Canadian Journal of Philosophy

I The Parameters of the Problem

Most of us would want to say that it is true that Socrates taught Plato. According to realists about past facts,¹ this is made true by the fact that there is, located in the past, i.e., earlier than now, at least one real event that is the teaching of Plato by Socrates. Presentists, however, in denying that past events and facts exist² cannot appeal to such facts to make their past-tensed statements true. So what is a presentist to do?

There are at least three conditions that would ideally be met in a satisfactory solution to this problem:

1. It must preserve our views about which statements are true and which false [End Page 1]
2. It must be transparent what the truthmakers are for those statements
3. It must accommodate the truth-value links between various times.

I shall survey two different families of proposals for the presentist's truthmakers, recent examples of each being advocated by Craig (2003) and Ludlow (1999), and show that they fail at least one of these three conditions. This is not entirely negative, for it shows us what an adequate solution to the problem would look like. I go on to show where presentists can find suitable objects that satisfy these conditions, and in this way give a clear statement of presentism, something that is lacking in the literature.³

II Priorian Presentism

The truth of our statements depends on the way the world is: contingent truths require truthmakers. Everyone should agree with this. What people disagree about is what truthmakers are. But so long as we all agree that truths depend in some way on the contents of the world (whether they are states of affairs, as in Armstrong's work, e.g., his 1997, or particulars, as in Lewis's 2003), then that is enough to generate the concerns of this paper.⁴ Thus, given that there are true past-tensed statements, these too require truthmakers. But where in the presentist's world are these to be found? An obvious response is to say that since only the present exists, these truthmakers are to be found in the present. But what sort of truthmakers could these be?

Suppose we take our cue from Arthur Prior, and use the sentential tense operators, P, N, and F (read 'It was the case that,' 'It is (now) the case that,' and 'It will be the case that,' respectively), to pick up the tense of an interior tensed proposition. However complex the tense of the initial proposition, it can be analysed as a basic present-tensed interior proposition together with the appropriate iteration of the tense operators. Since the interior proposition is always present-tensed, there is a [End Page 2] redundancy in the use of the operator N to signify the present tense. This leaves only the F and P operators as non-superfluous; we can always drop N when it appears outside the scope of P or F (see Kamp, 1971; McArthur, 1976). Because of this, Prior held a 'redundancy theory of the present' (Prior, 1957, 9-10; 1967, 8-10, 14-15; 1968a, 17-23; 1968b, 101): everything that is true at present, is true simpliciter; N is superfluous in the same way as 'It is true that' is redundant according to the redundancy theory of truth. Indeed, he says 'reality consists in the absence of a qualifying prefix' (1970).

However, this is not to say (pace Tooley, 1997, 166) that we should not think of it as being there, for, according to Prior:

means

where p is the proposition that Socrates is teaching Plato. This has to be correct because it is a present truth that Socrates taught Plato. Thus the tacit presence of an N outside the scope of all other operators implies that all tensed propositions are present-tensed; that is, implies, e.g., that the proposition that FPp is the proposition that NFPp; and the first stage of the presentist agenda to reduce past- and future-tensed propositions to present-tensed propositions is easily and naturally achieved.

But the essential question is this: what makes such truths true? What are the constituents of the facts that make them true? Prior denies that these questions need answering: for Prior, there is nothing more to say about the nature of time than is said by a perspicuous tense logic (e.g., Prior, 1996, 45). Nevertheless, I (with others, e.g., Smith, 1987, 188-91; Tooley, 1997, 166-7; Le Poidevin, 1998, 38-9) feel that more needs to be said, especially since the issue of truthmaking is independent of the theory of truth: truthmaking arises just as much for deflationary theories of truth as it does for the correspondence theory (as Lewis, 2001 and Smith, 2003 show). Thus, even if Prior adopts a deflationary theory of truth (and the present) this will not help him avoid the issue of truthmaking; he still needs to say more.

But if we press for an ontology, Prior is ultimately left in an uncomfortable position on one of the prongs of a dilemma. For given his doctrine that propositions are themselves 'logical constructions' out of the objects they are about (Prior, 1971, ch. 1), how is it possible for the proposition that Socrates taught Plato to be true? Which particulars can be invoked as the constituents of such a fact? Not Socrates or Plato — they don't exist. Nor can we invoke a present past-Socrates — what a mysterious object that would be! The alternative is to invoke the primitive [End Page 3] present fact that Socrates taught Plato. But without being able to say how this fact is structured (for its constituents are certainly not Socrates or Plato), this move is far from satisfactory. Thus either option leaves us with an obscure ontology.

We might think that these are just problems peculiar to Prior's position and of which later presentists must surely have said something satisfactory. But it appears not. Craig (2003) carries on in the same vein. He claims that reference to past objects is possible, since 'the proper name "Socrates'' expresses an individual essence of Socrates rather than denotes nonconnotatively the actual object Socrates and so does not require Socrates to exist in order for the name to refer' (395). This move, following Plantinga's work on proper names (e.g., Plantinga, 1974, 71-81; 137-44; 149-63), would allow propositions like Socrates taught Plato to be about Socrates and Plato because 'Socrates' and 'Plato' express the essences of Socrates and Plato, and thus refer to them. Now, this sounds incomprehensible to those who subscribe to the view that in order for genuine reference to an object to take place, the object must exist: for according to this view, if Socrates does not exist, 'Socrates' does not refer. But Craig is at pains to draw a distinction between 'reference' and 'correspondence' (394). Correspondence is a relation between world and word and thus requires an object to fall under a name; but reference, we are told, doesn't, since it has 'to do with how terms serve to pick out individuals' (395), something which, it is claimed, doesn't require the existence of the individual. But even if we concede this point, it hardly addresses the question of which objects are required for truthmaking, a quite separate issue. If it isn't the concrete object that figures in truthmaking, is it the 'individual essences' expressed by the proper names, essences of objects which exist even though the concrete objects themselves don't? No, for whatever these 'individual essences' are supposed to be, Craig only uses them to argue for the possibility of 'reference' to past objects, not in order to spell out the truthmakers for past-tensed propositions. So what are we left with? For Craig, what makes the proposition Socrates taught Plato true is the 'tensed fact' (400) that Socrates taught Plato. End of story. And quite literally so, because nothing more is said about this fact. It certainly cannot have Socrates and Plato as constituents (regardless of whether 'reference' can be made to them), since it is a present fact that Socrates taught Plato and, at present, they don't exist. And it is no good complaining that they did exist, because we want to know what does exist to make the proposition true now. (Note that this is not the unjustifiable requirement that what makes propositions true at a particular time must also be located at that time; since what makes the proposition Socrates taught Plato true now, according to the tenseless view, is a fact located in the past, and this seems an entirely reasonable view. The requirement is rather that there be something in [End Page 4] existence to make the proposition true; so this cannot be something involving Socrates and Plato, for the presentist, regardless of whether they did exist.) We are simply not told what the constituents of these facts are or how they are structured. It seems that no real progress has been made on this issue with this version of presentism since Prior's work.

But the view that the present contains within it all of the facts required to make past-, present-, and future-tensed propositions true creates another, less obvious, concern. We can represent times as sets of present-tensed propositions. Suppose p is a true present-tensed proposition. Now, merely from considering the truth-value links which must hold across times, the following set of true present-tensed propositions must also hold of that time: {p, FPp, PFp, ...}.⁵ But what guarantees that a later time would preserve these links? Given the present time represented by {p, FPp, PFp, ...}, there must be a later time represented by {Pp, PFPp, PPFp, ...}. But how can a Priorian Presentist even guarantee this, let alone explain it? Of course, PFPp must hold in the future, if p holds at present: every adequate theory of time must have this as a consequence or be rejected. But there is no mechanism in this version of presentism to guarantee it. It is no good just appealing to the tense logic and then interpreting this in a deflationary way, since, as noted above, this semantic issue does not address the ontological issues at stake. The ontology, at least as far as Craig is concerned, is such that the present time comprises certain tensed facts, including those about what will happen and what has happened. We can add that all times are contingent and distinct entities. But then it follows, if we take the Humean stance, that the truth-value link cannot be guaranteed by any necessary connections between such times or the tensed facts of which they are comprised (at least at the ontological level of tensed facts, if not the semantic level of tense logic). Furthermore, for that matter, there cannot be any transtemporal relations to link times together on this view: other times don't exist; and it is hard to see how the ontological content of the present time can in itself legislate how other distinct entities, other times, can be comprised. Yet somehow it must — for if the content of the present time does not legislate this, what, according to this view, does? — since it is incredible to think that there could be missing tensed facts from, or additional tensed facts in, various times. The truth-value links have [End Page 5] somehow to be a feature of how the facts are structured (as they are on the tenseless theory of time, for example, and as they are on the view I present below). But because no mechanism is in place to preserve the truth-value links and the possibility of a violation of them is opened up, this version of presentism should be rejected.⁶

III Reductive Presentism

A second shot at the thought that the present itself can supply all of the required truthmakers is the Reductive version of presentism. Ludlow (1999, ch.9) is the most recent substantial work on time that advances this view, and Le Poidevin (1991) (a tenseless theorist) recognises this option:

What makes a certain statement about the past true ... is the evidence that at present exists.... This is possible in virtue of the fact that there are present facts which derive their character from causal connection with past states of affairs, and which determine (at least to some extent) the character of the future.... [This] is essential to the ... (presentist's) position.

And

The extent to which the principle of bivalence is violated by statements about the past or future depends, for [the presentist], upon how much causal determinism he is prepared to allow. In a fully deterministic universe ... all future- and past-tensed statements have a determinate truth-value, as this is guaranteed by present fact. But in an indeterministic universe ... many statements about the future must for [presentism] lack a truth-value.

Although it is obvious from the above discussion that, contra Le Poidevin, this story is not 'essential to the [presentist's] position,' it is a position that has been held for a variety of reasons (see, e.g., Łukasiewicz, 1970, 127-8 and Wheeler, 1978, 41, who thinks that the two-slit experiment in quantum physics vindicates this conception). But it is not a position which has been fully endorsed by any serious philosopher, although, of course, Dummett has played with this view for years (see Dummett, 1969, and his latest, 2003). Ludlow (1999, 162) claims that [End Page 6] although this position is not 'an inevitable consequence of the A-theory,' it is 'a possible avenue of investigation.' So, although this is only a tentative endorsement, it is important to briefly show how just a short walk down this supposed avenue reveals it to be more of a cul de sac.

There are two broad categories of theories of laws of nature, which I shall label 'Humean' and 'non-Humean.' Consider the Humean theory, which states that laws consist in nothing more than this: whenever there is an F, there is a G, i.e., (∀x)(Fx ⊃ Gx). The first issue to address here is the 'whenever' quantifier. If it just ranges over times that exist (i.e., the present time!), then any present conjunction of F with G will constitute a law, which is unsatisfactory. But, more damagingly, if it is restricted to times that exist (the present time), then laws cannot do the job for which they were intended. For according to this account, they do not extend beyond the present, and thus cannot determine the truth-values of statements about the past and the future. We should, then, take it that 'whenever' ranges over present times as well as those that have existed and will exist. The problem now is: what makes it true that all Fs are, have been, and will be Gs? For there is nothing we could appeal to in the present. We certainly can't appeal to deterministic laws of nature to ground what has happened and what will happen, since this just lands us in a circle, for the whole problem was to find something in the present that could ground what that law is. And the present state of the universe by itself cannot determine what has happened and what will happen because the present state of the universe is compatible with infinitely many different mutually contradictory pasts and futures: it is only the state of the universe at a given time together with deterministic laws of nature which can determine what has happened and what will happen. But there is nothing in the reductive presentist's world which can ground such a law. It is not as if the law can be seen as an extra ingredient in the present, which can help determine one course of history over another, for remember that we are talking about a Humean conception of law, and according to this conception, there is nothing more, ontologically speaking, to a law over and above what does actually happen. This leads us to consider an alternative option, namely to appeal to primitive facts: things just did and will occur in this sort of regularity. But this then renders the excursion into Reductive Presentism entirely redundant if we adopt primitive facts: we may as well be Priorian Presentists. Thus, a Humean account of laws will not give Reductive Presentists what that they would hope for.

What about non-Humean accounts? Suppose we thought that laws of nature are some kind of necessary connection between universals. According to Armstrong, the law that Fs are Gs is some sort of necessitation relation N between the universals F-ness and G-ness. N(F, G) is a contingent nomic necessity that may not hold across all possible worlds, but if [End Page 7] it actually holds, then N(F, G) ⇒ (∀x)(Fx ⊃ Gx). For various reasons, Armstrong (1983, 86-99) argues that N(F, G) is a first-order relational universal that is instantiated by particular states of affairs Fa and Ga. Furthermore, according to Armstrong's Principle of Instantiation (e.g., 1983, 82) universals must be instantiated in order to exist. It follows that given that there are only present states of affairs in the Reductive Presentist's world, any instantiation of a law can only take place in, and be true of, the present. Thus such laws of nature cannot tell us about what happens at other times, and thus cannot ground determinate truth-values for past- or future-tensed statements. (This is particularly problematic given Armstrong's view that the necessitation relation does not supervene on the state of the world, and can change in what it relates from world to world. For even if N(F, G) holds of the present time, there is nothing to say it will hold across all past and future times. Yet that was what the Reductive Presentist hoped the necessitation — although it is hard to see how we can still call it 'necessitation' given these considerations — would guarantee.)

This well-known and well-worked-out non-Humean theory was worth mentioning in order to illustrate the general problem with which Reductive Presentists have to deal: how can the laws reach beyond the present? There are, of course, non-Humean accounts that argue that laws can exist without being so instantiated (e.g., Tooley, 1977; Mellor, 1980). But a presentist who appealed to these accounts of laws of nature would either have to find something in the present to ground these laws — and it's difficult to see what this could be — or they must appeal to facts outside the present, in which case they've conceded that not all of the truthmakers can be found in the present, which was their initial contention. The notion of determinism will not help the presentist on the issue of securing determinateness of truth-value.

Needless to say, Reductive Presentism cannot have determinate truth-values in an indeterministic world. Yet even in an indeterministic world, we still want to say that at least past-tensed statements have determinate truth-values. We want to say that it is either true or false that a certain brontosaurus had two plants for breakfast, regardless of whether there are any traces of this fact in the present, and regardless of whether the present state of the universe and the laws of nature can determine this fact. To confuse determinateness with determinism is an offence. Reductive Presentism thus violates condition (a) of section I.

Furthermore, Reductive Presentism, however much it initially seems to, doesn't in fact meet the truth-value links problem. First, consider Russell's hypothesis (1921, 159-60): what is to say that the world has not just come into existence complete with fossil records, memories, and other such causal traces? Russell intended this to illustrate the limitations of our knowledge, but pre-theoretically, it seems like a genuine possibility [End Page 8] for how the world could be. Suppose it is true, then Reductive Presentism gives the wrong answers, for it states that certain past-tensed statements are true when they shouldn't be. Of course, nobody thinks that hypothesis is true; but Russell's hypothesis doesn't have to be actually true to cause trouble; that it is true that is it possible is enough to show that Reductive Presentism has a problem accounting for truth-value links. And we couldn't rule out Russell's hypothesis by appealing to actual laws of nature that wouldn't allow for such a complex environment in such a short space of time, for as I have argued above, the grounding of those very laws is called into question on the Reductive view. That is, in a nutshell, truth-value links are supposedly underpinned by causal links: present facts are effects of previous causes and themselves causes of later effects. But this relies on what it is in the present that grounds this causal link: for it to be true that a given fact is an effect of a previous cause, we need the present fact that it is the effect of a previous cause to make this true, since without this fact there is nothing to link the supposed 'effect' to anything that came before, or the supposed 'cause' to anything that comes after. (And we certainly need this, given the many different pasts and futures compatible with the present state of the universe.) But what sort of fact could this be? Taking it as a primitive present fact lumbers us with the Priorian Presentist's facts and all their attendant problems. We are no better off after all.

We now have an appreciation of what any successful version of presentism must hope to achieve, and the pitfalls it must avoid. The version of presentism I am about to propose meets these conditions. It also meets the challenges that some (e.g., Oaklander, 2002; 2003) throw down for presentism to tackle, namely how presentists can help themselves to the notion of earlier than without having to invoke real relata, and how presentism can distinguish the past from the future. It should also be attractive to those who see close analogies between time and modality, and who prefer ersatz theories of possible worlds (such as Adams's 1974) over genuine modal realism (such as Lewis's 1986), for the position I will defend is essentially this: all of us should agree that Socrates taught Plato, i.e., that the proposition that Socrates is teaching Plato was, at some time, true. This, to most people, is so obvious as to not be worth stating. So I ask them to keep this in mind during the more technical exposition of the position which is about to follow. For I say we should take the 'i.e.' seriously; that is, that what makes it true that Socrates taught Plato is the existence of a proposition that states this is the case for some time in the past, where a time is a set of propositions that states the other truths about what happens at that time. For obvious reasons, I shall call this position 'Ersatzer Presentism.' [End Page 9]

IV Ersatzer Presentism⁷

a) An improvement over Prior's Presentism: e-propositions and u-propositions

I go some way with Prior (1967, 79-82; 1968a, 122-6) in constructing times from certain sets of present-tensed propositions.⁸ However, my account differs in important respects. According to Prior, the present moment should be equated with the conjunction of all those propositions which are presently true, i.e., true simpliciter; and generalising, he equates any time with a conjunction of all those present-tensed propositions which would be true at that time.

My conception differs, first, in that I distinguish present-tensed propositions that contain either P or F operators, which I shall call 'embedded propositions' (e-propositions), from those that do not, which I shall call 'unembedded propositions' (u-propositions). (Compare, for example, the e-proposition: It is now the case that it was the case that Socrates is sitting (i.e., NPp), with the u-proposition: It is now the case that Socrates is sitting (i.e., Np), or simply: Socrates is sitting (i.e., p).)

As we have seen, Priorian Presentism required the present to be equated with e-propositions in order for the present to supply all of the truths we believe there are, but that it was rather mysterious what the truthmakers were for these propositions, and neither could it account for the truth-value links between the different times. The theory to be developed avoids these difficulties.

b) Defining times, and the E-relation

I propose we construct times using maximally consistent sets of u-propositions, which intuitively we can see as those u-propositions that are true [End Page 10] at that time. These propositions I take to give a complete, maximally specific description of what is true at that time. But more needs to be added: they, at least, need to be ordered by an 'earlier than' relation (what I will call an 'E-relation'), in order for the ersatz time series to be structurally similar to a real time series, so it can be taken to be a sufficient substitute. That is, we can introduce the ordered triple <T, E, t>, where T is a set, E is a relation on T, and t∈T.⁹ Intuitively, T is the set of times, E is the 'earlier than' relation, and t is a particular time. Times I take to be more than sets of present-tensed propositions: first, they consist of sets of u-propositions; second, they also contain a 'date.' That is, I take times (at least for the time being) to be ordered pairs of the form t = <μ, n∈ >, where μ is a set of u-propositions and n∈ is the date. Times can now be defined as those ordered pairs of the form t = <μ, n∈> that are members of the class of sets of ordered pairs of the form t = <μ, n∈> that are E-related. Actual times are those times that are members of the class of times that are actually E-related. (What this amounts to is explained below.)

Defining times in the way I have done requires us to alter our terminology slightly. Lewis (1986, 140) has complained that ersatzism concerning possible worlds commits us to saying that since worlds are, say, sets of propositions, the concrete object that we are part of — 'the world,' speaking with the vulgar — cannot be said to be a world. Similarly, since, according to Ersatzer Presentism, times are abstract objects, the present time is not something we inhabit. But rather than introduce a new term, or typescript, for what I am calling 'times,' I prefer to keep this useful terminology. The only differences it makes are that we must be said to inhabit the concrete realisation of the present time, and that Ersatzer Presentism is the view not that only one time exists, but that only one time has a concrete realisation. This is a mere nominal difference, equivalent to saying that presentists do not believe in any times other than the present; and all is as it should be.

The E-relation is not the genuine earlier than relation since it does not relate spatio-temporal objects, but it does represent the earlier than relation in the way it relates times. The properties of the E-relation match whatever we take to be the properties of the genuine earlier than relation. This allows presentists to have a time series related by 'earlier than' without being committed to the existence of real, or rather, concretely realised relata, something anathema to presentism. Ersatzer Presentism thus bypasses the problems that other presentists get into when they do not [End Page 11] take such relations as basic, and try to define them in terms of tenses (see, e.g., Oaklander, 2002; Mellor, 2003, 236-7).

Various metrical and topological features of time can be represented in the structure of the abstract objects. For example, if we take time to be linear, the E-relation will have the properties of being irreflexive, asymmetric and transitive; if it is circular, it will be an equivalence relation, and times form an equivalence class; if time is continuous, the times are assigned real numbers as dates; if discrete, then only those times that have dates n∈ will have instantiations; if it is infinite in both directions from the present, then the dates will have the standard order type of infinite time (ω*+ω); if time is circular then the dates will be of the form of a periodic function (for some period with value P, t(x+P) = t(x)), and so on.

But despite the fact that these properties of time can be represented in this way, presentism has a very good reason for adopting a branching topology; that is, where the E-relation is a one-many relation in the direction from the present to future (the direction in which the dates increase in magnitude), but only a one-one relation in the direction from the present to the past (the direction in which the dates decrease in magnitude). The reason is simple: we all need a way of distinguishing the past from the future. Because presentism treats the past and the future as ontologically on a par, in the sense that it denies that there are any concrete truthmakers located there, the branching structure of times is an obvious way in which presentism can differentiate between past and future. Furthermore, it accounts for the platitude that the past is 'fixed' and the future is 'open,' something that any good theory of time should. Thus, if we follow my proposal, presentism does have at its disposal means for distinguishing the past from the future.

c) Truth-conditions, truthmakers, truth-at-a-time, and truth simpliciter

I also distinguish truth-at-a-time from truth simpliciter.¹⁰ Truth simpliciter is an absolute, not time-relative, notion, whereas truth-at-a-time is time-relative: all propositions at a time are true relative to it, but only those propositions which are true at the present time are true simpliciter. So, where the propositions involved are any atomic propositions, be they from propositional or predicate calculi, we have: [End Page 12]

1. 'Socrates is sitting' is true-at-a-time iff there is a time, i.e., E-related order pair <μ, n∈> such that μ includes the u-proposition that Socrates is sitting, i.e., 'p' is true-at-a-time <T, E, <μ, n∈>> iff p∈m.

And similarly,

2. 'p' is true-at-a-time <T, E, <μ, n∈>> iff p∉m.¹¹
3. 'p&q' is true-at-a-time <T, E, <μ, n∈>> iff p, q∈m.

Whereas:

4. 'Socrates is sitting' is true simpliciter iff Socrates (i.e., an actual, concrete, flesh and blood Socrates) is currently sitting.

Furthermore, it is not only present-tensed u-propositions that can be true simpliciter; past-tensed propositions can be too. The difference is that whereas present-tensed u-propositions are made true simpliciter iff corresponding actual concrete facts are presently realised, past-tensed propositions are made true simpliciter by something entirely different.

I propose that we treat past- and future-tenses as statements about the present; or rather, about where our present's corresponding time is in relation to the others. Since these statements are really statements about how times, i.e., sets of E-related ordered pairs, are ordered, we have the following truth-condition:

'It was the case that p' is true simpliciter iff p is a member of a set μ of u-propositions that is the first element of an order pair <μ, n_i∈> actually E-related to the currently instantiated ordered pair <n, n_j∈>, where n is the set of u-propositions that is true simpliciter, and n_i<n_j.¹²

Note that the 'actual' here is not superfluous: it has a technical use. Since time has a branching structure, then time actually has a branching structure and all possible times on each of these branches are in some sense of [End Page 13] 'actual' actually E-related. Yet I require only some times as truthmakers for past- and future-tensed statements, namely those that correspond to what we would ordinarily want to call the actual history of the world. Thus, when I use the phrase 'actually E-related' I mean that there is a time which is E-related to the time that is true simpliciter, i.e., that is actually concretely realised. This grounds which E-related branch we should use for finding truthmakers for past-tensed statements, since only one E-related branch is accessible from the concretely realised time in the direction towards the past.

In short we have the following story about truthmakers:

• What makes u-propositions true simpliciter are actually realised concrete facts
• What makes past-tensed propositions true simpliciter are actually E-related ordered pairs of u-propositions and dates.
• What makes 'p' true-at-time-t is the fact that p∈m, where t = <μ, n∈>.

It now remains to complete the story by dealing with quantification. First, as noted in n.2, presentists should read the existential quantifier tenselessly. I also take it that '(∃x)Fx' should be read in the standard objectual way. Thus, '(∃x)Fx' is true simpliciter iff there is at least one object, x, in the domain of quantification that is F. Despite being read tenselessly, '(∃x)Fx' might change truth value from time to time because of the continually changing concrete facts and subsequent change in the domain of quantification. Thus:

'P(∀x)Fx' is true at <T, E, t> (and is true simpliciter iff t is true simpliciter) iff (∀x)Fx is a member of every μ element of times earlier than t (understood in the ersatz way), where '(∀x)Fx' is true-at-a-time <T, E, <μ, n∈>> iff (∀x)Fx ∈m.

'P(∃x)Fx' is true at <T, E, t> (and is true simpliciter iff t is true simpliciter) iff (∃x)Fx is a member of some μ element of a time earlier than t (understood in the ersatz way), where '(∃x)Fx' is true-at-a-time <T, E, <μ, n∈>> iff (∃x)Fx ∈m.

d) The future

An interesting question that arises for Ersatzer Presentism is accounting for future-tense propositions. Can the sentence 'It will be the case that p' be true simpliciter? According to this theory, [End Page 14]

'It will be the case that p' is true simpliciter iff p is a member of a set μ of u-propositions that is the first element of an order pair <μ, n_i∈> actually E-related to the presently realised ordered pair <n, n_j∈>, where n is the set of u-propositions that is true simpliciter, and n_i > n_j.

However, 'It will be that p' is neither determinately true simpliciter nor determinately false simpliciter, given the branching structure, when p is contingent. Furthermore, consider the truth-conditions for future-tensed quantified statements:

'F(∀x)Fx' is true at <T, E, t> (and is true simpliciter iff t is true simpliciter) iff (∀x)Fx is a member of every μ element of times later than t (understood in the ersatz way), where '(∀x)Fx' is true-at-a-time <T, E, <μ, n∈>> iff (∀x)Fx ∈m.

'F(∃x)Fx' is true at <T, E, t> (and is true simpliciter iff t is true simpliciter) iff (∃x)Fx is a member of some μ element of a time later than t (understood in the ersatz way), where '(∃x)Fx' is true-at-a-time <T, E, <μ, n∈>> iff (∃x)Fx ∈m.

Given the branching future, 'F(∀x)Fx' is only true simpliciter when '(∀x)Fx' is true at all times later than the present, i.e., on all possible branches. And 'F(∃x)Fx' is only false simpliciter when '(∃x)Fx' is true at no time later than the present, i.e., on no possible branches. That they have determinate truth-values only in these extreme circumstances, however, must only really be treated as a consequence of the branching view, not a criticism of it. But I should discuss here Lewis's (1986) argument against branching time, since it is based on a common misconception it. Lewis writes:

The trouble with branching exactly is that it conflicts with our ordinary presupposition that we have a single future. If two futures are equally mine, one with a sea fight tomorrow and one without, it is nonsense to wonder which way it will be — it will be both ways — and yet I do wonder. The theory of branching suits those who think this wondering makes sense only if reconstrued: you have leave to wonder about the sea fight, provided that really you wonder not about what tomorrow will bring but about what today predetermines.

Lewis is wrong, however, to conclude that a branching future commits one to holding 'it will be both ways.' He is confusing the fact that there are many future branches to be realised as of the present time, with the fact that only one of them will be realised. After all, that is the only way to make sense of the claim that future contingent statements have [End Page 15] indeterminate truth-values¹³ — if it will be both ways, then both of the statements are true. But according to the branching view, future contingents are presently indeterminate precisely because only one branch of the possible branches will get realised, but that there is no branch in particular that is presently that branch. Thus it makes good sense to wonder now which of the unrealised branches will become the future. Thus branching does not conflict with our ordinary presupposition that we will have a single future, although it does conflict with the idea that we have a single future. The former presupposition is something we surely do not want to reject — it is close to being a platitude. But it is far from clear that the latter presupposition has such a status, and arguably is contrary to most people's commonsense view of the future: many people think, in a clear-headed way, that pace Lewis, we don't have a single future, but we will have one. And, moreover, this has nothing to do with what is 'predetermined': as already noted above (III), determinism is neither here nor there on the issue of the determinateness of the future. Lewis's argument is thus far from compelling.

V Branching time for presentists

It just remains to fill out the story concerning the branching structure and show how Ersatzer Presentism can account for the truth-value links between times. Consider:
[End Page 16]

At the present, there is only one past accessible to it and no unique future. The set-up is such that all possible futures are represented and accessible from each node on a particular branch, so whichever time gets realised next will be represented somewhere on one of the future branches accessible from it: the branching structure gives us completeness.

a) A problem with truth-value links?

But a problem arises. Suppose that p₀ is true simpliciter, i.e., that time t₀ is currently realised. Now suppose that the proposition p₁ is made true simpliciter next. What guarantee have we that it is p₁ at t₁ that has been realised rather than, say, p₁ at t₃? If there is nothing to constrain which time is realised, it seems that we are committed to saying that the concrete facts that have just 'become' realise all of these times equally. But if this is the case, then there are branches that state

is the history of the world, whereas others state

is, whereas others state that

is, and if p₁ is true simpliciter, it seems that there is nothing to choose between these options. Something needs to be done to preserve the truth-value links.

The problem here arises because of the sparseness of the concrete present: it is not sufficiently rich to determine everything that needs to be determined in order to defend a plausible view of time. But this is no great news; that's exactly what I have been arguing so far, and precisely the reason for invoking more than the concrete present facts as truthmakers for tensed statements. This was, after all, what confined the previous versions of presentism to an implausible position. But once we remember that Ersatzer Presentists have more artillery at their disposal, this problem has an obvious solution. For according to the theory here proposed, the sets of propositions follow the ordering of the dates; so although the content of the proposition p₁ is the same wherever it occurs as a member of a time, it is also associated with a date. This allows us to stipulate that when p₀ is realised at time 0, the next set to be realised must by definition be a set that is indexed by 1 (or whatever number sequence or metric we choose to be appropriate), and so on. This is no more mysterious than holding that 2nd January follows the 1st January, [End Page 17] something that all sides accept. So there is a natural and non-arbitrary way of specifying which time it is that has been realised. The truth-value links remain intact.

b) A further problem with truth-value links for branching time presentists?

However, there is another potential objection. Suppose that it is <p₁, 1> that is realised after <p₀, 0>, rather than, say, <p₁, 3>, due to the fact that <p₁, 1> is indexed by 1 and <p₁, 3> is indexed by 3, and 1 is the least available number greater than 0. Now suppose that proposition p₂ is made true simpliciter by the next concrete facts to 'become.' What guarantee have we that it is the p₂-set in the history

that is realised, rather than the p₂-set in the history

Indeed, since there is nothing in the present to determine which one is realised (both p₂-sets are, after all, indexed by 2), it seems that we are committed to saying that both have equal claim to have been realised, and our truth-value links again break down.

This objection again relies on there not being enough in the present to determine which of the sets is realised. But, again, this is not the theory that I am putting forward; there are more facts available to determine which p₂-set is realised. For once we have in place the fact that it is <p₁, 1> that is realised after <p₀, 0>, there is a fact of the matter as to which branch has been realised, namely the <<p₁, 1>, <p₀, 0>> history rather than the <<∼p₁, 1>, <p₀, 0>> history. It follows that it must be the p₂-set accessible from the <p₁, 1> that is the next to be realised, since it is a fact that that branch is realised and not the <∼p₁, 1> branch.

This can be represented formally, rather than diagrammatically, by adding an accessibility relation defined over times, and adding a 'positional' element to times. Times are now ordered triples of the form <μ, n∈, σ>, where σ is the position of the time in the branching structure, something given by specifying the sequence of times along its past. For discrete time, this can be achieved simply by including the last time realised as the third element (since this time will specify its past, and so on). Thus, for the case of discrete branching time, time n+1 has the form t_n+1 = <μ, n+1, t_n>, where t_n = <v, n, t_n-1>, and so on. The accessibility relation R for <T, E, t, R> can be defined as follows: for all times, t_n+1 is accessible to t_n iff t_n is a member of t_n+1, i.e., [End Page 18]

We can also generalise to continuous time. The reason that we must say something slightly different for continuous time is that there is no 'next available date' or 'last time realised' to be the positional element. But all that needs to be said to constrain the order of realisation is first to say that the order of realisation must follow the order of the real numbers as given by the greater than relation. Second, it may not be possible to specify a particular time as the positional element of a time, but it is possible to specify the position using the sequence of times leading up to that time (which I represent as 'σ_<a'), and then represent times as follows:

t_a = <μ, a∈, σ_{< a} >

So generalising to include continuous time, the accessibility relation R for <T, E, t, R> can now be defined as follows: for all times, t_b is accessible to t_a iff t_a is included in the third element of t_b, i.e.,

(∀t)(R (t_b, t_a) ≡ ((t_a⊂σ _{< b}) & (t_b = <v, b∈, σ_{< b} >))).

c) Tying up loose branches: one last objection met

Now that the metaphysics and semantics have been given for Ersatzer Presentism, I should answer one possible objection: Why is it that a certain atomic proposition p appears in a given E-related time if not because it was the case that p? But if this is the explanation, if this is how the story ultimately bottoms out, then how is Ersatzer Presentism any better off than Priorian Presentism?

There are a few things to say in response. First, whether a given proposition appears in a time is simply a brute fact. Or rather, since all possible sets of propositions appear on the branches, it is a necessary truth that such propositions appear in some of the times, and it is a brute fact that one of them gets realised. This is no objection since it is no more mysterious than concrete facts being realised at the times they are according to tenseless theorists. The real bite of the question comes from explaining why the various times are E-related. But on reflection it is more of a toothless suck than a vicious bite. For according to Ersatzer Presentism, what makes 'It was the case that p' true is an actually E-related ordered triple, whereas according to the tenseless theory, what makes it true is an actually earlier than-related concrete fact. Now to ask why these ordered triples are actually E-related is about as fair as asking why the concrete facts are actually earlier than-related in the tenseless theory, i.e., not at all — they just are. In this sense, then, all theories take it as a brute fact that it was the case that p; the advantage over Priorian [End Page 19] Presentism is that these other theories have an account of what this fact looks like, be it an E-related abstract structure or an earlier than-related concrete one.

VI The advantages of Ersatzer Presentism

Ersatzer Presentism, then, can meet the three conditions that any satisfactory account of time should: it allows us to state truths about the past; it wears its ontological commitments on its sleeve; and it ensures that truth-value links are preserved. There are two further advantages: first, it is often argued that there is no substance to the tensed-tenseless debate — that once we recognise the confusion over tensed and tenseless readings of various quantifiers and copulas, the tenseless theorist's 'does exist in the past' is not substantially different from the tense theorist's 'existed' — they are the same theory under different descriptions. This charge certainly cannot be levelled against Ersatzer Presentism. According to Ersatzer Presentism, the constituents of past and future times are not spatio-temporally or causally related objects, unlike past and future times according to the tenseless theory. The constituents of ersatz past and future times are propositions, numbers and sets of propositions and numbers, and so can only have those relations that can hold between sets, propositions, and numbers. These conceptions of past and future times are so radically different that it would stretch the meaning of the word 'same' to claim that they are essentially the same theory under different descriptions. Unlike the tenseless theory, according to Ersatzer Presentism, there really is no time like the present.

The second advantage is that many have complained that there is no good motivation for becoming a tense theorist, let alone a presentist. Tooley (1997, 250) has argued that before we do commit ourselves to a tensed theory, there is a 'need for metaphysical argument.' Here I claim the argument in favour of Ersatzer Presentism over its rivals is ontological parsimony. The tenseless theory, for instance, postulates appropriately related past and future concrete objects as truthmakers for various tensed statements; but this is something Ersatzer Presentism can discard. Neither does Ersatzer Presentism postulate mysterious facts in the present to make past-tensed statements true. Indeed, the discussion indicates that a theory that just rests solely on concretely realised present facts will not adequately account for all the truths we need. So the appeal to more objects other than concretely realised present facts is the necessary concession presentism must make to be viable. But it is still a highly attractive theory if you already believe in abstract objects (propositions, numbers, and sets), which many of us do. Thus it isn't an issue of quantitative versus qualitative parsimony (see Bacon, 1995, 87; Lewis, [End Page 20] 1973, 87 for this distinction). The argument is that Ersatzer Presentism is quantitively more parsimonious than the tenseless theory; that is, the set of objects that Ersatzer Presentism postulates is a proper subset of those objects that the tenseless theory postulates. But quantitative savings are still significant: everyone agrees that one can have too much of a good thing. Thus, if you prefer ontologically economical theories without having to live on a shoestring, then Ersatzer Presentism is the theory to adopt.¹⁴