I present, first, an introduction, extended no more than is necessary, of the notion ‘both earlier and later’, then a more general introductory note, the proof and what follows, and some explanatory notes. The proof, I believe, is sound, though undoubtedly surprising at first sight, seeming to get three, namely, past, present and future, from two: earlier and later. In fact the three are got from a changing two – not the unchanging order of earlier-later, it turns out, but a present and future becoming a past and present – change, pace Sydney Shoemaker, being of the essence of time.Footnote ¹ But this itself calls for immediate comment. I have argued that time can be considered change itself in the abstract, pace both Shoemaker and Aristotle.Footnote ² Noting that Kant calls change, ‘under the concept of alteration (Veränderung), “a combination of contradictorily opposed predicates in one and the same object,”’ something that can happen only in temporal succession, and suggesting that the notion of change, for both Kant and Aristotle, can be simplified to that of ‘both A and not-A’, again a matter of temporal succession, I go on to say:

If that is so, then what is earlier and what is later are not in some changeless ‘block-universe’ state. The meaning of ‘earlier-later’ contains this notion of change, of existence and non-existence, which underlies the other, purely logical, function of dyadic correlatives, that ‘earlier’ and ‘later’ share with ‘left’ and ‘right’, ‘smaller’ and ‘greater’ and ‘precedes’ and ‘succeeds’. And this produces what at first sight seems a paradox, but on further inspection is not: earlier and later, as ontologically such, is change, impermanence – but as depicting the logical relation of order of change, ‘earlier-later’ expresses something unchanging, permanent. And this in turn indicates, as we shall see, that there need be less of a quarrel than there has been thought to be between impermanence and permanence in the matter of time.

The proof, I believe, will stand up logically, but even if so, given the definitions it depends on, it does require that we defend those definitions, especially that of ‘present time’, and the definitions in turn require that we pay attention to what is to be taken as general and what as individual – a very important consideration, it turns out.

1. General Introductory Note

In 1908 J. Ellis McTaggart, in a now-famous article in Mind, entitled ‘The Unreality of Time’, distinguished between what he called the A series and the B series, the first purporting to express the structure of time as past-present-future, the second the structure of time as earlier-later. Arising from that distinction, accepted by a good many subsequent philosophers as genuine, there is a continuing dispute as to which of these two expresses the true structure of time, since quite different conclusions seem to follow from taking the one way or the other, the A series seemingly lending itself to a view of time as constantly changing and the B series seemingly lending itself to a view of time as a non-changing, static structure. My purpose here is not to go into the details of the two views or of the dispute between them, all of that being very well known,Footnote ⁴ but to make the radical argument, in terms of establishing a logical equivalence, that the dispute between the two views is unnecessary, the distinction on which it is based false. It should be clear that this, if it is well reasoned, is a very serious claim and merits the more attention because the dispute between the two schools is taken so much for granted.Footnote ⁵ If the claim is not well reasoned, then showing so should at least be a service to both of the opposed parties and allow the dispute itself to continue towards some more enlightening resolution.

Since I aim to establish an equivalence I should start with one of the two views, defining the terms of the other in the terms of it. I could then, without more ado, simply start from the earlier-later side and produce definitions of ‘past’, ‘present’ and ‘future’ accordingly. If such definitions should be acceptable and the equivalence in terms of them established, that should be sufficient to show that one could at least not dismiss, from the B-series, earlier-later point of view the logical coherence of the other, past-present-future point of view, whatever one might think of a genuine A series; but indeed, if the equivalence should be real, then the one point of view, being logically substitutable for the other, must have exactly the same logical force.

But it is worth saying here why a start from the earlier-later side seems preferable, the more so as devotees of the past-present-future side may be tempted simply to dismiss as prejudiced any approach which does not from the very first seem to give equal consideration to their view. It is a question of the easier path to a logical argument, which of course the attempt to establish an equivalence has to be. Time as past, present, future seems, at least initially, if we want to start from it, to exist within somewhat uncertain logical boundaries. It seems to be not quite clear just what we are dealing with from the logical point of view. Many supporters of the earlier-later view, following an argument of McTaggart's, think that the past-present-future view is self-contradictory.Footnote ⁶

And, from that latter point of view itself, are we dealing with something triadic or not, either a triadic relation or some triadic arrangement of properties? That is, as we shall read ‘triadic’ here, if there is a present, must there be both a past and a future or only either a past or a future? It has been said before now that the so-called A series is not really a series,Footnote ⁷ so just what it is, logically, seems not quite clear. And also there seems to be at least some awkwardness about defining ‘earlier’ and ‘later’ in terms of ‘past’, ‘present’ and ‘future’. Richard Gale tries just such a definition but seems to himself forced to admit that his attempt may be circular. He concludes, though, that ‘our reductive analyses of B-relations [in terms of ‘earlier-later’] to A-determinations [in terms of ‘past-present-future’], even if circular, are of great importance, showing that there could not be a B-series unless there is also an A-series.’Footnote ⁸ That the earlier-later approach requires the reality of past-present-future time obviously I agree, if I think that earlier-later and past-present-future are logically equivalent, even if I quibble over calling the latter the A series. It does, at least prima facie, seem possible to define ‘earlier’ and ‘later’ in past-present-future terms, but those definitional accounts seem roundabout and complex. For example, ‘“e is earlier than f” is true if and only if: e is past and f is present or is present and f is future or e is past and f is future or e is more past than f or f is more future than e.’Footnote ⁹

As opposed to the seeming logical murkiness of time as past-present-future, time as earlier-later is a perfectly good example of a series with a generating relation that is connected, transitive, irreflexive and asymmetrical – the last property following from the other three.Footnote ¹⁰ It is perfectly regular from a logical point of view. Furthermore – an important point, which always seems to intrude if one insists that there really are two rival series for the structure of time – the terms ‘past’, ‘present’ and ‘future’ seem to signal some relation to the earlier-later series, since ‘past’ always signifies, no matter what else, something earlier, and ‘future’ something later,Footnote ¹¹ whereas ‘earlier’ and ‘later’ themselves tell us nothing about whether the thing referred to is past, present, or future.Footnote ¹² Again, the present, simply as a time, is always earlier or later than some other time. The concepts ‘earlier’ and ‘later’ are very persistent and hard to get rid of – the more colloquial ‘before’ and ‘after’ being simply, as temporal, substitutes. For all these reasons, therefore, the earlier-later structure is certainly the easier starting-point for logic, and we may look for definitions of ‘past’, ‘present’ and ‘future’ in terms of it.

2. Proof

Assume that ‘earlier-later’ expresses the order of time. Define ‘past time’ as ‘any time earlier than present time’, ‘future time’ as ‘any time later than present time’ and ‘present time’ as ‘any time t at time t’ – ‘at’ here having the usual significance of temporal position, as in ‘the alarm went off at seven o'clock’. This significance of temporal position, it should be pointed out, implicitly involves earlier and later, since it is in terms of these that temporal position is determined, so that the definition of ‘present time’ is then also in terms of ‘earlier’ and ‘later’.

The statement ‘Earlier and later if past, present and future’ follows immediately from the definitions since past time is always earlier and future time always later. But the statement ‘Earlier and later only if past, present and future’ – that is, ‘If earlier and later, then past, present and future’ – also follows. For consider any two times, t ₁ earlier than t ₂. Then, at t _1,t ₁ itself is, according to the definitions, present and t ₂ is future, and, at t ₂, t ₂ itself is present and t ₁ is past. So past, present and future, all three, are given by those two times, though not in the usually recognized arrangement of three, that is as past and future to, earlier and later than, the same identical present – which arrangement is however, according to the definitions, permitted where there are three times, the middle one being taken as present. But even with just two times all three classes occur, the change from the position at the earlier time to that at the later requiring the original present to become past and the original future to become present. At least this much change, it seems, is essential to time.Footnote ¹³ So the statement ‘Earlier and later only if past, present and future’ is true, as is the statement ‘Earlier and later if past, present and future’, and then of course the conjunction of the two, ‘Earlier and later if and only if past, present and future.’ The so-called A series and the B series are equivalent.

We shall have to say more about the order of past, present and future, but if the reasoning above, from the definitions, holds, then it follows that McTaggart and some who followed him were mistaken in taking the A series to be a distinct, differently behaving temporal series from the B. It also follows that McTaggart and some who followed him in making that false distinction were mistaken in thinking past-present-future time to be self-contradictory. It is exactly as self-consistent as earlier-later time.

3. Explanatory Notes

That ‘earlier-later’ expresses the order of time is, of course, recognized by McTaggart, at least to the extent of his bestowing the name ‘B series’ on the prima facie series of temporal positions in that order.Footnote ¹⁴ ‘Past time’ and ‘future time’ as defined in terms of that order are hardly problematic. It is certainly true that past time is always earlier than present time and future time always later – that, a significant fact, persists no matter how we want to regard past and future. It is going to be the definition of ‘present time’ that will be of particular interest – and most particularly whether the expression ‘time t at time t’ is meaningful or sufficient for such a definition.

It is of course true that, if ‘at’ has the usual significance of position, then time t is at time t, and ‘time t at time t’ is meaningful. But that very significance of position indicates that there are also other times – and at one of those other times any particular time t which is not at that one is past or future, that is, earlier or later than that one considered as present. If time t ₂ is later than time t ₁, then, though time t ₁ is at time t ₁ and time t ₂ at time t ₂, at time t ₂ time t ₁ is past, that is, earlier – at two o'clock one o'clock is past. Similarly Cambridge is at Cambridge and not north, south, east or west of there, but at London Cambridge is north. There is, however, a difference of ontological origin as regards the positional significance of ‘at’ in the temporal and in the spatial sphere, though each may play the same, more abstract, logical role in a dyadic relation. Each particular time seems, as a matter of experience, to occur only once.Footnote ¹⁵ Each particular time, then, if it is not the final element of the series – a possibility to be explained later – is replaced by another time, another temporal position. This is also the necessary condition of the three-out-of-two situation, the three being in fact a changing two, present and future to past and present, the logical relation of earlier and later remaining constant. The spatial position, which can be revisited, seems to involve no such necessary replacement, except in unidirectional movement from one spatial position to the other, which movement involves time.

Given the temporal positional significance of ‘at’, therefore, and given that that significance indicates other positions, other temporal ats, ‘time t at time t’ seems a good definition for ‘present time’ – and it is hard to think what another definition, as opposed to individual identification and description, of it might be. It certainly seems true that some particular time is present time – ‘the present’, we tend to say, signifying individuality – at that time itself.

But is ‘time t at time t’ a sufficient definition, as representing the class of present time itself? The present seems to have an especially vivid kind of reality – which is a strong reason why some philosophers think that the earlier-later account of time does not suffice, being too general to catch that vivid reality. But here we must make a very important observation – one that seems obvious enough but, as may happen with the obvious, one not taken enough into explicit as opposed to vaguely implicit account by some who follow McTaggart in asserting an A series. It is true that such vividness is not characteristic of the general; it is characteristic of the individual. The definition of ‘human’ does not pick out, say, McTaggart or Queen Anne or the person reading this; the notes characterizing such humans are, from the logical point of view, matters of description rather than of definition. So the characteristics of earlier and later, as general, do not depict the vivid, real present. But neither then do the definitional characteristics of present, any present, as general, depict that vivid, real present. For, and this is extremely important, the terms ‘past’, ‘present’ and ‘future’ can be taken either in the general sense, as designating any past, present or future time, or in the sense of indicating some individual past, present or future, falling under such a general designation. In the latter case we often use ‘the’ to signal individuality as opposed to generality – the present, the past, the future. It is to such individuals that the vividness belongs, not to the general definitions. And to insist that any use of the terms ‘past’, ‘present’ or ‘future’ respect such vividness is to commit an error in the logical field: it is to confuse description with definition. To put this in another way: if one insists that such vividness be respected in any use of ‘past’, ‘present’ or ‘future’ one is, logically, depriving oneself of the possibility of using these terms in the definitional sense, which does not exhibit such vividness – one should speak only individually, that is, historically, of the past, the present or the future, and never of past, present or future as such. And that, of course, would make general argument about them impossible.Footnote ¹⁶

If we do regard the generality of the definitions, then ‘time t at time t’ seems to be a sufficient definition of ‘present time’. It applies to every present time and to nothing else. So, for example, the present time – then – of the death of Queen Anne was that time at that time, which is, of course, historically, at this present time, past. Again, we may speak of the present instant, minute, day, year, century, millennium, and of each of them it is true that it is the time of that instant, minute, day, year, century, millennium at that time, at least in the sense of being at some temporal position contained within it.

But does not ‘time t at time t’, precisely as general, apply to every past time and every future time? Yes, but not, according to the definitions, as past or future simply but precisely as present then – that is, at that time. Every time is, in its time, present, but that does not mean that ‘present time’ is meaningless or trivial. The positional, temporal ‘at’, which signifies relativity to other positions which are replaced by one another, other possible temporal ats, ensures meaningfulness and non-triviality. And just what is present time at any such time is identified as an individual by description as well as being determined as present, as falling into the class of presents, by definition.

By ‘description’ then I mean definite description in the logical sense. It pertains to individual identities rather than to general classes, which are designated by definition. So ‘the author of Ulysses’ is a definite description applying to one unique individual human person, and ‘James Joyce is the author of Ulysses’ is an identity relating a name to that definite description. Similarly, ‘the author of Finnegans Wake is the author of Ulysses’ is an identity relating one definite description to another. The use of the definite article here is symptomatic of the reference to a unique individual, and though the article can be used otherwise, as in ‘the bat is a mammal,’ it is usually quite clear from the context how the usage is to be taken – ‘the bat is a mammal’ being quite different from ‘the bat is injured.’ In ‘the present’, ‘the past’ and ‘the future’ such a specific or generic use is possible, as in ‘the present is time t at time t’, but it is clear that ‘the present is the time of my reading this’ denotes not such a general use but rather an individual identity. It is clear, that is, that the present here is not any time of my reading this but this time –just as what I am reading is a this, and my reading it is this individual instance of my reading it. The time in this instance is a unique individual time the individuality of which is indicated not by general definition but by a name or definite description. So also in ‘the past contains the death of Queen Anne’ and ‘the future contains my own death’ past and future are referred to as individuals – collective individuals, it may be, but individuals none the less. It is an interesting fact that all of time, the complete extension of what is contained under the appropriate definitions, can be summed up in three particulars, the past, the present and the future. None of this is changed by the fact that each such individual reference is to a different, by kind or degree, past, present or future, the definitions under which these designations fall being, respectively for past, present and future, exactly the same in each case. This being a different individual again indicates the importance of change for time.

It appears, though, that in the case of time there is a peculiar temptation to take the arrangement of individuals under the definitions of ‘past’, ‘present’ and ‘future’ – that is to say, to take history, actual or potential – as though it were another kind of logic, a rival for the arrangement of temporal order to the logic of earlier and later. And that, of course, history is not. To confuse history with logic here is in effect to confuse the individual with the class, the description with the definition, the concrete with the abstraction from it. To make that confusion is to move uncritically, and unjustifiably, between ‘present’, say, as identifying this unique now and ‘present’ as a tag for any present whatever. The uncritical wrong move, which, if used in logical argument, constitutes an equivocation, is of course facilitated by the fact that the same term is used both for the class and for the individual member, not to mention for the individual member both as singular individual and as collective, so that confusion becomes possible.

A source of confusion here – the noting of which does not depend on our proof – in both McTaggart and those who follow him, is to take notice of the arrangement of individual times as an arrangement of impermanence, which then seems to set that arrangement up in opposition to the permanent order of earlier and later. What is now past, present or future is something that is constantly changing, in kind for present and for at least immediately proximate future and past, and at least in degree for more remote future and past. This constant change is acknowledged by McTaggart in his setting up the so-called A series, clearly as a kind of rival to the B series, as the mere labels ‘A’ and ‘B’ indicate. ‘If M’, he says, ‘is ever earlier than N, it is always earlier. But an event, which is now present, was future and will be past.’Footnote ¹⁷ This seemingly fundamental opposition, based on permanence and impermanence, as considered in terms of what appear to be numerically the same temporal positions, has led to the now extended history of the vexed question which of the two opposed series, the impermanent or the permanent, gives the true structure of time. As Mellor, in his 1998 book, Real Time II, puts it: ‘the A-times of events change and their B-times do not. That we have seen to be the defining difference between A- and B-series’ (my italics).Footnote ¹⁸ But it is taking that difference – which is obviously real enough as something remarked about time as we observe it prima facie – as defining, that is, as marking in some sense two fundamentally different possibilities for the structure of time itself, that makes for the long-lived, and I believe mistaken, controversy between two schools, and has led to an enormous proliferation of clever, sophisticated and much reiterated argument on both sides.Footnote ¹⁹ It is all on the basis of an unexamined, and mistaken, assumption.

The assumption is that that permanence versus that impermanence for the same temporal positions must be rivals. And that assumption is false. As I pointed out in ‘McTaggart's Argument’, what McTaggart and his followers, on both the A and the B sides, seem not to have noticed is that the basis for judging the impermanence in the so-called A series is not the same as that for judging the permanence in the B, so that the two are not necessarily opposed. In the B series M is compared to another entity, N, with which it is in the permanent relationship of earlier, whereas in the ‘A series’ M, let us say, is compared only to itself, in terms of its three different states, future, present and past. There is no basis therefore for saying that the permanence in question is in any way directly opposed to the impermanence in question. M as future, present, past successively is still permanently earlier than N as also future, present, past successively. The impermanence and the permanence, not on the same basis, can live together. It is somewhat like saying that ‘All humans are equal’ on the basis of a common humanity, and ‘No two humans are equal’, on the basis of personal gifts, can be true together.Footnote ²⁰

As ‘All humans are equal’ and ‘No two humans are equal’, on different bases, can be true together, and a supposed opposition would be merely apparent, merely from the use of opposed terms where the basis of the seeming opposition was not examined, so also with the seeming opposition of the non-permanence in the so-called A series to the permanence in the B. The non-reality of the opposition is in fact signaled by the persistence of the notions of earlier and later in those of past and future.

What this persistence shows is that the order of the impermanent past-present-future is the permanent earlier-later order. The past is always earlier than the present, the present later than the past, if there is one; the future is always later than the present, the present earlier than the future, if there is one. That ‘if there is one’, in the case of either past or future, is necessary to add if there is a possibility of a first or a last member of the earlier-later series. If there is the possibility of creation from nothing, or perhaps given big bang, there seems to be a first member, a first event and so a first point in time, call it instant, moment or whatever. In this case, given the continuing of the series, it would of course be a time t at time t, that is, a present. But there would be no earlier time, no past but only a future. Similarly, if a last member of the temporal series is, by whatever means, possible, its time would also be a time t at time t, that is, a present time. But there would be no later time, no future but only a past. What we would have then is not present and therefore past and future but only present and therefore past or future or both, according to the law of logical disjunction. That is as regards a single present. If however we think in terms of at least two presents, one earlier than the other, as required by the earlier and later positions, all three classes of past, present and future occur in those two times in a logical conjunction – and in their usual arrangement where there are three times, the middle being taken as present. It is at least logically conceivable that all time should comprise just two positions, one earlier one later, only one single change ever occurring – and there would still occur all three classes of past, present and future, though not in the usually expected individual arrangement of three, as we saw in our proof. That past was earlier than present and future later would nevertheless be respected. But that all three classes of past, present and future should arise from just two times, one earlier and one later, has to be necessary if indeed past-present-future is equivalent to earlier-later. What all this seems to indicate is that past-present-future is not a triadic but only a dyadic arrangement.

And that arrangement is the earlier-later one, the generating relation of the series. Present is earlier than future as past is earlier than present; present is later than past as future is later than present. This, because there are three rather than merely two terms involved, may in some conventional sense be called a triadic relation,Footnote ²¹ but – and the distinction for us here is again very important – it has a very different structure from a triadic relation such as giving, in which any one term necessarily requires as correlatives the other two, and without any reiteration of the relation. A gift requires both a giver and a receiver, and the relation between giver and gift is not identical with that between gift and receiver. There is no repetition, as in past earlier than present, present earlier than future, and there is the requirement of three poles, giver-gift-receiver, rather than two, an earlier and a later. A logic of gift and therefore either giver or receiver is not what is exhibited here, whereas that of present therefore either past or future is, or can be, as we saw. Past-present-future is, therefore, a three-term arrangement within the dyadic earlier-later series rather than a triadic arrangement in the strong sense of having three poles – at least that is the way we read ‘triadic’ for our purposes here. This again shows justification for our deriving in the proof the three-termed, which is not three-poled, ‘past-present-future’ from two times, t ₁ and t ₂, and leads again to our equivalence.

There is, on this account, no reason for denying that there is a perfectly particular past, present and future, or that the description of each changes with any change whatever. Equally, there is no reason for denying that ‘earlier’ and ‘later’ express adequately the nature of temporal order, since ‘past’, present’ and ‘future’ are defined by means of those terms. The difference is that between the descriptively vivid individuals and the non-vivid generalities under which they are defined. The particular descriptions change, whereas the definitions remain the same; the definitions are permanent whereas the descriptions are not. Similarly, the order, expressed by the defining terms ‘earlier’ and ‘later’, of the ever changing descriptions remains permanent. That permanence, of definition and order, and impermanence, of description, can be true together. The ‘A series’ is consistent with, not the rival of, the B series. The ‘A series’ may in some looser sense be said to reside within the B series, though, strictly, the two are equivalent.

Each present, each time t at time t, is a different individual at each different time t, though retaining exactly the same definition. That same individual time at some time earlier than the time which it itself is at – that is, from the reference point of an earlier time – is future, and, at some time later than the time which it itself is at, is past, according to those respective definitions, just as at London – that is, from the reference point of London – Cambridge is north.

‘Past’, ‘present’ and ‘future’, in their general, definitional, sense, are a degree more abstract than ‘the past’, ‘the present’ and ‘the future’, in their particular, vivid, descriptive, sense. And ‘earlier’ and ‘later’ are a degree more abstract than ‘past’, ‘present’ and ‘future’ in their definitional sense, since the latter are defined in terms of the former. In turn, ‘precedes’ and ‘succeeds’ are a degree more abstract than ‘earlier’ and ‘later’, in the sense that ‘precedes-succeeds’ is taken, if only nominally and conventionally, to express neutrally the logical relation which ‘earlier-later’, in terms of time, ‘left-right’, say, in terms of space, and ‘smaller-greater’ in terms of number, equally express – all abstracted from their differentiating characteristics. Space and number and time are, logically, all dyadic series in that sense, even if their ontologies should be very different from each other. It is a question of the levelling power of abstraction, of its abstracting, precisely, from either less abstract generalities or non-abstract particularities.Footnote ²²

The meaning of the terms ‘earlier’ and ‘later’ point to the fact of change, as opposed to their being mere ciphers, with goodness knows what meaning, with the same logical force as ‘right’ and ‘left’ and ‘smaller’ and ‘greater’, and, in the more abstract expressions, ‘precedes’ and ‘succeeds’. The differences as well as the similarities have to be accounted for, and the ontology of time is quite different from that of space or that of number – whatever the latter might ultimately be. The meaning of ‘earlier-later’ carries of course the same connected, transitive, irreflexive and asymmetrical dyadic logical structure as that of space and number, which may, in the case of all three, be generally expressed as ‘precedes-succeeds’, but the meaning of ‘earlier-later’ carries also a natural reference to change which ‘left-right’ and ‘smaller-greater’ do not, and which ‘precedes-succeeds’ abstracts from.Footnote ²³

Time tends to be observed in terms of change – heartbeats or thoughts or states of consciousness replacing one another are good examples. One individual thing, and the individual time which it is at, which it occupies, replaces another, all of which is a matter of impermanence. On the other hand, the abstract order of that impermanence, having been, precisely, abstracted to a class, namely earlier-later – which contains present, future and past as classes – is always the same; it is permanent. So, though the individuals, together with the individual present time of each, constantly change, that abstract order of change remains the same, permanent. History does not have to be a rival to logic. The individuals' impermanence and the general, abstract permanence of the order of that impermanence can live together without contradicting each other; past-present-future time, as constant individual change, exists as the ontological foundation of its abstraction, earlier-later time, and exists within that abstraction, as the individual human is the ontological foundation of the abstraction ‘human’ and exists within that abstraction, that class, the collective individuals past, present and future being the complete extension of their class just as all individual humans together are the complete extension of the class ‘human’.

This is on a par with what Aristotle in the Physics says about geometrical properties and the very physical bodies from which they are abstracted. ‘For natural bodies,’ he says, ‘have surfaces and volumes, and lines and points, with which the mathematician deals.’ He explains:

In the same way permanent earlier-later time is ‘separated’, abstracted from impermanent, individual, past-present-future time, ‘and it makes no difference, nor does anything false come from [its] being separated.’ Individual past-present-future time exists within, and consistently with, abstract, general earlier-later time, just as physical bodies exist within and consistently with the non-physical, mathematical, ideal space. The impermanence in the individual times is ontological, the permanence in the abstract time logical, and there is no conflict between them, as there is none between the individual physical bodies and abstract space.

Any present is a time – that is, is earlier or later. There is of course no earlier without a later and vice versa, the two being correlative. This means that if there is any present at all, there is at least one other. The different presents we may regard as individual members of the species denominated ‘present’, each individually distinct from every other but all falling under the same definition. In the statements in the proof, ‘At t _1,t ₁ itself is … present and t ₂ is future’ and ‘At t ₂, t ₂ itself is present and t ₁ is past’, the t ₁ and t ₂ referred to are, of course, different presents, each required as a different time by the other, and the future and past designations follow according to the definitions, all three classes occurring in only two individual times, each of which individual times, however, changes its class – from present to past and from future to present.

Therefore, ‘Earlier and later if and only if past, present and future’ is true. Past-present-future time and earlier-later time are equivalent, and, given one, you are given the other.

It follows, necessarily, that McTaggart and others were mistaken in trying to distinguish a past-present-future A series as something different from, not equivalent to, the earlier-later B series. It then also follows that his and others' reasoning, based on that false distinction, that past-present-future time is self-contradictory, is also mistaken. That the latter logically must be so is all that we establish here, not how it is so – which requires a detailed examination of the actual words of McTaggart and perhaps those of others.Footnote ²⁵ Past-present-future time is exactly as self-consistent as earlier-later time. We must be careful, though, about when, with the past-present-future terms, we are referring to individuals and when to the classes of which they are members.

Our proof also shows us why the quarrel of those, following McTaggart in their opposed ways, has gone on for so long. All of the disputants – accepting McTaggart's distinction as unproblematic – would initially take the so-called A series as a kind of rival to the B series. Each side then has something true to say about its own favoured view of time, but is mistaken in trying to eliminate the other as a rival. The B side, often because of a desire to conform to the ‘block-universe’ view of relativity theory, would like to eliminate past, present and future and reduce the relationships of earlier and later to some ‘spatialized’ – in reality, simply more abstract, logical – status, as though ‘earlier’ and ‘later’, though obviously temporal terms, as McTaggart recognized, had no really ontological temporal status – a status which makes time a single dimension even in string theory, where very many dimensions can be theoretically contemplated. The A side, on the other hand, would urge that the permanent order of earlier and later is not sufficient to represent the changing condition of time and not sufficient to represent the peculiar vivacity of the present, as though historical description trumped logical definition. Both sides, our proof shows, are mistaken. If we have earlier and later, we have past, present and future and vice versa. The two situations are equivalent, past, present and future, as individualized, being the full extension of the term ‘time’, the intension of which is caught both in the relational expression ‘past-present-future’ as generalized and in the more abstract relational expression ‘earlier-later’. What, it seems, we can say is that past, present and future have a perfectly good ontological grounding, however that is to be worked out in detail, and that ‘earlier-later’ is a perfectly adequate logical, abstract expression of the temporal order contained in past, present and future.Footnote ²⁶ There is really in all of this, which is perfectly well recognized in our ordinary speech about time, nothing to be reformed.

If this argument is correct, the consequences are very significant. What is required is not only a rethinking of the whole controversy of the so-called A and B series but a rethinking of the nature of time itself – not indeed as we normally think of it but as we analyze that perfectly correct thinking, so that there remain no paradox in the analysis. What that rethinking would amount to is a much larger project, of which, I believe, I have some notion,Footnote ²⁷ but consider it sufficient here to establish only that the project must be undertaken.Footnote ²⁸