Let’s apply this admittedly complex set of considerations to the terrorist case, by evaluating the case with respect to the two types of tensed physical-modal propositions I hold to be inequivalent. We’ll first analyze the idea that the facts of the case allow us to conclude today that ~◊P 1E. Note that here the modal operator is given wide scope over the only explicit tense-operator in the proposition. This fixes the time at which and with respect to which we evaluate the modal at now. It fixes the situation in the context of which, together with the physical laws that govern the way the world works, we evaluate the physical modal, at the situation that obtains now. What is one component of the physical situation that obtains now? It is that there are fewer than 20 rads of radiation on the Amherst College campus. We have seen that, given that (P1E → (R>20)), which I have charitably granted to Taylor amounts to the same as □(P1E → (R>20)), the fact that ~(R>20) obtains today is physically incompatible with there having been a nuclear explosion on campus yesterday. We can thus say, from the vantage-point of the evaluative moment now, taking note of the physical situation that obtains now, that, given the character of the world-situation now, viz., the low radiation on campus, and given the fact that, had there been an explosion yesterday, there “surely” would be high radiation today, it is today physically impossible that a nuclear explosion did occur yesterday, (~◊P1E). Giving Taylor the benefit of the doubt at every modal turn, we can accept the conclusion that □(P1E → (R>20) implies (~(R>20) → ~◊P1E).
Let’s now determine whether the facts of the terrorist case could in any way allow us to conclude today that P1~◊E. See first that an explicit indexed tense-operator designating a point in the past is in this proposition given wide scope. This fixes the time with respect to which we evaluate the modal at the time designated by P1, namely yesterday. It fixes the situation in the context of which, together with the physical laws that govern the way the world works, we evaluate the modal, at the situation that obtained at the interval designated by P1, namely yesterday. So we are now concerned, not with what is physically possible to have occurred in the past given the physical situation that obtains now, but rather with what was physically possible yesterday given the physical situation that obtained yesterday.
Now, the relevant features of the physical situation that obtained yesterday are things like the nuclear weapon being fully functional, the triggering mechanism being operational, the terrorist being fit and healthy and alert and able to move his finger and not constrained. These look to be the things that determine whether the nuclear explosion is possible at P1, and it seems plausible to say, under some natural view of a causally-connected world, that these things are, if anything, actually functions of situations that obtained at times prior to P1 and stood in appropriate causal relations to the physical-situation-at-P1. (Please keep this in mind: to say that E-at-P1 was possible-at-P1 is to say basically that there was nothing in the situations obtaining prior to P1 that rendered E-at-P1 impossible; thus physical possibility is beginning to be understood as a relation between situations through time, as a “diachronic” relation of compatibility between sets of conditions. This will turn out to be very important.) Thus it looks sensible to say that the only circumstances not obtaining at P1 that might possibly affect the modal character of E-at-P1 are those that obtained prior to P1 and had a causal influence on the situation-at-P1. The modal character of E-at-P 1 cannot be a function of the presence or absence of high radiation today, because the presence or absence of high radiation today was obviously not a part of the physical situations that obtained at P1 and prior to P1 and that together with physical laws determined what was physically-possible-at-P1. The fact that there is low radiation today did not yesterday affect the terrorist’s ability and freedom and opportunity to press the trigger and so cause the explosion any more than the absence of a battle today yesterday bore on the admiral’s freedom and power to give order O if he chose. This again is because what is physically-possible-at-a-time is determined by general laws and by the physical situations that obtain at or before that time (ultimately I am going to argue that what determines p’s possibility-at-tn are only the situations that obtain prior to tn, not at tn), and I assert that temporally posterior consequences of events are very obviously not part of the physical situations that obtain at the times that bear on the possibilities of those events.
It might be objected that I am either covertly rejecting presupposition 1 and denying LEM/PB as applicable to future-tensed propositions, or else simply begging the question against Taylor: if it was indeed true at P1 that there would not be high radiation today, this fact seems to affect the possibility-at-P1 of E occurring at P1. This I do not accept. That certain propositions might possess a certain designated semantic value at P1 is not a part of the physical situations that obtain at and before P1 and that determine what is physically-possible-at-P 1. To deny this fact is, I think, simply to confuse and reverse the relationships between the truths of propositions and the states of affairs in the world that make them true. That “There will not be more than 20 rads on the Amherst campus tomorrow” is true at P1 is a function of what happens at P1, given that (P1E → (R>20)); what happens at P1 is here a function of what is physically possible at P1 and what agents do at P1; but what is physically possible in a situation-at-time is determined by general laws and by compatibility with certain other situations-at-times standing in the appropriate causal relations, and what agents do is determined by such things as their characters and motives and their causal relations to other things and agents. At no point in this network is what happens or can happen a function of what propositions are true; rather it is the other way around. If the fatalist still disagrees, he is invited to present an argument for the very strange idea that what is physically possible in the actual concrete world depends on the semantic properties of abstract entities. (And please note that this cannot simply be Taylor’s argument again: for, first, his argument was that the actual occurrence or non-occurrence of events in the future can affect what is physically-possible-now, which is a very different claim; and, second, I think I’m presenting an analysis that casts significant doubt on the very things Taylor claims to have proven, and on his proof itself.)
So far I have tried to provide some motivation and common-sense evidence for the claim that the apparent force of Taylor’s fatalistic argument hinges on confusions about what exactly its legitimate conclusion is—that it hinges on our not noticing what I have called the Taylor inequivalence. What we now require is the introduction and characterization of a formal device, the beginnings of a system, under which the inequivalence can be demonstrated and accounted for in a rigorous way.
V. A FORMAL DEVICE FOR REPRESENTING AND EXPLAINING THE TAYLOR INEQUIVALENCE: FEATURES AND IMPLICATIONS OF THE INTENSIONAL-PHYSICAL-MODALITY SYSTEM J.
It’s hoped that the terrorist case, and the application of its analysis to the original Taylor problem, provides some motivation for (and enhances the attractiveness of) an attempt to show that tensed physical-modal propositions of different explicit-operator scope are not always equivalent and should not always be treated as such. That there are fewer than 20 rads on the Amherst campus today means that, given the situation today and the sufficiency-relation that obtains between nuclear explosions and radiation, it is not now possible that an explosion occurred yesterday, not now possible that the terrorist did yesterday press the trigger. But clearly we do not want to say, the unpleasantness of the case aside, that a feature of the physical situation today alone was also a feature of a physical situation that obtained yesterday or before yesterday and constrained the terrorist’s freedom of action yesterday. That is, we do not want to say that yesterday it was not possible for the nuclear explosion to occur in Amherst, not possible then for the terrorist to press the trigger. But it is precisely the second sort of conclusion that Taylor and his defenders would have us draw from the facts of the case, because only a conclusion of the second sort would constitute or imply fata
lism. It is precisely the second sort of conclusion I wish to resist, by claiming first that only the first sort of conclusion is in any way “forced” upon us by the very most Taylor-ish interpretation of the terrorist case, and then that the first sort of conclusion is not equivalent to the second. Obviously, my claims, and so my resistance, will be much stronger if I can provide formal reasons for thinking that propositions such as MT1’ and MT2’ are not equivalent, if I can point to a rich and interesting and non-ad hoc device for showing that they are not. The introduction, characterization, defense and application of such a device will constitute most of the rest of this essay.
A device for formalizing, representing and interpreting tensed physical-modal propositions will be a type of “intensional” semantics, a semantics designed to accommodate considerations both of modality and of time. I plan first to give a bit of diagrammatic demonstration of the way I propose to understand such propositions as P1~◊O and ~◊P 1O. I will make some use of the “possible-worlds” semantics of Kripke, and of Montague’s important work in intensional logic. This is so even though Kripke and Montague deal formally only with logical modalities, and thus understand possibility in terms of a synchronic relation between alternative, simultaneous possible “worlds” that stand in appropriate relations (while I will be arguing that physical possibility is best understood as a diachronic relation of compatibility under causal laws between sets of conditions as the condition-sets stand in appropriate relations through time). I’ll use them because, particularly in terms of graphic representation, the important Kripke and Montague models have features that will serve even my very different purposes very nicely. Some reasonable familiarity with the Kripke and Montague models will be assumed: I will explicitly introduce only (simplified versions of) the features that bear directly on what I wish to do.
Kripke understands the semantics of possibility in terms of a set K of possible worlds. Since the modalities Kripke is concerned with are alethic, K is the set of all worlds which are not logically inconsistent. A very important relation R between possible worlds, called an accessibility relation, is introduced to denote the relation of relative accessibility among the members of the set K: if of worlds W and B we can say that WRB, this means that W is “accessible” from B. Requirements for the accessibility relation obtaining between worlds can be strengthened or weakened to yield different modal systems and models. A reflexive and transitive relation R yields the modal system S4, a stipulation that R be reflexive, symmetric and transitive yields the different system S5, and so on.34 For a simple and intuitive representation of Kripke’s device, we can assume that every member of K (with K of course being nondenumerably infinite) is accessible from every other member.
Modal operators can thus be understood as working in the following way. ◊p is true in the actual world iff p obtains in some world accessible from the actual world, here any member of K. So to determine the truth of ◊p, we examine what I ask us to pretend are all the members of K, with capital letters standing for possible worlds, of which W will designate the actual world:W(p)
A
B(p)
C
D
E
F(p)
G
H
to determine whether any of the worlds include p. Since as we can see at least one world does, ◊p is here true. To determine the truth of the simple proposition p, we look to see whether p obtains in that member of K which is the actual world, W. p does here obtain in W, so p is here true. □p is true just in case p obtains in every possible world accessible from the actual world. Here p does not obtain in every world, so □p is here false.
Richard Montague’s intensional logic enriches the one-dimensional Kripke access-plane, (simplistically) represented above, by introducing a time-axis. This lets us speak, not just of possible worlds, but of possible worlds-at-times, or “indices”:
(Note here that ideally in the Montague diagram the number of time coordinates is densely infinite and between every two discrete moments there is a third moment.)
Montague’s device allows us here to visualize an analysis of a proposition containing both tense- and modal operators. Let t3 take the place of our old metrio-index operator P1. The proposition t3◊p will be true just in case p obtains in some accessible member of K-at-t3. The truth of t3p will depend on p obtaining at index 〈W,t3〉, or W3. The truth of t3□p will depend on p obtaining in all accessible K-member worlds-at-t3, at all 3-indices, where an index is simply an ordered pair 〈world, time〉. Note again that here possibility is still understood in terms of a synchronic relation between worlds: p is possible in some world-at-tn iff it is actual in some other world-at-tn. Possibility is here conceived as a relation between alternative worlds at the same time; where, as I have said, this essay’s analysis of physical modality will understand physical possibility in terms of a relation between physically compatible situations through time, joined in the appropriate causal relations.
The reason why I propose to understand possibility in a fundamentally different way from, say, Montague, is that Montague is concerned in his analysis only with alethic modalities, □p and ◊p, not with physical modalities, □p and ◊p. I hold that tensed physical-modal propositions require a very different kind of analysis from that appropriate for tensed alethic-modal propositions. Again, my reason is that (situational) physical modalities enter into relationships with time that alethic modalities just do not. See for instance that under the above representation of Montague’s intensional device, we are unable to analyze the two propositions (IV-1) and (IV-2); the closest we can come is an analysis of two tensed alethic-modal propositions:V-1) t3(~◊O)
andV-2) ~◊ (t3O).
The reader should be able to see that (V-1) and (V-2) are actually equivalent under a Montague-type analysis, even given the intuitive scope-rules on this essay’s pages 32 and 33 [see this volume, page 171—eds.]. Since the tense-operator has wide scope in (V-1), to determine the truth of the proposition we go to the time-coordinate t3 and examine the set of worlds-at-t3 for the presence of O. In the above Montague-type grid, since O does not appear at any 3-indices, (V-1) is true. Since the modal operator has explicit wide scope in (V-2), we go in Montague’s model “first” to world-set K, and see whether O perhaps appears in any of the worlds ... at t3, which means we are again simply examining the same set of worlds-at-t3. The evaluative procedure is thus the same for (V-1) and (V-2) here, and the two propositions come out equivalent in this model, true under exactly the same conditions.
Montague’s is the most complete and satisfactory intensional semantics on the market today, but I hold that it is not appropriate for this essay’s analysis of the Taylor problem. As I’ve said, this is because Montague—and Kripke—semantics are designed to handle logical modalities. The relations of logical modalities to times are (comparatively) simple. The relations of physical modalities to times are not. A Montague-type semantics does indeed provide us with an elegant way to evaluate modalities with respect to times, but it is neither concerned with nor equipped to take formal account of the difference, absolutely vital in physical-modal problems, between: (1) evaluating a modality at a time (i.e., the time with respect to which the modality is evaluated); and (2) evaluating a modality-at-a-time (i.e., the time, the temporal interval, to which the modality is asserted to apply) at a time (i.e., the time with respect to which the modality-at-a-time is to be evaluated).
Here we require a physical-modal semantic device that can take account of the fact that the time with respect to which a modality is evaluated, and the physical situation that obtains at the time with respect to which the modality is evaluated, can have an effect on that evaluation. We need, in a satisfactory semantics of physical modality, to be able to distinguish between an evaluation at t4 of a modality asserted to obtain at t3, and an evaluation at t3 of a modality asserted to obtain at t3; between an evaluation at t3 of a modality, and an evaluation of a modality-at-t3. That these really need to be disti
nguished has, I hope, been made apparent by this essay’s analyses of the Taylor problem and the terrorist case so far. In, for example, the Taylor problem, the difficulty is precisely that not only is a physical modality asserted to obtain at a certain time-and-situation (today, on the deck of the destroyer, the admiral “cannot” give order O), but that the character of this modality, the truth-value of the modal proposition, is affected by another, distinct time-and-situation (tomorrow, which we’re somehow able for evaluative purposes to see, there is no sea-battle), and that Taylor proposes to have us ignore this sort of cross-situational modal effect. What we need is an intensional system in which we can show that the claim: “At t3 it is physically possible that p, given the situations that obtain at and before t3,” is at least in some cases perfectly compatible with the claim: “Given the situation that obtains at t4, at t4 it is not physically possible that p-at-t3.” Under the existing intensional semantics, these claims are in fact not compatible (indeed, the second claim would in the existing systems make sense at all only if understood as a kind of re-phrasing of a denial of the first). We require a system equipped to show that sometimes what is possible-at-time-tn relative to one time-and-situation is in fact not possible-at-time-tn relative to some other time-and-situation.
Here, initially presented in an intuitive, diagrammatic way, are some features of a device I regard as appropriate for coherently interpreting tensed physical-modal propositions: the physical-modality system J. It differs at this point from a Kripke-/Montague-type device in only a couple of ways. First, the set of possible worlds-at-times (or world-situations-at-times) is restricted to the set of all possible world-situations in which all and only those physical laws which govern our actual world also apply. Second, the primitive accessibility relation R, as it applies to distinct worlds, is to be understood as corresponding to diachronic physical compatibility between causally-connected worlds, between worlds at different times. This has the seemingly counterintuitive result that no given world-at-time appears to be accessible from itself, but, as we’ll see, system J allows us to preserve the tautologicality of “p implies possibly-p” and “necessarily-p implies p,” and thus retains all the desirable features of a system in which R is reflexive. The obvious importance of time to this analysis means that I will be concerned from here on with relations between indices (worlds-at-times), and not worlds per se.
Fate, Time, and Language Page 18