World Enough for Time

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Worlds Enough For Time Author(s): John Bigelow Reviewed work(s): Source: Noûs, Vol. 25, No. 1 (Mar., 1991), pp. 1-19 Published by: Wiley Stable URL: http://www.jstor.org/stable/2216090 . Accessed: 07/03/2013 03:32 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Wiley is collaborating with JSTOR to digitize, preserve and extend access to Noûs. http://www.jstor.org This content downloaded on Thu, 7 Mar 2013 03:32:21 AM All use subject to JSTOR Terms and Conditions

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possible worlds and time

Transcript of World Enough for Time

Page 1: World Enough for Time

Worlds Enough For TimeAuthor(s): John BigelowReviewed work(s):Source: Noûs, Vol. 25, No. 1 (Mar., 1991), pp. 1-19Published by: WileyStable URL: http://www.jstor.org/stable/2216090 .

Accessed: 07/03/2013 03:32

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

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Wiley is collaborating with JSTOR to digitize, preserve and extend access to Noûs.

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Worlds Enough For Time JOHN BIGELOW

LA TROBE UNIVERSITY

Time passes. Time passes. This is surely the most important thing about time. Yet it is also the most mysterious. The more you think about the passage of time, the less coherent it seems. I will describe one way we might try to make sense of the passing of time.

What I propose, more precisely, is an account of what it is that we believe time to be like. I will not give an argument to prove that we are correct in our conception of time as something which passes. Einstein's theory of relativity suggests that, as a matter of fact, our naive conception of time is deeply mistaken. I will not argue that the folk theory of time can be defended against physics: only that it can be defended from purely philosophical objections, from allega- tions of internal incoherence, from arguments which seem to show that folk theory can be shown on its own terms to be inconsistent.

I have long been intrigued by J.M.E. McTaggart's notorious argument for the Unreality of Time (see McTaggart, 1908, 1927, 1934, and Broad, 1938, for a full exposition and critique, and Dummett, 1960 for a sympathetic exposition and defense, and Gale, 1967 for bibliographic leads). On my reading, the argument hinges on two plausible premises. The first is that it is one of the essential properties of time that it is something which passes. The second is that the notion of the passage of time is incoherent, and involves either a logical contradiction, or a vicious circularity, or a vicious regress. Together, these premises seem to entail that time is unreal- that there is no time, that nothing ever really occurs before or after anything else, but only appears to do so.

Many are less bold than McTaggart. They think time is real. Yet they have a hard time dispelling the notion that time is essen- tially something which passes. So they search for some watered down, logically consistent sense in which time can coherently be described

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as "passing". And they dismiss it as an illusion, a bewitchment of the intelligence by language, to think that time passes in any more substantial sense than that. This view threatens to become no more than verbally distinct from McTaggart's. It merely asks us to apply the term "time" to something which, McTaggart can grant, does appear to be time, but which he is unwilling to dignify with that title, for want of a sufficiently robust sense in which it may be said to "pass."

What does the passage of time amount to? In plain terms, it amounts to at least the following. Every past time was once future, became present, and has now become past. Everything in the future will one day become present, and then it too will become past. And lastly, what is now present was once future, and will soon be past. This last is the one I will dwell most upon. It is something which can be of comfort, when things are going badly; and yet it can also prompt a subtle, elusive kind of sadness, when things are otherwise going best of all.

McTaggart's opponents often claim that everything which is essential to the passage of time is captured by assertions like the ones above. Any residue is mere metaphor. And they claim that such assertions can be consistently analyzed, in a way which makes them clearly true. The standard analysis offered is sometimes called the indexical or token-reflexive theory of tenses (see Russell, 1940, 1948, Reichenbach, 1947, Smart, 1949, 1955, 1964, Goodman, 1951, Williams, 1951, Quine, 1960). The indexical theory runs roughly as follows.

An assertion that something is past, when made at a time t, is equivalent to the claim that that thing is true at some time before t. Similarly, an assertion that something was so, when made at a time t, is equivalent to the claim that it is so at some time before t. Notice that this analyzes the notion of the past by utilizing a con- textually supplied moment of time, plus the earlier/later, temporal ordering. Analogous accounts can be given for words like 'present', 'now', 'future', and for the future and other tenses.

This analysis of tenses, and 'past', 'present', and 'future', does enable us to give some sort of account of the so-called passage of time. It enables us to give an analysis of claims like:

What is present was future, and will be past.

What does such a claim amount to? When uttered at any given time, the indexical theory makes it equivalent to something like:

What is simultaneous with this time is later than some time which is earlier than this, and is earlier than some time which is later than this.

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And this, of course, is true. If this is all the passage of time comes to, then time passes. And so, it is concluded, the so-called passage of time is no threat to the reality of time.

But really, is this all the passage of time amounts to? On the indexical analysis, attempts to describe the passage of time yield true assertions, but pretty vacuous ones. How could anyone be com- forted, or saddened, by the kinds of assertions which the indexical theory offers as an analysis of the passage of time? A McTaggart must respond by urging that there really ought to be something more to the passage of time than is captured by the indexical analysis.

Or, more forcefully, a McTaggart sympathizer can complain that the indexical analysis is misdirected from the start. It analyses the passage of time by appealing to an earlier/later relation. But this relation is presupposed to be a temporal ordering. Not every relation with the logical properties of the earlier/later relation, not every linear ordering, counts as a temporal ordering. There is an important difference between, for instance, the higher/lower rela- tions and the earlier/later relation. And what, we may ask, makes the earlier/later relation a distinctively temporal ordering? It is precisely the passage of time which is required to make the earlier/later relation a temporal one. In other words, in order for the earlier/later relation to be a temporal relation it must be defined in such a way as to ensure that when what is earlier is present then what is later is still future, and when what is later becomes present then what is earlier will be past-that is, the earlier/later relation is a temporal one only if it concerns something which passes. And this entails that it is the earlier/later relation which must be defined by appeal to the notions of past, present and future, and not the other way around.

Questions of conceptual priority are always a delicate matter. It is often unclear whether one thing should be analyzed in terms of another, or whether the second should be analyzed in terms of the first, or whether it makes any difference which way the analysis runs. In the present case, however, it is easy to sympathize with McTaggart's dissatisfaction with the indexical analysis of time and tense. So it is worth seeing whether we can take, as the more primitive notions, those of the passage of time, of past, present and future, and not those of earlier and later, before and after. I will try to make sense of the passage of time without presupposing a temporal, earlier/later ordering. And I will try, in fact, to define earlier and later in terms of the passage of time, rather than the other way around.

By reordering the analysis of earlier and later with respect to past, present and future, I will vindicate one of the elements in

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the passage of time, as involving more than mere metaphor. It may nevertheless be the case that indexical theories are right to construe the passage of time as involving some degree of metaphor. When we speak of time "passing", using an active verb ip the present continuous tense, we do seem to be speaking of a dynamic process rather than a static structure. And this may involve a degree of metaphor. The theory I will arrive at will be one which can be described in ways which make it seem like a "static" structure, and hence in ways which make it seem to miss the point of the passage of time as a "dynamic" process. (See McCall, 1966, 1976, 1984, for an example of a defense of a dynamic conception of the passage of time.)

I allow to the indexicalists that there may be some aspects of the "dynamic" conception of time which are best construed as metaphor. When we speak of time passing, we do seem to imply that it is changing in a manner which is closely analogous to that in which other things change, as for instance when a cake rises. When things change, it makes sense to ask about the rate at which they are changing. When geese are passing overhead, it makes sense to ask how rapidly they are passing. Talk of time passing thus seems to- imply that it makes sense to ask how fast it is passing, and to expect a non-trivial reply. Yet the only reply I can see is the highly trivial one, that time necessarily passes at the rate of one second per second. The silliness of any such measure of a rate of passage for time seems to signal something incoherent, or metaphorical, about the "passage" of time. Perhaps that incoherence could be evaded, and a fully fledged "dynamic" theory rendered consistent; I do not prove that this cannot be done. I leave it open to the resolute dynamist to supplement the theory I will offer, in a way which makes it rest on less of a static structure, and more of a dynamic process. I am skeptical. But in the light of such dynamic intuitions, I refrain from asserting that the theory I offer captures all the significant intuitions about the passage of time. I aim only to defend some of the core conceptions invoked by the idea of the passage of time.

In particular, what I aim to show is that it is an important and non-trivial fact about time, that what is present was future, and will be past. This is, I take it, at least part of what is involved in the passage of time. It is this which I take to be the coherent, non-metaphorical core behind the belief that it is an essential feature of time that it is something which passes. The only way in which this coherent core can be sustained is by defining the relations of earlier and later by reference to such facts as that what is present was future, and will be past; this is the lesson to learn from McTag- gart. Earlier and later must be analysed in terms of the passage

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of time, not the other way around. And therefore the passage of time, in turn, must be analysed and shown to be consistent, without presupposing the ordering of events under the relation of earlier to later.

The trouble with the passage of time, non-indexically construed, is that it seems to require that each moment is past, and is present, and is future, in some sense. Yet, McTaggart assumes, these three properties are contraries. Necessarily, what is past is not future. So it is a contradiction to say that what is present is also future and is also past.

The only way to sublimate this contradiction is by saying that each moment is past and present and future relative to different points of reference. What are these points of reference? The first natural guess is to take them to be moments of time. So we deflect the contradiction by saying that what is present was future and will be past-and we parse these tenses on the indexical model. But, as McTaggart points out, this presupposes what we are trying to explain.

Yet all is not lost. We just need to find points of reference other than moments of time. I nominate possible worlds. Not only can contrary properties attach to the same object at different times, but also in different possible worlds. Something may be small at one time and large in another; but similarly, it may be small in one possible world and large at another. (Which is just to say that, although it may actually be small, it could have been large.) Possi- ble worlds can be used to keep contrary properties out of each other's hair.

Can we apply this strategy to time? The passage of time seems to require each moment of time to be past and present and future- but of course it must be so relative to different points of reference. What if these points of reference are taken to be possible worlds?

It sometimes happens, in certain novels, that the novelist writes her heroine into an impossible situation, and then announces, without apology, that "With one bound Jill was free!", or something of that nature. It may seem as though the introduction of possible worlds frees us from McTaggart's impossible situation in a similarly unbelievable manner. Yet the introduction of possible worlds is not as arbitrary as it may seem. The very way in which McTaggart poses the difficulty about the passage of time contains the seeds of the modal solution. The passage of time involves the truth of such claims as that what is now past was once present. If it was present, then clearly it is possible for it to be present, and that means (in other words) that there is a possible world in which it is present. Which possible world is that? Could it be the actual world? No,

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the event which is past in the actual world could not be present in the actual world, since (as McTaggart assumes) being past and being present are contrary properties. Hence the possible world in which the event that is actually past has the property of being present must be some possible world other than the actual world. Thus McTaggart's assumptions entail the modal solution. Possible worlds are not like an unexpectable bolt from the blue that cuts McTag- gart's knot in a flash. What McTaggart has given us is a slipknot which contains its own, modal, resolution.

Let us say, then, that the things in the actual world, the world we are in, have the property of being past, or of being present, or of being future. Suppose the things there were, are, or will be are represented by letters of the alphabet, and that ones which are past are written in boldface type, whereas ones in the future are written in italics, and the one which is present written as an upper- case letter. Then the things there were, are, or will be may be represented by some such sequence of letters as:

abcdefghijKlmnopqrstuvwxyz.

We could add to this the idea that pastness and futurity can come in degrees. One thing may be more distantly past than another. This is not to be interpreted as meaning just that there is a relation between the two things, one being earlier than the other: this tem- poral ordering is to be grounded in the fact that one is more distantly past than the other. Each has a degree of pastness, and it is in vir- tue of their different degrees of pastness that one is earlier than the other. We may imagine that things further and further in the past are represented in deeper and deeper shades of blue. Conversely, we may imagine things in the future represented in shades of pink, so that things further and further in the future are represented in deeper and deeper shades of pink.

The idea of degrees of pastness and futurity can then be used to justify an artifact of the representation of the world given above: the order in which the letters are written. The letter a is written to the left of b, and this suggests that I am presupposing that a is earlier than b. Yet it is important that the theory should not at this stage presuppose a temporal earlier/later ordering. So it is im- portant to note that a is written to the left of b, not because of a temporal relation between a and b, but because a has the property of pastness to a greater degree than b. Similarly, z is written to the right of y, not because of a temporal relation between y and z, but because z has the property of futurity to a greater degree than y does.

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In fact, it is not necessary, for the purposes of the modal theory, to assume degrees of pastness and futurity. In the absence of such degrees, it is even more important not to read too much into the presentation of events in the world as linearly ordered. We should not read the order imposed on the letters representing things which are past as presupposing some linear ordering of those things. Rather, the placing of a to the left of b should be taken as merely mnemonic. It anticipates a structure which will emerge only from examination of other possible worlds. It reflects the fact that, for instance, there is a possible world where a is present and b future, and no world in which b is present and a future. But this is to anticipate the explanation to be given at greater length below. For the moment, note only that the boldface type for past events is to represent a property they all share in common, and the italic typeface for future events is to represent a property which they all share in common, but the order in which the past and the future events are written does not presuppose temporal earlier/later relation on the events in the world.

The properties of pastness (to a given degree), presentness, and futurity (to a given degree) need to be carefully construed, if the modal theory is to discharge the explanatory task which has been set for it. They cannot be taken to be the properties of being related in a certain way to a contextually supplied moment of time. Then the theory would collapse into the indexical theory which was described earlier. That theory may-yet be true, but the current project is to seek for a theory which lets time pass in a more robust sense than the indexical theory permits. Thus, for present purposes, one strategy would be to take the properties of pastness, presentness and futur- ity to be intrinsic properties. If we construe these properties to be "intrinsic" ones, this means that we will be taking say the property of pastness to be one which something possesses independently of the existence of properties or relations of any things other than itself.

Strictly speaking, it is not necessary to take all three properties, pastness, presentness and futurity, to be intrinsic properties in that sense. We could take only presentness to be an intrinsic property. And we could construe pastness and futurity as relational proper- ties, properties concerning the manner (and the degree) to which things are related to the present. It is important, however, to insist that at least the property of presentness is an intrinsic one. (Or at least, if it is to be a relational property, it should involve only relations to things outside the totality of events which occur in time. McTaggart assumes that presentness is the property of being pre- sent to something, but to something nontemporal.) Things which

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are present have it; things which are past or future do not. Things which are past or future do in some sense have the property of be- ing "present relative to themselves". But things which really are present do not have merely the property of being present relative to themselves and to one another. If that is all that the property of being present amounted to, the theory would reveal itself as merely another presentation of the indexical theory of time. We must in- sist, for the purposes of the theory, therefore, that things which are really present do have some property in virtue of which they are present, and in virtue of which they are all present relative to one another.

The requirement that the properties of pastness, presentness and futurity be intrinsic, or that at least the property of presentness be intrinsic, does not sit comfortably with Einstein's theory of relativ- ity. Simultaneity, we are told, and hence it would seem also present- ness, are always relative to some "frame of reference". Relative to one frame of reference, two events may both be present; yet relative to another frame of reference, one will be earlier than the other, and hence they cannot both be present. Hence, it seems, presentness cannot be intrinsic.

Yet if we take presentness to be relational, to be relative to a frame of reference, we are well on the way back to the indexical theory of time, which the modal theory is intended to supplant. Is there any way of relativising presentness to frames of reference, without transforming the modal theory into an indexical one?

Yes, there is. We must distinguish between simultaneity relative to a frame of reference, and presentness relative to a frame of reference. Two things may be simultaneous relative to a frame of reference, but both may be past-and they may both be past even relative to that same frame of reference. For each frame of reference, there will be the properties of being past, present and future. The property of presentness, on its own, cannot be taken as intrinsic, unrelativized to any frame of reference. But consider the property of being present relative to a given frame of reference. There is a sense in which that property may be intrinsic-intrinsic, that is, to the pair of the thing in question and the frame of reference. Within a frame of reference, some things are present and others are not, and their possession of this property of presentness relative to the frame of reference need not be construed as simply their being simultaneous with some contextually supplied further entity. That refusal to relativize to some contextually supplied entity in addition to a frame of reference will be enough to prevent the theory from collapsing into an indexical theory.

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Einsteinian relativity theory forces difficult revisions of the modal theory of time, but it does not automatically rule it out of court. For simplicity, however, it is worth ignoring relativistic complica- tions and presenting a modal theory of nice, tidy, linear time. Leave relativity theory for another time. In any case, the modal theory is intended primarily as an account of how we at least sometimes think about time, and not as an argument to show that it is right to think about time that way. If the modal theory could not be adapted to relativity theory, that would show that it is mistaken, but it would not show that we never think about time that way.

Returning to the simple, linear view of time, let us see how we could use the postulated properties of pastness, presentness and futurity to explain the passage of time. The passage of time, as argued above, is to be understood as involving the truth of claims like:

What is present was future, and will be past.

Instead of giving an indexical analysis of this, let us give a modal analysis. Let us represent tenses as kinds of modal operators. In the actual world, what is present is not future. But what is present in the actual world may be future in some other possible world. What is present in the actual world could have been future. To say something was future, then, will be to claim that it is future in some other possible world. Similarly for the future tense: to say something will be past is to say that it is past in some other possible world. Then the claim that we are trying to analyse, that what is present was future and will be past, entails a modal claim of the form:

What is actually present could have been future and could have been past.

For such modal claims to be true, we will require that the very same thing which exists in the actual world, with the property of being say present, may have different temporal properties in another possible world. Leibniz and Lewis have held that two different worlds must always contain different individuals ("a different act-a dif- ferent Adam!"). Such a view would impose a need for substantial reformulation of the theory that "the same" thing which is present in one world is past in another. A similar revision would be called for on any metaphysical theory which prevented the relevant tem- poral entities from existing in more than one world. Consider for instance the recently mooted theories of events which require that for two events to be numerically identical, they must have the same causes and effects. Such theories would make it difficult to make the required strict identifications of events across possible worlds.

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Such difficulties, however, are not artifacts of the modal theory of time, but are very general difficulties for both the logic of modalities, and for the possible worlds metaphysics which seeks to illuminate modalities. It is not necessary for the modal theory of time to take sides on the issue of the identification of individuals or events across possible worlds. Although I shall speak of "the same" thing existing in several worlds, it should be understood that I am not thereby committed to a rejection of the view that, strictly speak- ing, what we are dealing with is not really numerical identity across possible worlds, but some weaker relation involving a sufficient degree of qualitative similarity. The claim that what is present was future, and will be past, is to by analyzed, according to the modal theory, as entailing that:

What is actually present could have been future and could have been past,

but this, in turn, might be analyzed either with or without the assumption of a strict identity between things in different possible worlds. For simplicity, I will speak as though the same thing, whether individual or event, that has one temporal property in the actual world, might also have differing temporal properties in other possi- ble worlds; but remember that this mode of speech may be open to a variety of disputed analyses.

The modal theory is not yet complete. When we say what is actually present could have been future, we need an account of what sort of "could" is required here. It cannot be just the "could" of logical possibility, nor can it be just the "could" of physical possibility. It will assume the usual framework for the analysis of modal claims: possible worlds and accessibility relations defined be- tween possible worlds (Hughes and Cresswell, 1968). Something is "possible" when it is true in some "accessible" world. Different kinds of possibility are grounded in different accessibility relations. We need to discover the special accessibility relation which grounds the passage of time.

Let us survey possible worlds, and note which of the things in them are past, which are present, and which are future. Call something a temporal object in a world when it has any one of the three properties past, present or future in that world. Then given a world u, suppose there to be some other world, v, which contains exactly the same temporal objects as u. And let us suppose that these worlds meet the following three conditions:

(i) everything which is past or present in u is past in v; (ii) everything which is present or future in v is future in u; (iii) everything which is future in u and not future in v is present or past in v.

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When these conditions are met, we may say that u is "in v's past", and that v is "in u's future", and that v is "accessible from" u. For any world, we may take "its future" to be the set of worlds which are "in its future". Similarly, for any world, we may take "its past" to be the set of worlds which are "in its past".

The theory which emerges will be formally equivalent to struc- tures discussed in tense logic (foundational work in tense logic is well illustrated in Prior, 1957, 1958, 1959, 1967, 1968, 1977). The notational difference between tense logic and the modal theory is not in itself important. Yet the notational difference does signal a difference in philosophical interpretation. The use of modal ter- minology enables us to assimilate the formal tricks of tense logic into the more familiar ontological realm of individuals, properties and relations, together with modalities. (See Hughes and Cresswell, 1968, and Lewis, 1986, for more on the framework within which I am proposing to absorb the passage of time, as formalised in tense logics.)

To illustrate, suppose the actual world, @, to be represented by a sequence of letters, boldface for past things, italic for future things, as explained earlier:

abcdefghijKlmnopqrstuvwxyz.

Now consider another possible world, u, which contains the very same things, except that some of the things which are past in the actual world are future in this world:

abcdefGhi/klmnopqrstuvwxyz.

This is a world in the actual world's past. In contrast, the world v:

abcdefghijklmnoPqrstuvwxyz

will be one of the worlds in the actual world's future. Further, v is in u's future, and u is in v's past. There will be a series of worlds, each containing the same things, and differing only in which of those things are past, which are present, and which are future. This series of worlds can be defined using only the properties of pastness, presentness and futurity. We do not need to presuppose any tem- poral ordering relation, of earlier/later.

If there are such worlds, then we can give a modal account of the passage of time, one which does not appeal to any prior assump- tion of a temporal earlier/later relation. The passage of time is described in sentences like:

What is present was future.

The truth of such claims is, as McTaggart claims, a precondition for any earlier/later relation to count as a temporal relation. So we

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need to explain the truth conditions for such sentences without presupposing the earlier/later relation. Here is how we can do so.

First, we may account for tenses. We may say something was true in a world w just when it is true in some world in w's past. And something will be true when it is true in some world in w's future. The past tense is then a modal operator analogous to "possibly": something was true when it is true in some accessible world, where the worlds accessible from w are the worlds in w's past. Similarly, the future tense is a modal operator analogous to a different "possibly": something will be true when it is true in some accessible world, where the accessible worlds from w are the worlds in w's future.

Now we can analyse the claim, for instance, that what is pres- ent was future. This claim is true in a world w just when everything which is present in w is future in some world in w's past. An analogous account can be given for the idea that what is present will be past, and for other such articulations of what is involved in the passage of time.

In effect, the modal theory explicates the passage of time in terms of counterfactuals. What is present was future, because it would have been future if past things were present. This may cast a dif- ferent light on the sadness or comfort one can get from the passage of time. One may feel sad, for instance, at the passage of time, for the same sort of reason that one may feel sad at the thought of what might have been.

It is then possible to define an earlier/later relation by using the properties of pastness, presentness and futurity. Let us say, first, that a thing a in a world w is "currently" earlier than another thing, b, just when either a is past and b is present or future in w, or else a is present and b is future in w. (We could extend this definition, if we were to appeal to degrees of pastness and futurity: a is cur- rently earlier than b when a has pastness to a greater degree than b or when a has futurity to a lesser degree than b. But the tech- nique which follows makes it unnecessary to appeal to degrees of pastness and futurity.) The relation, "currently earlier than", can then be extended to a general earlier/later relation, as follows:

In any world w, a thing a is earlier than a thing b just when a is currently earlier than b, either in w itself, or else in some world in w's past or future.

Provided we assume the existence of properties of pastness, present- ness and futurity, and provided we assume that what is actually present could have been past or future, we can then give a non- circular definition of temporal ordering by the earlier/later relation.

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We thus have a way of meeting McTaggart's challenge. We can explain what makes the earlier/later ordering of events a temporal ordering, by showing that it derives from the prior, specifically tem- poral properties of pastness, presentness and futurity.

The temporal ordering we define in this way will have a number of logical properties. The question arises, which logical properties are possessed by the earlier/later relation? The answer will depend on which possible worlds there are, and more especially, which worlds there are in any given world's past or future. I will assume that certain minimal conditions must be met, if there is to be time of any sort at all in a world w. I will assume, for instance, that if a thing a is present in world w, then there is a world in w's past for which a is future, and there is a world in w's future for which a is past. The existence of such worlds is essential for time to be something which passes, so if the modal theory is to work at all, we must assume that there are at least these worlds.

There are many other hypotheses which could be considered, about what worlds there are in a given world's past or future. If the right hypotheses are made, then the temporal ordering we define from them will have various of the logical features which we are used to assuming time to have in the actual world. In particular, there are conditions on the worlds in w's past and future which will ensure that the earlier/later ordering of things in w will be a linear ordering. This would mean that the earlier/later relation is irreflexive (so that if x is earlier than y then y is not earlier than x), and transitive (so that if x is earlier than y and y than z, then x is earlier than z), and connected (so that if x is earlier than y and z is not earlier than y then x is earlier than z), and dense (so that if x is earlier than z then there is a world y which is later than x but earlier than z).

However, it is not necessary here to insist that any temporal ordering would have to be a linear ordering. The logical possibilities of branching or circular or discrete time should not be ruled out without good argument. For instance, suppose that we have two things a and b in a world w. And suppose that in some world in w's past we have a past and b present, while in some other world, a world in w's future say, we have b past and a present. Then the definition of earlier/later given above will instruct us to say both that a is earlier than b and that b is earlier than a. That sounds very bad. Yet it is the sort of thing which we would have to admit if time were cyclic. A thing might be both earlier and later than another, in the way one thing can be both east and west of another on the surface of a spherical earth. (Incidentally, cyclic time also threatens McTaggart's assumption that the properties past, present

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and future are contraries. This would complicate, but not under- mine, the arguments motivating his rejection of standard theories of time, and motivating the current construction of a modal theory of time.)

Hence, the modal theory of time permits hypotheses which will generate a tidy, linear ordering of events as earlier and later, but it also permits other hypotheses about how many worlds there are, and these other hypotheses will generate other logical properties for the earlier/later relation.

There are further possibilities which are left open by the modal theory. The first hypothesis to consider is the one which deviates least from the orthodox space-time metaphysics of our time. This is the one described earlier, in which the actual world contains things with the property of being past, and ones with the property of be- ing present, and ones with the property of being future. There are worlds which exactly match the actual world, except that what is past or future in the actual world is present in them. These worlds may be represented by sequences like the following:

abcdefghijKlmnopqrstuvwxyz abcdefghijkLmnopqrstuvwxyz abcdefghijklMnopqrstuvwxyz

abcdefghijklmnopqRstuvwxyz

Another hypothesis is worth exploring. It could be claimed that the actual world contains things which are past and present, but does not contain any things which are future. Future things do not exist at all. On this hypothesis, the relevant possible worlds may be represented by sequences like the following:

abcdefghijK abcdefghijkL abcdefghijklM

abcdefghijklmnopqR

If this is what worlds there are in the actual world's past and future, then the modal theory will be one in which there is, in an impor- tant sense, a genuinely open future. On this view, something may have the property of being future in a world, without existing in that world. It has the property of being future in w not because it exists in w, but because it exists in some other world which is importantly related to w. On this theory, being future is a property which is a close cousin of the property of being non-existent.

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A further hypothesis is open to investigation, based on the metaphysics urged by Storrs McCall, of branching futures. On this hypothesis, the temporally "accessible" worlds from the actual world can be represented by some such sequence as the following:

abcdefghijKlmnopqrstuvwxyz. Xjtzvo7rPoruvXxb&. ...

M1J0npCTY+XtM t~ blbto ...

abcdefghijkLmnopqrstuvwxyz MIOllpCTYcPXtLL' it Ut%%blb9b OJI ...

abcdefghijklMnopqrstuvwxyz.

. . .and so on.

In each of these possible worlds, there exist certain things which are past, and there also exist all the things which are as we might say possibly future. In the world in which K is present, there exist various future sequences, including an italic one, a greek one, and a cyrillic one. These are represented in separate rows, but this struc- ture in the representation is merely mnemonic, and does not signal any prior presupposition of a branching temporal structure which is intrinsic to the events in the world taken by itself. The branching structure in the representation of the first possible world merely an- ticipates the contents of the worlds which lie in its future.

In the first world represented above, the one in which K is pre- sent, all the italic, cyrillic and greek things exist, and have the prop- erty of being future (together with many more future things in- dicated by the ". . . "s). However, the worlds in the future of that world contain fewer things. The first world in its future is the one in which L is present and K past. The greek events which were future in the previous world do not exist at all in this second world. The (possible) future event X does not exist, there was a choice be- tween the italic I and the greek X, and the I does exist in this world, while the X does not. All that exist in this world are the italic and the cyrillic future things, together with all the future things elided by ". . . ". The second world represented above counts as the "next" one after the one in which K is present; but this does not presup- pose a temporal ordering which specified that the future event 1 is the one which is earlier than all the other future events. It is the "next" one, because it is the one, of all the worlds in the first world's future, which differs least from it in how many future events it con- tains. It lacks only the greek letters, whereas all the other future worlds lack the greek letters and more besides.

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The third world represented above lies in the future both for the first and for the second of the worlds represented. In this world there are no greek or cyrillic future things, only italic ones, together with whatever further future things were elided in the previous worlds by ". . .": there was a choice in the second world between the italic m and the cyrillic M, and in the future world the italic m is the one which exists. There are many more "branchings" which have not been represented above. Each world contains a vast number of future things, and each world in the future of another contains fewer future things. The things the one world has and the other one lacks are like a "branch" that has "fallen off" the tree of time. Branching time is another possibility which is open to the modal theory.

There is yet another possibility worth mentioning. In possible worlds semantics, as developed by Kripke, Montague, Lewis and Cresswell, it is possible for a thing to have a property in a world without existing in that world. A property of a thing is taken to be a function from each individual to the set of possible worlds in which the individual has that property (or equivalently, a function from each possible world to the set of individuals in that world which have that property). There is no set-theoretical difficulty in defin- ing a function which maps an individual onto a set of possible worlds in which it does not exist (or which maps a possible world onto a set of individuals which do not exist in that world). Thus for in- stance a thing has the property of being nonexistent in precisely the worlds in which it does not exist. By drawing on this option, we may frame the hypothesis that the actual world does not contain existing, branching futures, nor even a linear actual future, but that future things do not exist in the actual world at all. Hence we can present a metaphysics of an open, or rather an empty, future, as described earlier.

Pursuing this further, we could posit that past things, as well as future ones, do not exist at all in the actual world. All that ac- tually exists is the present. Things have the property of being past, not by existing in the actual world and having the property of pastness, but by existing in some other possible world which is "in the actual world's past". Something which is past in w is something which did exist in w: but this means, not that it does exist in w with a special property, but that it does exist in some other world which is related in a distinctive way to w, a world in w's past. In some ways, this seems right. Standard theories of time make it true, for instance, that there are dodos (though it is allowed that there are no dodos which are simultaneous with the theorist's theoris- ing). But it is plausible to object that it is strictly false that there

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are dodos, unless this is misconstrued as simply meaning that there were dodos (and this is not what "there are" really means at all). There are no dodos. There were dodos, because if some past time had been present then it would have been the case that dodos exist. Yet in the actual world it is not the case that dodos exist. This hypothesis, of the "emptiness" of both the past and the future, por- trays the worlds in the actual world's past and future as a sequence of simple presents:

K L M

For each of these worlds, the things in their future and their past do not exist in that world itself. The things in a world w's future or past have the properties in w of being future and past, in virtue of existing in other worlds than w.

This conception of worlds, as truncated to include only the pre- sent, requires us to rethink the modal theory of time in a number of ways. For instance, in defining an accessibility relation between possible worlds, I spoke of comparing possible worlds and noting which of the things in them are past, which are present, and which are future. One world will be "temporally accessible" from another only when they contain the same things and differ only in a systematic transposition of the properties of pastness, presentness and futur- ity. This conception of accessibility will have to be modified if worlds are construed as containing only things present. We will have to require "temporal" accessibility to depend on a systematic transposi- tion of the properties of pastness, presentness and futurity, with respect to the worlds in question, and with respect not only to the things which those worlds contain, but also with respect to things which have properties with respect to those worlds even though they do not exist in them. With care, however, it can be seen that the definitions of "temporal" accessibilities, and the rest of the modal theory of time, can be recast under the hypothesis that only things present actually exist.

Which hypothesis should we accept? Does the actual world con- tain both existing past, and existing future things? or existing past but not existing future things? or neither? or a single string of past things and branching strings of possible future things? I think the first hypothesis is the right one, but I will not defend it here. The first task we should undertake is not that of deciding which hypothesis is true, but simply that of understanding some of the ways in which

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it is possible to think about time-whether or not such ways of think- ing turn out to be defensible in the long run.

To sum up, I am claiming that we can use three properties, past, present and future, to define a temporal ordering, provided there are enough possible worlds in which the properties of past, present and future are redistributed in various ways. I have not given arguments to show that there are the required sorts of possi- ble worlds. Nor have I said anything about how the properties of past, present and future are to be understood. They seem, in fact, very likely to be unanalyzable and utterly mysterious. What I am claiming, primarily, is that if time is to be real, or at least if it is to be something which really passes, then there must be possible worlds and properties of the kind posited by one of the hypotheses described above. If you can argue against the admission of such properties, then those arguments can be used to support McTag- gart's conclusion that Time is Unreal. I am strongly inclined to that conclusion, though I do not defend it here. However, that con- clusion will then rest, not on the internal incoherence of the very idea of Time, as McTaggart urged. Rather, 'if Time is Unreal, that is because there are no such intrinsic properties as those of pastness, presentness and futurity, with the modal characters described in the modal theory.

Let us return to McTaggart's classic argument. He distinguished what he called an A-series from what he called a B-series. The A- series is ordered using the properties past, present and future (and degrees thereof. The B-series is ordered using a temporal earlier/later relation. McTaggart claims that in order for the B-series to exist at all, it must be definable in terms of the A-series. Yet, he argues, that cannot consistently be done. Hence all that really exists in the world must be, at most, some ordering of things, a "C-series", which-like spatial and many other orderings-fails to qualify as genuinely temporal. Time is unreal.

The crux of McTaggart's argument is that we cannot consistently define the B-series from the A-series without vicious circularity or regress. It is this claim which is challenged by the modal theory of time. Time is real; or at least, time is no less real than possible worlds.

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