Philosophia

, Volume 38, Issue 2, pp 229–241

McTaggart’s Paradox and Crisp’s Presentism

Authors

    • Department of PhilosophyUniversity of Michigan-Flint
Article

DOI: 10.1007/s11406-009-9222-4

Cite this article as:
Oaklander, L.N. Philosophia (2010) 38: 229. doi:10.1007/s11406-009-9222-4

Abstract

In his review of The Ontology of Time, Thomas Crisp (Notre Dame Philosophical Reviews, 2005a) argues that Oaklander's version of McTaggart's paradox does not make any trouble for his version of presentism. The aim of this paper is to refute that claim by demonstrating that Crisp's version of presentism does indeed succumb to a version of McTaggart's argument. I shall proceed as follows. In Part I I shall explain Crisp's view and then argue in Part II that his analysis of temporal becoming, temporal properties and temporal relations is inadequate. Finally, in Part III, I shall demonstrate that his presentist ontology of time is susceptible to the paradox he so assiduously sought to avoid.

Keywords

McTaggartPresentismTimeErsatz B-relationsTemporal becoming

In his review of The Ontology of Time, Thomas Crisp (2005a) argues that Oaklander’s (2004a, b, c) version of McTaggart’s paradox does not make any trouble for his version of presentism. The aim of this paper is to refute that claim by demonstrating that Crisp’s version of presentism does indeed succumb to a version of McTaggart’s argument.

The standard way of understanding McTaggart is first to assert that events require the A-properties of pastness, presentness and futurity in order for there to be change, and then claim that those properties are incompatible: if an event is present then it cannot be past or future; if it is past then it cannot be present or future and if it is future then it cannot be present or past. Yet every event has all three incompatible characteristics and for that reason change is contradictory, and since time requires change it too is contradictory and hence unreal.1

Within the A-camp there have been basically two ways of responding to this objection.

First, one calls attention to the fact that no event has incompatible A-characteristics simultaneously or timelessly, but has them successively or at different times. There are, however, two very different versions of this first response. One version specifies the times when an event has various tensed properties using tensed language. Thus, it is not the case that every event is now (read “simultaneously”) past, present and future or tenselessly (read “timelessly”) past, present and future, but rather an event/thing/moment or whatever is your preferred temporal object, is now present, was past and will be future, or is now past and was present and still earlier was future, or is now future and will be present and still later will be past. The other version specifies the times when an event has incompatible tensed properties tenselessly, using dates. Thus, for example, the contradiction is avoided by claiming that say; event e is (tenselessly) future at time t, present at time t1 and past at time t2.2

Second, and this is the response typically put forth by presentists, McTaggart’s argument cannot get off the ground since time and change do not require all three A-properties, but only requires, at most, the non-relational temporal property, presentness. However, if there are no A-characteristics or only the property of presentness, then the incompatible properties problem cannot arise.

While the presentist account is adequate as far as it goes it does not go far enough or as some including myself have argued, it is a non-starter since it is nothing more than a verbal solution to the ontological problem of time, which is to clarify the ontological status of the tenses, temporal relations and temporal becoming. Thus, a critic will ask: What are propositions that seem to make reference to past and future objects about? What is the ground of their truth? Furthermore, what is the ground of earlier/later than relations between and among temporal objects? Indeed, what are the “times” that stand in such relations, and how are we to understand temporal becoming—the change events/things/moments undergo as they move from the future to the present and from the present into the past?

To his credit Crisp is sensitive to these questions facing presentism for in his review he claims that,

Oaklander, rightly in my view, complains that there is something deeply unsatisfying about versions of presentism that invoke primitive tense operators and then say nothing about the ontology that underlies them. So, he wants to know, what is the ground or truth maker for statements of the form ‘WAS(S)’? What is the bit of reality that makes it true, say, that WAS (dinosaurs exist)? I agree with Oaklander that these questions need an answer…. (2005a)

Crisp’s answers to these questions are intriguing. First, he rejects the A-properties pastness and futurity, and claims that they are eliminable in terms of (tenseless) ersatz B-relations which obtain between and among times construed as abstract objects, i.e., maximally consistent sets of (tenseless) propositions: “intuitively, an abstract representation of an instantaneous state of the world” (2007: 99). Given Crisp’s characterization of the idea of an abstract time I think I am justified in adding that Crisp’s propositions are all tenseless. Since my criticisms of Crisp crucially depend on this interpretation I shall defend it below.

Unlike some presentists (Craig 1998, 2000), Crisp even reduces the property presentness or being present to the non-temporal property being true. Moreover, he offers an account of temporal becoming according to which tenseless propositions change their truth value, and on his version of presentism, the typical gloss of the thesis “only the present exists” is taken to mean that “there is no temporal distance between any two things in our widest domain of quantification” (2007: 102). In what follows I shall in Part I explain these different aspects of Crisp’s view and then in Part II argue that his analysis of temporal becoming, temporal properties and temporal relations is inadequate because it is contradictory or circular. Finally, in Part III I shall demonstrate that his presentist ontology of time is susceptible to the paradox he so assiduously sought to avoid.

Part I Crisp’s Presentism

We can begin our summary of the main elements of Crisp’s ontology of time by saying that he reduces monadic A-properties pastness and futurity to primitive B-relations or, more accurately, primitive ersatz B-relations between “the present time” and earlier and later times, and the property being present to the property being true. As he puts it in his review:

'WAS(x)' and 'WILL(x)', we can suppose, are predicates, expressions that combine with a singular term t to yield a sentence that expresses a proposition about the referent of t. Both predicates, we can suppose, express properties. The property being present, we can say, is just the property being true — the property a proposition has iff it is true. 'WAS(x)', then, expresses the property being past, the property a proposition has iffdf. it is true at a time earlier than whatever time happens to be the present time — i.e., iff it is entailed by a time in the ersatz B-series earlier than the present time. Mutatis mutandis with 'WILL(x)'. (2005a)3

I shall return to Crisp’s account of the tenses when I also consider his paper, “Presentism and the Grounding Objection” (2007), for now, what needs to be clarified is his notion of “times” and “ersatz B-relations.”

Crisp eliminates B-relations between concrete objects, which relations according to his ontology do not exist, and replaces them with primitive, ersatz B-relations between times that do exist. An ersatz B-relation is “a primitive, unanalyzable relation on abstract times that is transitive, irreflexive and asymmetrical and orders the times (better: some of the times) into an ‘ersatz’ B-series representing the history of the world” (2005a).

Calling a relation between abstract propositions a “temporal” or earlier than relation is obfuscating. Admittedly, the relation Crisp defines purports to have the same logical properties as a B-theoretic temporal relation, but so does greater than and no one would claim that greater than is a temporal relation. Crisp claims that his ersatz B-relation is “the relation we learned at mother’s knee when we learned that last Tuesday is earlier than today” (2007: 102). I disagree. What we learned when we learned what “earlier than” represents is grounded in our experience of one event occurring before another, and not by contemplating a relation between abstract objects. In fact, his analysis of presentism even makes use of a B-theoretic notion of the earlier than relation for he says, “When the presentist says that everything is present, what she really means is that there is no temporal distance between any two things in our widest domain of quantification. (What is temporal distance? The general idea should be clear enough. Suppose eternalism is true. Then the temporal distance between Lincoln’s assassination and now is a little over 139 years” (Crisp [2007]: 102–03; some italics added). Since the notion of temporal distance incorporates a B-theoretic earlier than relation, I still do not see how he has made sense of a temporal relation between atemporal objects or “times.”

Regarding “times” he says:

A time, let us say, is any proposition that satisfies the following schema:

x is a time = df. For some class C of propositions such that C is maximal and consistent, x is the proposition that ∀y (yC ⊃ y is true), where (i) a class C of propositions is maximal iff, for every proposition p, either p or its denial is a member of C, (ii) a class C of propositions is consistent iff, possibly, every member of C is true, and (iii) all verbs in the schema are tenseless. (2005a)

We shall see that it is the last condition—that all verbs, and hence, all propositions in the schema, are tenseless—that gets him into trouble. But is Crisp committed to maintaining that all propositions are tenseless? Clearly, if Crisp is not intending to say that all the propositions in the scheme are tenseless then his formulation here is very misleading. Nevertheless, a critic might object that I am overlooking the possibility that there can be sentences that contain only tenseless verbs, but that nevertheless express tensed propositions. And indeed, it should be noted that a sentence containing only tenseless verbs does not entail that the sentence expresses a tenseless proposition, since a sentence can combine a tenseless verb with tensed adjectives. This is so, for example, in the sentence "The end of World War II is (tenselessly) past", which expresses a tensed proposition, the tenseless verb notwithstanding.

In support of my interpretation note that Crisp proposes to reduce the property of being present to the "non-temporal property being true." Given that reduction, and the related accounts of the properties of being past and being future, it seems that there aren't any tensed adverbs that are such that, when combined with tenseless verbs, will result in a sentence that expresses a tensed proposition. For that reason, henceforth I shall assume that the textual evidence supports my interpretation of Crisp as claiming that all propositions are tenseless and as defining abstract times in terms of propositions related to sets of tenseless propositions.

Although all propositions in the schema are tenseless, Crisp is emphatic that tenseless propositions can change their truth value, and indeed must change their truth value if there is to be time and temporal becoming (2005a). However, if there is to be time and becoming, that is, if propositions must change their truth value, then tenseless propositions must also be tensed, which is contradictory. Of that too, more later (see section Part II A Critique of Crisp's Presentism).

Another feature of Crisp’s presentism is his analysis of temporal becoming. According to it,

to say that presentness moves along the ersatz B-series is to say something like this:

(*) The B-series is such that (i) one and only one of its members tαhas the property being present, (ii) for every time t1 in the series such that t1is earlier than tα, WAS[t1hasbeing present], and (iii) for every time t2 in the series such that t2is later than tα, WILL[t2hasbeing present]. (2005a)

With this background we can turn to a critique of Crisp’s version of presentism.

Part II A Critique of Crisp’s Presentism

The first problem with Crisp’s version of presentism, one that permeates his entire system, is that his view of “times” as a certain class of tenseless propositions is inconsistent with his tensed account of time.4 To see why suppose we begin by asking what, then, is a tenseless proposition and how does it differ from a tensed proposition? The answers are crucial since Crisp believes that the A-properties of being past, being present and being future, and thus the tensed propositions that they enter into are not fundamental, but can be reduced to or eliminated in terms of ersatz B-relations and the tenseless propositions that they enter into.

Crisp draws the distinction between tenseless and tensed propositions in the following passage that I shall quote at length:

We begin by taking tenselessness seriously. Call the properties being past, being present and being future “A-properties.” A tenseless proposition, let us say, is a proposition p such that it is possible to grasp or conceive of p without thereby grasping or conceiving of an A-property. …

I’ll assume there is such a thing as the tenseless ‘is’—henceforth, ‘IS.’ A few words about the meaning of ‘IS.’ When we say that x IS F, we indicate thereby that the two-term instantiation relation connects x and the property F, but say nothing about whether x’s possession of F is past, present or future. This as opposed to the present-tensed ‘is’: when we say that x is (present-tensed) F, we indicate thereby that x’s possession of F is present—that x has F at the present time. …

One “takes tenselessness seriously,” let us say, iff one thinks there are tenseless propositions. In what follows, I shall suppose there are. I shall suppose henceforth that expressions like ‘[∅x] ‘[∀x ∅x]’ (where does not express an A-property) denote such propositions. (2007: 98–99; some emphasis added.)

Thus, on Crisp’s account of tenselessness, a tenseless proposition p does not indicate when its subject possesses a property, and it is possible to grasp or conceive of p without thereby grasping or conceiving of an A-property. I shall argue that this conception of tenselessness renders Crisp’s version of presentism inconsistent.

Recall that in Crisp’s analysis of temporal becoming, the first clause reads: “The B-series is such that (i) one and only one of its members tαhas the property being present” (2005a). Since, (i) contains the proposition [tαhas the property being present] it cannot be grasped without conceiving of an A-property and thus is clearly a tensed and not a tenseless proposition. However, the proposition [tαhas the property being present] is also a tenseless and not a tensed proposition not only because he says that “underlined verbs and copulae are tenseless” (2005a). (since that does not rule out the possibility that tenseless verbs are being combined with tensed predicates to generate sentences that express tensed propositions), but also because Crisp reduces the property of being present to the non-temporal property of being true, thereby ruling out the possibility just mentioned. Thus, Crisp’s account of temporal becoming is contradictory since it implies that one and the same proposition is both tensed and tenseless.

The confusion between tensed and tenseless propositions and his shifting from one to the other masks many other flaws in Crisp’s version of presentism. Consider, for example, his analysis of the tenses and reduction of tensed properties. Crisp claims that one and only one time tα in the ersatz B-series is “the present time” (or has the property being present), by reference to which, together with ersatz B-relations, WAS and WILL be are defined or analyzed.

Thus, for example, WAS [t1hasbeing present] would be analyzed as:
  • (A) [t1hasbeing present] is true at a time earlier than whatever time happens to be the present time.

Since [t1hasbeing present] can only be true at t1 and t1 is earlier than whatever time happens to be the present time, it follows that
  • (A’) [t1hasbeing present] isearlier than whatever happens to be the present time

and since times, including presumably the present time, are relations between propositions, we could express (A’) as:
  • (A’’) [t1hasbeing present] isearlier than [tα and tα is the present time].

Crisp’s presentist reductive analysis of the tensed properties of being past and being future is logically equivalent to his analysis of the tense operators WAS and WILL, although this may not initially be apparent. Crisp claims that

We can now see how the presentist can do without primitive pastness, presentness and futurity. She need simply take the foregoing earlier than relation as primitive, and say that a time is past iff df. it is earlier than the present time, that a time is future iff df. it is later than the present time, and that the present time is df. whatever time happens to be true. (2007: 104–105)5

Suppose t1 is past, then t1 is earlier than the present time. Given Crisp’s account of temporal becoming, if t1 is earlier than the present time, WAS [t1has being present], but then the analysis of pastness or, t1 is past, can be expressed as:
  • (A) [t1hasbeing present] is true at a time (t1) earlier than whatever time happens to be the present time (tα)

or as
  • (A’’) [t1hasbeing present] isearlier than [tα and tα is the present time]

that is the same as the analysis of the tense operator WAS.

However, the notion of “the present time” in Crisp’s reductive analysis of WAS (and being past) and WILL (and being future), is not the same as his notion of “the present time” (or being present) in his analysis of temporal becoming. In his reductive analysis of the tenses and the tensed properties of being past and being future, “the present time” is a tensed notion—the time that is NOW true or NOW exists with a capital E—but in his analysis of temporal becoming “the present time” is a tenseless notion—the time that has (tenselessly) the property of being present. Crisp needs “the present time” to be given both a tensed and a tenseless interpretation, but it can be given neither.

In order to ground the first clause (i) in his account of temporal becoming “the present time” must be tensed so that one and only one time is present. However, if “the present time” is tensed, then Crisp’s reductive analysis of the tenses and consequently his eliminative analyses of A-properties (in particular being present) are circular. On the other hand, if “the present time” or the property being present is tenselessly instantiated by times, then his account of temporal becoming fails since (i) is false, and Crisp could only think otherwise by confusing [tαhas the property of being present] with its present tense counterpart [tα is NOW present].

To see what is involved in these claims and to defend them, note that Crisp analyzes the tensed proposition WAS [t1 has the property of being present] in terms of the ersatz B-relation “earlier than” to “the (tensed) present time,” i.e., NOW, which is circular since he claims to eliminate the property of presentness.6 To avoid the circularity he shifts, in his analysis of temporal becoming, to interpreting “the present time” as tα, the time that has (tenselessly) the property being present. The proper analysis of WAS [t1has the property being present] is, therefore, the tenseless proposition:
  • (B) {[t1has the property of being present] is earlier than [tαhas the property of being present]}7

In that case, however, there is no basis for claiming that tα is the one and only time that has the property being present, or that tα is the present (tensed) time. Since (B) (and the propositions contained in it) is tenseless, it says nothing about when tα or t1has the property being present, and nothing about whether tα or t1 is past, present or future. The only difference between t1 and tα is that t1is present earlier than tαis present, but that ersatz B-fact does not in itself indicate whether t1 or tαdid, does or will have the property of being present. Thus, it is false that one and only one time has the property of being present.

I am not arguing that since every time is present sometime, all times are present at every time; a thesis which is palpably absurd. Rather I am claiming that since on Crisp’s view all times are present at the time they are regardless of what time it is, there is no basis or ground in the ersatz B-series for picking out one and only one time that has the property of being present to the exclusion of all earlier and later times that are (tenselessly) also present at their own respective time, and there is no time that NOW has the property of being present since in Crisp’s ontology nothing is irreducibly NOW. Thus, his account of temporal becoming fails or falls into inconsistency and circularity.

Crisp could reply to the charges of inconsistency and circularity by claiming that since the property of being present “is just the property being true” (2005a), the proposition [tαhas the property being present] is, when properly analyzed, the tenseless proposition [tαhas the property being true]. Since the proposition [tαhas the property being true] can be grasped without conceiving of an A-property it is not both tensed and tenseless, and since it does not contain the tensed property being present it sought to analyze, it would appear that both inconsistency and circularity are avoided.

I shall argue, however, that here as elsewhere appearances are deceiving since eliminating being present in favor being true does not ultimately avoid either difficulty of contradiction or vicious circularity, but merely shifts them from A-predication to truth predication. That is, [tαhas the property of being true] just like [tαhas the property of being present] must be both a tensed and a tenseless proposition and cannot be either. To eliminate the A-property of being present from his ontology, Crisp defines “the present time” as the time that “happens to be true.” To avoid circularity, “happens to be true” must be a tenseless predication. However, it cannot be predicated tenselessly, for if tαhas (tenselessly) the property of being true, then given his analysis of the tenses, WAS [t1has the property being true] is reducible to:
  • (C) {[t1has the property of being true] is earlier than [tαhas the property of being true]}

and there is nothing in that complex proposition that accounts for tα rather than t1 being the one and only member of the ersatz B-series that has the property of being true. Consequently, his analysis of WAS (and WILL) fails, for if there is no uniquely present or true time, then what is earlier (past) or later (future) than the one and only true time has no ontological ground.
Furthermore, we now can see the problem involved in Crisp’s claim that tenseless propositions can change their truth value. Crisp says:

I think the presentist should hold that there is one and only one ersatz B-series … but that it does not include all the abstract times among its members. It counts among its members only some of the abstract times—those that did, do or will represent the world. (2007: 104).

In order for Crisp to defend “an A-theoretic or dynamic view of time—the view that, necessarily, what exists and what is true is constantly changing” (2007: 104), the ersatz B-series must be tensed, that is, have members that did, do or will represent the world, and thus have members (propositions) that were true, are now true and will be true. However, if the true time is tenselessly true, then it cannot also be NOW true and thus, there is no basis for distinguishing those members of the time series that did, do or will represent the world, and that is a disaster. If propositions cannot be true or false in a tensed sense they cannot change their truth value, which is essential to Crisp’s account of time and change.

Crisp is clear that in order for tenseless propositions to change their truth value they must possess tensed truth and change with respect to their tensed truth value. Consider the following passage:

Note well that to say of a predication that it's tenseless is not to say that it is always true if true or always false if false. Presentists who think there are such things as tenseless predications will think of them as changing their truth values over time: e.g., [Bush is president] was false, is now true, and will be false again.” (Crisp 2005a)

In that case, however, truth cannot also be tenseless. If tα is (tenselessly) true, then it cannot also be the case that tα is now true, but it must if what exists and what is true is constantly changing, that is, if tenseless propositions are to change their truth value. For that reason, Crisp’s presentism is inconsistent.

We just saw that the problem of explicating “WAS” and “WILL” requires that one and only one member of the ersatz B-series be the (tensed) present time. We now see that the problems of explicating the difference between those times that “did” “do” or “will” represent the world and of grounding change also require that one and only one member of the B-series be the (tensed) true time, but given that A-propositions are reducible to tenselessly true propositions, there is no such time. Thus to ground an A-theoretic distinction between past, present and future times in an ersatz B-series, and an A-theoretic or dynamic view of time [tα is true] must be tensed and cannot be tenseless, but in order to eliminate A-properties in terms of B-relations [tα is true] cannot be tensed, but must be tenseless and that is a contradictory. 8

Crisp’s two meanings of “present” encapsulate these contradictions. On the one hand, he maintains that presentism is the view that everything is present1 meaning that “there is no temporal distance between any two things in our widest domain of quantification” (2007: 102). How, then, can abstract times be related by the earlier/later than relation if nothing is at a temporal distance from anything else? Crisp’s answer is as follows:

[I]f she believes in abstract times, she’ll think there’s another sort of presentness—presentness2 let us say—such that only one among the infinity of times—the present time—has this property. What is this presentness2? I’ll suggest below that a time is present2 iff it is the true time, the time which has the property being true (abstract times, recall, are just propositions). (2007: 103)

Thus, although all times are present1, only one time is present2. Crisp’s distinction between the present1 time and the present2 time cannot be sustained because it equivocates on the tenseless and tensed inherence of being present and being true. To see how and why suppose we ask: How can he account for one and only one among the infinity of times having the property of being present or reductively, being true (and thus representing the world at an instant)? Surely, if truth is tenseless so that [t1is true] is a tenseless proposition, then there is no ground for one and only one time being the true time or the present2 time and thus for everything being present1.

The argument is familiar. If WAS [t1is true] then [t1is true] is true at a time1 earlier than whatever time, tα, happens to be the present or true time. It follows, as I have argued, that [t1is true] is earlier than [tαis true]. From that one cannot infer that t1 rather than tα is the present2 (or the true) time. Every time is (tenselessly) true at the time it is so that t1is true at t1is earlier than tαis true at tα. This is not to say that all times are the timelessly, simultaneously or eternally true (or present2) time, but that no time is the one true time or the present2 time.

Thus, truth must be tensed. However, assuming that truth is tensed the analysis of WAS [t1 is (present tense) true] would be:
  • (D) {[t1 is (present tense) true] is (present tense) earlier than [tα is (present tense) true]}

In that case, however, there is still no ground for t1 rather than tα being the one and only member of the B-series that is (present tense) true. Moreover, if what is now true represents an instantaneous state of the world, (D) would represent an instantaneous state of the world and that brings up one of the basic problems with presentism, namely, its inadequate account of temporal relations. How can an instantaneous state of the world be what is represented by a proposition that contains an earlier than relation between present tense propositions? Alternatively, if [t1 is (present tense) earlier than tα] represents a non-instantaneous state of the world, then it would appear that it represents that t1 is at a temporal distance from tα and thus it is not the case that everything is present1, that is, presentism is false.

To summarize my argument, Crisp wants truth (being present) to attach to each individual set of those members of the ersatz B-series that did, do and will represent the world history, that is, he wants present tense truth (or presentness) to attach to times. His account can only do so by moving in an equivocal way between reliance on the tenseless "has" as in "tαhas the property being present" that is tenselessly true, and a tensed sense of "has", because then one tα is now present (or true), and the other ti’s in other propositional sets are (present-tense) false. Crisp has to secure "moving truth” from individual set to individual set in the ersatz B-series, but he does this by incoherently arguing that each is both tensed and tenselessly true—and also false since each set "loses" tensed truth. It is such a tenseless temporal order that Crisp depends on to even try to make sense of tensed truth within the B-series, but how a tenseless ersatz B-series can make sense of tensed truth is an unanswerable question for Crisp’s presentist ontology that leads directly to a version of McTaggart’s paradox as I shall show in the next section.

Part III McTaggart and Presentism

In the previous section I have demonstrated that Crisp is committed to maintaining both that there is a single ersatz B-series whose terms each have (tenselessly, at the time they do regardless of what time it is), the property of being present or reductively, being true, and in addition has one and only one member that is (now) present or is (now) true. Those two commitments present Crisp with the following dilemma. What is NOW true changes its existential status (and thus what is tenselessly true changes its truth value), or it does not. If what is NOW true does not change its existential status and what is NOW true represents an instantaneous state of the world then Crisp’s version of presentism eliminates time and change since no proposition changes its truth value and there is no temporal becoming. The following diagram illustrates what I mean:
https://static-content.springer.com/image/art%3A10.1007%2Fs11406-009-9222-4/MediaObjects/11406_2009_9222_Figa_HTML.gif

Diagram I

On the other hand if, as Crisp does in fact maintain “necessarily, what exists and what is true are constantly changing” (2007: 100), his version of presentism is contradictory. For if what is NOW true changes as presentness moves along a single ersatz B-series of tenselessly true times then each term is (tenselessly) and NOW true, and the direction of time and change is lost (as in Diagram II).
https://static-content.springer.com/image/art%3A10.1007%2Fs11406-009-9222-4/MediaObjects/11406_2009_9222_Figb_HTML.gif

Diagram II

Moreover, given that the position of the NOW determines whether a time is past, present or future, if every term in the B-series is NOW true, then every term of the ersatz B-series is NOW past, present and future. More specifically, since t1 is earlier than t2, (the present or true time), t1 is past. Similarly, since t1 is also later than (the present or true time) t0, t1 is future, and since t1 is also simultaneous with (the present or true time) t1, t1 is present. Accordingly, since each term is NOW true, it follows that t1 and every other time is NOW past, present, and future, which is a contradiction.

To avoid the contradiction, the members of the first order ersatz B-series must be NOW true at different times, and thus a second order ersatz time series is required (as in Diagram III):
https://static-content.springer.com/image/art%3A10.1007%2Fs11406-009-9222-4/MediaObjects/11406_2009_9222_Figc_HTML.gif

Diagram III

It should be obvious, however, that this second order time series will face precisely the same problems as the first order series, for we must ask: What is the relation between the terms in this series of ersatz B-series? If it is a non-temporal relation then, since every term in the first order time series is (tenselessly) and NOW true, the contradiction that the second order ersatz series was introduced to avoid is unresolved. If the terms, B1a, B1b, B1c … B1i, in the second order series are related by an ersatz temporal relation, then we are off on a vicious infinite regress. For if one and only one member of the second order B-series is NOW true then there is no change (as in Diagram IV):
https://static-content.springer.com/image/art%3A10.1007%2Fs11406-009-9222-4/MediaObjects/11406_2009_9222_Figd_HTML.gif

Diagram IV

If every term in the second order B-series is NOW true we have a contradiction that cannot be avoided by appealing to times in a third order B-series at which times in the second order series are NOW true.

A critic may argue that the application of McTaggart's argument to Crisp's view crucially requires that if truth is tenseless then every abstract time is true, which is inferred from the claim that tenseless propositions do not change truth value.9 In reply I would claim that I do not argue in this fashion which makes it looks as if I am begging the question by assuming rather than arguing that tenseless propositions cannot change their truth value. The order of my argument is just the opposite. To spell it out more clearly: If truth is tenseless and there are no tensed propositions (theses Crisp is clearly committed to given his reductive analysis of the tenses and A-properties) it follows that no propositions are NOW true. Rather, every abstract time (“recall, are just propositions” (2007:103)) in the ersatz B-series is true at some time earlier or later than every other time (or set of propositions) in the ersatz B-series (assuming, of course, that there is no first or last moment of time and there is no recurrence). Thus, there is no basis for maintaining that one and only one time is the present time or the true time; the time that NOW represents the world, or the time is NOW true. Since the only way what exists and what is true could be constantly changing would be for propositions to constantly change their tensed truth (from will be true to now true to was true), if tenseless propositions could change their truth value then they would have to be both tenselessly and tensedly true. Since that is a contradiction, tenseless propositions cannot change their truth value, but Crisp’s theory of time requires that tenseless propositions can change their truth value. It follows, therefore, that his theory of time is inadequate. This is the structure of McTaggart’s paradox: there is no change or if there is change, then there is no time, since time and change are incompatible.

Crisp’s reductive analysis of tense has been shown to be a failure. His appeal to times as abstract objects (sets of maximally consistent propositions) forming an ersatz B-series does not provide an adequate account of temporal becoming since there is no ground for one and only one member of the series having the A-property being present (because there is no such property) or the non-temporal property being true (since every time has that property at the time it is). The attempt to combine tenseless truth with tensed truth inevitably results in contradictions and an inadequate theory of time. I conclude, contrary to what Crisp asserts that McTaggart’s paradox (or a suitably applicable version of it) does indeed make serious trouble for the version of presentism he puts forth.

Footnotes
1

For a complete discussion of my interpretation of McTaggart’s positive and negative accounts of time see (Oaklander 2004d).

 
2

See, Michael Tooley (1997: 323–329) and (2010) for a defense of this way out of McTaggart’s Paradox. Note that the presentist cannot take this way out for two closely connected reasons: first, for presentists the basic sentences and the propositions they express do not contain dates and second, there are no past or future dates or times.

 
3

In the papers I am discussing Crisp refers to singular terms, but he does not tell us if he is thinking only of singular terms that refer to presently existing objects, or whether, for example, "Abraham Lincoln" is also a singular term for Crisp.  In other words, is Crisp employing non-referring singular terms?  If so, how are such terms to be understood?  As disguised definite descriptions? Crisp addresses some of these concerns in (2005b).

 
4

This point depends crucially upon the claim that Crisp is analyzing abstract times in terms of propositions related to sets of tenseless propositions. I have argued, however, that interpretation of Crisp is the right one.

 
5

A referee pointed out the following difficulty with Crisp’s analysis here. Crisp analyzes the proposition that a time, X, is present in terms of the proposition that X is true.  But given that the universe could repeat itself, this doesn't look like a satisfactory analysis, since in a universe that repeated itself, more than one time could, so to speak, be present at a given time, since on Crisp's definition of times, there could be two or more times that consisted of precisely the same set of propositions.

 
6

In (2005b) Crisp does claims that the present time is the time that is NOW (or has NOWNESS), but he abandons that view in (2005a and 2007) since he wants to completely eliminate A-properties from his ontology in order to give what he believes is an adequate account of temporal relations and to answer the grounding objection.

 
7

(B) follows from Crisp’s analysis of WAS(x) given that tαis the present time is interpreted as tαhas the property of being present (which it is in his account of temporal becoming).

 
8
Crisp objects to my use of McTaggart to refute presentism because of unwarranted assumptions he says I make. For example,

Pick two times t1 and t2 such that t2is later than t1 and suppose that t1hasbeing present and that WILL[t2hasbeing present]. Oaklander’s argument seems to presuppose that a proposition like [t1hasbeing present and WILL[t2hasbeing present]] is true only if the relevant t1 and t2bothhavebeing present. But why think that? (2005a)

The answer as follows: Consider the proposition [t1hasbeing present and WILL[t2hasbeing present]] Since in both sentences “has” is underlined, it follows that each conjunct in the proposition expressed is tenseless. Thus, if we suppose that t2 is later than t1, then the proper analysis of the proposition in question is that [t2hasbeing presentis true at a time later than the present time (t1)] is equivalent to [t2hasbeing present later than t1hasbeing present]. Admittedly, it does not follow that t2 and t1 are both present simultaneously or timelessly, but neither does it follow that either time is NOW present or always present, but only that t2 and t1havebeing present one later than, the other earlier than the other. This conclusion wrecks havoc on Crisp’s presentism for then there is no ground for distinguishing the present time from the past or future time and thus no way of distinguishing the times that did or will represent the world from those times that NOW represent the world. Without such an account however, there is nothing about his view that renders it an A-theory of time and since it is clearly not a B-theory of time (since he rejects B-relations between temporal items), there is nothing about his account that renders it an ontology of time.

 
9

This objection was raised by one of the referees.

 

Acknowledgements

I wish to thank V. Alan White and Joshua Mozersky for the very helpful discussions I had with them, and two anonymous referees for their constructive comments.

Copyright information

© Springer Science+Business Media B.V. 2010