CHAPTER IX

Temporal Asymmetry

§1. Actions Versus Sequences of Conditions. Armed with the principle that temporal separation requires action and with the distinction between action as a component of entities–progressive action–and action as a condition of entities–complete action–we can now show just how action enters into the foundations of temporal relational properties.

I shall try to show, in the next section, that action is asymmetrical. This means that if any condition is the result of an action, then this action is not a result of that condition. So if time can be founded by action, then both relational properties corresponding to symmetrical temporal relations and relational properties corresponding to asymmetrical temporal relations can be founded by action. Not only will relational properties corresponding to at a different time from and between be founded, but also those corresponding to the asymmetrical relation prior to. Accomplishing the task for prior to automatically accomplishes it for the symmetrical relations. To complete the program of the action view of time, it suffices to carry it out for prior to.

Over and above these temporal relations, there are of course the time modes, past, present, and future. It is a mistake, though, to suppose that these have foundations in entities. An entity is said to have a time mode not because of any special property it has of itself or because of any property it has in view of there being minds. Rather, the claim that an entity is past is made true merely by its occurring before a given time, the time of the claim. Sentences contain various devices that help us interpret what propositions they express in certain circumstances. Tenses are one such device. A tense in a sentence indicates that the proposition expressed by that sentence is about a time to be fixed relative to the time of utterance of the sentence. To treat time modes as realities is to project the tensed character of sentences that express propositions true of an untensed reality onto that reality.¹ Since the action view of time is intended to be applicable only to temporal relational properties, time modes, not being properties at all, fall outside its scope.

Suppose action does account for temporal asymmetry. It turns out then that it is an a priori matter, not a matter of empirical confirmation, that there are events that have asymmetrical temporal relatedness. This can be seen as follows. First, there must be events that are temporally related. Otherwise, there would not be the necessities across time required by the practice of action on prior experience. Second, if there is temporal relatedness, there must be actions. For without a ground in action for temporal relatedness, the relatedness would have to be seen as derivative from relations themselves. But this would be incompatible with necessities across time. Adding to these two premisses the supposition that an action and its result are the basis for temporal asymmetry, we arrive at the conclusion that there must be events that have asymmetrical temporal relatedness. In short, there are events that are separate in time, and, since action is the basis for time, these events will be asymmetrically related. In saying that this is a priori, I mean that there is no human practice, judgment ineluded, whose presuppositions are compatible with denying that there are events that have asymmetrical temporal relatedness.

This view fails to accord with those accounts of temporal asymmetry that ground it on factors less basic than action. A physical basis for temporal asymmetry has been sought in (1) the expansion of the universe and (2) the tendency of systems branching from the main body of a larger system to increase their entropy. Doubtless, if these possibilities prove to contain difficulties, reference will be made to (3) the fact that a neutral meson in an antisymmetric state may decay in such a way as to violate charge-parity symmetry. To right the balance required by special relativity of a charge-parity-time symmetry, the temporal converse of such a decay cannot be realized in nature. Meson decay thus becomes a basis for temporal asymmetry in nature.² There is the additional possibility that temporal asymmetry might stem from (4) the irreversible way in which a measuring apparatus affects the quantum mechanical psi-function characterizing a physical system.

None of these factors of scientific interest needs to be present, however, in a universe in which action is the basis of temporal asymmetry. Such a universe need not expand; it may lack the kind of micro-structure that would permit the application of a statistical entropy concept; there may be no mesons in it; and, finally, it may obey classical rather than quantum physics. Since action is part of the required ontology, temporal asymmetry will characterize time in any universe coherent with human practice. But the proposed scientific factors are not part of the required ontology. So if one of them were the genuine basis for temporal asymmetry, it wou!d be an a posteriori matter whether temporal relatedness is characterized by asymmetry.

If the basis for temporal asymmetry is right under our noses, why have so many powerful minds looked for it so far afield? It surely cannot have escaped their notice that action is at least a prima facie candidate. Why did it seem a bad choice? The answer lies in a metaphysical preconception that has rarely been questioned by scientifically minded philosophers. This preconception is the Ockhamite view that action is merely a temporal sequence of property conditions, that is, of the havings of properties. Let us call such a sequence, with the addition of a result of the action, an ‘action-result״ sequence. Of course, temporal relations between conditions in the sequence are required until a ground for them has been proposed. But these relations cannot be grounded in action since they are part of what action is reduced to in the Ockhamite view.

What will be the basis for an asymmetrical temporal relation in the sequence to which the action is reduced? There are two approaches to this question. According to the first, the basis must be found in the property conditions of the sequence. If we do not consider the contents of these conditions, but consider instead the conditions just as conditions in a sequence, there is no basis for saying that conditions at one end of the action-result sequence are before or after those at the other end. Indeed, there is no basis for saying which end of the sequence contains the result. To establish a basis for these distinctions one must look to the contents of the conditions. In some sequences, a condition at one end is such that conditions like it–those with similar content–can only occur before, never after, conditions like the one at the other end of the sequence. This irreversibility of the seqeunce can be appealed to as the basis for the asymmetrical temporal relations in it and as the basis for calling certain conditions results. A low entropy state, for example, is prior to a higher entropy state if it is the case that the increase of entropy is irreversible.

In the Ockhamite view of action, we are to say that, in a world of exclusively reversible processes, ‘the assertion that a given motion of a ball was ‘from A to D’ rather than ‘from D to A’ expresses not an objective physical relation between the two terminal events of the motion, but only the convention that we have assigned a lower time-number to the event of the ball’s being at A than to its being at D."³ This cannot be disputed if the motion–an action in the sense of praxis–is nothing but the sequence of events terminated at the ends by the ball’s being at A and its being at D. If the motion is but a sequence, there would be an objective basis for the distinction between the beginning and the end of the motion only if some difference between members of the sequence were irreversible.

However, the appeal to irreversibility is unnecessary if the action is not reducible to a sequence of property conditions. It is solely the Ockhamite prejudice that entails a search for facts like (1)–(4) in order to find a basis for temporal asymmetry. Irreversibility is used here in a reduction of time to certain differences between conditions in sequence. The differences might be those between successive entropy states. But since the differences are in sequence, they will be associated with the actions of coming to and ceasing to obtain, as we saw in Chapter VIII, §5. So action is available even here to ground temporal asymmetry. It is only the mistaken Ockhamite view of action that motivates the appeal to irreversibility.

According to the second approach, the basis for the temporal relation in the sequence to which an action is reduced is a causal relation. On the atomistic view, there are temporal relations, and hence no necessary connections across time. Those who try to replace temporal relations by causal relations agree with us in rejecting temporal atomism. One causal theorist expresses his dissatisfaction with temporal atomism by noting that ‘Hume begins his definition of cause by saying, ‘The cause is an object precedent . . . to another’, without appearing to care to know what the expression ‘precedent’ signifies."⁴ But digging behind the temporal relation leads the causal theorist only to another relation: . . for us on the other hand those facts that are conditions of others in a group are said to precede them, and the others are said to follow the first, where these expressions signify no more than the relation of occasional causality.”⁵ But behind the problem of time is the question of how events can have relational properties, whether temporal or causal. This problem is not advanced by introducing a causal relation ‘as a primitive relation for the purpose of then defining temporal betweenness.”⁶ In my view, events have both temporal and causal relational properties on the basis of action. To hold for anything less is to accept the network view that posits property conditions set in isolation by external relations. Ontologically, matters are as hopeless if the relations are causal as they are if the relations are temporal.

Granted that the Ockhamite view is unacceptable, do actions provide what is needed for temporal asymmetry? It will be seen that they do only if they have what I shall call a vectoral character.

§2. The Vectoral Character of Action. To say that action has a vectoral character is to say two things about action. First, it is an objective fact and not a matter of convention that certain conditions are the result of a given action. If being at B is a result of motion from A, then no mere switch of conventions or of perspectives will make being at A the result of this motion. Second, no action is the result of any condition that is itself a result of that action. This does not preclude interaction between systems,⁷ for it allows the action to be influenced by some condition, other than its result, that is nonetheless a condition of the system that ultimately has this result. So, in all, an action is vectoral since its pointing toward any of its results is an objective fact, and since none of its results point toward it as their result. The first vectoral aspect can be called the “objectivity of results” and the second the “asymmetry of action.”

Why are both objectivity and asymmetry needed? Suppose, on the one hand, results are objective but actions are not asymmetrical. It is an objective fact that condition b is a result of action a. But it is possible that a is also a result of b. Action and result can, then, no longer ground the asymmetry of time. Suppose, on the other hand, actions are asymmetrical but results are not objective. The asymmetry of action is then like that of the relation left of. This relation is not grounded in its relata but depends on a point of view. Under one convention or under one point of view, b is a result of a, whereas, under another, a is a result of b, which is no longer a result of a. Under each convention or perspective taken by itself, the action-result structure is asymmetrical in exactly the way left of is asymmetrical under any given perspective. But since there are several conventions or perspectives, it is not an objective feature of any given event to be a result. Consequently, the temporal asymmetry based on the asymmetry of action would not be objectively grounded. Yet our hope is to give prior to an objective grounding. To realize this hope, I must show that actions are indeed asymmetrical and results are indeed objective.

To show that actions are not asymmetrical, one attempts to show that a closed chain of action is possible. If such a chain is not possible then closed time–time represented in the form of a circle–is not possible. The priority relation is symmetrical for closed time. If, along one part of the circle, a is prior to b then there is another part of the circle along which b is prior to a. The failure of previous attempts to deny the possibility of closed time has been observed by Grünbaum.⁸ But he does not note that such attempts fail because they do not ground time in action. The impossibility of closed time must be viewed as a consequence of the impossibility of a closed action chain.

Suppose a is a complete action whose result is b. Suppose also that b itself is a complete action that, through various intermediate results, has the original action a as a result. Now the general feature of a result is that it involves dependence on an action. Since b is a result, it depends on the complete action a. Of course, this is not to say that a is a sufficient condition for b. It is obvious that, in a closed action chain such as this, any action depends on itself. For every action is, indirectly at least, a result of itself. Since b has a as a result and since b was in the first place a result of a, a is self-dependent. How, though, can a result depend on itself? Even a necessary being would not depend on itself, but would instead depend on no being. An individual could depend on itself in the sense that one of its components depends on another of its components. But it is strict self-dependence that is in question here, a state of affairs that is as absurd as owing yourself money in one and the same account. One condition owes nothing to itself. Otherwise, it would be (inconsistently) both itself and a condition other than itself.

Let us now turn to the objectivity of results. Does it not fail in an obvious way in certain cases? When an action-result sequence– a sequence of property conditions associated with an action where the sequence includes the result–is a reversible sequence, there seems to be no objectivity of results. For if this sequence of property conditions is reversible, there is no objective basis for the condition we happen to call a result being a result rather than an initial condition. Only when the action-result sequence is irreversible is there an objective basis for the condition called the result being a result. There is, then, no general objectivity of results, and hence not all actions are vectoral.

This objection holds only when actions are reduced to sequences of property conditions. To show this, let me begin by saying something about the reversibility of sequences.⁹ Suppose a sequence of conditions can be described by saying that condition a is before condition b and condition b is before condition c. In respect to this description, the ‘converse’ description is the proposition that a is after b and b is after c. One simply replaces temporal relational terms by their converses to obtain the sentence expressing the converse description. In Ockham’s view, a given description and its converse purport to describe distinct actions since actions are sequences of conditions, and one gets a distinct sequence when the relation before is replaced by after. Nonetheless, the two sequences, and hence for Ockham the two actions, are not in all cases equally possible, for some sequences are irreversible:

(1) A sequence of conditions is ‘irreversible’ if, though this sequence is possible, the converse description in respect to a description of this sequence is not a description of a possible sequence.

However, a sequence is reversible if it is possible and if a sequence fitting the converse description in respect to a description of it is also possible.

If a sequence is reversible, there is no objective basis in the conditions making up the sequence for saying one of them is before another rather than after it. There is no basis since both arrangements are in themselves possible. Consequently, there is no objective basis for saying that one condition is a result rather than an initial condition. In the sequence view of actions, results do not have objectivity when the sequences are reversible.

Suppose that, as I have argued, actions are not sequences of property conditions. Must, then, actions that have associated reversible sequences lack a vectoral character? I think not. A converse description of a sequence of property conditions associated with throwing a ball is still a description of a sequence associated with the same action, the throwing. The description does not become one of catching, for it was not a description of anything more than a sequence. Of course, the throwing cannot be associated with a sequence in which the initial condition involves the ball being in the air and the final one involves its being in the hand of the pitcher. But that is no more remarkable than that converse descriptions of some sequences of properties cannot be true.

Here we have a new kind of irreversibility, irreversibility not simply for sequences but for sequences with actions. A sequence with an action is irreversible if, though the sequence with this action is possible, this action with a sequence fitting the converse description of the original sequence is not possible. Clearly, every sequence cum action, whether the sequence by itself is reversible or not, is irreversible.

Why not talk about the converse description of the sequence cum action, rather than merely about the converse description of the sequence alone? After all, it would seem that irreversibility for a sequence cum action should be defined relative to the wider description. This will change the description by including in its scope not just the property-condition sequence but also the stages of the complete action. The converse description of the sequence cum action will, for some division of the complete action into stages, have ‘before’ in place of ‘after,’ not just between references to property conditions but also between references to stages of the complete action.

This does nothing to change the preceding conclusion that every sequence cum action is irreversible. The reason is that the converse description only concerns the temporal relations between stages of a complete action. It dos not refer to stages distinct from the ones originally referred to. Hence it does not refer to a complete action, or even a progressive action, other than the one associated with the sequences originally described.

For example, suppose I report that Jones’s wind-up was prior to his delivery of the ball. Here the wind-up and the delivery can be considered stages in a pitch. The converse description of this division into stages of the complete action–the pitch–is that Jones’s wind-up was subsequent to his delivery. The stages referred to, on any division, remain the same under the converse description, since, by hypothesis, only the temporal relational terms have been replaced by their converses.¹⁰ Of course, if the pitch is only a sequence of property conditions, the converse description will not refer to the same stages, since the stages will be different when the temporal relations between their associated property conditions are inverted. But it is not open to us to accept this Ockhamite view of action.

It follows, then, that every sequence cum action is irreversible relative to any division of the corresponding complete action into stages. For:

(2) A sequence cum action is “irreversible” relative to a division of the corresponding complete action into stages if, though the action with this action-result sequence is possible, either the action-result sequence that fits the converse description or the sequence of stages of the complete action together with a result (relative to the given division) that fits the converse description is not possible.

How can this be applied to the question of the objectivity of results? In the sequence view of actions, the irreversibility of an action-result sequence is a basis for the objectivity of results. For, since the converse description of an irreversible sequence cannot be satisfied, a condition that is a result could not trade roles with the condition that is the initial one in the sequence. Similarly, when actions are treated as irreducible, the irreversibility of a sequence cum action is the basis for the objectivity of results.

Sequence cum action irreversibility implies that either the converse description of the action-result sequence or the converse description of a sequence of stages of the complete action taken together with the result cannot be realized. Either, then, the sequence of property conditions is irreversible. Hence a result in the sequence could not be an initial condition. Or the sequence of action stages followed by a result cannot occur in reverse order. Otherwise, in a pitch the delivery might be before the wind-up. So a result of such an action could not be an initial condition of it; that would require that the stages could also occur between inverted time relations.

This means that even the motions of classical mechanics are, together with their sequences, irreversible. Of course, the sequences by themselves are reversible. The sequence of conditions associated with a ball’s rolling from m over a smooth surface to nis reversible. But it is an objectively based fact that the result is the condition the ball has of being at n. For the converse description–that the moving from m is after the moving to n–is unrealizable. Action even in the uneventful universe of mechanics is vectoral; it involves the surge toward results that has time as a product. Attempts to ground the temporal relation prior to on entropy increase, cosmological expansion, meson decay, or the collapse of the wave packet to an eigenstate under measurement are based on the assumption that, if sequences of property conditions were reversible, there would be no objective difference associated with that between before and after. This assumption is false since actions are not sequences of property conditions and since the objective difference between action and result can be associated with that between before and after.

Still, before would not be asymmetrical if a closed action chain were possible. The irreversibility of action, and hence the objectivity of results, would establish only that there is a difference between before and after along a given path in a closed time. It would be an objective fact that a is before b along the path in the circle. It would not, without assuming the asymmetry of action, show that there is no other path along which a might be after b instead of before b. Thus, the need for both objectivity of results and asymmetry of action.

§3. Results of Actions. Before appealing to results in order to ground time, the relation result of must itself be grounded.

It is not a mere tautology that actions have results in the sense of terminal results. For initially, at least, the suggestion that there might be actions that never end is plausible. Closer consideration is needed to show that such actions are not possible. For the doing of an action is not a multiplicity of entities. It is a single condition corresponding to a single progressive action. However, if it were a multiplicity, it could even be an infinite multiplicity. There is no reason why such a multiplicity, when ordered in a sequence, should ever end. But, in fact, there is a unitary doing of an action corresponding to each progressive action. This will entail that the doing of an action needs to be bounded. If the action went on without end, there would never be the unitary entity that is the doing of it. The doing of it would, as it were, be on the way but would never arrive. A result is then essential to an action, for only if there is a result is there one of the boundaries needed for there to be the doing of an action. Without a limit, an action would be incomplete and would lack the unity essential to any entity. Doing an action is then called a complete action in recognition of the fact that it is an entity only because there is a result.

Disagreement over this point arises immediately upon taking the Ockhamite view of action. For if doing an action is a multiplicity of property conditions in sequence, doing an action is not a single entity. There is no objection to an unending multiplicity, since a multiplicity is not a genuine entity. Thus the Ockhamite will insist on the possibility of actions that never end. This criticism is misplaced if, as I have claimed, doing an action is not a multiplicity.

The unitary character of actions has the further consequence that Zeno’s paradox of the dichotomy¹¹ does not arise for uninterrupted motion. The problem is how one can walk to the end of a certain distance since doing so requires doing an infinity of actions. One must go half the way, then half of what remains, and so on without end. But if walking is an action, there is only one doing of walking corresponding to the one progressive action of walking. There is not an infinity of walks to be done in order to get to the end. The stages of the walk defined by the dichotomy are only components of the complete action and are thus not distinct from the single complete action. But saying there are unitary actions does not imply there is a temporally shortest action. An action can be of any finite duration.

Of course, if the walker were what Grünbaum calls a staccato walker–one who rests, but for progressively shorter times, at each division defined by the dichotomy–the walker would reach the end only by doing an infinity of walkings.¹² Zeno’s problem clearly arises here, even though in my view it did not arise for the uninterrupted walker. The durations of the walks and of the rests between the walks become shorter and shorter, so that the sum of the times of the walks and the rests is finite. If the staccato walker can complete the infinity of walks, then the unitary theory of action was not needed in the case of the uninterrupted walker. But can the walker complete the infinity of walks?

The problem is not the mathematical one of showing that the sum of the durations is finite. It is, rather, the physical one of showing how the walker can perform an action in all of those durations with the result that he or she is at the end. The dichotomization yields no last duration. But can a last action be dispensed with if we assume that the walker gets to the end? The walkers being at a point beyond that where he or she is after any one of the walks in the infinite sequence is, in this case, not happenstance but is, rather, a result of walking. The only alternative to saying that it is a result of walking is saying that the staccato walker got to this point not by walking or any other action, even though he or she was not at that point earlier. This possibility is clearly not available to us since a walker who makes it through the infinity of walks will not be just anywhere–as would be possible if where the walker is does not result from an action of walking along the specified distance–but will be at the end of the distance walked.¹³

Yet there seems to be no action in the sequence that has being at the end as a result. Indeed, since there is no last walking, there is no action which results in being at the end. So, for a walker who arrives at the end of the specified distance, an infinity of acts of walking in traversing that distance in a finite time is impossible. Otherwise, there would be a result that is the result of no action. If the walker simply vanished upon completing the infinity of acts there would be no problem, but we assumed the walker made it through to the first moment after the walks.

What this shows is that the view of actions as single entities rather than multiplicities is indispensable if Zeno’s problem is to be avoided or even solved for the uninterrupted walker. To suppose that actions are multiplicities of actions is to allow that the uninterrupted walker completes an infinity of walks. Considération of the staccato walker makes clear that this is impossible. For being at the end is a result of some walk, yet it can be the result of no walk if there is no last walk.

I have argued that since doing an action is a single entity it is complete and that it is complete by having a result. A result serves to limit a corresponding action. Suppose x has progressive action A and that a is x’s condition of having A. Then let b be a result of a, where b is y’s having component B. There are two aspects to the result of relation, and we should consider each in getting foundations for this relation.

First, there is the limit aspect. A complete action has a result that plays the role of a limit. Second, there is the dependency aspect. A result depends on a complete action; it comes about because of this action. As for the limit aspect, a result, in a given context that will include other limiting factors, is a sufficient basis for the unity of the complete action. This suggests taking as one of the foundations y’s component > (B–a is a unity), where the > is the factive conditional connective introduced in Chapter IV, §3. In effect, where b is a result in a given context, that By makes it true that the complete action a has unity. I use the factive rather than the relevant conditional connective since the connection depends on the specific context. In another context, a similar result might limit a quite dissimilar action.

As for the dependency aspect, the appropriate connection seems again to be the factive conditional connective. If b is a result of aג then b depends on a in the sense that (Ax > By). In the given context, that x does A makes it true that y has B. Since the factive conditional need not be causal, this dependency need not be causal, and hence not all actions need be causal. Pushing is causal though seeing is not. As a second foundation, we then have y’s component (Ax > B-). (We could just as well have made this foundation x’s component (.A_ > By).) It remains only to bring in the action itself. The progressive action A of x is, then, the third and final foundation for result of.

There are two other types of entity that are often called results, but neither has the role of a limit. First, there are ‘intermediate results,’’ such as being at the half-way point before the walk is over. Though the intermediate result satisfies the criterion of dependency, it limits only a stage of and not the entire action. It can limit a stage only if there is a complete action that has stages. As a limit and hence as a result, it depends on there being a limit and hence a result of a complete action. Second, there are “pari passu results,” such as the motion of the surface of a cushion as a hard ball is pressed into it. Such a result is a complete action that must itself have a limit. Moreover, its limit will also be the limit of the action on which it depends. Of itself, it does not play the role of the limit of the pressing. When I speak of results, I shall not intend to include either intermediate or pari passu results. This is a significant restriction in respect to temporal asymmetry since neither type of result is posterior to action.

§4. The Foundations of Temporal Asymmetry. To provide adequate foundations for temporal asymmetry, it suffices to provide foundations for four different cases.

(I) In all those instances of temporal priority in which the priority stands between a complete action–the doing of A, say– and a result–the having of B, say–the progressive action A and the component B of the entity with the result are the foundations of the priority. Priority in such cases is to be called “action-result” priority. As we shall see, these are the basic cases of priority. The foundations here are components of individuals, even though the temporal relational properties for which they are the foundations are properties of conditions of individuals, that is, of complete actions and results. The dependency of conditions on things with them is merely emphasized by the fact that the foundations of the relational properties of the conditions are components of those things.

(II) Suppose an individual must lose one component, B, in order to gain another, C. If it changes from being B to being C, that is, from the condition b to the condition c, then b is prior to c. But b need not be a complete action at all. So the priority need not be action-result priority. It may, nonetheless, depend in an indirect way on action-result priority. For suppose there is a complete action a that not only has casa result but also has being not-i? as a result. Clearly, these action-result pairs provide a basis for the priority of b in respect to c. Since, by hypothesis, B and C are incompatible, the action must counteract the presence of B in order to have c as a result. Let us then call b a “counter condition” in respect to c whenever, for c to be a result of some action, being not-B must be a result of that action. Thus b being prior to c is a case of “counter-result” priority. The foundations of b’s counterresult priority to c are very simply the foundations of a s actionresult priority to being not-B and of its action-result priority to c.

Suppose an individual did not have B before having C. Even so, a might have casa result, and B might be incompatible with C. In this case, one would not have to say that being not-B is a result of a. Why not? Simply because B was not present and thus did not have to be countered by a. Counter-result priority comes in when, since not-B is a result, B had to be countered. Discussion of a negative condition is feasible here in view of our earlier admission of negative properties.

(III) What is to be done with the temporal priority between two conditions separated by a chain of many actions? This calls for linking together many priorities, each of which is either an actionresult or a counter-result priority. Suppose m₁ is a complete action or a counter condition and m₂ is the associated result condition. Likewise, suppose m₂is a complete action or a counter condition and m₃ is the associated result condition, and so on down to m_n. Then we say that the sequence m₁, m₂, m₃, . . . , m_n is an “action chain” and that the members are “links.”

Now suppose conditions a and b are such that there is an action chain terminating in a and b in which chain a is not a result of any link; no link is a counter condition in respect to a; no link is a result of b; and b is not a counter condition of any link. Under these suppositions, a is prior to b. We can call this “chain” priority, of which action-result and counter-result priority are special cases. The foundations of as chain priority to b are the foundations of the action-result priorities that are ultimately involved in the chain. I say “ultimately” since there may be counter-result priorities immediately involved. Moreover, action chains cannot be closed. This is a direct consequence of the asymmetry of action when action-result priority is the only priority in the chain (§2). By a more complicated reasoning, a closed action chain is excluded by the asymmetry of action even when there is also counter-result priority in the chain. But I shall omit this reasoning here.

An action chain need not be a chain of conditions of a single individual. A result in the chain may belong to an individual different from the one the action belongs to. This distinguishes my view from that of Leibniz, for whom temporal priority was grounded in the activity of a monad, which does not act on other monads. By its activity a monad develops itself into an ordered sequence of states; “... a body is not only at the actual moment of its motion in a place commensurate to it, but it has also a tendency or effort to change its place so that the succeeding state follows of itself from the present by the force of nature.”¹⁴ Without action between monads, temporal priority between states of distinct monads could not be grounded in action. Either there is no such relation or it is based on something other than the action of created monads. However, in my view, distinct individuals are linked in time only through their actions on one another. Asymmetrical temporal relatedness between conditions of distinct in dividuals is on an equal footing with that between conditions of the same entity. There is no fundamental importance in the dis־ tinction between “external” time grounded in action on the world and “internal” time grounded in self-development.

What about individuals that are not related at all by action? On the one hand, there is no reason to suppose that if there were such isolated individuals their conditions would be temporally related. On the other hand, it is doubtful that there are individuals unconnected, directly or indirectly, by action. The smallest mote, we are told, exerts a gravitational pull on the most distant star. Nonetheless, if one were to consider non-interacting worlds, separate time systems would be appropriate to them.

There is, though, a more subtle objection. Suppose Jones and Smith pass through childhood without acting on one another. Then, when both are adults, Jones does something that affects Smith. This action sets up a link between their histories that allows certain conditions of one of them to be prior to certain conditions of the other. Jones’s action that provides the link is, say, chain posterior to his first hammering a nail. The effect in Smith is, we suppose, chain prior to Smith’s wedding. So Jones’s hammering is chain prior to Smith’s wedding, as is plain in Figure 1. There are some interesting consequences of the fact that neither acted on the other during childhood and of the fact that it was Jones who acted on Smith and not Smith on Jones.

Figure 1: A one-way connection leads to gaps in temporal relatedness.

First, Smith’s beating on a drum when he was a child has no temporal relation to Jones’s early hammering. There is simply no chain of action from the one to the other. Second, Smith’s drumming is not even prior to Jones’s dying, since Smith never acts back on Jones. Thus in our uncompromising action view of time, there is no basis for temporal relations in such cases. It will not do to appeal to temporal measurement to put these events in a temporal order with one another. For if a clock is attached to Smith and another is synchronized with it and sent on a voyage to Jones, the two individuals are connected by acts of synchronizing and moving clocks.

Our example is unrealistic, for if Jones acts on Smith, then in all likelihood Smith would act on Jones. Only if one overlooks the unrealistic nature of the example will one find it objectionable that our view, as applied to the example, yields so many temporal gaps. But if it is still felt that there is a difficulty, it is tempting to try to overcome it by appealing to possibilities.¹⁵ At the time Smith beat his drum he could instead have done some different action that would have had a result in Jones, a result that would be chain prior to Jones’s hammering. If there is this possibility, then surely we can say that the drumming is prior to the hammering. But what is needed to have such a possibility?

What difference would there be between a case in which the connecting action is possible and a case in which it is not? Smith’s drumming and Jones’s early life up to the hammering might be indistinguishable in the two cases. The difference could only be that, when the action from Smith to Jones is possible though not actual, there must be an actual action-result priority from an event of Smith’s to an event of Jones’s. Specifically, either an actual “slow” action–a sound signal, say, represented by 1 in Figure 2–prior to Smith’s drumming has a result coincident with Jones’s hammering. This would make it possible for a “fast” action –a light signal, say, represented by 3–coincident with the drumming to have a result coincident with the hammering. Or an actual fast action–represented by 5–posterior to the drumming has a result coincident with the hammering. This would make it possible to reach the hammering from the drumming with a slow action–epresented by 3. In either case, a possible action that is not actual presupposes an actual one because it presupposes an actual action-result priority. So the appeal to possibilities is of only limited usefulness in filling the gaps in temporal relatedness.

Otherwise, to say that an action is possible along one route, even when there is no other route with the same end points along which action actually occurs is to hypostatize possibility. That is, it treats possibilities as independent of factors they depend on. It is possible for a poker to burn a hole in wood only because wood fibers oxidize. It is not because the poker has this possibility that there is this fact about the nature of wood. Similarly, it is not possibilities for action that are the basis for time. Instead, such possibilities depend on temporal relatedness. An action that does not occur, but that could occur, could occur precisely because some other action did occur that provided the temporal basis needed for a possible action.

Figure 2: 3 is a possible action only if it is a side of a “triangle”–1 2 3 or 3 4 5–whose other sides are actual actions.

(IV) We thus have a fourth case of priority. If an action chain from a to b is possible, then a is prior to b. This can be called ‘possible chain’ priority. It exists only when it can be represented as the third side of a ‘triangle’ whose other sides represent actual chain priorities. There are many foundations for the possible chain priority of a to b. First, there are the foundations of any two such actual chain priorities. Second, there is the conditional property of the agent that, if there are action chains corresponding to these actual chain priorities, there is a kind of action such that an action of that kind could connect a with b.

Does it follow that before an individual performs an action it has no possibility for any action? It would seem so since the possibility for action depends on temporal relatedness which, in turn, depends on actual actions. Yet if actions are impossible, as they seem to be for any individual that has yet to act, nothing will ever be done. In short, there will be no time.

There is an obvious mistake behind this objection. The objector assumes that for it to be possible at t₁ for an action to take place at t₂ the conditions of its existence must exist already at t₁. In particular, since among the conditions of its existence are certain relational properties of priority, these properties, too should exist at t₁. But we are concerned about the possibility att₁of an action at t₂ and not of an action at t₁ Thus the requirement that there already be a priority and hence an action is unnecessary. It suffices for the possibility of an action that there be priority when the action is done. And this condition is satisfied whenever the action is done.

Everything would be easy if events were located in a network of temporal relations. Possibilities for action could then be based on actualities by being based on the temporal relations in the network. They would not depend on actions being done in any part of the network. But since belief in such a network is incoherent with human practice, we cannot appeal to it to support possible actions. So the possibility of any action must depend either upon the doing of that action or upon the doing of other actions that provide the two actual sides of an associated temporal triangle. Two individuals belong to the same universe, not because they are in a spatio-temporal network but because one acts on the other.

§5. The Impossibility of Counterdirected Action. Even though action has a vectoral character, might it not be the case that some actions point in one direction whereas others point in another? Nothing we said in arguing for the vectoral character of action implies that all actions cohere in pointing in one direction. But if the direction from one action to its result is opposite to the direction from another action to its result, the attempt to found temporal asymmetry on action seems to face a serious obstacle. For such an attempt would then lead us to say, in some cases, that though a given event is before another, an event simultaneous with the first is after, not before, an event simultaneous with the second. This is an intolerable consequence for any theory of time. Can it be avoided on the action theory of time?

The possibility of counterdirected action has been conjectured in various areas of physics. First, the sequences of conditions associated with processes governed by classical electromagnetic theory are reversible. This means that an oscillating electrical charge could have either a ‘retarded’ or an ‘advanced’ potential associated with it. But it has been conjectured more recently that a charge might have both a retarded and an advanced potential. A charge would then act to have posterior results by the retarded potential, and prior results by the advanced potential.¹⁶ It would not only set off oscillations in other charges at later times but also set off oscillations in other charges at earlier times. It would then have counterdirected actions.

Second, it has been conjectured that there are particles– tachyons–moving faster than the speed of light. Since they move faster than light, they move in space-time between points whose time order can be reversed simply by changing the frame of reference.¹⁷ This reversal is a direct consequence of the relativity of simultaneity. It then follows that since they never move slower than a speed greater than that of light, their emission and absorption by atoms can be reversed merely by changing the frame of reference in which they are considered. An emission of a tachyon in one frame of reference would be an absorption in another. The result of an action in one framework would be before that action in another framework. An atom that might be a source of a tachyon would then act with posterior results in one framework and with prior results in another. Again counterdirected actions seem possible.

The following objection has been raised to such conjectures. It would follow from these conjectures that a result could exist without the action from which it results. Suppose there is an antecedent effect of the acceleration of a charge. This effect could be taken as a signal to trigger a series of events resulting in the charge’s failure to accelerate at all. So the effect occurs without its cause. The same paradox confronts the tachyon. One could intervene to prevent what in one framework is the causal action. So in that framework there would be a prior result of an action that does not occur.

However, there are limits to intervention intended to prevent actions posterior to their results. Hesse notes that when a result occurs prior to its associated action, there may be no possibility of detecting the result in time to intervene and prevent the posterior action.¹⁸ This will be true if the dimensions are small enough to make quantum uncertainties an important factor. Quantum uncertainties can then restrict somewhat the threat of paradox resulting from counterdirected action. In a micro-process, the amount of energy absorbed as a prior result of a posterior action will be small. The time of absorption will, by the uncertainty principle, be correspondingly large. But this absorption was to be the signal for intervening. There will not then be time to intervene and prevent the causal action. Since all counterdirected actions might be limited to the quantum domain, a different attack on the possibility of counterdirected action is needed if we are to be able to accept the action theory of time.

One can conceive of two types of counterdirected action. There are the counterdirected actions on different halves of a closed action chain. Counterdirected actions of this type are possible only if there can be self-dependent actions. Since, as we saw in §2, there can be no self-dependent actions, there are no counterdirected actions of this type. There are also counterdirected actions that run simultaneously. In a closed action chain, by contrast, counterdirected actions are not running concurrently; they are in distinct parts of time. I shall try to show that the second type of counterdirectedness implies the possibility of self-dependent action also. Thus there can be no counterdirectedness of the second type either. To show this it will be necessary to discuss simultaneity, which, like other temporal relations, has its basis in action. This discussion divides into two parts, one of simultaneity among conditions of one individual, and the other of simultaneity among conditions of distinct individuals. No special treatment is given here of spatial relations, but through priority and simultaneity, an action theory of space can be constructed.¹⁹

(I) As regards simultaneity among conditions of one individual, it is important to observe that an individual is the same particular as any component of it. Some of its components will, however, be had before others. So it in no way follows from the fact that an individual is not distinct from each of two components that the having of one is simultaneous with the having of the other. If not, though, what is the basis for simultaneity within an individual?

The actions, properties, and parts of an individual are alike in being components, but there is an important difference between actions and other components. An action will be what it is on the basis of at least some of the properties and parts of the individual with that action. The walking of a one-legged person is not similar, in the previous sense of exact similarity, to the walking of a twolegged person. The actions of reflecting white light from square and triangular surfaces are not similar either. By saying that a component is a ‘basis’ for what an associated action is, I mean that if the component were not present, the action would not be similar to the actions that it was originally similar to. Such a basis is clearly not a causal one, for a dissimilar cause could still have an exactly similar action as effect.

This notion of a basis is directly related to that of simultaneity. The components that shape what an action, however caused, is to be are ones that an individual has while it is acting. So if a component of an individual is a basis for what an action of the individual will be similar to, then having the component and doing the action are, at least in part, simultaneous.* Two conditions that are partially simultaneous to a given complete action need not, by this account, be partially simultaneous to one another. But since they could be partially simultaneous, our task is unfinished since as yet there is no account of how they could.

Among actions are the actions of losing and gaining components. Associated components will determine what such actions are similar to. Losing weight by disease and by loss of a limb are dissimilar actions of losing weight. Magnetizing a paramagnetic substance, in which the presence of atoms acting as magnetic dipoles is the important factor, is not similar to the action of magnetizing a diamagnetic substance. Thus where A and B are components of an individual j, if A (or B) is a basis for what ןs losing or gaining B (or A) is similar to, then being A is partially simultaneous with being B.

One might wonder whether, beyond these two cases, there are yet others of partial simultaneity for a single individual. One will so wonder if one is convinced that any two conditions of an individual must be either concurrent or successive. Yet this conviction stems from the view that over and above individuals there is a network of temporal relations that requires either concurrence or successiveness. I am content to say that strictly isolated components of a single individual are not involved in definite temporal relations with other components of this individual. The most one can say is that the individual has these components along with the others while it exists. If being A is neither a basis for what being B is nor a basis for what gaining or losing B is and, conversely, if being B is neither a basis for what being A is nor a basis for what gaining or losing A is, then if being A is neither prior nor posterior to being B, the temporal relation of being A to being B is indeed indefinite.

(II) Now consider simultaneity among conditions of distinct individuals. The motivating idea here is that, since actions are prior to results, simultaneous conditions are not connected by an action chain. If we give up this requirement of unconnectedness, we are faced with the prospect of treating simultaneity as an ungrounded relation. If we conform to it, we shall suppose that conditions of two entities are simultaneous only when they lie between an action of the one entity on the other and an action of the second back on the first. There is then no simultaneity between entities that do not interact. There is sufficient interaction in the world to make simultaneity a common sort of relatedness.

Suppose an individual i acts to produce a result in j and that, after the coming into being of the result in j, j itself acts back on i. Any condition of j after the coming into being of the result in j of the action by i and before the action by j itself is simultaneous with some part of any condition that i has that lasts from the time of its action on j until the change due to the action from j. Hence, the one condition is partially simultaneous with the other. The relations of priority employed here depend on both the relation of chain priority and on the relation of partial simultaneity within individuals. Thus the condition of j that is after the coming into being of the result in j brought about by i need not have chain posteriority to this coming into being of the result. It need only have chain posteriority to some condition of j that is partially simultaneous with coming into being of the result.

It is now time to apply this notion of partial simultaneity in order to show that actions cohere by pointing in the same direction. Consider the two individuals i and j. The complete action c_j of i is, we suppose, counterdirected to the complete action Cj of j, as in Figure 3. To suppose this is to suppose, since closed

Figure 3: Counterdirectedness with simultaneity leads to self-dependence.

time was excluded, that i and j belong to a common time in which the directions of their actions can be compared. Thus there will be actions of each of these individuals with results in the other. Otherwise, there would be no simultaneity, partial or complete, between the conditions of the two, and thus no basis for comparison of the directions of their actions. But partial simultaneity and counterdirectedness generate a difficulty when taken together. To see this, notice that the following situation, described in Figure 3, is always possible.

The complete action a_j of j has a result b_i in i. Leta_j be a basis, along with at least one other action, for partial simultaneity between the beginning of a_j itself and a_i. Also, b_i is to have c_j as a result and b_i is to be a basis, along with at least one other action, for partial simultaneity betweenc_j which is to be a result ofa_i,and the beginning of c_j. Thus a_i and c_j are counterdirected since a_i is partially simultaneous with the beginning of a result of c_j– that is, with the beginning of a_i–and the beginning of c_j is partially simultaneous with a result of a_i–that is, with c_j–but not with a_i. There is then counterdirectedness since, though a_i is before c_j and the end ofc_j is after the beginning of c_j,a_i and the end of c_j are partially simultaneous as are c_j and the beginning of c_j but not a_i and the beginning of c_j. But what has been allowed by all this is for a_i to be self-dependent, and as we saw, no entity can be self-dependent. Complete action a_i depends by counterdirected action on c_jand c_j depends, indirectly, by forward action on a_i. Since such a situation can always arise under the above supposition, the possibility of counterdirected action must be rejected.

This criticism can be seen to apply to the case of a tachyon only after we have asked what it is that the transformation of coordinates reverses. But before applying that criticism, we shall propose an interpretation of tachyons that renders them unproblematic as regards time.

The time order of the conditions associated with a tachyon can be reversed by changing the coordinate system. But this only means that a sequence of conditions has a different temporal order in different coordinate systems. Recall that a converse description of a sequence is not a converse description of an associated action. Correspondingly, transforming the coordinates to obtain a reverse time order will not yield a counterdirected action. One will merely obtain an emission of a particle that is after its absorption. It was then incorrect of us to say that an emission in one system would be an absorption in another. The threat of counterdirected action is thus set aside. What remains is the absurdity of an emission occurring after the particle is already distant from the emitter.

This situation will not be absurd for the Ockhamite view of action. In this view there is no emission over and above the sequence of states of the atom and the tachyon. In particular, in the reverse time order obtained by transforming the coordinates there is no emission that occurs after the tachyon is distant from the emitter. There is only the sequence of positions of the tachyon.

However, in the view adopted here that actions exist along with sequences of property conditions, it is indeed impossible to have the reverse sequence associated with the original action. Precisely what an action contributes that a sequence need not contribute, because it may be a reversible sequence, is direction.²⁰ Still it is undeniable that the existence of tachyons would imply the possibility of a reverse sequence. Since this sequence does not imply a reverse action, we must conclude, to avoid absurdity, that the emission and the absorption of tachyons are not genuine actions at all. They and tachyons with them are to be interpreted as mere sequences of property conditions. We thereby avoid having either an action in one coordinate system that is counter in direction to that in another or a single action that can be both before and after its result.

This reductionist approach of ours to actions associated with tachyons is mandatory if even more awkward consequences are to be avoided. First, since a tachyon is a ‘space-like’ entity, the emission and the absorption of a tachyon can, by an appropriate choice of coordinate system, be simultaneous. This conflicts with the principle that an action and its result are grounds for priority. However, if, as I am claiming, the life of a tachyon does not involve the actions of emission and absorption, then tachyons are no exception to this principle that action implies priority.

Second, with the aid of a tachyon one can close an action chain, if its emission and absorption are genuine actions. In an appropriate coordinate system, the absorption of a tachyon will antedate its emission, and its absorption can then trigger its emission, on which the absorption depends. We again get the specter of selfdependency found in the general case of counterdirectedness. If, however, the life of a tachyon involves only a sequence of conditions and neither an action of emission nor an action of absorption, that part of the chain described as an emission is no longer an action; the objectionable self-dependency of closed action chains is thereby avoided.

§6. The Branch Hypothesis as Superfluous. The action view of time rests on two principles. The first is that action is vectoral, which was justified in §2. This principle implies that there is a distinction between an action and one of its results that is not eliminated simply by a change in the order of the sequence of associated conditions, and that the action is not a result of any of its results. The second principle is that actions are coherent, which was justified in §5. This principle states that all actions point in one direction; that is, given a certain action and one of its results, there is no other action in a relation of partial simultaneity with this result of the first action but not with the first action itself whose result is in a relation of partial simultaneity with the first action. I wish now to show that these two principies are relied on in using the so-called branch hypothesis of statistical physical theory to give a foundation to temporal asymmetry. Since they alone are sufficient for the action view of time, the branch hypothesis must then be superfluous in giving a foundation to temporal asymmetry.

Suppose you are blindfolded, and a cup of coffee is placed before you. The blindfold is removed, and you look at the cup. A cloud of milk is rolling over somewhere in the middle of the cup with a layer of black coffee still on top. You are now faced with the question of the origin of this state of affairs. You reply immediately that someone has just poured milk into the coffee. Naturally you would not reply that the state of imperfect mixture had been preceded by one in which milk and coffee had been thoroughly mixed together. However, if the cup and its contents are treated as an isolated system, then the response should be different. The response should be that it was more probable that the state of imperfect mixture was preceded by a state of more perfect mixture than by a state of less perfect mixture.²¹ It is clear then that the rationale for your response is to be constructed by appealing to the assumption that the cup is very likely a system that was recently distinguished from its surroundings because it was affected by a special action. Thus we say the cup of coffee ‘branched off’ from a system that included it through the action of pouring milk. Associated with this is the further assumption that a branching that leads to imperfect mixture began as an even less perfect mixture. In other words, since less homogeneity corresponds to lower entropy, the imperfect mixture, which is a low entropy state, is assumed to be a state of a system that has recently branched off in an even lower entropy state from a larger system.

Taking our cue from this reconstruction of the natural response to how coffee and milk came to be half-mixed, a general hypothesis can now be proposed that applies to any system in the universe. The hypothesis is that (a) there are many systems branching off from larger systems, and (b) the initial micro-states of similar branch systems with the same initial entropy are a random sample of all the micro-states associated with that entropy.²² To this hypothesis, which does not follow from the laws of statistical physical theory, we add a consequence of that theory. Consider a permanently closed system over a long time span. Such a system will (c) have near maximum entropy most of the time, and (d) at other times it is overwhelmingly more likely that the system will be in a local minimum for entropy than on an upgrade, downgrade, or local maximum.²³

Suppose, on the one hand, that a system branches off in a state of less than its maximum entropy. Now if (d) holds true for the lessthan-maximum entropy states of a single closed system, then, by the principle (b) of randomness, (d) will also hold true for the lessthan-maximum entropy states of a multiplicity of branch systems.²⁴ That is, a low entropy state of a system that has just branched off from a larger system is most likely a local minimum in that system. Hence, it will be followed by a higher entropy state, in all likelihood. But unlike the case of the permanently closed system, it will not be preceded by a higher entropy state. Being a branch system, it simply has no antecedent existence. If, however, (b) does not hold, our transferring (d) from application to closed systems to ensembles of branch systems would not be legitimate. For if the branch systems of a given less-than-maximum entropy were not random samples of the micro-states associated with that entropy, there need not be a general trend among them to higher entropy.

If, though, the micro-states are random samples, what holds for such states in permanently closed systems will hold true in an ensemble of branch systems. But given both (b) and (d), it follows that branch systems in less-than-maximum entropy states have a greater probability of increasing than decreasing their entropy. In view of (a), it can be concluded that among systems in less-thanmaximum entropy states in the universe the majority are branch systems. So from (a), (b), and (d) it follows that the majority of systems in less-than-maximum entropy states increase their entropy.

It remains to consider, on the other hand, systems branching off in near equilibrium state, that is, in a state of near-maximum entropy. This time our problem is to transfer (c) from application to permanently closed systems to branch systems, making use again of the randomness principle (b). According to (c), if a permanently closed system is near equilibrium it is likely to stay there, for it is there most of the time anyway. Suppose a system branches off in a near equilibrium state. By (b), its micro-state is a random one of the micro-states that can constitute such an equilibrium. An ensemble of such branching systems will, since they are a random selection from the micro-states, tend to do what permanently closed systems tend to do in equilibrium, which is to remain in equilibrium. So from (a), (b), and (c), it follows that a majority of nearmaximum entropy systems tend to remain in that state.

By (a)-(d) the majority of systems are such that, if they change their entropy, they increase it. This is then a basis of a physical sort for temporal asymmetry. For earlier and later can now be distinguished on the basis of lesser and greater entropy for the majority of systems that change their entropy.

This provides a magnificient demonstration of the probabilistic irreversibility of sequences of entropy conditions of non-equilibrium systems. That is, it shows that if (a)–(d) are true, then a converse description of the entropy sequences probably associated with such systems is not likely to describe actual sequences. But as we pointed out, temporal asymmetry need not be based on the irreversibility–and we now add the probabilistic irreversibility–of sequences of property conditions. It is certainly well to know that there are sequences that are most likely irreversible. But the character of action suffices to found the asymmetry of time.

(I) The vectoral character of action is relied on in this statistical account. That is, the branching and the blending back into a main system are not supposed to change into one another just by a redescription in converse temporal relational terms. The branching involves an action with the blending back containing its ultimate result. The pouring of the milk is the branching, and the result is the perfect mixture that exists when drinking the coffee blends it back into the main system. If the blending back could itself be the point of branching□\if the result could cease to be a result–then by applying (b) to it this branching would most likely result in a maintenance of the equilibrium of the perfectly mixed coffee.²⁵ This contradicts the result of treating the pouring as the point of branching. For when the pouring is the branch point, tracing backward from the drinking would most likely lead to a decrease of entropy. Because of this conflict of results, the whole attempt to establish an entropie asymmetry would have to be scuttled. To avoid the conflict is merely to recognize the objectivity of results.

This is all on the supposition of the randomness principle (b). What now if (b) does not apply when the merger point is treated as a branch point? If it does not apply here, how can it apply in the above demonstration? And if it does not apply there, the whole attempt fails.

Clearly, then, (a) itself must introduce a vectoral notion of action by introducing the notion of branching off. However, if actions lacked a vectoral character, as they would if they were mere sequences of conditions, then a branching could be at either end of a branch system that an arbitrary time ordering happens to put first. Entropy increase would not then be an objective consequence of branching, but a consequence of describing the system in one time order rather than its converse. In one time order there would be the likelihood of an increase; in the other, though, there would be the likelihood of maintaining equilibrium.

(II) The coherence of action is also relied on. Let it be granted that the majority of branches are such that, at a remove from the branch point, the branch system is in a higher entropy state if it branched in a non-equilibrium state. If there is counterdirected action, then branching can lead to systems developing in opposite directions. Looking at time in either direction may then reveal an equal distribution of branches in which entropy falls to the branch point and of branches in which entropy rises from the branch point.

So in laying down (a) it is presupposed that the majority of branchings cohere. That is, if t₁ and t₂ are the distinct times of two branch points and if t₁’ and t₂’ are the times of conditions along the respective branch systems, then it is not the case both that the interval t₁t₂ is totally included in the interval t₁’t₂’ and that the latter is totally included in the former. But unless one can show that the majority of branchings cohere without also showing that all actions must cohere, as we have shown, the statistical account of temporal asymmetry presupposes that all actions cohere. I am concerned here not merely with the possibility of counterdirected actions that are as far apart as different galaxies,²⁶ but also with the possibility of counterdirected action in small localities as well.

It is clear, then, that the statistical account of temporal asymmetry must assume at least statistical variants of the two principles behind the action view of time. It must assume the vectoral character of the majority of actions and the coherence of the majority of actions as applied to the special action of branching. But even these statistical variants would suffice to ground temporal asymmetry. For the direction of the temporally later would be the direction of most results as judged from corresponding actions.

Thus the addition of special postulates of branching and randomness and the reliance on statistical physical theory for (c) and (d) are unnecesary for temporal asymmetry, however helpful they might be in establishing the quite different matter of the temporal irreversibility of sequences of conditions. Under the sway of the Ockhamite view of action, advocates of the statistical approach to temporal asymmetry will be reluctant to recognize the cogency of the action view of time. But it must be brought home to them that the hypothesis of branching harbors the very notion of action that Ockham rejected.

* Conditions are partially simultaneous when a stage of one is simultaneous with a stage of the other. But here there is no need for the notion of simultaneity, since the problem of counterdirectedness can be handled by that of partial simultaneity.