(c) Aristotelian Society 1998.
Aristotelian Society Supplementary Volume 72 (1998), 29-43
For my comments at the 1998 Joint Session symposium on Mr Lucas'
reply to this paper, see my
Transcendental Tense: Reply to Mr Lucas.
Kant's Critique of Pure Reason (1781 - all references are to the Kemp Smith translation listed) takes our knowledge of time to be transcendental, by which Kant means that it is
occupied not so much with objects as with the mode of our knowledge of objects in so far as this mode of knowledge is to be possible a priori (p. 59).
That is, Kant takes our knowledge of time to be
(a) a mode of knowledge of objects that do not include time, or times, and
(b) a mode of knowledge that is possible a priori, i.e. which need not be derived from experience.
Why should we accept these claims? Suppose for example that at some time T I know, because I can see, that it is raining, i.e. raining now. Why is the time T, or the present, not an object of which I thereby know that here there is rain at it: why is this knowledge only a mode of knowledge of rain? And what, if it is, makes this mode of knowledge possible a priori?
Part of the answer lies in Kant's view that time is 'not an empirical concept that has been derived from any experience' (p.74). There is something to be said for this. Suppose I see that a digital clock shows first 1 and then 2 seconds past midnight. I do not have three experiences here, one of each of these events - the appearance of a '1' and a '2' - plus an experience whose content is that '2' is later than '1'. My only experiences are of '1' and '2'. What fixes the time order I thereby perceive these events to have is not a third experience but the time order of the two experiences I do have: the event I see later ('2') is the event I thereby see to be later.
While there are some apparent exceptions to this (Dennett 1991 p.191), it is certainly the general rule. It raises the question of how I know that I see '2' later than I see '1', but that we need not discuss. For what matters here is not how we do know the time order of our experiences, and of the other events we thereby perceive, but the obvious fact that we are not given this knowledge by a special kind of experience of succession; since there is no such thing.
Thus understood, this consequence of Kant's claim that our knowledge of time is not derived from experience may well be true. But Kant claims more than this, and in particular that without a concept of time we could have no experiences at all. By this he means that all our experiences are experienced as being in time; just as, when an experience is of something - e.g. of a clock showing '1' - what it is of is experienced as being in space. This is what he means by calling time 'the form of inner sense', i.e. of our experiences, and space, which he also takes to be an a priori concept, 'the form of outer sense', i.e. of whatever (if anything) our experiences are experiences of.
So far perhaps so good. I say 'perhaps' because we might well wonder if a concept of time is really necessary for experience. Might not animals, for example, have experiences without any concept of time? They might indeed, if a concept has to be something which its owners are or could be conscious of having. But here I shall grant, for the sake of Kant's argument, that a concept need only be a component of beliefs, or of other propositional attitudes, and that these need not be conscious. For then, even if a mere vegetable could have experiences without having any concepts, I do not think any agent could.
My reason for thinking this is that I, like others (e.g. Davidson 1963), take actions to be events or facts whose causes include some of the agent's desires and beliefs, conscious or otherwise. This being so, agents do need concepts in my weak sense. In particular, as we shall see, they need a concept of time in order to make them capable of timely actions, i.e. actions whose success in getting agents what they want depends on when those actions are done. And as we must all be capable of some timely actions - like drinking, eating and escaping predators when necessary - in order to survive, we all need a concept of time. So if, as Kant says, we cannot derive this concept from experience, it must be a priori.
So far then, let us assume, so good. But this hardly entails Kant's other claim, that time is not an object. By this he means that
time is not something which exists of itself, or which inheres in things as an objective determination ... [for] ... Were it self-subsistent, it would be something which would be actual and yet not an actual object. Were it a determination or order inhering in things themselves, it could not precede the objects as their condition, and be known and intuited a priori by means of synthetic propositions (p. 76).
In other words, Kant rejects both Newton's view that time is an entity in its own right, which could exist even if nothing else did, and Leibniz's contrary view of time as the temporal order of objects, which could not therefore, in Kant's words, 'precede the objects as their condition'.
Kant reject these views because, he says, those
who maintain the absolute reality of space and time, whether as subsistent [Newton] or only as inherent [Leibniz] ... have [either] to admit two eternal and infinite self-subsistent non-entities [i.e. Newton's absolute space and time; or - Leibniz] are obliged to deny that a priori mathematical doctrines have any validity in respect of real things (for instance in space) (pp.80-1).
The stock reply to Kant's point against Leibniz is that, as we now know, a priori mathematical doctrines, like the axioms of Euclidean geometry, do not 'have any validity in respect of real things': the geometry of space, or of spacetime, is not knowable a priori. And one reply to Kant's point against Newton is that an absolute spacetime may well be a self-subsistent entity which, on a natural reading of general relativity, interacts causally with the objects it contains, its local curvature affecting their inertial properties and being affected in turn by how their masses are distributed (Mellor 1980).
Such replies to Kant are well known and I shall not discuss them further. The Kantian objection to real time I do need to discuss is the one in his first antinomy (pp. 396-7), where he argues that time can neither lack nor have a first moment. It cannot lack a first moment because without such a moment an infinite sequence of moments, coming into existence by becoming present and then passing away, must have been completed before the present moment. But this Kant says is impossible, because one cannot reach the end of an infinite sequence. And time cannot have a first moment, To, since To could only have come into existence, by becoming present, if there had been an earlier moment when To did not exist. But then To would not have been the first moment after all.
The importance of these arguments here is that they both assume that time flows, i.e. that times, and objects located at them, successively possess the transient monadic properties of being future, present and past, properties which I shall follow the custom of calling 'tenses'. (These properties must not of course be confused with the variations in the spelling of verbs in English and some other languages that grammarians also call 'tenses'.) For if, as I (in my 1981) and others have argued, there are no tenses in this sense, and time does not flow, then neither the thesis nor the antithesis of Kant's antinomy poses any problem at all. For then there can easily be infinitely many moments before any moment, since there is no infinite process, of them becoming first present and then past, to be completed. But equally, there can easily be a first moment, since it, like any other moment, can exist without coming into existence from an earlier state of non-existence. In short, Kant's first antinomy is obviously no threat whatever to the view that time is real but non-flowing, whether it be a Newtonian thing-in-itself or a Leibnizian ordering of objects by an unchanging later than relation.
Upholders of tensed views, who think there is a flow of time that takes objects from the future to the past via the present (with or without taking them in and out of existence), may of course be able to meet one or both of Kant's arguments. But that is another issue, which I shall not discuss. For even if a tensed view of time can cope with Kant's first antinomy, it faces other problems: like McTaggart's (1927 ch.33) proof that the flow of time entails a contra diction; and the difficulty of reconciling it with Einstein's special theory of relativity (Tooley 1997 ch.11). These and other objections to such views provide my first reason for taking a tenseless view for granted in what follows. My other reason is a desire to encourage those still oddly reluctant to accept such a view by showing how it can make more sense of much of what Kant says about time than Kant himself does. That is the main object of this paper.
Let us start by looking more closely at the concept of time that agents need in order to make them capable of timely action. As Perry (1979) and others have shown, this capacity requires a concept not just of time but of tense. Take an action a, my turning on my television set to catch a televised event e. I have long known e's location in McTaggart's B series, i.e. its tenseless time Te. But that knowledge will not make me act until I come to believe that it is now Te, i.e. that e is now present. This is the present tense belief b whose acquisition at any tenseless time Twill cause me to do the action a at T.
Yet only if I do a at Te will this action succeed. That is, only then will it achieve the object of my desire to see e - because only then is my belief b, that e is present, true. (To simplify the argument I shall ignore the short time it takes e's image to reach my screen and for me to act on my beliefs.) This is an instance of the general fact, invoked by so-called 'success semantics' (Whyte 1990), that truth is the property of beliefs which makes the actions they combine with desires to cause succeed, i.e. which makes them achieve the objects of those desires. This is the fact that enables a tenseless view of time to make better sense than Kant does of much of what he says about time.
To show how, I must first sketch what, on my tenseless view of time, makes my belief b true at Te (for more details see my 1981 ch.2). It cannot be the fact that, at Te, e has the property of being present, for in my view there is no such property. What on my view makes b true at any tenseless time T is e's occurring at T; a truth condition which, unlike that of any tenseless belief, varies with T - and does so in a way that we can use to define its temporal content. For if b's truth condition did not vary with T in this particular way it would not be a present tense belief. If, for example, e's occurring at T made b true at all times later than T, b would be the belief that e is past; just as, if that fact made b true at all earlier times, b would be the belief that e is future. And so in general: the temporal content of any tensed belief can be given by a function from any tenseless time T to its tenseless truth condition at T.
This idea is however open to an objection that has now become too widely accepted to pass without rebuttal. Suppose my function gives b's content, i.e. tells us what anyone, at any time, believes by believing that e is present. Then you and I can believe the very same thing and yet my belief be true and yours false - as will happen if I believe that e is present at Te and you believe this at any other time - and that seems absurd. This is why Kaplan (1989 §VI) calls b's function from times to truth conditions its 'character', reserving the term 'content' for a single truth condition, like e's occurring at a specific T. But there are objections to this too. An important one here is that it begs the question against advocates of tensed time, who think that b's content is a single truth condition - that e is present - but one that only obtains at Te. And even on a tense less view of time, calling b's truth condition at T its 'content' at T carries a high price, since it entails that
(i) b has the same content at T as the tenseless belief that e occurs at T,
(ii) b's content varies with T and so changes every instant, and
(iii) to know b's content, i.e. what I believe by believing that e is present, I must know what time it is.
The absurdity of these consequences is what makes me prefer to say that the content of a tensed belief like b is given by the function from T to its truth condition at T. (This does not of course make me identify b itself with the tenseless belief that b is true only at Te; on the contrary, it stops me doing that, as we shall see below.) This terminological preference is reinforced by the fact that, on any view of time, it meets the desideratum that a belief's content be the primary (i.e. truth-related) meaning of sentences that express it. It does so because the temporal meaning of 'e is present' is clearly given, for any time T, whether tenseless or tensed (e.g. yesterday), by stating the function from T to its truth condition at T, namely that e occurs at T. For to know this meaning, i.e. to know when to use this sentence, and what temporal inferences to draw from its use, we need only know that it is true when and only when e occurs, whether we use tenseless or tensed terms to say when that is. Similarly for all other tensed sentences: the aspects of their meaning that relate to truth, and should therefore be the contents of the beliefs they express, are obviously given by the functions from any time to their truth conditions at that time.
Similarly too for other so-called indexical sentences, like 'Here we are in England or 'I am a philosopher'. The truth-related meanings of these sentences are also functions whose values are truth conditions, like the function from any place S to the condition that S is in England, or from any person X to the condition that X is a philosopher. Letting these be the contents of those beliefs may admittedly make me believe truly what a French deconstructionist just over the Welsh border could believe falsely; but this is a small price to pay to stop me needing to know exactly where I am to know what I believe by believing I'm in England, or who I am to know what I believe by believing I'm a philosopher.
One last point of terminology. I propose to call these contents of temporally, spatially or personally indexical beliefs 'propositions'. Many philosophers, like Lewis (1979), will not do this, precisely because, on a tenseless view of time (and on its spatial and personal analogues), they do not correspond to single truth conditions. But 'proposition' is a term of art, and I prefer the older art that applies the term to whatever a belief's content may be. This art begs fewer questions than one that artfully confines the term to a favoured candidate, e.g. a set of possible worlds, for being the content of a belief. This is why I prefer to call the contents of all beliefs 'propositions', and then, for the reasons just given, to identify all of them with functions from times, places and people to their truth conditions at those temporal, spatial and personal loca tions. Non-indexical propositions are simply the constant functions, i.e. those that have the same truth conditions as values for all these arguments.
From all this three relevant things follow. First, since the tenseless truth conditions of all seriously tensed propositions (as opposed to tautologies, contradictions and propositions which say that something always was, is and will be so) vary with time, these propositions differ from all tenseless ones. Thus the truth condition of the proposition that e occurs at Te is always the same, namely that e does occur at Te. So if this proposition is ever true, it is always true; whereas the proposition that e is present is true only at Te. This is why no seriously tensed proposition P can be identical with any tenseless proposition P': P and P' cannot be identical if there are times when P is false and P' true.
That is my first point: a tenseless view of time need not, and mine does not, entail that seriously tensed propositions are identical with tenseless ones or - to put the point in linguistic terms - that seriously tensed sentences can be translated by tenseless ones. On the contrary, my view entails that they cannot, and hence that no seriously tensed belief can be identified with any tenseless belief.
The second relevant consequence of my view is that all agents need some seriously tensed beliefs. I noted earlier that the success of many actions, including those on whose success we depend for our survival, depends on when they are done. But then it follows, from the link between the truth of beliefs and the success of the actions they cause, that all agents need beliefs that are true only at certain times, i.e. seriously tensed beliefs. This does not of course ensure that we will only have such beliefs when they are true: the perceptions that generate tensed beliefs, like looking at a clock to see the time, are as fallible as those that generate any other beliefs. But it does mean that, if only to survive, we must have generally reliable ways of getting tensed beliefs when they are true and of losing them again when they are false. Our mental machinery must include something like a clock. That is my second point.
The third and most important consequence of my view is that, despite its requiring agents to have irreducibly tensed beliefs, many of which are true, it still makes time tenseless. It does this by showing how all tensed beliefs are made true or false at any time by purely tenseless facts, including facts about when those beliefs are held. By so doing it shows not only why we need, but how we can have, true tensed beliefs in a tenseless world. And by doing this it gives the only explanation known to me of how true beliefs enable timely action.
Where does this tenseless tale leave Kant's view of time? The short answer is that it makes much of what Kant says about time true of tense. Take my tensed knowledge of the televised event e. An hour before e I know that e is future. When I see e occur, I know (if e lasts longer than the time its image takes to reach my screen) that it is present. An hour later I know that e is past. What, on the view I have just outlined, is the temporal aspect of these three pieces of knowledge - of e's successive futurity, presentness and pastness - knowledge of?
This knowledge, like all knowledge that is not just know-how (e.g. knowing how to ride a bicycle), I take to be reliably caused true belief, e.g. the belief that e is present. So if I get this belief at Te, when it is true, and do so because a reliable clock tells me it is now Te, then I know that e is present. Similarly for my earlier knowledge that e is future and my later knowledge that e is past. These are all cases of knowing that P, for some true tensed proposition P which, as we have seen, differs from any tenseless proposition.
Yet I agree with Kant that none of this tensed knowledge is, in his sense, knowledge of any temporal object. For as time is for Kant, so tense is for me 'not something which exists of itself [the future, the present, the past], or which inheres in things as an objective determination [of things as being future, present or past]'. This is why I believe of tense what Kant says of time in the rest of the same sentence, namely that it 'does not ... remain when abstraction is made of all subjective conditions of its intuition' (p.76). Or in other and perhaps even plainer words, the world contains no tenses: neither particular times like tomorrow, today or yesterday, nor such properties of objects as being future, present or past. Tense is just a way of representing the temporal locations of objects, a mode of knowledge that all agents need to use but which, as Kant says, 'in itself, apart from the subject, is nothing' (p. 78).
So far so good for the first of Kant's claims, taken as a doctrine about tense. Our tensed knowledge is not knowledge of tenses as objects. What about his second claim? Is this tensed mode of knowledge possible a priori, i.e. not necessarily derived from experience? The answer to this question depends on what 'experience' means here. If it means 'experience of tense', and we cannot experience what does not exist, then on my view of time our tensed mode of knowledge cannot be derived from experience and so must be a priori. But if 'experience' can mean 'experience of time', this remains an open question.
Let us take these options in turn. First, it is worth stressing that, contrary to much received wisdom, the tenses of objects, even if they do exist, are not perceptible as such. This assertion may seem absurd. When we see objects, do we not see them as tensed and, in particular, as present? That is, do the objects we see not cause us to have present tense beliefs about them? Well, often they do - although not I hope when astronomers see events which they know are happening light years away and hence years past. But certainly, as most earthly objects take longer to change than light takes to reach us from them, our eyes can mostly give us true present tense beliefs about what they show us. I do not deny that. What I deny is that anything ever looks - or sounds, or tastes or feels - present, as opposed to past or future. If it did, a replay of my televised event e could look wrong by making e look present when it should look past. But it couldn't, because there is no such thing as looking present, as opposed to looking past, or looking future.
The fact is that even if objects do have tenses, these are not properties we can perceive as we perceive the colours, shapes and temperatures of objects. My seeing an object as present, or as past, is always an interpretation, based on some feature of it that is perceptible, such as its being a glow-worm as opposed to a star. This being so, we have good reason to regard tense as transcendental in Kant's sense, even on a tensed view of time. For even if tenses exist, we cannot acquire our concept of tense by being taught to recognise instances of pastness, presentness or futurity that are perceptible as such, for there are none.
So reading 'experience' as 'experience of tenses' makes both of Kant's claims true of tense: our tensed knowledge is not knowledge of tenses as objects, but a mode of knowledge of other objects; and it is a mode of knowledge that is not derived from any experience of the tenses of objects, for we have none. And the tenseless explanation of these otherwise mysterious facts is very simple: our indispensable and irreducibly tensed beliefs are both made true, and caused in perception, by tenseless facts. This distinction, between tensed beliefs on the one hand, and their tenseless truth-makers and causes on the other, seems to me far clearer and more credible than Kant's own distinction, between merely empirically real tensed phenomena and real and tenseless but otherwise unknowable noumena. This is why I say that a tenseless view of time can explain far better than Kant himself why much of what he says about time is true of tense.
Yet about time itself most of what Kant says is false. First, not all our temporal beliefs are tensed: many are tenseless, like my belief that a digital clock's '2' is later than its '1'. And this belief, unlike a tensed one, can easily be made true, and be reliably caused, by the perceptible tenseless fact that '2' is indeed later than '1'. So it can easily amount to knowledge of that fact. We could therefore acquire our tenseless 'later than' concept by learning to recognise instances of the later than relation; and this suggests that our 'later than' concept may not be possible a priori in Kant's sense. We must of course be innately capable of acquiring it, but whether we could do so without perceiving some of its instances remains an open question.
To answer that question, we must first ask what links our 'later than' concept to our tensed concepts of future, present and past. The links are obvious: to be future is to be later than the present, and to be past is to be earlier than it. But which comes first? May not our tenseless 'later than' concept depend on our tensed ones, so that what 'later than' really means is, roughly, 'more future or less past'? No, for the following reasons.
First, although McTaggart does not say so, we use the same 'later than' concept to order his tensed A series locations (like yesterday, today and tomorrow) as we use to order his tenseless B series locations (like 10, 11 and 12 July 1998). Tomorrow, for example, is a day later than today in the very same sense in which 12 July is a day later than 11 July. What distinguishes the A and B series is not a different relation but different relata. Objects and times are ordered in the B series by how much later or earlier they are than each other; in the A series, by how much later or earlier they are than the present. This is why the later than relations that fix where objects and times are in the B series also fix where, at any instant, they are in the A series. The A series is therefore not independent of the B series; it is simply the B series plus an ever-changing present. This is what makes our tensed concepts of future and past depend on our 'later than' concept, and not the other way round. 'Future' does just mean 'later than the present', and 'past' does just mean 'earlier than the present'.
What of the present itself? I have said that we have an innate and indispensable ability to form irreducibly tensed beliefs, an irreducibility which reduces, as we have just seen, to that of the present. But this does not make the present indefinable in tenseless terms. On the contrary, as we have also seen, any tensed belief's irreducibility to any tenseless one can easily (and I think only) be explained by its content being given by a function from any tenseless time T to its tenseless truth condition at T. This makes a present tense proposition one whose truth condition at any time T is that something (like the event e) is simultaneous with T - simultaneity being just one more tenseless temporal relation. The contents of present tense beliefs are thus just as easily definable in tenseless terms as are those of past and future tense beliefs.
However, to explain our ability to form tensed beliefs, we must do more than give their contents in tenseless terms. We must also say how they get those contents. What, at any tenseless time T, makes present tense beliefs refer to T, in the sense of making T part of their tenseless truth conditions at T? The answer (given in more detail in my 1989 §§3-5) lies in two facts cited earlier, namely that
(1) my beliefs only cause me to act when I have them, and
(2) truth is the property of beliefs which ensures that the actions they combine with desires to cause will succeed.
Consider again my present tense belief b that the televised event e I want to see is on now. (1) tells us that only when I acquire this belief, at some time T, will it cause me to turn on my television. (2) then tells us that this action of mine will succeed if (and in this case only if) the belief b that causes it is true at T, i.e. if e occurs at T. This is what puts the time T at which I get my belief b into b's truth condition at that time, namely that e occurs at T. In other words, what enables my belief b to refer at any time to that time - which is what makes it a present tense belief - is the fact that I will only act on this belief when I get it.
This is why a present tense belief needs no special mechanism to make it refer to the time at which it is held. Its temporal contiguity with the actions it causes is all it needs to index its temporal reference in this way. That, I have argued elsewhere (Mellor 1989), is why we need no special theory of reference for present tense beliefs, any more than we need one for first-person beliefs, or for beliefs about what is here. For just as the near-contiguity of our actions to their mental causes makes my beliefs cause me to act only when I have them, so it also makes them only cause me to act, and only here, i.e. where I have them. Causal contiguity is all it takes to enable the truth-conditions of any of these indexical beliefs to include when, where or by whom they are held.
But what then distinguishes beliefs of these three indexical kinds - now-, here- and I-beliefs? All of them, I maintain, have contents given by functions from when, where and by whom they are held, to truth conditions, at those temporal, spatial or personal locations, which include one or more of those very locations. Causal contiguity, by linking any belief to all three of these locations, can index it to any of them. But what then fixes which of these locations - if any - it is in fact indexed to, thereby fixing whether it is a now-, here- or I-belief?
The answer lies in the different causal links which beliefs of these three kinds have to other beliefs. Take my beliefs that I live in Cambridge and that I am Hugh Mellor: these two I-beliefs dispose me to believe that Hugh Mellor lives in Cambridge. And as for me, so in general. Let 'X' be any non-indexical name or definite description of a person. Then, for any F, to believe 'I F' and 'I am X' is to be disposed to believe 'X Fs'. So no belief, of anyone who believes 'I am X' for some X, which does not, for any F, dispose that person to believe 'X Fs', can have a content of the form 'I F'. That is, it cannot be an I-belief.
But it can be a now- or here-belief. Take my belief that my televised event e is on now. This does not combine with my belief that I am Hugh Mellor to dispose me to have any belief of the form 'Hugh Mellor Fs'. But it does dispose me to believe, of whatever tenseless time T I believe to be present, that e is on at T. And similarly, mutatis mutandis, for my here-beliefs.
Thus what distinguishes now-beliefs from here- and I-beliefs is their being linked to non-indexical beliefs about times as opposed to places or people. So to have distinct beliefs of these three kinds I must be able to draw a generic distinction between times, places and people; and this means being able to distinguish the different relations that differentiate entities of those three sorts. Thus I must be able to distinguish the later than relation, which differentiates times, from relations like North of, which differentiate places, and child of, which differentiate people. In short, I need a 'later than' concept that differs from any spatial or personal analogue of it. And how I could acquire it except by learning to recognise perceptible instances of the later than relation, I cannot imagine. This is why I think that our 'later than' concept is one we can only acquire by being shown some of its instances.
Yet this concept, and hence our concept of the present, may still be a priori in Kant's sense. Consider again my seeing '2' succeed '1' on a digital clock, which I noted in §I needs no special experience of succession over and above my experiences of seeing '1' and seeing '2'. This implies that I can perceive that one thing is later than another without having any single experience which has this, or any other, temporal content. So perhaps I need never have any such experience. Perhaps all I need, to acquire the 'later than' concept, is to have experiences - any experiences: since, whatever they are, they will come in a self-intimating time order. In other words, the fact that the later than relation is Kant's form of inner sense may suffice to enable us to distinguish it from any spatial or personal relation. And that in turn may suffice to distinguish our now-beliefs from our here- and I-beliefs, i.e. to give us a concept of the present - and hence, since all other tenses are definable by how much later or earlier they are than the present, all our other tensed concepts.
That is how, with a little charity, a tenseless view of time can admit that our concept of tense may be, in Kant's sense, possible a priori, i.e. may not have to be derived from the content of any experience. More importantly, and with no charity at all, a tenseless view can show that and how our tensed knowledge is not knowledge of tenses as objects, but is simply an irreducible mode, that all agents need, of knowledge of the temporal whereabouts of real objects, including tenseless times. By showing this it shows how, while time itself is real, the apparently ever-changing tenses of objects that constitute time's apparent flow are, as Kant would put it, transcendentally ideal and only empirically real - i.e. not real at all.*
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Lewis, D., 1979: 'Attitudes "De Dicto" and "De Se"', Philosophical Review 88, 513-43.
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Mellor, D. H., 1981: Real Time, Cambridge: Cambridge University Press.
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* Earlier versions of this paper have been given at intervals over several years to meetings at King's College London and at Cambridge, Birmingham and Nottingham Universities, and also at conferences in Copenhagen, Leeds, Edinburgh and Santa Barbara. I am most grateful to those who took part in these occasions (especially to Tony Brueckner, who replied to my talk at Santa Barbara) and also to private discussions of my penultimate draft with Alexander Bird and Paul Noordhof.