The lndexical Fallacy in McTaggart' s Proof of the Unreality of Time

E. J. LOWE

Events, as McTaggart pointed out, may not only be described as being earlier and la ter than one another (andas such constituting the 'B series') but also as being past, present and future (and as such constituting the 'A series'). 1 B-series sentences do not alter in truth value with time: if 'e1 is earlier than e2' is ever true, it is always true, whereas 'e happened yesterday', iftrue toda y, will not be true tomorrow.2 McTaggart holds that the reality of time demands the reality of the A series (the B series alone will not suffice) because, he considers, change is essential to time but can only be explained by reference to the A series. Suppose it is held (as Russell held) that my poker's changing from being hot to being cold consists merely in the fact that it is hot at one time and cold at a later time. According to McTaggart, this is inadequate, because although we may similarly say that my poker is hot at one end but cold at the other, this fact about it, which is formally analogous to the previous one, does not imply change. 3 There must in addition be something which distinguishes time from space in this regard, and only participation in the A series seems to suffice. So, if the A series is unreal, so are the B series and time in general. (Hugh Mellor has argued recently that a further relevant difference is that objects like pokers are 'wholly' present at any time at which they exist, but not necessarily at any place at which they exist, because such objects ha ve spatial but not temporal parts. 4 Although the merits of this suggestion are not something 1 mean to discuss here, my feeling is that while the doctrine about parts is correct, it

1 See J. M. E. McTaggart, The Nature of Existence, Cambridge, Cambridge University Press, 1927, Vol. 11, Chap. 33· The advent of relativity theory has taught us to allow that there may be more than one B series, corresponding to different 'frames of reference' in motion with respect to one another. However, as others have pointed out, the important thing is that for each B series there is a unique A series corresponding to it, and vice versa; and for our purposes we can assume that we are restricted to just one frame of reference. (McTaggart himself allowed the possibility of such plurality: see ibid., § 323.)

2 1 should explain that expressions definable in terms of'past', 'present', and 'future' are themselves to be accounted A-series expressions. Thus 'yesterday' qualifies, because it is definable as 'a day earlier than the present da y'. This example should make it clear, incidentally, that the relations 'earlier' and 'la ter' (and relations definable in terms of them, such as 'a da y earlier') do not exclusive! y appear in B­series sentences; rather, the special feature of the latter sentences is the absence from them of any A­series expressions. Events are ordered in the B series by the earlier/later relation without reference (however indirect) to the present moment.

3 This objection, though not explicitly expressed by McTaggart, may be extracted from what he says in op. cit., § 3 r6.

4 See D. H. Mellor, Real Time, Cambridge, Cambridge University Press, 198r, p. rrr.

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does not constitute a relevant difference between time and space, i.e. one which explains why time, but not space, is the dimension of change.)

The structure of McTaggart's argument against the reality of time is, then, as follows:

(1) Time essentially involves change.

(2) Change can only be explained in terms of A-series expressions.

(3) A-series expressions involve contradiction and so cannot describe reality.

(4) Therefore, time is unreal.

Of course, not all philosophers accept ( 1 ), since sorne claim to be able to make sense of the notion of time in an unchanging world; but 1 shall not be questioning (1) here. Most philosophers who oppose McTaggart's conclu­sion would, 1 think, challenge either (z) or (3), or both-1 have yet, of course, to examine McTaggart's reasons for holding (3), which 1 shall find wanting. An interesting position is held by Mellor, who accepts (3) but not (z). Mellor holds that McTaggart succeeds in demonstrating the unreality

. of tense (there are, for Mellor, no tensed facts), but not that of time, since he holds (as 1 have already remarked) that change can be explained without reference to the A series. Another interesting feature of Mellor's position is that, while rejecting the notion of tensed facts, he insists that the possession of tensed beliefs is indispensable for rational action, since without them agents cannot know when to act appropriately. 5 While admiring the ingenuity ofMellor's position, 1 have no sympathy for it, because instead of accepting (3) but not (2), 1 accept (2)-or at least find it very plausible-but not (3).

What exactly are McTaggart's reasons for holding (3), that the A series involves contradiction? McTaggart starts with the observation that past, present, andfuture are incompatible predicates. An event which is past, for example, is ipso Jacto neither present nor future. But, he says, change consists precisely in future events becoming present and present ones becoming past. 6 So, every event must, after all, be future and present and past, and so have incompatible predicates ascribable to it. (E ven ifthere is a first event or a last event because time has a beginning or an end, such an event must still ha ve two of the incompatible predica tes ascribable to it; so let us ignore this complication.) Now, the obvious rejoinder to this, McTaggart realises, is to point out that we do not have to say that any event is simultaneously past, present, and future, which would indeed be contra­dictory. We can say instead that an event which is present will be past and was future. However, he argues, the same threat of contradiction then arises at this higher level of second-order tenses. For instan ce, to say that an event

5 Ibid., pp. 78 ff. 6 SeeMcTaggart,op.cit.,§3II.

64 E.]. Lowe

wasfuture is to say that it isfuture in the past (where this last 'is' may be read as tenseless). But this same event is also past in the past, which is again inconsistent. To this the response might once more be made that such an event is not simultaneously past in the past and future in the past: rather, what was future in the past will be past in the past. However, repeated recourse to this manreuvre of resorting to higher-order tenses in order to evade the threatened contradiction can only serve to generate a (vicious) infinite regress, and so the manreuvre can by no means resolve the original difficulty arising with the first-order tenses.

Michael Dummett has perhaps expressed this crucial part of McTag­gart's argument in the most succinct way to date. 7 He points out that there are nine second-order tenses, which may be represented as follows:

{ past } present future

in the { past } present future

However, three of these, namely

{ past } present future

in the present

are, he says, just equivalent to the three first-order tenses, so that if there is a threat of contradiction on the first level it cannot be evaded merely by moving up a level.

Let us look at McTaggart's reasoning a little more closely. We attempt to avoid the initial threat of contradiction, it is said, by stating that an event that is present (i.e. is present in the present) will be past (i.e. is past in the future) and was future (i.e. is future in the past). (Of course, the parenthetical uses of 'is' in the previous sentence should again be seen as tenseless.) But is this move to second-order tenses the proper or only response to make, or does it just invite trouble? Does the notion ofhigher­order tenses, strictly speaking, even make sense at all? (The presence of pluperfect and future perfect constructions in European languages should not be seen as decisive on this point; nor should we be too deferential to the deliverances of so-called 'tense logic'.) Consider this: is it true to say of a future event that it will be present (is 'present in the future')? One might on first reflection be inclined to say yes, but in fact the answer is surely no. What may be correct is something significantly (though not unmistakably) different, namely, that if e is a future event, then there will be a time when the sentence 'e is present' is true (expresses a true statement). Similarly, rather than saying of a past event e that it was present ( or is 'present in the past'), we should at most say that the sentence 'e is present', though now

7 See M. A. E. Dummett, 'A Defence ofMcTaggart's Proof ofthe Unreality ofTime', reprinted in his Truth and Other Enigmas, London, Duckworth, 1978.

The Indexical Fallacy in McTaggart's Proofofthe Unreality ofTime 65

false, was true. 1 shall suggest that by emphasizing distinctions like these we may avoid the entanglements in which McTaggart tempts us to become ensnared.

At this point it is helpful to accentuate a feature of A-series expressions which McTaggart himself stressed less than do many modern philosophers of time, namely, their indexicality (or 'token-reflexiveness'). 'e is present' means, of course, 'e is happening now', and 'now' may usefully be compared with other indexical expressions like 'here' and '1'. The truth conditions of utterances containing indexicals are context-dependent. Thus, the utterance of a token ofthe sentence 'e is happening now' is true if and only if the token is uttered ata time t such that e is happening at t (where the last three occurrences of'is' are to be read tenselessly, of course). Similarly, the utterance of a token of the sen ten ce 'e is happening here' is true if and only if the token is uttered at a place s such that e is happening at s. This is not to imply something undoubtedly false, that 'now' means 'the time of this utterance', or that 'here' means 'the place ofthis utterance', or, again, that '1' means 'the utterer of this utterance'.

Now, does McTaggart's argument simply turn on a blunder in the logic of indexicals? Dummett, who endorses McTaggart's argument, thinks not. He at least seems to think that it is legitimate to say concerning a future event e, which is not happening now, that e will be happening now, or is happening now in the future, since he seems to think it equally legitima te to say that an event e, which is not happening here, but is happening over there, is happening here over there. Dummett writes: 8

Every place can be called both 'here' and 'there', both 'near' and 'far', and every person can be called both '1' and 'you': yet 'here' and 'there', 'near' and 'far', '1' and 'you' are incompatible. lt would be no use for an objector to say that London is nearby far away, but far away nearby, or that it is 'here' there but 'there' here, since it can also be called 'nearby nearby' and "'here" here', and so on. Similarly, it would be no use an objector saying 'You are "you" tome, but "1" to you', because everyone can be called both '"you" to me' and '"1" tome'.

Note that Dummett is not contending that what the imagined objector says is illegitimate ( certainly no more illegitimate than statements made with first-order indexicals), only that it is 'no use' for the purposes of removing the initial incompatibility. (One complication in Dummett's treatment is his frequent placing of indexical expressions in quotation marks, sometimes nested ones, in a not altogether consistent manner. It is because 1 am not quite clear as to the purpose behind this practice that 1 am somewhat tentative in attributing to him the opinions stated just before the quoted passage.) Dummett's overall position vis-a-vis McTaggart appears to be a qualified endorsement of each of claims (1) to (4) above; and the reason why he sees no danger of an extension of McTaggart's argument to the unreality

8 Ibid., p. 353·

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of space or personality is simply that he accepts in their case nothing corresponding to daims (1) and (2), although he does accept something corresponding to claim (3). 1 speak only of a qualified endorsement because Dummett nevertheless questions what he takes to be McTaggart's implicit assumptions about the nature of 'reality'.

1 cannot, however, concur with Dummett's defence ofMcTaggart on this point. 1 think that McTaggart does commit a simple, if understandable, blunder in the logic of indexicality. lt is not legitimate to say that an event e, which is not happening here, but is happening over there, is happening here over there. All we can say really is that if e is happening there and not here, then an utterance over there ofthe sentence 'e is happening here' is true. lt does not follow that e is happening here over there-which is a blatant con­tradiction. Now, similarly, when 1 say that e will happen, 1 am not implying that e is happening now in the future, though 1 am implying that in the future it will be possible to make a true statement by saying 'e is happening now'.

But how, exactly, does this bear on McTaggart's argutüent? Well, cer­tainly, 'e is present', 'e is past', and 'e is future', said simultaneously, express contradictory statements. But what about the claim that what is future will become present and then past (so that every event is past, present and future)? This is simply false, or, more strictly, incoherent. What should be said is that if e is a future event, i.e. if e will occur, then it will be possible to express a true statement by means of the sentence 'e is present', or 'e is happening now'. lf it is asked whether we can now say something which expresses now what that sentence will then express, the answer is yes: for the statement that 1 now express by saying, for instance, 'e will happen tomorrow' is the same as will be expressed tomorrow by saying 'e is happening today'. By analogy, the statement that 1 express here by saying 'e is happening over there' is the same as is expressed over there by saying 'e is happening here': for speakers uttering these sentences in the respective locations will, clearly, be in agreement as to what is the case, and to this extent may be said to be expressing the same statement. (lf'statement' is not liked in this context, 'proposition' or any other suitable favoured term may be substituted; the terminology is unimportant so long as it enables us to refer to what is invariable as between two such speakers.)

Thus, 1 agree with McTaggart in so far as 1 accept claims (1) and (2), and disagree with him in rejecting claims (3) and (4): but, furthermore, 1 do not accept his elaboration of (2), that is, 1 do not accept that change is to be explained in terms of A-series expressions in the way he suggests, since 1 do not accept that change is to be explained in terms of future events becoming present and so on. Just what explanation 1 do want to adopt 1 shall discuss in a moment. But before doing so, 1 want to mention another rebuttal of McTaggart's argument that has been advanced recently, by Richard Sorabji.9

9 See R. Sorabji, Time, Creation and the Continuum, London, Duckworth, 1983, p. 68.

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Sorabji's response to the initial threat of contradiction posed by McTaggart is to say that we can distinguish the times at which an event is past, present, and future by reference to dates, rather than, as McTaggart invites us, by resorting to (second-order) tenses. Rather than saying, of a present event e, that it is future in the past, past in the future, and present in the present, we may say, for instance, that it is present in 1986, future in 1985, and past in 1987, and thus not past, present, and future at any one date. This response presupposes that dates can be identified independently of A-series determinations. 1 very much doubt whether this presupposition is warranted. However, Mellor (who is the more immediate object of Sorabji's attack) accepts it, for reasons which 1 shall not examine here, but which are connected with bis belief that the earlierflater relation between events can at least sometimes be observed-when we observe movement, for instance-quite independently of whether the events in question are characterized by us as past, present, or future. 10 Nonetheless, Sorabji's response is inadequate, even waiving this issue, given that 'e is present' means 'e is happening now': because it simply is not, timelessly, true to say that e is happening now in 1986, where e is a present event-said in 1987 this is simply false. Said in 1987, 'e is present or is happening now in 1986' would imply (justas it does now, in 1986) 'It is now 1986', and said in 1987 this last would, of course, be false. Moreover, for the same reasons, it never has been, is not, and never will be true that e is future in 198 5 or past in 1987, where e is a present event (one that is happening now, in 1986).

Sorabji is making essentially the same indexical mistake that 1 have attributed to McTaggart, Mellor, and Dummett. If e is a present event (i.e. is happening now, in 1986), we should not say that it is a future event in 1985, but, at most, that in 1985 it was possible to express a true statement by means of the sentence 'e will happen in 1986/next year'. The mistake consists in forgetting the uneliminably indexical nature of A-series expres­sions, at least while they are being used as opposed to being mentioned: philosophers writing on time are constrained to use these expressions in the same context-dependent way in which ordinary humanity is-they, too, cannot escape their own temporal perspective, however much they may be tempted to suppose that they can view things sub specie aeternitatis. They are thus tempted to think of events as being presentfor certain persons, pastfor others, and future for yet others. But when I use the expression 'present', or 'now', 1 can no more use it to refer to another temporal perspective than I can use '1' to refer to another person. This is because what I can use such indexical expressions to refer to is not just up to me, but is constrained by my circumstances; the context of use of such an expression helps to determine its reference, my wishes notwithstanding. (This is not, of course, to deny that I can now refer to persons and times to which other persons at other times would refer by the expressions '1' and 'now', only to insist that 1

10 See Mellor, op. cit., pp. 24 ff.

68 E.J.Lowe

must use appropriate indexical expressions todo so-the propriety being determined by my context of use.)

1 return now to the problem of the nature of change. How are we to explain or describe it, given that we cannot say with McTaggart that change consists in future events becoming present and present ones 'receding' into the past? How can we capture the notion of the 'flow' of time without falling into paradox or absurdity? One's first thought, in an attempt to avoid the indexical fallacy 1 ha ve identified in what McTaggart himself says, might be to propose that change simply consists in the existence of such facts as that what can now be truly stated by saying 'e will happen tomorrow' will tomorrow only be truly stated by saying something different, namely, 'e is happening toda y'. But nothing like this will do by itself, for such facts only reflect the indexical character of A-series expressions, and so fail to distinguish time from space as the dimension of change. Thus it is equally a fact that what can here be truly stated by saying 'e is happening here' can yonder only be truly stated by saying 'e is happening over there' or something equivalent: but this carries no implication of change.

However, one important difference between time and space in connection with the problem of change may be that, soto speak, we cannot choose when we are whereas we can-at least to sorne extent-choose where we are. Again, there is no temporal analogue to the sort of truth that can be stated by saying '1 have been here before'-'1 am now elsewhere' is just absurd. The relevan ce of these facts to the problem of change ( or, as 1 would prefer to call it, the problem ofjlux orflow, since my concern is with an intrinsic feature of time rather than with a feature of objects which they possess by virtue of inhabiting time) is that it is the ineluctability of time that chiefly moti vates the simile of time as a river which 'like an ever-rolling stream, bears all its sons away'. It is not just that time has a direction, for space too might ha ve been anisottopic. Rather, it is that each one of us has no choice but successively and unrepeatedly to adopt each of a fixed sequence oftemporal perspectives, whereas there is no similar constraint on our spatial perspectives. The same spatial perspective may be adopted at many different times by the same person, and the temporal order in which spatial perspectives can be adopted by the same person is a matter open to that person's choice.

To this it may be objected that while it is true that the same temporal perspective may not be adopted analogously at many different places by the same person, none the less the spatial order in which temporal perspectives are adopted by the same person is analogously a matter open to that person's choice-so that what has been said so far does not capture the alleged 'ineluctability' of time. This objection is sound and shows that more care is needed in formulating the relevantly different facts about spatial and temporal perspectives. The idea we must try to articulate is that a person's route through space is open to choice in a way that his route through time is

The Indexical Fallacy in McTaggart's Proof ofthe Unreality ofTime 69

not. But, of course, every route is at once a route through space and a route through time: routes are just ordered sequences of space-time positions, of the form ((s¡, t 1), (s 2, t 2), ••• , (sn, tn)). 11 The relevant difference between space and time, then, comes clown to this, that in all the possible space-time routes a person may take the order of temporal positions will be the same, while the order of spatial positions may vary (and, indeed, such routes may differ in respect of all oftheir spatial positions except, perhaps, the first-a fact that only partly reflects the isotropy and multidimensionality of space in contrast with the directedness and monodimensionality of time). Thus, if ((s¡, t 1), (s 2, t 2), ••• , (sn, tn)) is a possible route for a given person, any other possible route for that person must be ofthe form ((-, t 1), (-, t2), •• • ,

( -, tn)) but it need not be of the form ((s¡, - ), (s 2, - ), ••• , (sn, - )). (One crucial determinant of this result is the monodimensionality of time; but that is not all there is to the matter, since those routes are ruled out in which successive space-time positions contain the same temporal co-ordinate­time will not 'stand still'. Part of m y suggestion is, therefore, that it is beca use we can make sense of the notion of 'staying at the same place' but not of'staying at the same time' that we conceive of time rather than space as the dimension of change or flux.)

A problem arising from what has just been said is that if it succeeds in capturing what it is that makes time but not space the dimension of change or flux, it might appear todo so without invoking A-series expressions, inas­much as 'routes' ha ve been characterized in terms of space-time positions without any clear implication that these positions have to be referred to by means of A-series terminology. lfthis is so, McTaggart's claim (2) has been undermined, despite my earlier support ofit. M y answer is that 1 conceive of a 'route' not as something viewed sub specie aeternitatis but as a sequence of spatio-temporal perspectives. To think of what a route is therefore requires of us an effort of imaginative projection: we must conceive of what it would be like ('from the inside') to experience the world from perspectives other than here and now. In other words, we must understand what it would be to use the words 'now' and 'here' correctly at other times and places-times and places which 1 cannot now refer to other than by recourse to other indexical expressions. But, very arguably, such an understanding is essential to an understanding of the very meaning of words like 'now' and 'here' (how else could we communicate across time and space?), and so cannot be beyond us.

It might be thought that an alternative (or indeed additional) way of trying to capture the nature of change or flux consistent with the views that 1 ha ve already expressed would be to build on the idea that, while time is real, the future (and perhaps also the past) is not. The suggestion might be that time-and more particularly change or flux, which is, we have claimed, essential to it-is real precisely because the future is not real, or, more

11 For simplicity, 1 ignore the fact (if it is a fact) that space and time are continuous rather than discrete.

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precisely, beca use it is not yet real, but will be. This, of course, would allow us to distinguish time relevantly from space, inasmuch as we should not analogously wish to claim that only here is real, notyonder. While 1 ha ve some sympathy with this line of thought, 12 it opens too many questions to be pursued here ... examination of it must await another time and another place.

University ofDurham so Old Elvet Durham DH1 3HN

E. J. LOWE

12 lt may be that our inclination to say that the past is not 'real' in the way that we conceive the present to beis connected with our sense ofthe 'irretrievability' ofthe past. Assuming that time is not circular, we cannot later return to an earlier time, in the way that we can by traversing space return to the same place. If we did conceive time to be circular, we would, 1 think, ha ve less inclination to regard the past (and indeed the future) as 'unreal'. But ifthis is so, then it will not after all doto attempt to capture the notion that time is the dimension of change or flux by emphasising the 'reality' ofthe present-for there would still be flux if time were circular.