∈W}, otherwise ⟦TimeProp(Φ)⟧ = ∅. As Fine (1977: 167) puts it, ‘the instant-­‐propositions and the world-­‐propositions are the temporal and modal cross-­‐sections, respectively, of the two-­‐dimensional instant-­‐
world manifold’. To eliminate world and time quantifiers we proceed as follows: First, relabel the variables so that no index is shared between variables of different types. Second, eliminate ‘At’ in favor of ‘Present’ and ‘Actualized’, as described above. Third, replace each subformula of the form ∀wi(…) with ∀pi(WorldProp(pi)→…), each subformula of the form ∀ti(…) with ∀pi(TimeProp(pi)→…), and each subformula of the form Present(ti) or Actualized(wi) with pi. It is straightforward to verify that, on the symmetric model theory, the resulting sentence is logically equivalent to the original. Note however that these equivalences are extremely fragile, and do not hold in any of the variant model theories considered in this paper.16) Of course, the fact that Perpetuity is logically equivalent to propositional eternalism in the symmetric model theory does not, on its own, constitute anything like an argument against Perpetuity that could be set against its seemingly obvious Schaffer 2012), the present operator-­‐theoretic formalism provides some natural tools for those who wish to resist it. 16 CITATION DELETED FOR BLIND REVIEW shows that no reduction of quantification over times to propositional quantification, tense and modal operators is tenable if we accept either Perpetuity or contingency in the composition of the time series. 9 truth. Perhaps model-­‐theoretic elegance is some sort of guide to truth; but there are reasonably elegant model theories which validate Perpetuity without validating propositional eternalism. For example, Montague (1973) uses W×T models, but gives a clause for □ equivalent in our notation to the following: ⟦□Φ⟧ = {⟨w,t⟩: ⟨w′,t′⟩∈⟦Φ⟧ for all w′∈W and t′∈T } In a footnote to Montague’s clause, his editor Richmond Thomason writes ‘Here, □ is interpreted in the sense of “necessarily always”’ (Montague 1974: 259). In fact, however, Montague explicitly states that ‘necessarily’ is to be translated as ‘□’, and we see no reason not to interpret him homophonically. In what follows, we will call the result of revising the symmetric model theory’s clause for □ as above, and correspondingly changing the clause for @ to ‘⟦@Φ⟧ = {⟨w,t⟩: ⟨α,η⟩∈⟦Φ⟧}’, the ‘Montagovian model theory’. (The clauses for explicit quantification over times can also remain unchanged. In §VI below we will consider how ‘possible worlds’ might work in such a setting.) Given the availability of both the symmetric and Montagovian model theories, considerations of model-­‐theoretic elegance provide little if any support to either side in the debate about Perpetuity. But we can use the symmetric model theory to extract some interesting deductive arguments against Perpetuity, and indeed a certain metaphysical vision inconsistent with Perpetuity. The next four sections will discuss these arguments, and in so doing flesh out the metaphysical vision behind them. III Let an eternal proposition be one that is necessarily either always true or always false. The symmetric model theory validates the principle that everything supervenes on the eternal, in the following sense: 10 Supervenience: Every true proposition is necessitated by some true eternal proposition. ∀p(p → ∃q(q∧□(Aq∨A¬q)∧□(q→p)) The combination of Supervenience with propositional temporalism uncontroversially entails the falsity of Perpetuity. For Let p be some temporary truth—by propositional temporalism, there is at least one such truth. By Supervenience, p is necessitated by some eternal truth q. So q→p is a counterexample to Perpetuity: it is necessary, but not eternal, since it is false whenever p is false. So we could argue against Perpetuity by appeal to the combination of Supervenience and propositional temporalism. But what could motivate a propositional temporalist to accept Supervenience? One could of course derive Supervenience from some other supervenience claim, to the effect that everything supervenes on propositions of a certain type F, together with the claim that all type-­‐F propositions are eternal. (Possible candidates for ‘type-­‐F’: propositions about microphysics, spacetime-­‐
theoretic propositions, “fundamental” propositions….) But how would one motivate the thesis that for every truth there is a type-­‐F truth that necessitates it without also motivating the stronger thesis that for every truth there is a type-­‐F truth that always necessitates it? We obviously cannot, for example, motivate the weaker supervenience thesis on the grounds that every truth is a type-­‐F truth. We could try appealing to the thesis that every truth is in some sense ‘metaphysically dependent’ on a type-­‐F truth. But this would require an exotic conception of metaphysical dependence, according to which dependent truths are necessitated by, but need not be eternally implied by, the truths upon which they depend. We know of no views of this form, and have trouble understanding what the underlying vision could be. In addition to their strangeness, such views involve an asymmetric treatment of tense and modality (since claims of metaphysical dependence are said to have modal consequences but to lack the analogous temporal consequences), and are to that extent in tension with a central motivation for the symmetric model theory. 11 In what follows, it will be most convenient to engage with the denial of Perpetuity directly, rather than by way of Supervenience; but each of the arguments against Perpetuity that we consider below can fairly straightforwardly be reconfigured into an argument for Supervenience. IV A more interesting argument against Perpetuity turns on the following initially attractive principle, which is also valid according to the symmetric model theory: NOW Every proposition is such that, necessarily, it is true just in case it is now true. ∀p□(p↔Np) Given NOW, the falsity of Perpetuity follows from propositional temporalism together with the following uncontroversial principle about ‘now’: RIGN Every true proposition is always now true. ∀p(p→ANp) For suppose p is a temporary truth. By RIGN, the proposition that p is true just in case p is now true is also a temporary truth, since it is false whenever p is false. But by NOW, this biconditional is necessarily true, and is therefore a counterexample to Perpetuity.17 Those who accept NOW do not, of course, think that the proposition it expresses is always true. Nevertheless, presumably anyone who accepts NOW will accept the following general principle, according to which it is necessary that each time has at itself the modal status that NOW attributes to the present time: 17 Note that NOW entails Supervenience, since it entails that every true proposition p is necessitated by the eternal truth that p is true now. 12 NOW+ Necessarily, at each time t, every proposition is such that, necessarily, it is true just in case it is true at t. □∀t At t ∀p□(p↔At t p)18 As Fine (1977: 169) puts it: ‘At each time, the same present runs through each possible world’. NOW+ is also valid in the symmetric model theory once it is supplemented with explicit quantification over times. As defenders of Perpetuity, we of course deny NOW. Since it was once the case that the proposition that there are dinosaurs was true without being now true, we conclude that it must be possible for it to be true without being now true. So we think NOW is in need of an argument. We will now consider and reject two arguments for it. First: NOW can be derived from the following claims: (a) ‘necessarily’ commutes with ‘now’ (i.e. the propositions that are now necessarily true are exactly those that are necessarily now true); (b) every proposition is such that necessarily, now, it is true if and only if it is now true. For given (a), (b) entails that every proposition is now such that necessarily, it is true if and only if it is now true, which entails NOW by the principle that every proposition that is now true is true. We have a few doubts about (b).19 But more importantly, we see no compelling grounds to accept (a), understood as a claim about metaphysical necessity. While it might be true that ‘necessarily now Φ’ and ‘now necessarily Φ’ are equivalent in typical contexts, it is also clear that in typical contexts, prefixing a context-­‐sensitive sentence with ‘now’ will lead us to favor resolutions of its context-­‐sensitivity on which it expresses a non-­‐eternal proposition—otherwise, the ‘now’ would be pointless. But according to us, all claims of metaphysical necessity are eternal: if true they are necessarily true, hence always true; if false they are necessarily false, 18 Or for fans of the approach discussed in footnote 15: □A↑ ∀p□(p↔↓ p). 1
1
19 See the discussion in §VII of the inference from (3) to (4). 13 hence always false.20 So in typical contexts it will not be natural to interpret the ‘necessarily’ in ‘now necessarily Φ’ as expressing metaphysical necessity. For this reason, one should be wary of drawing any conclusions about metaphysical necessity from our intuitive reactions to the schema ‘Necessarily now Φ just in case now necessarily Φ’.21 Second: one could derive NOW from the following principles about counterfactuals: (i) Every proposition is such that, if it had been true, it would have been true now; (ii) No contingent proposition is such that if it had been true, a contradiction would have been true. 22 To avoid this argument, proponents of Perpetuity will need to deny (i) or (ii). We feel pretty comfortable denying (i). We do seem to feel free to insert and delete ‘now’s in the consequents of ordinary counterfactuals. But most of the counterfactuals we ordinarily consider have antecedents which could be true now, so it is tendentious to abstract from this practice a rule which licenses such insertions and deletions even in counterfactuals whose antecedents could not possibly be true now. (For example, ‘now’ seems not to be redundant in ‘If we were at a philosophy conference in the year 2500, we would probably not be talking about any of the issues that people now regard as 20 This follows from Perpetuity given the axioms ‘4’ and ‘5’ of the modal logic S5, which respectively say that what is necessarily true is necessarily necessarily true and that what is possibly necessarily true is necessarily true. 21 It would be a bit uncomfortable for us if, outside of special contexts like that of philosophy, modals in ordinary language always commuted with ‘now’: one might then worry that metaphysical necessity as we conceive it does not really count as a kind of necessity in the ordinary sense of the word. But there are cases where ordinary modals do not commute with ‘now’. Consider for example the use of ‘might’ on which it concerns the epistemic state of some salient person other than the speaker. If the salient person is not sure of the time, we can say things like ‘Although she knows that it is raining, it might not (as far as she knows) be raining now, since she is still trying to find out what time it is’. ‘When’ and ‘whenever’ also have readings that are modal as well as temporal. Imagine the following conversation between physicists discussing the implications of general relativity: A: ‘When(ever) there is no matter, spacetime is flat’; B: ‘No, there could be gravitational waves’. 22 Let > abbreviate the counterfactual conditional. By (i), ∀p(¬(p↔Np) > N¬(p↔Np)). By the commutativity of N with truth-­‐functional connectives, ∀p(¬(p↔Np) > ¬(Np↔NNp)). By the redundancy of multiple ‘now’s, ∀p(¬(p↔Np) > ¬(Np↔Np)). So ∀p(¬(p↔Np) > ⊥). So by (ii), ∀p□(p↔Np). Thanks to REMOVED FOR BLIND REVIEW for suggesting this argument. 14 important.’23) Analogously, in evaluating ordinary counterfactuals we freely help ourselves to the actual laws of nature24, but it would be very tendentious to extract from this practice a general principle to the effect that the actual laws would have been true no matter what.25 V A third argument for temporary necessities appeals to the following two premises, both of which are valid according to the symmetric model theory: ACT Every proposition is such that, always, it is true if and only if it is actually true. ∀pA(p↔@p) RIG@ Every true proposition is necessarily actually true. ∀p(p→□@p) ACT and RIG@ jointly entail that whenever p is temporarily true, the proposition that p is actually true is a counterexample to Perpetuity—necessarily true, by RIG@, and yet sometimes false, by ACT. 23 REMOVED FOR BLIND REVIEW suggested that one might account for the non-­‐
redundancy of ‘now’ in this example by claiming that at the level of logical form it occurs both under a tense operator and under a modal operator. While this might be correct, it is hard to see how such a view could avoid positing such operators in all counterfactual conditionals, in which case we see no way of sustaining the combination of (i) and (ii). 24 See REMOVED FOR BLIND REVIEW. 25 Note too that there is no need for defenders of Perpetuity to make a once-­‐and-­‐for-­‐
all choice between (i) and (ii): one could also reasonably maintain that (i) holds in some contexts while (ii) holds in others. Counterfactuals are, after all, notoriously context-­‐sensitive. It is not implausible to suppose that that because of this context-­‐
sensitivity, ‘would have been true no matter what’ applies to different propositions in different contexts. 15 We suggest that this argument rests on an equivocation. We do not deny that both ACT and RIG@ have readings on which they express obvious truths.26 But sentences involving ‘actually’ have a number of different readings in English, and we are skeptical that there is any joint resolution of the context sensitivity in ACT and RIG@ on which the argument from them to the denial of Perpetuity is sound.27 In some cases—e.g. in ‘I could have actually run you over!’—‘actually’ appears to be completely inert from a semantic point of view, serving merely to flag some sort of emphasis or contrast. On this reading, the following principle is true ACT□ Every proposition is such that, necessarily, it is true if and only if it is actually true But on this interpretation of ‘actually’, RIG@ is false, since together with ACT□ it would entail that every truth is a necessary truth. Our opponents might object to our attempt to assimilate ACT to ACT□ by pointing to certain contrasts between the ways in which ‘actually’ embeds in temporal versus modal environments. Consider: (1) a. b. The climate could be warmer than it actually is. The climate will be warmer than it actually is. 26 However, see Yalcin forthcoming for reasons to doubt that RIG admits any such @
reading in English. 27 There are a number of ways in which this multiplicity might be generated. On one view (Crossley and Humberstone 1977), the word ‘actually’ is ambiguous, having a ‘rhetorical’ use on which it is semantically inert, and a separate ‘logical’ use characterised by RIG@. On a second view (Correia 2007; inspired by Vlach 1973), ‘actually’ is more flexible, and has the effect that the sentence it embeds is evaluated as if it occurred at some wider scope in the sentence; on its most natural implementation, this view treats ‘actually’ as a bindable operator, introducing structural ambiguities analogous to those introduced by variables. On a third view, ‘actually’ is strictly speaking semantically inert, although it may provide clues to the resolution of certain structural ambiguities which are present in the ‘actually’-­‐free sentence (see Yalcin forthcoming for some considerations which favour this view). 16 (1a) is fine28, whereas (1b) sounds odd, at least out of the blue. (Our informants said things like ‘That is not the right way to say it English—“actually” should be replaced by “now”’.) So our opponent might argue as follows: (1b) is infelicitous because it is false on all readings; the falsity of (1b) is an instance of a more general principle from which it follows that (ACT) is true on all readings; hence, since it is agreed that (RIG@) is true on at least one reading, it follows that there is at least one reading on which the argument against Perpetuity goes through. We think that this argument fails at the first step (which is not to concede that the rest of the argument is unproblematic). Although we agree that there is a contrast between (1a) and (1b) as regards the ease of accessing true readings, we think that with the right kind of setup—for example, when a certain fiction, misapprehension, or only recently ruled out hypothesis about the current climate is salient—(1b) can sound fine. What makes us particularly confident that (1b) admits a true reading is reflection on the results of deleting the word ‘actually’ from (1a) and (1b): (2) a. b. The climate could be warmer than it is The climate will be warmer than it is (2b) clearly has a true reading, and in view of the equivalence of (2a) and (1a), it is hard to see how the addition of ‘actually’ in (1b) could prevent it from having a parallel reading. Of course, (2b) has a trivially false reading too. But this may be the exception rather than the rule, since when the present tense occurs in a past-­‐tense environment (with or without an ‘actually’ operator) a reading analogous to the true reading of (2b) seems to be forced – e.g., ‘It used to be colder than it (actually) is’.29 The infelicity of (1b) discourse initially is an instance of a general feature of ‘actually’ that has nothing to do with temporal environments. In response to the 28 If you are having trouble accessing the required non-­‐epistemic reading of ‘could’ in (1a), consider instead ‘He could work harder than he actually does’. 29 See Wehmeier 2004 and McKay 2013. 17 question ‘What do you do for a living?’, it would be odd to reply ‘Actually, I am an accountant’. This is because ‘actually’ is normally used to signal something surprising, or that something is being corrected, or the like – when it is not clear from context how it could play any of these roles, it is generally infelicitous. In light of this generalization, the striking fact to our mind is that (1a) is felicitous out of the blue – and, more generally, that ‘actually’ embeds more happily under certain modals than in other environments. But we will not speculate about the explanation of this fact, since we see no reason to think that it bears on the question of this paper: it is completely unclear how rejecting Perpetuity might afford an explanation of the puzzling contrast concerning the environments in which ‘actually’ can felicitously occur discourse initially. One might reply that even if the use of @ that makes RIG@ unambiguously true is a philosopher’s invention, it is intelligible and useful. We agree; but clearly we have no business having pre-­‐theoretic judgments about the truth of ACT when @ is introduced by the stipulation that RIG@ is true. Another way to introduce @ as a piece of philosopher’s jargon is to appeal explictly to the metaphysics of possible worlds: @p =df p is true at α, where α is the actual world. But on this interpretation, the right way to assess ACT and RIG@ is again not on the basis of their pre-­‐theoretical plausibility, but as part of some more general background theory about the metaphysics of possible worlds. In the next section we explore the connection between Perpetuity and the metaphysics of possible worlds. VI Philosophers writing about possible worlds often take the following principles for granted: Leibnizian Possibility: A proposition is possibly true if and only if it is true at some possible world. 18 Conjunction: The conjunction of two propositions is true at a possible world if and only if both of those propositions are true at that world. Negation: The negation of a proposition is true at a possible world if and only if that proposition is not true at that world. Historicity: No two possible worlds agree on all eternal propositions. All of these principles can be expressed in the formal language of §II and are valid according to the symmetric model theory. And they are, uncontroversially, jointly inconsistent with the combination of Perpetuity and propositional temporalism. (Proof: Let h be the conjunction of all eternal truths, and suppose a certain proposition p is sometimes but not always true. Then h∧p and h∧¬p are both sometimes true. By Perpetuity, both are possibly true. By Leibnizian Possibility, there is a world at which h∧p is true and a world at which h∧¬p is true. So by Conjunction, there is a world at which h and p are true and a world at which h and ¬p are true. By Negation, the latter is a world at which p is not true. Thus there are two worlds at which h is true, contradicting Historicity.) If propositional temporalists want to maintain all four principles, they can do so by adopting a metaphysics of possible worlds on which what is true at a world is a temporary matter. A proposition is necessary just in case it is true everywhere in the pluriverse. But the state of the pluriverse is changing, so some of the propositions that are necessary today will be contingent or impossible tomorrow.30 We are unmoved by this argument. Like many other metaphysicians, we think it is dangerous to let one’s opinions about modal questions be driven by one’s metaphysical theory about the nature of possible worlds, rather than the other way around. Since the “changing pluriverse” picture is inconsistent with Perpetuity, at least one of its constituent doctrines has to go. But which? 30 This is not the only way of reconciling propositonal temporalism with the principles. One could instead adopt a view on which facts about what is true at a given possible world are both eternal and necessary, but which things are possible worlds is a temporary matter, so that any given possible world is only possible for an instant. 19 One option is to be a propositional eternalist, in which case one can unproblematically hold on to all four of the above principles; indeed, we suspect that propositional eternalism has been a tacit presupposition of much of the literature on the metaphysics of possible worlds. Propositional temporalists, on the other hand, must, given Perpetuity, choose between three ways of thinking about worlds and truth at a world. On the first conception, we think of worlds as corresponding to maximally compossible classes of propositions. On this conception, Historicity fails while the other three principles hold. On both the second and third conceptions, worlds correspond to a maximally compossible classes of eternal propositions, so that Historicity is preserved. When, at a certain world, a proposition is sometimes true and sometimes false, the world is not enough to settle its truth value: we also need to know which time is present. The differences depend on how we understand truth at a world. On the second conception, we take the propositions true at a world to be the members of the corresponding class of eternal propositions. This lets us preserve Conjunction but requires giving up Leibnizian Possibility (since some non-­‐eternal propositions are possibly true) and Negation (since the negation of every non-­‐eternal proposition is non-­‐eternal). On the third conception, the propositions true at a world are those that are compossible with the corresponding class of eternal propositions. On this conception, Leibnizian Possibility holds but Conjunction and Negation both fail (since every possibly-­‐true non-­‐eternal proposition will be such that, for some possible world, both it and its negation are true at that world). Moreover, on this view there will also be contingent propositions that are true at all worlds (because they are necessarily sometimes true), in violation of the following principle: Leibnizian Necessity: A proposition is necessarily true iff it is true at every possible world. Our own feeling is that in view of the potential for equivocation between these competing conceptions, it would wise to avoid ‘world’-­‐talk altogether in theorizing about temporary matters. But if forced to choose, we prefer the first option 20 (dropping Historicity), since the other options involve a seemingly arbitrary choice between Leibnizian Possibility and Leibnizian Necessity.31 Each of these three options corresponds to a natural way of introducing ‘world’ quantifiers into the Montagovian model-­‐theory. On the first conception, we assign world variables members of W×T and give the following clause for ‘At wi’: ⟦At wi Φ⟧ = {⟨w,t⟩: ⟦wi⟧∈⟦Φ⟧} On the second conceptions we assign world variables members of W and the clause is ⟦At wi Φ⟧ = {⟨w,t⟩: ⟨⟦wi⟧,t⟩∈⟦Φ⟧ for all t∈T} On the third conception, world variables are again assigned to members of W but the clause is ⟦At wi Φ⟧ = {{⟨w,t⟩: ⟨⟦wi⟧,t⟩∈⟦Φ⟧ for some t∈T} 31 It is worth mentioning a fourth possible view of truth at a world, according to which a proposition is true at a world if and only if it would have been true had all the members of the corresponding class of eternal propositions been true. This view vindicates Conjunction. Moreover, given Conditional Excluded Middle and the principle that only counterfactuals with metaphysically impossible antecedents are vacuously true, it allows us to preserve Negation. However, given Perpetuity and propositional temporalism, it violates Leibnizian Possibility; and when combined with Conditional Excluded Middle, it also requires giving up Leibnizian Necessity. 21 But to reiterate, given that ‘possible world’ is a philosopher’s term of art, we don’t see any substantive issue here.) Where does this leave the argument against Perpetuity from ACT and RIG@ on the interpretation considered at the end of the previous section, where ‘@’ means ‘At α’ where α names the actual world? Logically, there turn out to be surprisingly few connections between the possible-­‐worlds role described above and those principles. But our attitude that judgments involving modal and tense operators should drive our judgments about the metaphysics about possible worlds extend to ACT and RIG@ on this interpretation, so we find that argument no more compelling than the main argument we have been addressing in this section. VII In the last four sections we have defended Perpetuity from a range of arguments. But even though these arguments are, as we have seen, dialectically weak against those (like ourselves) who find Perpetuity obvious, their premises together constitute a unified Perpetuity-­‐denying metaphysical vision. In this section we will go on the offensive and provide a positive argument against that vision, focusing on NOW (which is central to the view). Our argument will be based on the claim that it is contingent what times there are. This is a plausible idea; moreover, its plausibility is independent of that of Perpetuity, and so it is dialectically appropriate as an argument for that principle. Indeed Fine, immediately after presenting the symmetric model theory, suggests that the account is ‘perhaps over-­‐simple’, and that a ‘more sophisticated’ account would allow for contingency as regards what times there are (Fine 1977: 167). We will begin by laying out an argument against NOW based on the strong premise that any given member of the time-­‐series could have been absent from the time-­‐series (i.e. never present). Here is the argument: (1) Every time t is such that possibly, t is never present. (2) So the present time is possibly never present. 22 (3) So the present time is not necessarily present. (4) So the present time is not necessarily such that all and only true propositions are true at it. (5) So it is not necessary that every proposition is true if and only if now true. (6) So not every proposition is, necessarily, true if and only if now true. (6) is the negation of NOW, so proponents of NOW will have to get off the boat somewhere. We will first consider views that resist one of the inferences that bridge the gap from (1) to (6). After that we will discuss the two options for resisting (1): denying that there is any contingency at all in the composition of the time-­‐series, and saying that while some times could have been absent from the time-­‐
series, the present time is not among them. The latter option is interesting in part because it is what one will end up with if one modifies the symmetric model theory in the most natural way to allow for contingency in what times there are. We will conclude the present section by explicitly presenting this modification, along with the corresponding (and equally straightforward modification) of the Montagovian model theory required to make it consistent with contingency in the time series. The first order of business is to defend the inference from (1) to (6). Each step warrants discussion. Let us consider them in turn. From (1) to (2): Someone might resist here on the grounds that there aren’t any times, so that (1) is vacuously true while (2) is untrue. But this is a radical view indeed, given the pervasiveness of quantification over times in ordinary discourse. Note that this point is accepted by those like Prior and Fine who think that quantification over times is in some sense not “basic”: their view is that such quantification must and can be made legitimate by providing a reductive analysis (Fine 1977: 136), such as the analysis in terms of propositional quantification and standard modal and temporal operators discussed in §II. Moreover, those who prefer to eschew explicit quantification over times in favor of operators for “temporal anaphora” (of the kind discussed in footnote 15) can translate our argument into their favored ideology. Given that defenders of NOW must accept the 23 intelligibility of the “now” operator, it is hard to see what principled grounds they could have for denying the intelligibility of such operators more generally.32 From (2) to (3): This inference seems obviously valid. For something to be never true just is for it to be something that is not true, has never been true, and will never be true. Thus nothing could possibly be both present and never present. If you disagree with us about this, we need never say ‘never’ again: we invite you to replace all instances of ‘never Φ’ in the above argument with ‘¬Φ ∧ H¬Φ ∧ G¬Φ’. This will make the step from (2) to (3) uncontroversially valid, without doing anything (as far as we can see) to undermine the plausibility of (1) or of the inference from (1) to (2). From (3) to (4): Say that a time is accurate just in case all and only the true propositions are true at it. On one attractive view, to be a present time just is to be an accurate time, in which case (3) and (4) come to the same thing. However, all the inference requires is that accuracy necessitates presentness, and this claim follows from all reasonable accounts of the relationship between presentness and truth at a time. There are a number of options for spelling out this relationship. On one natural view, for p to be true at t is for it to be the case that always, if t is present, p is true. On another natural view, for p to be true at t is for it to be the case that sometimes, t is present and p is true. On a third view, for p to be true at t is for it to be the case that p would be true if t were present.33 These views give divergent answers to the question what would be true at a certain time t if t were never present. On the first view, all propositions would be true at t; on the second, no propositions would be true at t (just as nothing would be happening in Oxford if Oxford failed to exist); the third view leaves us free to be more discriminating. But all three of these views entail that accuracy necessitates presentness, and thus 32 Some philosophers do not deny that there are any times, but insist that there is only one of them (the present). Close inspection will reveal that our argument does not, in fact, need to assume that it is possible for there to be any non-­‐present times. 33 The first two views both entail that ‘At t’ fails to commute with negation in modal contexts if it is contingent what times are sometimes present. The third view, by contrast, allows us to preserve such commutativity so long as we are willing to embrace Conditional Excluded Middle. 24 vindicate the inference from (3) to (4).34 Our own view is that, while it may be vague which account of truth at never-­‐present times is correct, it is definitely the case that accuracy necessitates presentness.35 From (4) to (5): This inference follows from the principle that the present time is such that necessarily, a proposition is true at it if and only if it is true now. The only reason we can think of for rejecting this principle would involve a fairly thoroughgoing skepticism about talk of times. And those who hold such a skeptical view can replace quantification over times with operators for handing temporal anaphora, as discussed above in connection with the inference from (1) to (2). From (5) to (6): This inference is of the form ¬□∀pΦ ⊦ ¬∀p□Φ, which is valid if and only if the propositional version of the Converse Barcan Formula (∀p□Φ→□∀pΦ) is valid. The validity of this general form will thus be denied by those who think that there could have been propositions which do not actually exist. In principle, they might think that the present time is necessarily accurate with regard to all actually existing propositions (as required by NOW) while denying that it is necessarily accurate simpliciter, on the grounds that there could have been propositions that do not actually exist with respect to which it was inaccurate. However, this combination of views is implausible for reasons independent of the Converse Barcan Formula. Consider the proposition, concerning the present time, that it is inaccurate—i.e. that either some true proposition fails to be true at it, or some proposition that is true at it fails to be true. According to (5), this proposition 34 Suppose t is accurate: (i) every proposition true at it is true, and (ii) every proposition not true at it is not true. Since it is always the case that t is present if t is present, the first analysis of ‘At t’ entails that the proposition that t is present is true at t, and hence true by (i). Since it is never the case that t is present and t is not present, the second analysis of ‘At t’ entails that the proposition that t is not present is not true at t; so by (ii) it is not true; so by double negation elimination t is present . Finally, since t would be present if t were present, the third analysis of ‘At t’ also entails that the proposition that t is present is true at t, and hence true by (i). 35 We also think it should be uncontroversial that presentness entails accuracy. On the first and second analyses of ‘At t’, this follows from the principle that “times are present only once”—i.e. that if sometimes t is present and p is true, then if t is present then p is true. On the third analysis, it follows from modus ponens for counterfactuals (which entails that if t is present, and p would be true if t were present, p is true) and the ‘and-­‐to-­‐if’ rule for counterfactuals, which entails that if p is true and t is present, p would be true if t were present. 25 is possibly true. And since this proposition actually exists, NOW entails that necessarily, it is true if and only if now true. Therefore (5) and NOW together entail that the present time could have been, not just inaccurate, but inaccurate at itself. By itself this is not an absurd result—for example, it follows from the view discussed above according to which, each time is such that necessarily, if it is never present,every proposition is true at it. But given (2), that view entails that the present time could have been such that contradictions were true at it, which is obviously inconsistent with NOW. Might there be some other reasonable view on which it is possible for a time to be inaccurate at itself without having contradictions true at it? Not if truth at a time respects the following two principles: (i) ‘At t’ is closed under classical entailment; (ii) ‘At t’ is redundant when it occurs within the scope of ‘At t’ without any intervening modal or temporal operators. By (i), it is necessary that if anything at all is true at t, then the logical truth that ∀p(p is true ↔ p is true) is true at t. By (ii), it is necessary that if the proposition that ∀p(p is true ↔ p is true) is true at t, then so is the proposition that ∀p(p is true ↔ p is true at t)—i.e. that t is accurate. So necessarily, if the proposition that t is inaccurate is true at t, its negation is also true at t, and so by another application of (i), the contradiction that t is both accurate and inaccurate is true at t. Given the plausibility of (i) and (ii), we see no plausible way to resist the inference from (5) to (6). So, defenders of NOW need to reject (1). Is this plausible? We think not, on the following grounds: for each non-­‐initial time t, it is possible that history should have come to an end before t; the present time is non-­‐initial; so it is possible that history should have come to an end before it; so it is possible that it should never have been present.36 However, not everyone will be convinced by such blunt 36 One might resist this claim on the grounds that times have rich essences of certain sorts. For example, assuming time is in fact infinite in the future direction, one might think that it is part of the essence of every time that, if it is ever present, then the time series is infinite in the future direction. But such essentialist views support a very similar argument for (1), via the premise that time could have failed to be infinite in the future direction, and so provide no help to the defender of NOW. We will return to the question of whether times have such essences in discussing Die Another Day. 26 appeals to de re modal judgments about times, so let us explore in more detail views that accept NOW and deny (1). Such views fall into two camps: Tomorrow Never Dies: Every time is necessarily sometimes present. Die Another Day: Some times, but not the present time, are possibly never present. Let us consider these two options in turn. There are strong reasons to reject Tomorrow Never Dies that do not rely on any de re modal judgments about times. It suffices that there could have been fewer times than there in fact are. In support of this claim, we can argue as follows: the before-­‐after relation in fact has the structure of the real line, but it is possible that it should have had, instead, the structure of the integers, or that of the rationals. Since there are more real numbers than there are integers or rationals, the hypothesis that every time in the time-­‐series is necessarily in the time-­‐series is ruled out. There is a closely related argument which has the advantage of making no assumptions about the actual structure of time. Its premises are that it is possible that time has the structure of the reals; that it is possible that time has the structure of the integers or of the rationals; and that the proposition that every time is necessarily sometimes present is necessarily true if true at all. Why believe that the qualitative structure of time is contingent in these ways? The most obvious reason is that the structure of spacetime is contingent in parallel ways: spacetime could have been continuous, discrete, dense but countable, etc. Admittedly, such possibility judgments are controversial: they will be denied, for example, by those who identify metaphysical possibility with physical possibility. Moreover, it is a vexed question exactly what the connection is between the metaphysics of spacetime and the metaphysics of time. But it would be extremely surprising if there were no such connection: it is not a mere coincidence that spacetime and time are both continuous (or both discrete, etc., as the case may be). 27 The proponent of Tomorrow Never Dies might at this point employ a defensive maneuver familiar from Timothy Williamson’s defense of the claim that it is necessary what things there are (Williamson 2013). According to Williamson, there are a great many rather boring things in addition to the interesting ones with which we are familiar. On his view, there is something that could have been Wittgenstein’s child, since Wittgenstein could have had a child: but this thing is not a person, and in fact lacks all interesting “non-­‐modal” properties, such as mass, spatial location, etc. Williamson thinks that believers in contingent existence are by and large right about the ways in which it is contingent what interesting things there are, but fall into error by failing to take boring things into account. Defenders of Tomorrow Never Dies might try an analogous strategy, suggesting that we have fallen into error by confusing the true claim that it is a contingent matter which times are interesting with the false claim that it is a contingent matter which times are sometimes present. “Boring” times could perhaps be characterized as times at which all objects are boring, in the sense in which Wittgenstein’s possible children are boring according to Williamson.37 Since we are inclined to judge that there could have been sometimes-­‐present times that are actually never present, proponents of this strategy for explaining away our putative errors will have to say that there actually are sometimes-­‐present boring times, just as Williamson himself is committed to there actually being boring things like Wittgenstein’s possible children. This is a radical suggestion: have we not learned from physics that there always have been and always will be interesting things, like the physical fields in spacetime? Moreover, note that there is an important disanalogy between Williamson’s thesis that there are boring objects and the present proposal that sometimes there are only boring objects. Whereas Williamson denies that boring objects are spatially related to interesting ones, the present proposal is committed to giving some account of how the boring times are temporally related to familiar interesting times. Are they clustered at one or other end of the time series, or are they interleaved between the interesting times? (Have 37 Note that boring times cannot be characterized as times at which nothing whatsoever is true, since it is always true that all dogs are dogs. 28 you been interesting ever since you were born?) Such embarrassing questions lead us to conclude that even those who are sympathetic to Williamson’s modal ontology should find Tomorrow Never Dies unattractive. Let us now turn to Die Another Day. According to this view, although there are some times that are only contingently sometimes-­‐present, the present time is not one of them, since it is necessarily present. (Perhaps the present is the only time that is necessarily sometimes present, or perhaps certain other times enjoy this status, such as all past times.) On its face, this view is quite odd. Of course, in assessing claims to the effect that the present time enjoys this special modal profile, we must remember that, according to propositional temporalism, the present time is guaranteed to have an extremely special alethic profile: it is the only accurate time. And presumably proponents of Die Another Day will regard the modal specialness of the present time as a temporary feature of it, just like its alethic specialness (which it has only when it is present). Indeed, assuming they endorse NOW+ in addition to NOW, they will say that every time is modally robust when it is present. So, it is important to be careful. Nevertheless, even with the alethic specialness of the present squarely in mind, we still find it quite difficult to believe that its modal profile differs from those of other times in the particular way required by Die Another Day. While Die Another Day is thus hard to get used to, proponents of the view can take a quantum of solace in the fact that it is consistent with almost any claim about the possible qualitative features of the time-­‐series. For example, although Die Another Day is of course not consistent with the judgment that history could have been just as it actually was up to the year 1972 and then come to an end, it is consistent with the judgment that history could have played out just as it did in the pre-­‐1972 portion of actual history in all temporally intrinsic qualitative respects, and indeed in all respects specifiable without reference to specific times. Similarly, Die Another Day is consistent with almost any amount of contingency as regards the cardinality of the time-­‐series. For all the view says, there could have been many more or many fewer sometimes-­‐present times, so long as the present time was 29 among them. This is an important advantage of Die Another Day over Tomorrow Never Dies. However, it is worth noting that proponents of Die Another Day can admit contingency in the structure of the time-­‐series only insofar as they reject the following plausible essentialist thesis about times: necessarily, a time in a time series with a certain structure could not possibly be part of a time-­‐series with a radically different structure.38 If this thesis is true, and the actual time-­‐series is continuous, then none of its members, including the present time, could belong to a discrete time-­‐series—and so, by Die Another Day, time could not have been discrete. Die Another Day has some even less palatable consequences. While all opponents of Perpetuity think that some propositions are only temporarily necessary, the examples we have considered so far have all involved non-­‐eternal propositions, such as the proposition that it is raining if and only if it is raining now. Given Die Another Day, however, some eternal propositions must also change their modal status. One example is the proposition, concerning the present time, that it is sometimes present. According to Die Another Day, this proposition is necessarily always true, and so eternal. However, assuming that the present time is not always modally robust, the proposition is only temporarily necessary. The strangeness of change in the modal status of eternal propositions manifests itself vividly as a change in the truth-­‐value of counterfactuals with eternal antecedents and consequents. Since proponents of Die Another Day are willing to allow for possibilities in which certain non-­‐present times are never present, they should be open to the truth of the following counterfactual: if the mass-­‐density of the universe were above x in the year 2014, then the universe would undergo gravitational collapse, and time would come to an end before the year 3,000,000. The antecedent and consequent of this counterfactual are eternal. But at times after 3,000,000, the consequent will be necessarily false, for the same reason that it is now necessarily false that history came to an end before 2014. Since the antecedent will still be possibly true, the counterfactual will then be false. By our lights, this variation in 38 This form of essentialism is reminiscent of, though far weaker than, the metric essentialism about spacetime defended by Maudlin 1990. 30 the truth-­‐values of counterfactuals with eternal antecedents and consequents looks like reason enough to reject Die Another Day. We should make it clear that we are not suggesting that there is any formal incoherence in Die Another Day. In fact Die Another Day is the view one gets if one modifies the symmetric model theory of §II in the most obvious way to accommodate contingency in the time series. On this modification, we will equip our models with a distinguished “domain” D ⊆ W×T, where intuitively ⟨w,t⟩∈D just in case t belongs to the time-­‐series of w. We require that the interpretation function ⟦∙⟧ map every formula to a subset of D and that ⟨α,η⟩∈D, and we modify the clauses for □, G, H and the time and world quantifiers as follows: ⟦□Φ⟧ = {⟨w,t⟩∈D: ⟨w′,t⟩∈⟦Φ⟧ for all w′∈W such that ⟨w′,t⟩∈D} ⟦GΦ⟧ = {⟨w,t⟩∈D: ⟨w,t′⟩∈⟦Φ⟧ for all t′∈T such that t<t′ and ⟨w,′t⟩∈D} ⟦HΦ⟧ = {⟨w,t⟩∈D: ⟨w,t′⟩∈⟦Φ⟧ for all t′∈T such that t′<t and ⟨w,′t⟩∈D} ⟦∀tiΦ⟧ = {⟨w,t⟩∈D: ⟨w,t⟩∈⟦Φ⟧′ for all ⟦∙⟧′ differing from ⟦∙⟧ at most on ti such that ⟨w,⟦ti⟧′⟩∈D} ⟦∀wiΦ⟧ = {⟨w,t⟩∈D: ⟨w,t⟩∈⟦Φ⟧′ for all ⟦∙⟧′ differing from ⟦∙⟧ at most on wi such that ⟨⟦wi⟧′,t⟩∈D} (Note that since we are preserving the symmetric character of the model theory, in allowing contingency as regards what times are sometimes present we also allow temporariness as regards which worlds are possibly actualized.) The difficult questions concerns N, @, ‘At ti’ and ‘At wi’. How, for example, should we evaluate NΦ at ⟨w,t⟩ when ⟨w,η⟩ is not in D? As we saw in connection with the inference from (3) to (4), there are a number of options for how to think about what would be true at a time t if t were never present. The first two views we considered—that all propositions would be true at t, and that none would—correspond respectively to the following clauses for ‘At ti’: ⟦At ti Φ⟧ = {⟨w,t⟩∈D: ⟨w,⟦ti⟧⟩∈⟦Φ⟧ or ⟨w,⟦ti⟧⟩ ∉ D} ⟦At ti Φ⟧ = {⟨w,t⟩∈D: ⟨w,⟦ti⟧⟩∈⟦Φ⟧} We can give parallel clauses for N, and (by symmetry) for ‘At wi’ and @. Both of these proposals validate NOW and NOW+, as well as ACT, Supervenience, and the theory of possible worlds considered in §VI. (We also considered a third option 31 according to which p is true at t iff p would have been true if t were present; how to implement this proposal model-­‐theoretically depends on one’s preferred way of modeling counterfactuals.)39 The Montagovian model theory admits of parallel modifications in order to allow for contingency in the time series. The central difference is just that the above clause for □ is replaced by ⟦□Φ⟧ = {⟨w,t⟩∈D: ⟨w′,t′⟩∈⟦Φ⟧ for all ⟨w′,t′⟩∈D} We modify the clauses for G, H, ∀pi, ∀ti and At ti exactly as we did for the symmetric model theory; we don’t modify the clause for @ at all. How we treat world quantification is a largely verbal issue as discussed at the end of §VI. VIII The previous section argued that the existence of contingency in the time series poses a serious problem for NOW, and thus for the larger metaphysical vision given expression by the symmetric model theory. In conversation, it has been suggested to us that opponents of Perpetuity should retreat from NOW—which says in effect that the present time is necessarily accurate—to the following weaker claim, which says that the present time is, necessarily, accurate if ever accurate: 39 What if we broke this symmetry and left the clauses for modal operators in their original form? To make sense of this view without making the modal operators trivial we would need to allow the interpretation function ⟦∙⟧ to assign arbitrary subsets of W×T, not just subsets of D. One problem with this approach is that it blocks the intersubstitutivity of ‘always’ and ‘at every time’, making the former stronger than the latter. Suppose that ⟨w,η⟩ and ⟨w,t⟩ ∈D but ⟨α,t⟩ ∉ D, and ⟦p⟧={⟨α,t⟩}. Then ◇S@(p ∧ ∀t(At t(¬p)) is true in the model although ◇S@(p ∧ A¬p) is false. (1) is still inconsistent in this variant model theory, but if we replace the tense operator ‘never’ in (1) with an explicit quantifier over times we get something consistent: every time t is such that possibly, there is no time at which t is present. However it is hard to see that the advantage of being able to say this outweighs the disadvantage of drawing a distinction between ‘never’ and ‘at no time’; moreover, the view does nothing to avoid the objections to Die Another Day presented above.
32 NOW′ Necessarily, if it is sometimes the case that every proposition is true if and only if now true, then every proposition is true if and only if now true. □(S∀p(p↔Np)→∀p(p↔Np)) NOW′ is consistent with the possibility that the present time is absent from the time series (and hence never accurate). Moreover, NOW′ is just as good as NOW for the purposes of arguing against Perpetuity, since it is asserts the necessity of something that is clearly not always true. Although NOW′ is a natural way of minimally modifying NOW to avoid the argument of the previous section, we do not think that this fact suggests that NOW′ should be at all attractive to those who have been convinced by this argument. To give up NOW is to renounce the changing pluriverse picture, and the symmetric model theory along with it: what is left to support NOW′? Irrespective of its motivation, there are serious internal problems with the view that accepts NOW′ while rejecting NOW. Just as no-­‐one should accept NOW without accepting the more general NOW+ (as we argued in §IV), one should not accept NOW′ without accepting the following more general principle, according to which it is necessary that each time has, at itself, the status that NOW′ attributes to the present time: NOW′+: Necessarily, each time is such that, at it, it is necessary that if it is sometimes accurate, it is accurate. □∀t At t □(S∀p(p↔At t p)→∀p(p↔At t p)) However, given a modal logic that includes the B axiom (everything possibly necessary is true), NOW′+ is inconsistent with the claim that history could have ended before now. For suppose that it could have—i.e. that there is some set X of past times (times that are not present but once were present) such that possibly, always, some member of X is present. Then, by the necessary factivity of ‘always’, it is possible that some member of X is present. Given the modal rigidity of set-­‐
membership, it follows that for some t ∈ X, t is possibly present, and hence possibly 33 accurate. But by NOW′+, it is necessary that if t is accurate, then: necessarily, if t is sometimes accurate, it is accurate. So we can conclude that it is possibly necessary that if t is sometimes accurate t is accurate. Given the B axiom, this entails that if t is sometimes accurate it is accurate. But t, being past, is sometimes accurate (since it is sometimes present), and yet not accurate (since the false proposition that it is present is true at it): contradiction. Since we take B to be non-­‐negotiable, we can conclude that, given the contingency of the time series, NOW′ is just as untenable as NOW. IX Someone might agree with everything we have said up to this point while nevertheless regarding the dispute we have been engaged in as “merely verbal”. Perhaps our opponents mean something different from us by the term of art ‘metaphysically necessary’, such that (a) ‘All metaphysically necessary truths are always true’ is false as used by them, and (b) both our and our opponents’ ways of talking are “equally good”. In particular, one might suggest that our opponents use ‘metaphysically necessary’ to express the notion of immediate necessity, where this can be defined in terms of our notion of metaphysical necessity as follows: it is immediately necessary that Φ just in case the truth of Φ is a metaphysically necessary consequence of the truth about which time is present. (In symbols: □IΦ =df ∃t(Present(t) ∧ □(Present(t) → Φ)), where t is not free in Φ). For any formula Φ, let ΦI be the result of substituting immediate for metaphysical necessity in Φ (i.e. the result of replacing all occurrences of □ with □I, in accordance with the aforementioned metalinguistic abbreviation). We can now express the interpretative hypothesis as follows: For any Φ, Φ as used by our opponents is equivalent to ΦI as used by us. In favor of this hypothesis, one might appeal to the following facts: (i) Any formula Φ is logically equivalent to ΦI relative to the symmetric model theory, both in its original form and in the modified version presented in §VII. 34 (ii) PerpetuityI is uncontroversially false, since it entails that the present time is always present. (iii) SupervenienceI is uncontroversially true. For any proposition p, p is immediately necessitated by the proposition that p is now true (∀p□I(Np→p)), which is an ‘immediately eternal’ proposition (∀p□I(ANp∨A¬Np)). (iv) NOWI is uncontroversially true, since it is uncontroversial that the present time is such that necessarily, if it is present, then the true propositions are exactly those propositions that are true now. (Parallel reasoning applies to NOW+I.) Insofar as the symmetric model theory gives expression to our opponents’ views, (i) makes it natural to think of our opponents as recognizing only one notion of necessity where we recognize two. It might therefore be thought that we face an interpretative dilemma that should be resolved by considerations of charity; and given (ii)–(iv), such considerations seem to support the present interpretative hypothesis. We are unmoved by this mode of argument. Philosophers do regularly make mistakes about general metaphysical principles, and equate notions that are in fact distinct, so the fact that a certain interpretation avoids attributing such errors is very weak evidence for its correctness. But even if we were convinced that there was a practice afoot of using ‘metaphysically necessary’ to express immediate necessity, we would still emphatically reject the claim that this way of speaking was “just as good” as ours. For there are hypotheses about the possible structures of time that simply cannot be expressed in the language of tense operators, propositional quantifiers, and an operator expressing immediate necessity. For example, consider the following pair of hypotheses: H1 It is metaphysically necessary that time is dense. H2 Although it is only contingently true that time is dense, it is necessary that for each time t, either it is necessary that if t is ever present, time is dense, or else it is necessary that if t is ever present, time is discrete. 35 (Intuitively: necessarily time is either dense or discrete, and whether a time belongs to a discrete or a dense time-­‐series is an essential property of it.) H1 and H2 strike us as perfectly respectable competing hypotheses about the modal metaphysics of time, about which there could be a substantive debate. But there seems to be no way of conducting such a debate using immediate necessity as one’s basic modal notion. For one thing, H2I is flatly inconsistent and so clearly fails as a way of making sense of H2. Nor would it help to instead replace ‘metaphysically necessary’ in H2 with ‘always immediately necessary’, or ‘immediately necessarily always’, or with any finite string consisting of alternating ‘always’ and ‘immediately necessarily’ operators, or even with the infinite conjunction of all such strings. We will focus on this last proposal. Model-­‐theoretically, the effect of such a replacement is an operator corresponding to an accessibility relation over the domain which is the transitive closure of the disjunction of the temporal accessibility relation and the “immediate possibility” accessibility relation (the relation one possibility bears to another if and only if they agree about which time is present). Such an operator is not semantically equivalent to metaphysical necessity, since this accessibility relation need not be universal. In particular, H2 is only true in models where this relation fails to be a universal relation: if H2 is true, then no sequence of steps each of which either takes you to a different point in history or to a different history with the same present time will allow you to reach any of the possibilities in which time is discrete, whose existence is asserted by H2.40 By our lights, these expressive limitations constitute an important respect in which a language whose only modal operator is immediate necessity is inferior to ours. Moreover, this inferiority strikes us as a powerful consideration against the hypothesis that our opponents are in fact speaking such a language. But to be clear, we are not advancing this “expressive power” consideration as an argument for 40 Formally, it is straightforward to see that, relative to the Montagovian model theory, modified to allow for contingency in the time series in the way described in §VII, there is no formula Φ such that ΦI is logically equivalent to H1 or to H2, since we can construct pairs of models that agree as regards ΦI for every formula Φ, but disagree as regards H1 and H2. 36 Perpetuity. Our target is the view that the debate over Perpetuity is merely verbal: it is not the case that both “ways of speaking” are equally good, since if our way of speaking is good, then our opponents’ way of speaking is worse, since it is expressively impoverished relative to ours. We are not claiming that it is an important desideratum that a view be neutral as regards hypotheses like H1 and H2: on the contrary, we see settling such difficult questions as a virtue of a theory about time and modality. X Those who failed to find Perpetuity obvious at the outset might be disappointed that nowhere have we given a direct argument for the principle. In the absence of such an argument, they might think it would be prudent to remain uncommitted on the question. Setting aside our judgment of Perpetuity’s obviousness, such a hesitant methodology strikes us as unwise. In metaphysics, as in natural science, we ought to be looking for simple, strong, and unified packages of general principles. We have considered two such packages — our own, and the package of commitments validated by the symmetric model theory — and argued in §VII that the latter is seriously problematic. Perhaps there are other comparably systematic pictures of the interaction of time and modality on which Perpetuity fails, but we don’t know of any. Until someone defends such a view, or serious problems are raised for the view we have defended, abductive considerations tell strongly in favor of Perpetuity. 37 References Correia, Fabrice (2007): ‘Modality, Quantification, and Many Vlach-­‐
Operators’. Journal of Philosophical Logic 36: 473–88. Cresswell, M. J. (1990). Entities and Indices. Dordrecht: Kluwer. Fine, Kit (1977): ‘Prior on the Construction of Possible Worlds and Instants’. In Prior 1977. Reprinted in Modality and Tense, 133–75. Oxford: Oxford University Press. Kamp, Hans (1971): ‘Formal Properties of “Now”’. Theoria 37: 227–73. Kaplan, David (1989), ‘Demonstratives’. In Almog, J., Perry, J., and Wettstein, H. (eds.), Themes from Kaplan, 481–566. Oxford: Oxford University Press. Kaufmann, Stefan, Cleo Condoravdi and Valentina Harizanov (2006): ‘Formal Approaches to Modality’. In William Frawley (ed.), The Expression of Modality, 71–106. Berlin: Mouton de Gruyter. King, Jeffrey C. (2003): ‘Tense, Modality, and Semantic Values’. In John Hawthorne and Dean Zimmerman (eds.), Philosophical Perspectives vol. 17: Language and Philsophical Linguistics. 195–246. Oxford: Blackwell. Kurucz, Agi (2006): ‘Combining Modal Logics’. In P. Blackburn, J. van Benthem and F. Wolter (eds.), Handbook of Modal Logic, 869–924. Amsterdam: Elsevier. Knuuttila, Simo (1982): ‘Modal Logic’. In Norman Kretzmann, Anthony Kenny, and Jan Pinborg (eds.), The Cambridge History of Later Medieval Philosophy, 342–
57. Cambridge: Cambridge University Press. Maudlin, Tim (1990): ‘Substances and Space-­‐time: What Aristotle Would Have Said to Einstein’. Studies In History and Philosophy of Science Part A 21: 531–561. McKay, John (2013): ‘Quantifying over Possibilities’. Philosophical Review 122: 577–617. Montague, Richard (1973). ‘The Proper Treatment of Quantification in Ordinary English.’ In Patrick Suppes, Julius Moravcsik & Jaakko Hintikka (eds.), Approaches to Natural Language: Proceedings of the 1970 Stanford Workshop on Grammar and Semantics, 221–42. Dordrecht: D. Reidel. Reprinted in Montague 1974: 247–70. 38 — (1974): Formal Philosophy: Selected Papers of Richard Montague. Ed. and intro. by Richmond H. Thomason. New Haven, Conn.: Yale University Press. Ninan, Dilip (2010): ‘Semantics and the Objects of Assertion’. Linguistics and Philosophy 33: 355–80. A. N. Prior (1955): ‘Diodoran Modalities’. Philosophical Quarterly 5: 2015–13. — (1971): Objects of Thought. Ed. Peter Geach and Anthony Kenny. Oxford: Clarendon Press. — (1977): Worlds, Times and Selves. Ed. Kit Fine. London: Duckworth. Rini, A.A. and M.J. Cresswell (2012): The World-­‐Time Parallel: Tense and Modality in Logic and Metaphysics. Cambridge: Cambridge University Press. Salmon, Nathan (1983). Frege’s Puzzle. Cambridge, MA: MIT Press. Schaffer, Jonathan (2012): ‘Necessitarian Propositions’. Synthese 189: 119–62. Schlenker, Philippe (2006). ‘Ontological Symmetry in Language: A Brief Manifesto’. Mind and Language 21: 504–39. Thomason, Richmond H. (1984): ‘Combinations of Tense and Modality’. In D. Gabbay and F. Guenther (eds.), Handbook of Philosophical Logic, vol. 2: Extensions of Classical Logic, 135–65. Springer. Vlach, Frank (1973): ‘Now’ and ‘Then’: A Formal Study in the Logic of Tense and Anaphora. PhD dissertation: UCLA. Wehmeier, Kai F. (2004): ‘In the Mood.’ Journal of Philosophical Logic 33: 607–30. Williamson, Timothy (2003): ‘Everything’. In John Hawthorne and Dean Zimmerman (eds.), Philosophical Perspectives 17: 415-­‐-­‐65. Oxford: Blackwell. — (2013) Modal Logic as Metaphysics. Oxford: Oxford University Press. Yalcin, Seth (forthcoming): ‘“Actually”, Actually’. Forthcoming in Analysis. Zalta, Edward N. (1988): Intensional Logic and the Metaphysics of Intentionality. Cambridge, MA: MIT Press. 39 40