Logic, Thought, and Action 14

Chapter 14 AGENTS AND AGENCY IN BRANCHING SPACE-TIMES∗ Nuel Belnap Pittsburgh University

Abstract

Branching time puts an indeterminist causal structure on instantaneous but world-wide “super-events” called moments. This theory has “action at a distance” as an inevitable presupposition. In contrast, branching space-times puts an indeterminist causal order on tiny little point events. The benefit of branching space-times theory is that it can represent local indeterminist events with only local outcomes. The “seeing to it that” or “stit” theory of agency developed in Belnap, Perloff and Xu, 2001 employs branching time as a substructure, and thus has the following shortcoming: It is inevitably committed to an account of actionoutcomes that makes them instantaneously world-wide. This essay asks how the stit theory of agency can be adapted to branching space-times in such a way that action-outcomes are local.

The aim of this essay is to make some suggestions for the beginnings of a theory of agents and agency in branching space-times. The thought is to combine the ideas of agency as developed by Belnap, Perloff and Xu, 2001 against the relatively simple background of branching time with the richer notions of mere indeterminism as structured in the theory of branching space-times. My plan is to say a little about agency in branching time and a little about branching space-times, and then ask how the two can be brought together.

1.

Stit theory

In this section I offer some brief and general remarks in connection with “stit theory” as described at length in Belnap, Perloff and Xu, 2001

∗ An

earlier version of this essay appeared in the Journal of Sun Yatsen University (Social Science edition), vol. 43, 2003, pp. 147–166.

D. Vanderveken (ed.), Logic, Thought & Action, 291–313. 2005 Springer. Printed in The Netherlands.

c


292

LOGIC, THOUGHT AND ACTION

(henceforth FF).1 In the course of these remarks, I will occasionally refer to the following designedly boring example. Peter will drive his car to work one day hence.

[1]

What are the chief elements of research strategy for FF? The central aim is to use formal theory to help a little in understanding human agency as it works itself out amid the (we think) indeterministic causal structures of our world. We call it “stit theory.” It takes its name from the prominent use of a non-truth-functional connective: α sees to it that Q, which we abbreviate with [α stit: Q] In connection with stit, the analysis moves in several directions, as indicated in the remainder of this section.

1.1

Stit theses

In an effort to tie the formal grammar of stit to natural language, FF offers a series of so-called “theses.” The theses, which I paraphrase from the appendix to FF, are these: Agentiveness of stit thesis. English is ambiguous, and even the English “α sees to it that Q” is ambiguous, but [α stit: Q], in contrast, is designed to be unambiguous: It invariably attributes agency to α. Stit complement thesis. As a matter of grammar, Q can be any sentence whatsoever, including cases in which [α stit: Q] turns out trivially false, e.g. [α stit: the sun rises every morning]. Stit paraphrase thesis. An English sentence is an agentive if and only if it can be usefully paraphrased as a stit sentence, including the case in which Q is paraphrased as [α stit: Q], e.g. [Peter stit: Peter drives his car to work]. Imperative content thesis. The content of every English imperative is agentive, hence equivalent to a stit sentence; e.g., the declarative content of “Peter, drive your car to work” is [Peter stit: Peter drives his car to work]. 1 See

also publications listed there, and in particular, Horty, 2001.

Agents and Agency in Branching Space-Times

293

Restricted complement thesis. For many English constructions, e.g. “α promises that ,” it is illuminating to restrict their complements to future-tensed agentives, hence to stit sentences; e.g., Peter promises that one day hence [Peter stit: Peter will drive his car to work]. Stit normal form thesis. When worried about difficult questions of agency, it is generally useful to paraphrase each agentive as a stit sentence. Together these theses say that in thinking about the philosophy of action, it is helpful to use [α stit: Q] as a normal form for expressing that an agent does something — something that natural languages express in a bewildering variety of fashions. As applied to [1], this gives One day hence:[Peter stit: Peter drives his car to work].

[2]

Note that we have used “One day hence:” as a connective in order to express the desired tense structure with complete explicitness.2 This is important in being clear about indeterminism.

1.2

Stit theory and indeterminism

Stit theory assumes indeterminism. Why? Stit theory is equally prehumanist and pre-scientific. We take it as a fact that our agentive doings together with non-agentive happenings are enmeshed in a common causal order. The sense of “causal order” here should not be picked up from a determinist presupposition. The appropriate causal order is indeterministic. Nor does this involve “laws.” Why should it? Bare indeterminism is what is wanted: There are initial events in our world for which more than one future is possible. It is this simple concept of indeterminism that permits progress on the theory of agency. Thousands of pages written by philosophers about agency make (or seem to make) the contrary assumption of strict determinism. Hume and Kant are famous in modern philosophy for what has come to be called “compatibilism”, the doctrine that agency is compatible with strict and absolute and perfect determinism. On the other hand there are also hundreds of pages (I’m making up these numbers) assuming some form of indeterministic “free will”; one has to think only of William James and his rejection of the idea of a “block universe.” Rarely, however, is an indeterministic view of agency combined with a desire to let the enterprise be guided by a desire for mathematically rigorous theories. The aim of our research is 2 We

have dropped the “will” of [1] as logically redundant.

294

LOGIC, THOUGHT AND ACTION

to make an indeterminist account of agency “intelligible” (Kane, 1998). We try to pursue this aim by means of a simple and rigorous theory.

1.3

The metaphysics of agency

The simplest representation of indeterminism is this: A treelike structure that opens toward the future. That is, FF, following Prior and Thomason, represents indeterminism by means of a partial order in which there is no backward branching. We call the elements of the tree moments. A moment is an entire slice of history, so to speak, from one edge of the universe to the other. A moment is not a “time”; it is a concrete “super-event” (Thomson, 1977) caught up in the causal order, with its unique past and its future of possibilities. A maximal chain of moments is called a history, and represents one particular fully-detailed way in which our world might go or might have gone. Do not in your mind identify a single history with a “world”; instead, it is the entire treelike assemblage that is to be identified with Our World. Philosophical logicians generally use the term branching time for such a structure, and we shall do the same, even though “branching histories” would be less misleading. The right side of Figure 1 gives a picture. In FF we argue at length against the pernicious doctrine that one of the many possible histories through a moment has a special status as “actual” or “real.” FF shows that our understanding of assertion, prediction, promising, and the like is much improved if all histories through a moment are treated on a par, with none singled out. One reaches the “theory of agents and choices in branching time” by adding two elements to the bare theory of branching time. (1) Agent is a set whose members are considered to be agents capable of making choices and so acting. (2) Choice is a function that is defined on all moments m and agents α, and which delivers a partition of all the histories through m. The partition represents the choices open to agent α at moment m. We may call each member of the partition a choice that is available to α at m, or a possible choice for α at m. There is one serious theoretical constraint on Choice: If two histories h 1 and h 2 in the tree do not split from one another until after a certain moment m, then no choice available to α at m can distinguish h 1 and h 2 . This we call “no choice between undivided histories.” As we often say, you cannot make tomorrow’s choices today. When there are multiple agents under consideration, one naturally (?) assumes that the simultaneous choices of two agents α1 and α2 are radically independent; technically, we assume that every choice by α1 is consistent with every choice by α2 when their choices are simultaneous.

Agents and Agency in Branching Space-Times

295

That’s it. As you can see, our “metaphysical” theory of the causal structure of agency is about as minimal as can be.

1.4

The semantics of indeterminism

Simple as it is, the branching-time structure does not explain itself, especially not to philosophers who assume determinism as an unquestioned “fact.” One has to come to terms with what prediction, and more generally the use of the future tense and other kinds of statements “about” the future, mean in connection with the tree representation of indeterminism. This is a matter not of metaphysics, but of semantics. The problem is how to understand certain linguistic structures under the assumption that they are used in an indeterministic situation. The solution to that problem, including agency as a special case, lies in combining the Prior-Thomason semantics for branching time, in which truth is relativized to both moments and histories, with the Kaplan semantics for indexical expressions. The Prior-Thomason-Kaplan semantics makes the truth of e.g. [1] relative not only to the moment of utterance, but in addition relative to a moment-of-evaluation parameter needed for understanding tense constructions that take one away from the moment of utterance, and, crucially, relative also to a history-of-evaluation parameter that represents a way in which the future can unfold. The semantics is entirely two-valued, since relative to each moment of utterance, moment of evaluation, and history of evaluation, there is a definite truth value given to the example sentence. If a sentence is true [false] relative to every history through a given moment of evaluation, the sentence is said to be settled true [false [ ] at that moment. Picture [1] or [2] as asserted by someone on Monday in a situation such that whether or not Peter will drive to work one day hence is still an open question. The semantics, as explained in FF (and in more detail in Belnap, 2002a), makes it possible to say precisely: When asserted on Monday, [1] is neither settled true nor settled false. However, it is a settled truth on Monday that, no matter what happens, one of the following holds: Either on Tuesday it will be settled true that [1] was true at the moment of assertion (on Monday), or on Tuesday it will be settled false that [1] was true at the moment of assertion (on Monday). With slightly more brevity (but still with inevitable risk of confusion): The assertion of [1] is not settled one way or the other on Monday; but no matter what happens, on Tuesday it will be definitely settled whether or not the assertion was true at the moment of assertion.

296

LOGIC, THOUGHT AND ACTION

This confusing passage involves “double time references,” which are sorted out both in FF and in Belnap, 2002a. It may also be called “bivalence later” (Copley, 2000). The subtle part is that [1] is evaluated with respect to a Monday moment m 0 that is the moment of assertion, and with respect to each history h that passes through not m 0 , but instead through some Tuesday moment m 1 at which we are evaluating whether or not the assertion was true at the moment of assertion on Monday. In this way, the theory provides a framework for normatively evaluating assertions about the future in terms of what we may call “fidelity”: We say that an assertion is vindicated or impugned at some later moment depending on whether it is settled true or settled false at that later moment that the assertion was true at the moment of assertion.

1.5

The semantics of agency

On the basis of the Prior-Thomason semantics that relativizes truth to moment-histories pairs, we suggest two principal semantic analyses of stit. These analyses depend essentially on an indeterminist view of agency. Both share the idea that action begins in choice, and that there is no choice and therefore no action without the availability of incompatible choices. We work out the semantics of stit in two ways. Both involve a transition from an unsettled situation to settledness due to the choice of an agent. The achievement stit postulates that the relevant choice occurred at some temporal remove in the past of its settled outcome. The deliberative stit works through a concept of action based on the choice being in the immediate past of its settled outcome. In either case, the semantics says that [α stit: Q] is true at a certain moment m with respect to a certain history h provided (1) a prior (perhaps immediately prior) choice of α guaranteed the truth of Q, and (2) the choice was a real choice among incompatible alternatives. Put technically, the truth conditions for [α stit: Q] at a pair m /h when taken as the deliberative stit come to this: 1-1 Definition. (Stit) Positive condition. Q is true at m with respect to every history h 1 that belongs to the same possible choice for α at m as does h . (This is the part that says that α’s choice at m guarantees the truth of Q.) Negative condition. Q is not settled true at m: There is some history h 2 to which m belongs such that Q is false at m with respect to h 2 . (This says that α really does have a choice at m that is

Agents and Agency in Branching Space-Times

297

relevant to the truth of Q.) The semantic analysis that FF gives to the achievement stit is less satisfying. I nevertheless give a brief version here. First of all, one is required to add a “time” parameter (FF says Instant) to the underlying tree structure, with the assumption that all histories are temporally isomorphic. This amounts to the presumption that one can make sense out of comparing the “times” of incompatible moments. The achievement stit relies on this, letting the “positive condition” be, roughly, that [α stit: Q] is true at m /h iff Q is settled true at every moment that (1) is co-temporal with m and (2) lies on a history that is in the same choice for α at m as h .

1.6

Strategies

Stit theory allows a sophisticated but simple account of strategies, conceived of as a pattern of prescribed choices for future action. In this part of the theory, the principle of “no choice between undivided histories” in modeling agency is critical. The principle, you will recall, denies that our world allows us to choose today between histories that only divide tomorrow. In a nutshell: To choose a strategy is not to choose the later choices that define it. It follows that we must carefully distinguish the availability of a strategy from having a strategy. In other words, the stit theory of strategies makes it imperative that one distinguish “acting in accord with a strategy” from “acting on a strategy.” This is a special case of the following: The fundamental notions of stit and strategies may serve as a kind of foundation for intentional notions. If you start that way with the indeterminist causal structure of agency, you easily see how difficult it is to sort out the intentional notions in relation to causal notions in any way that keeps them sorted out, and you are more likely to avoid self-deception in appraising your theories. Almost breathlessly I have highlighted five parts of the stit theory of agency: the stit theses, how stit theory presupposes indeterminism, the metaphysics (causal structure) of agency as we see it, the difficult semantics of indeterminism in general, the truth conditions for stit, and, finally, the stit theory of strategies. It is this apparatus that I suggest should inform our discussion of how agency fits into branching spacetimes, and I shall presuppose it when I turn to this matter in §3. First, however, I drop agency in order to survey some main ideas of branching space-times.

298

2.

LOGIC, THOUGHT AND ACTION

Branching space-times

There is much less written on branching space-times (BST, as I shall say), and it is less accessible, in part because most essays on this topic were written with physical examples such as quantum mechanics in mind.3 Perhaps the easiest approach to BST starts from a Newtonian picture of the world of events.

Newtonian universe. Non-relativistic and deterministic: World The Newtonian universe has, as I see it, two features that = Line. are so fundamental that they can be described without advanced mathematics, using only the characteristics of the causal-ordering relation and its relata. First, the items that are related by the before-after causal order are momentary (= instantaneous) super-events: Laplace’s demon needs total world-wide information concerning what is going on at time t . Second, the causal order is strictly linear: For any two momentary events m 1 and m 2 , either m 1 lies in the causal past of m 2 , or vice versa: m 2 lies in the causal past of m 1 . It is the linearity of the causal order that answers to determinism, and it is the global conception of the items falling into a causal order that answers to non-relativistic “action at a distance”: An adjustment in positions or momenta here-now can immediately call for an adjustment over there in the furthest galaxy. The picture of the causal order in a Newtonian universe is therefore a simple line, with each point representing a world-wide simultaneity slice, all Nature at a certain time t . In such a universe there are not histories (plural), but only a single History, so that we may say that World = History; this makes the Newtonian universe deterministic. Furthermore, and independently, the relata of the causal ordering are momentary super-events; this make the Newtonian universe non-relativistic. With this in mind, we may say that on the Newtonian view, World = Line as on the left side of Figure 1. Branching space-times is to be both indeterministic and relativistic. Since the Newtonian universe is both deterministic and non-relativistic, it takes two independent moves to make the transition from the Newtonian universe to branching space-times. Branching time universe. Non-relativistic but indeterminisIn the first move away from Newton tic: World = Many Lines. 3

BST theory in the form deriving from Belnap, 1992 is discussed in the following places: Szabo, Belnap, 1996, Raki´ ´c, 1997, Belnap, 1999, Placek, 2000a, Placek, 2000b, Belnap, 2002e, Mueller, 2002, Placek, 2002b and Belnap, 2002b. Belnap, 2002d gives an overall view of both stit theory and BST theory.

Agents and Agency in Branching Space-Times

Figure 1.

299

Newtonian universe and Branching-time universe

we keep the relata of the causal order as momentary super-events, so that we remain non-relativistic. In order to represent indeterminism, however, we abandon linearity in favor of a treelike order. The result of this first transition from the Newtonian universe, when taken alone, is exactly what we have already discussed under the rubric “branching time.” In branching time there is indeed a single world, but instead of the equation World = Line, the world of branching time involves many line-like histories, i.e., many possibilities: Branching time is indeterministic. Since, however, we have kept the causal relata as momentary super-events, branching time remains non-relativistic: Splitting between histories in branching time has to be a world-wide matter of “action at a distance” since the consequences of the split are felt instantaneously throughout the farthest reaches of space. We may therefore say that according to branching time, World = Many Lines that split at world-wide momentary super-events as on the right side of Figure 1.

Einstein-Minkowski universe. Relativistic but deterministic: The other move away from Newton is that World = Space-time. made by Einstein in principle, and more explicitly by Minkowski. To obtain the Einstein-Minkowski universe from that of Newton, we keep determinism from the Newtonian universe; there is no trace of alternative possible futures. The change is rather that now the terms of the causal relation are no longer simultaneity slices (momentary super-events) that stretch throughout the universe. Instead, the fundamental causal relata are local events, events that are limited in both time-like and spacelike dimensions. When fully idealized, the causal relata are point events in space-time. This, to my mind, is the heart and soul of EinsteinMinkowski relativity. The move to local events is made necessary by Einstein’s argument that there is simply no objective meaning for a simultaneity slice running from one edge of the universe to the other. There is no “action at a distance”: Adjustments at e 1 influence only events e 2 in “the forward light cone” of e 1 , or, as will say, in the causal

300

LOGIC, THOUGHT AND ACTION

future of e 1 . I wish to urge that not only fancy Einstein physics, but even our ordinary experience (when uncorrupted by uncritical adherence to Newton or mechanical addiction to clocks and watches) shows us that events are not strung out one after the other. Take the event of our being here now at e 1 . Indeed some events lie in our causal future, so that there are causal chains from e 1 to them, and others lie in our causal past, so that the causal chains run from them to e 1 . But once we take local events as the relata of the causal order, there is a third category, always intuitive, and now scientifically respectable, since we have learned to be suspicious of the idea of (immediate) action at a distance. In this third category are local events e 2 that neither lie ahead of e 1 nor do they lie behind e 1 in the causal order. Letting < be the causal order relation, I am speaking of a pair of point events e 1 and e 2 such that neither e 2 <e 1 nor e 1 <e 2 , Instead, e 1 and e 2 have a space-like relation to each other. Neither later nor earlier (nor frozen into simultaneity by a mythical world-spanning clock), they are “over there” with respect to each other. Einstein makes us painfully aware that space-like relatedness is non-transitive, which is precisely the bar to the objective reality of momentary super-events. Events in their causal relation are not really ordered like a line. Our modern reverence for various parts of Newtonian physics and our related love of clock time delude us. Since the Einstein-Minkowski relativistic picture is just as deterministic as the Newtonian picture, there are no histories (plural), but only History, so that we have the determinist equation World = History. The difference from the Newtonian picture is with respect to an independent feature: A causally ordered historical course of events can no longer be conceived as a linear order of momentary super-events. Instead, a history is a relativistic space-time that consists in a manifold of point events bound together by a Minkowski-style causal ordering that allows that some pairs of point events are space-like related. Therefore, if we make the single transition from the Newtonian universe to that of EinsteinMinkowski, the result is that World = Space-time as on the left side of Figure 2.

Branching space-times: relativistic and indeterministic: World BST now arises by suggesting that the = Many Space-times. causal structure of our world involves both indeterminism and relativistic space-times; we are therefore to combine two independent transitions from the Newtonian universe. We can already make a certain amount of capital out of that suggestion. For indeterminism, we shall expect not World = History but World = Many Histories. For relativistic considerations, we shall expect that each history is not a line, but instead

Agents and Agency in Branching Space-Times

Figure 2.

301

Einstein-Minkowski universe and Branching-space-times universe

a space-time of point events in something like the Einstein-Minkowski sense. Therefore, in BST we should expect that World = Many Spacetimes as on the right side of Figure 2. Furthermore, just as histories in branching time (each of which is like a line) split at a world-wide momentary super-event, so in branching space-times we should expect that histories (each of which is like a space-time) should split at one or more point events. Technically, we represent our world as Our World, which is a set of (possible) point events, and we let e range over Our World. We need, however, more information about how the various histories (= space-times) fit together. What is analogous here to the indeterministic way in which branching-time structured individual Newtonian line-like histories into a tree? A crucial desideratum is this: The theory should preserve our instinct that such indeterminism as there is in our world can be a local matter, a chance event here-now that has no effect on the immediate future of astronomically distant regions of the universe. It needs zero training in mathematics to see that the theory of how BST histories fit together into a single world will be more complicated than the branching-time theory that arranged many lines into a single tree. I mention four key points underlying the theory of “branching space-times.”

302

2.1

LOGIC, THOUGHT AND ACTION

Consistency in BST

Histories are closely related to the ideas of possibility and consistency. A guiding idea here is that what allows two events to share a history, and therefore to be consistent, is that at least one event lies in their common future. As long as there is a standpoint in Our World from which one could truly say “both of these events have happened,” even if the two events are not themselves arranged one after the other, one may be confident that the two events can live together in single history. Peter’s (possible) driving to work in one village and Paul’s (possible) staying home in another village are consistent just in case there is some (possible) standpoint at which someone could truly say, using the past tense, that both events came to pass. Let us also turn this around: If two events are inconsistent, then no event can have both of them in its past. For example, although Peter’s choice to drive to work is altogether local, and although we cannot picture Our World as a tree, nevertheless, there is no standpoint anywhere in Our World that has in its past both of the inconsistent events represented by Peter’s having driven to work and his having stayed home. These inconsistent possibilities can and must lie ahead of the point at which he makes the choice, but they cannot both lie behind anything whatsoever. A single maximal consistent set of point events will be a space-time; we call it a history, we let Hist be the set of all histories in Our World, and we let h range over Hist.

2.2

Choice points local, not global

There have to be choice points, definite local events at which two histories split into radically inconsistent portions. It is presumably not true and we must not assume that when splitting occurs, it occurs in some magical world-wide way. When Peter is given the choice to drive or to stay home on a certain occasion, that occasion is confined in space as well as in time. His little bit of free will is local, not global. And the same might be true when the choice is only metaphorical, a matter of a random outcome of some natural event such as, perhaps, the decay of a radium atom in Paris. It might be that the decay is a strictly local matter, neither influencing nor influenced by contemporary happenings in, say, Manhattan. Whenever there is indeterminism, whether of choice or of chance, a good theory must give meaning to the difficult idea that the indeterminism is local, not worldwide. Important in BST theory are the same ideas of undividedness and of splitting that are so important for the FF theory of agency. For later reference we indicate some pieces of notation that are useful in speaking

Agents and Agency in Branching Space-Times

303

of these matters. 2-1 Definition. ((H ( (e) , h1 ≡e h2 , Πe h, Πe , and h1 ⊥e h2 )) H(e) is the set of all histories to which e belongs. h 1 ≡e h 2 means that histories h 1 and h 2 are undivided at e (they split at some point event that is properly later than e ). Πe h  = {h 1 : h ≡e h 1 } is the immediate possibility at e to which h belongs, and Πe is the set {Πe h : h ∈ H(e) } of all immediate possibilities at e . h 1 ⊥e h 2 means that h 1 and h 2 split exactly at e .

2.3

Prior choice principle

When I put the third point in everyday language, it sounds so obvious that you will yawn. And yet as far as I know a thoroughly controlled statement has never been made apart from the present theory of branching space-times. The BST postulate for locating choice points may be put informally in the following way: Whenever we find ourselves as part of some contingent event instead of in a history that is an alternative to the occurrence of that event, we may always look to the past for a choice that serves as a locus of the splitting. This is an axiom of BST theory; I call it the prior choice principle.

Example. Suppose that on a certain Tuesday Peter is in his office, having driven to work that morning. Think of this as a particular concrete event, with a definite causal past, and let the event be contingent, i.e., an event that has not been fated from all eternity. Take any history in which that event fails to occur, perhaps a history in which Peter spends the whole of Tuesday high in a tree in the Amazon jungle. The theory guarantees that if you look in the causal past of the given concrete Peter-in-office event (the one that we are supposing occurred), you will find a definite choice point at which things could have gone either in the direction of keeping the Peter-in-office event possible, or in the direction of keeping the Peter-in-tree history possible.4 You do not need to look in the future, and you also do not need to look far away at events going on “over there.” The point is that examination of the causal past of the Peter-in-office event suffices. (The theory does not presume to say if the choice point belonged to Peter or to a lion or to a bit of natural randomness or, perhaps, to some combination.)

4 The

theory will not let you exchange “event” and “history” here; precision of statement is essential.

304

2.4

LOGIC, THOUGHT AND ACTION

Funny business?

The fourth point records a recognition of an important way in which the theory should not be strengthened. The following is a principle that an unwary philosopher might easily be inclined to endorse.

Tempting principle (no funny business). If two choice points are related in a space-like way, so that the second is “over there” with respect to the first, then their respective choices are entirely independent of each other. This I call “no funny business.” Aristotle gives us a simple example of a causal story without funny business. He tells us how two market-goers meet at the market, as an “accidental” result of their choices that morning, made separately in far-away villages. Their individual choices, we all suppose, are bound to be totally uncorrelated, that is, independent, and that is what the tempting principle of no funny business says must be so. The theory of branching histories, however, resists this temptation. And it does not do so a priori. It does so because Our World seems, as a matter of fact, to contain violations of the tempting principle. With reference once more to Einstein, quantum mechanics seems to tell us that in fact it is possible for two utterly random choice points to be spacelike related, with no hint of a line of causal connection between them, and nevertheless fail to be independent. This is “funny business.” In Belnap, 2002e it is shown that this form of no-funny-business is provably equivalent to the following form: Every situation that is cause-like with respect to a certain outcome event lies in the causal past of that event.5 There is more. Reichenbach has taught us that whenever we find the long arm of coincidence stretching across space, it is in our nature to look for a common cause. Funny business precisely happens when there is objective coincidence — which is to say, a failure of independence — across space, without a common cause. Belnap, 2002c shows that two forms of “no common cause funny business” are equivalent to the two accounts of “funny business” of Belnap, 2002e. Since modern-day physics apparently says that funny business happens, it is good that the theory of branching histories has room for funny business — and indeed has the virtue of permitting us to offer a stable (albeit conjectural) account as to the difference between (1) mere indeterminism but with no funny business and (2) indeterminism with funny business.

5 If

you replace “every” by “some,” you have a different statement, one that is serious business instead of funny business, and one that is provably guaranteed by the aforementioned prior choice principle.

Agents and Agency in Branching Space-Times

2.5

305

Summary of BST

These necessarily too-brief points allude to a theory of branching histories that gives a satisfying account of how physical indeterminism can be local instead of global. It gives us an account of how choices and outcomes of natural random processes can affect only what lies in their causal future, touching neither their past nor the vast region of spacelike related events. But it does so in such a way as to allow plenty of room for individually random space-like-related processes to be, as the physicists say, “entangled.” Or, in the phrase I just used, the theory of branching histories helps us to come to terms with funny business. The result is that the theory of branching histories, in addition to helping us clarify ideas of action and agency, provides low-key suggestions for articulating some of the strangest phenomena uncovered by contemporary physicists. It does this by avoiding careless or fuzzy or sloppy formulations. It does this by insisting on a careful and rigorous account of what it is for indeterminism to be not immodestly global, but modestly local.

3.

Agents and agency in branching space-times

Since branching space-times is more realistic and more sophisticated than branching time as a representation of the indeterminist causal structure of our world, it is natural that one should consider agents and agency in branching space-times. The matter is hardly understood at all, which is precisely why it should be investigated. I offer some tentative thoughts on how some of the fundamental ideas might go.

Agents in branching space-times. First off, what shall we do with the concept of an agent? In FF the concept is represented by a set, Agent, and to be an agent, α, is merely to be a member of this set. Nothing is said about the “real internal constitution” of an agent beyond membership in the set Agent. Instead, the FF postulates describe agents only insofar as agents make choices. FF presumes that every agent has a choice at every moment (for technical convenience counting vacuous choices as choices), where the construction Choiceαm (h) represents the choice that agent α makes at moment m on history h . Since the choices of a set of agents Γ at the same moment are to be taken as simultaneous, it is natural that FF postulates such a set of choices to be independent: No combination of individual choices for the members of Γ is impossible. Behind these postulates lie the FF idea that, since moments are taken to be world-wide in extent, many agents can “occupy” a single moment. Recall §2: The chief difference between non-relativistic branching time

306

LOGIC, THOUGHT AND ACTION

and relativistic branching space-times lies in what “the causal order relation” relates. In the former case, moments are super-events taken to be “spatially” rich enough to be “occupied” by more than one agent. In the latter case, point events would seem to be so small as to admit the “presence” of at most one agent. For this reason it seems reasonable to begin by representing an agent as a set of point events, the set of point events that the agent may be thought of as occupying in the course of his or her life.6 Continuing to use Agent as the set of agents, we may therefore begin with the following. 3-1 Tentative postulate. (The agent as a set of point events) Every agent is a set of point events: ∀α[α ∈ Agent → α ⊆ Our World.7 When e ∈ α, we may say that the point event e is part of the agent α, or that α is located at or occupies e .8 Given that we are going to represent an agent α as a set of point events, what constraints make sense? In the beginning it seems best, since easiest, to think of the life of an agent in a particular history as a portion of a “world line.” 3-2 Tentative postulate. (Agents and world lines) The portion of the life of an agent in a particular history is a chain of point events: ∀α∀h[(α ∈ Agent & h ∈ Hist) → α ∩ h is a chain in Our World.9 Since point events in α can belong to many histories, and although the portion of α in each is a chain, it is easy to see (and is provable) that the entire set α will look like a tree, which accurately represents that there are alternative future possibilities for the life of α.10 I hope it is needless to say that I am claiming for this postulate only that its simplicity makes it a good beginning; it may well turn out to be better to represent an agent in a single history as a cloud of point events rather than as a chain. The thought is that nothing more that Tentative postulate 3-2 is

6 In

contrast, in branching time it would make no sense at all to represent an agent as a set of moments! 7 In this study I am not after a “fundamental” or “exclusive” ontology of agents. Here a representation counts as useful if its structure leads us to helpful theory. 8 None of these phrases is entirely happy, but given that in any event we are not trying to say what agents “really are,” perhaps it doesn’t matter. 9 A chain in Our World is a subset of Our World such that each pair of distinct members are comparable by the causal ordering relation. A chain may be empty. 10 I counsel you to have no patience with those who make fun of this picture by (falsely) describing it as saying that it is a possibility for tomorrow that Peter both be in his office and also stay at home.

Agents and Agency in Branching Space-Times

307

desirable for “first purposes.” For example, it may be that the track of an agent in any one history is dense or continuous; but at this stage of inquiry I doubt that it matters one way or the other. Let us think of the joint representation of two or more agents. Since point events are so small, it seems plausible that one should never have more than one agent located at a point event. Perhaps, then, we should enter the following. 3-3 Tentative postulate. (No agent overlap) Agents never overlap: ∀α1 ∀α2 [(α1 , α2 ∈ Agent & α1 =α2 ) →α1 ∩ α2 = ∅]. That requirement may, however, be too strong; it might be better to say only that no nonvacuous point event (no point event with more than one possibility in its immediate future) can be shared by two agents. The weaker restriction would not entirely forbid that the “world lines” of two agents intersect.

Choices and stits in branching space-times. The foregoing tentative postulates suggest (but certainly do not demand) that the choices open to an agent α at a point event e are exactly the same as the immediate possibilities at e in the sense of BST theory. In other words, there may well be no need for imposing a Choice function in addition to the possibilities definable from the BST causal ordering alone. Clarity will be heightened, however, if we introduce the Choice notation for agents as a separate primitive governing agents in branching space-times. 3-4 Notation. (Choice) Assuming that agent α occupies a point event e belonging to a history h , Choiceαe (h) is a new primitive to be read as “the choice available α at e to which h belongs.” Also Choiceαe , defined when e ∈ α, as {Choiceαe (h): h ∈ H(e) }, is to be read as “the set of choices available to α at e ,” and Choice is defined as that function defined for every agent/point-event pair α and e such that e ∈ α that delivers the set of choices available to α at e . Introducing Choiceαe (h) as a primitive leaves open the possibility that some interesting ideas turn out to need Choice as a separate concept. Since Πe (Definition 2-1) is the BST notation for the set of immediate possibilities at e , defined in terms of undividedness at e , the following postulate is to the effect that when a point event e is a part of agent α in history h , then there is no difference between the choice available at e for α that contains h , and the immediate possibility at e to which h belongs.

308

LOGIC, THOUGHT AND ACTION

α (h) and Π h) Let an agent α 3-5 Tentative postulate. (Choice ( e e occupy a certain point event e . The undividedness-at-e relation between histories of Our World determines what is immediately possible at e . We postulate that there is no difference between what is immediately possible at e and what α can choose at e : ∀e∀α∀h[e ∈ (α ∩ h) → Choiceαe (h) = Πe h ].

This postulate is not without content: It says that the choices concerning the immediate future that are available to an agent at a certain point event e are very finely articulated indeed. According to Tentative postulate 3-5, there is no finer articulation that Nature (or other agents) can impose beyond the choosing powers of the agent himself for his immediate future. On the other hand, and with equal realism, choices and happenings in the near vicinity of e can and doubtless will limit the powers of the agent concerning his non-immediate future. Joint agency, for example, will not concern what is immediately possible, as it does in FF; instead, the outcomes of joint agency will come to pass at some spatio-temporal remove from the choices of the various agents involved. Such is of the very essence of the relativistic flavor of agents in branching space-times. Already we can spell out a notion of agent responsibility for immediate outcomes along the lines of FF. Once we adapt to BST by letting truth be relative to pairs e/h (with e ∈ h ) rather than pairs m /h , the definition of the deliberative version of stit seems forced (compare Definition 1-1): [α stit: Q] is true at e with respect to h iff e ∈ α and 3-6 Definition. (Stit in BST) Positive condition. Q is true at e with respect to every history h 1 ∈ Choiceαe (h). (This is the part that says that α’s choice at e guarantees the truth of Q.) Negative condition. Q is not settled true at e : There is some history h 2 ∈ H(e) such that Q is false at e with respect to h 2 . (This says that α really does have a choice at e that is relevant to the truth of Q.)

Agent causation in branching space-times. There should be a rich possibility for notions of agent causation of outcomes adapted from Belnap, 2002b. Let O be a “scattered outcome event,” defined as a consistent set of outcome chains (chains that are nonempty and lower bounded). A “cause-like locus” for O is a point event e whose occurrence is consistent with the occurrence of O and which is such that what happens there makes a difference to the occurrence of O: ∃h [h ∈ (H(e) ∩ H O ) & h⊥e HO ]. Funny business happens when some

Agents and Agency in Branching Space-Times

309

cause-like locus for O is not in the causal past of O, and we know (or, via standard quantum mechanical puzzles going back to Einstein, we think we know) that funny business happens in the natural world. Whether choices of agents can exhibit funny business is presumably not something to be decided by fiat; nevertheless, it seems much the best to begin exploring agency in the absence of funny business, and so we have to be able to say what that means. I am far from sure of the best thing to mean by “no agentive funny business,” but the following, though probably inadequate, seems like a reasonable first suggestion to be pondered. 3-7 Definition. (Agentive funny business) A pair consisting of an outcome event O and a point event e counts as agentive funny business iff e is part of some agent α and e is a cause-like locus for O and e is not in the past of any part of O. 3-8 Tentative postulate. (No agentive-funny-business pairs) There are no pairs O and e that count as agentive funny business. It seems that Tentative postulate 3-8 suffices to rule out at least some cases of superluminal agent causation, and is in the vicinity of saying that the choices of agents make a difference only to the future. It is doubtful, however, that the postulate is strong enough to rule out all forms of agentive funny business; more analysis is needed than we have so far provided. A further strengthening might consider a joint choice initial involving an arbitrarily complex (but consistent) set of choice points, each of which belongs to some agent. It would seem that the whole complex should be independent of any joint choice initial, no matter how complex, and no matter if agentive or natural or mixed, as long as the purely agentive joint choice is space-like related to the other complex. Transitions are important in BST. We can be interested in an “effect” transition I → O, which is an ordered pair of an initial event I and a scattered outcome event O, with every member of the initial in the causal past of some member of the outcome. Such a transition is “contingent” if there is a dropping off of histories in the course of the transition, and in such a case we may with profit ask for a causal account of the matter. The answer is to be given in terms the set cc(I → O) of all of the causae causantes or “originating causes” of I → O. For this concept we first define the set cl (I → O) of “cause-like loci” for I → O as {e : ∃h[I ⊆h & h ⊥e H O ]}. These point events are exactly where “the action happens” in keeping O possible at the expense of ruling out altenative possibilities. Then cc(I → O) = {e → (H(e) ∩ H O ) : e ∈ cl(I → O)}; these are the transitions that “do the work.”

310

LOGIC, THOUGHT AND ACTION

In simple cases, which are the only ones that are so far well understood, we shall find all of cl (I → O) in the past of O, with no funny business. Often cc(I → O) will involve a mixture of agentive choices and non-agentive happenings; for example, the winning may have been partly caused by how the coin came up (e.g. heads came up), and partly by which bet was chosen (e.g. the bet was on heads). We know that given no funny business, the set of cc(I → O) form a set of “inns” conditions for I → O: Each member is an insufficient but non-redundant part of a necessary and sufficient condition (whence the acronymn “inns”) for the occurrence of the outcome O given the occurrence of the initial II. Nonredundancy means in particular that the complete causal story for I → O cannot leave out any member of cc(I → O). Furthermore, we can partition cl (I→O) into those cause-like loci that belong to α1 , α2 , etc., and those that do not belong to any agent. In this way we may expect fine control over our causal descriptions. For example, if cl (I→O)⊆α1 ∪ α2 , we can say that the two agents α1 and α2 were between them entirely responsible for the transition I→O.

Message-sending in branching space-times. Such joint responsibility is not of course the same as joint action. Our primitives, since being purely causal, do not permit discussion of factors such as “joint intention” that may be thought to underlie joint action. We can, however, say a little bit about the causal aspect of the communication that seems to be required for joint action. Look at Figure 3.

Figure 3.

Sending a message in branching space-times

What is represented causally is that α2 sends a message to α1 . Message-sending is agentive, and so there is a choice, in this case a choice by α2 whether to send (say) a red message or a blue message. There is indeterminism in the life of each agent, but only the sender is agentive in the matter; α1 is entirely passive. If the two are to “collaborate,” one would expect that a message sent by α2 to α1 would (not dictate but) limit the choices open to α1 at a subsequent choice point, depending on

Agents and Agency in Branching Space-Times

311

red vs. blue; and of course further collaboration would involve agency on the part of α1 in choosing a message to α2 . If an “intentional” story involving the minds of α1 and α2 cannot be told as an enrichment of the bare causal tale told by Figure 3, then it is hard to see how it could be useful. It is striking that Tentative postulate 3-8 says nothing about nonagentive funny business such as occurs, or seems to occur, in EPR-like funny business. I should think that it is essential to think through the combination of “no agentive funny business” with “some EPR-like nonagentive funny business.” Even physicists and philosophers of physics seem to distinguish between the independence constraints put on the chance outcomes of measurements on the one hand and on the deliberate settings-up of measurements. There is, however, too little talk about why there should be a difference, and its exact nature. It is plausible that the structure of agency in BST can help in thinking about this.

Choice and state in branching space-times. One last thought. In developing an evaluative theory of choices by an agent α in branching time, Horty, 2001 introduced the idea of a “state,” which Horty defined in effect by freezing the simultaneous choices of all agents except for the target agent α. Since in branching time the simultaneous choices of distinct agents are always independent (every combination is possible), he was in a position to consider the familiar two-dimensional diagram in which rows represent the choices available to α, columns represent the various possible “states” (independent of the choices of the agent), and the intersections are labeled with evaluative information. In transporting this idea to BST, one must consider from the beginning that space-like relatedness in BST, unlike simultaneity in branching time, is definitely not a transitive relation, any more than is the same relation in Minkowski space-time. It follows that a choice by an agent α can well be spacelike related to each of two choices by a second agent, those two choices being causally ordered (successive), so that although each is independent (by space-like relatedness) of the choices of α, they are not independent of each other. Worse (if that is the right word), branching space-times allows that choice points for other agents, even though space-like related to the given choices of α, are outright inconsistent with each other. One is therefore going to have to put in extra work — and work that should be carried out — in order to locate a satisfying account of “state” that is eligible to do the same work as Horty’s account in branching time. It seems to me that in working through this nest of considerations, one would naturally be led to a useful theory of “games in branching space times” that would take with utmost seriousness the causal structure of

312

LOGIC, THOUGHT AND ACTION

the players and the plays in a fashion that sharply separates (as von Neumann’s theory does not) causal and epistemic considerations. It is one thing for two choices to be independent of each other, and a different thing for there to be certain relations of (say) ignorance. So much is clear. It is not of course self-evident that the difference makes a difference. If it does not, however, that should be the result of serious reflection, not an unconsidered presupposition.

4.

Summary

I have tried to sketch some of the main ideas of stit theory, drawing on FF, and I have given a kind of overview of branching-times theory as discussed in the publications listed in note 3. Then I have raised but definitely not answered some questions concerning how to use these materials in order to fashion a theory of agents and their choices in branching space-times. I have suggested with all too much brevity some possible lines of investigation.

References Belnap N. (1992). “Branching Space-Time”. Synthese 385–434. — (1999). “Concrete Transitions.” In Actions, norms, values: Discussions with Georg Henrik von Wright. G. Meggle, Ed. Berlin: Walter de Gruyter. 227–236. A “postprint” (2002) may be obtained from http://philsci-archive.pitt.edu — (2002a). Double Time References: Speech-Act Reports as Modalities in an Indeterminist Setting. In Wolter et al, 37–58. A preprint of this essay may be obtained from http://www.pitt.edu/∼belnap — (2002b). A Theory of Causation: Causae causantes (Originating Causes) as inus Conditions in Branching Space-Times. A preprint of this essay may be obtained from http://philsci.archive.pitt.edu. — (2002c). No–Common-Cause EPR-Like Funny Business in Branching Space-Times. Forthcoming in Philosophical studies. A preprint of this essay may be obtained from http://philsci-archive.pitt.edu — (2002d). Branching Histories Approach to Indeterminism and Free Will. This essay may be obtained from http://philsci-archive. pitt.edu — (2002e). EPR-like ‘Funny Business’ in the Theory of Branching SpaceTimes. In Placek and Butterfield 2002, 293–315. A preprint of this essay may be obtained from http://philsci-archive.pitt.edu Belnap N., Perloff M. and Xu M. Facing the Future: Agents and Choices in our Indeterminist World. Oxford: Oxford University Press. 2000.

Agents and Agency in Branching Space-Times

313

Copley B. (2000). “Conceptualizing the Futurate and the Future”. In MIT working papers in philosophy and linguistics: The linguistics / philosophy interface, Bhatt R., Hawley P., Kackl M. and Maitra I. eds. Cambridge MA.: MIT Press. 45–72. Horty J.F. (2001). Agency and Deontic Logic. Oxford: Oxford University Press. Kane R. (1998). The Significance of Free Will. Oxford: Oxford University Press. Muller ¨ T. (2002). Branching Space-Time, Modal Logic and the Counterfactual Conditional. In Placek and Butterfield, 273–291. Placek T. (2000a)“Stochastic Outcomes in Branching Space-Time: Analysis of Bell’s Theorem”. British journal for the philosophy of science, 51 :445–475. — (2000). Is Nature Deterministic?. Krak´ o´w: Jagiellonian University Press. — (2002b). Partial Indeterminism is Enough: a Branching Analysis of Bell-Type Inequalities. In Placek and Butterfield 2002, 317–342. Placek T. and Butterfield J. (20020). Non-Locality and Modality. Dordrecht: Kluwer Academic Publishers. Rakic´ N. (1997). Common-Sense Time and Special Relativity. Ph. D. thesis. University of Amsterdam. Szabo L. and Belnap N. (1996). Branching Space-Time Analysis of the GHZ Theorem. Foundations of physics, 26, 8:989-1002. Thomson J. (1977). Acts and Other Events. Ithaca: Cornell University Press. Wolter F., Wansing H., de Rijke M. and Zakharyaschev M. “Advances in Modal Logic”. World Scientific Co. Pte. Ltd, 2002:3. Singapore.