Now is the time

Abstract

The aim of this paper is to consider some logical aspects of the debate between the view that the present is the only ‘real’ time, and the view that the present is not in any way metaphysically privileged. In particular I shall set out a language of first-order predicate tense logic with a now predicate, and a first order (extensional) language with an abstraction operator, in such a way that each language can be shewn to be exactly translatable into the other. I shew that this translation is preserved at the metalinguistic level, so that equivalent truth conditions can be defined in a tensed metalanguage or an indexical metalanguage. I then make some remarks about the connection between proofs of relative consistency and metaphysical truth; and some historical remarks about Arthur Prior's use of formal logic in expressing his presentist views.

Notes

¹This paper was presented at the 51st annual conference of the New Zealand Division of the Australasian Association of Philosophy in Hamilton, New Zealand, in November 2004; and as the opening invited address at LOGICA 2006 at the Hejnice monastery in the Czech Republic in June 2006. I am grateful to the Czech Academy of Sciences for support for attendance at the latter conference, and to participants on both occasions for helpful comments.

²Alternative names for the presentist/eternalist distinction are A-time/B-time (or A-series/B-series) and tensed/tenseless (or tensed/detensed). I avoid the A/B terminology since it is at least arguable that McTaggart's definition of A-series makes it contradictory (see Mellor 1998: chap. 7] and footnote 8 below); and I avoid the terms ‘tensed’ and ‘tenseless’ for the presentism/eternalism distinction since I want to use these terms for a classification of linguistic expressions.

³Though even this might be questioned by what Ludlow [2004] calls a ‘Very Serious Tenser’. Ludlow points out [27] that the ‘x’ in the English phrase ‘for some x’ stands for a predicate and that ‘every is mortal’ is not grammatical English. So if (according to alethic presentism) all truth is present truth, and if there can only be present truths about what presently exists then ‘for some x’ must refer only to what presently exists. If we then add a principle which Ludlow 2004: 26] calls ‘Seriousness about Semantics’, which entails that you can only understand a formal language if you can translate it into a natural language then ∃x must also refer only to what presently exists. In terms of the present paper of course Ludlow's article is concerned with the difference between Priorian presentism and alethic presentism, and not, as I am, with the difference, if there is one, between alethic presentism and indexical eternalism.

⁴I use V_μ ambiguously here. In L _p , V_μ(α,t) is a time-relative assignment for α in L _p ; while in L _i , for α in L _i , V_μ(α) is not. But each is uniquely determined by the same ⟨T,<,D,V⟩. Theorem 1 relates to truth at an index in a single (though arbitrary) model. Jerry Seligman has pointed out to me that nothing is said about logical validity. By analogy with modal logic one can define a frame as a structure ⟨T,<,D,t*⟩, i.e., as a model without V but with a ‘designated’ time t*. Validity on a frame is of logical interest, but for the purposes of this paper theorem 1 is only needed for a particular model which in some sense formalizes a particular interpretation for a natural language.

⁵ M(N t∧α) says that α is true at t. Note that the ‘now’ predicate N should not be confused with the ‘now’ operator of Kamp 1971. That operator requires a double indexed semantics, and was concerned with sentences like:

 (i) One day everyone now alive will be dead.

Nevertheless, the ability to express the truth of α at t enables L _p to handle Kamp's operator, since (i) can be formalized in L _p as:

 (ii) ∃t(N t∧F∀x(M(N t∧ alive x) ⊃ dead x)).

⁶The token-reflexive theory, as advocated for instance in Mellor 1981: 36] goes like this. Although the content of a sentence may be ineliminably tensed the content of a token is tenseless. (15) can be turned into a counterexample in the following way. Imagine two utterances:u ₁: That's overu ₂: Thank goodness for that.Presumably the ‘that’ in u ₂ refers to the content of u ₁, but the content of u ₁ is ineliminably tensed, since the utterer of u ₂ is not thanking goodness that the event in question is finished before a certain time, or that its occurrence precedes u ₁. This and other objections caused Mellor to revise his semantics in Mellor 1998, though like Sider he takes an indexical semantics to support eternalism.

⁷Lewis's full sentence is: ‘I know what to say if I want to make believe formally that shapes are relations rather than intrinsic properties, but I know better’. It seems to me that Lewis is perhaps a little disingenuous here, given that the formal ‘make-believe’ is the old method of extension and intension, which has been adopted in our day in formal semantics on the basis of Carnap's 1947 use of the distinction. Many metaphysicians appear to think that a relational solution is absurd, and some seem not even aware of the intension/extension distinction. Thus Craig 2000b: 185] goes so far as to say: ‘I am not aware of anyone who has explicitly propounded a relationalist solution to the problem of temporary intrinsics’. In this connection it is worth mentioning the work of Richard Montague. For the modal case, Montague 1974: 152] writes as follows: ‘A property of individuals is a function having I as its domain and subsets of U as its values’. Here I is ‘the set of all possible worlds’ and U is ‘the set of all possible individuals’. In a later article, Montague 1974: chap. 7] writes: ‘It is a slight oversimplification to call the members of I possible worlds. In connection with tensed languages, for instance, it is convenient to take I as the set of all ordered pairs consisting of a possible world and a moment of time’. This does perhaps assume that times are admitted as genuine entities, and this appears to trouble Sider 2001: 110f]. Lewis's discussion of the temporal case is found in [Lewis 1986: 202 – 4]. More recently Egan 2004: 54f] has defended the ‘relational’ account. Lowe 1998: 128] agrees with Lewis that temporary intrinsics are not relations, but claims that the time relativity enters into the connection between the universal and the object. One might describe this by saying that instantiation is instantiation at a time, where the very same non-relational universal can be instantiated at one time by an object, but not at another.

⁸I have in mind Mellor 1981; 1998; Le Poidevin 1991. Le Poidevin [24] says ‘each tense operator is applicable to every true proposition: Pp & Np & Fp’. Where N is a sentential operator Np is equivalent to p, and so Le Poidevin appears to be claiming that p⊃ (Pp & p & Fp) is valid in tense logic. Calling Mellor and Le Poidevin ‘followers’ of McTaggart needs qualification. McTaggart 1927: chap. 33] argues that the contradiction described in the text is essential to time itself, and therefore that time is unreal. Mellor and Le Poidevin hold that while McTaggart's arguments against the metaphysical reality of tense are valid his argument that the contradiction is essential to time are not.

⁹Maybe if he had dealt with things in the standard way he would not have been the genius he was. The excitement and frustration was also part of his nature as a supervisor. I used to meet him once a fortnight with a list of questions. I don't believe he ever answered any of my questions, but instead would expand on something in what I said that caught his fancy. My job was then to take this and see how to integrate it into my work!

¹⁰I seem to recall, though I cannot document this, that Prior called a symbol of this kind, whose arguments are an individual symbol and a wff, a ‘connecticate’.