And since the feature B (which equals not-B′) does obtain in the actual world yesterday, we get to conclude under J that (VI-6) and thus (VI-5) are here indeed true, and that the corresponding “fatalistic” conclusion—that past events for which our present actions are in some sense “sufficient” are nevertheless not in our control—accordingly goes through. This argument appears to be sound when analyzed under the rules of J.
We thus appear to be in a position to reject Taylor’s claim, with respect to his two different arguments and their two very different conclusions, that: “These two arguments are formally identical, except only for tenses.... If then either argument is a good one—and surely the first (the past-argument) is—then the other is just as good, no matter how anyone might feel about its conclusion.”41 We can say with Charles Brown that the two arguments are in fact not formally identical, because they equivocate between the two different senses of “sufficient”; order O is causally sufficient for B, but headline H is only an infallible indication of B, the same way we saw a fire “ensure” the presence of fuel. More rigorously, we can say that this essay’s physical-modality device gives formal reason to think that the two arguments are in fact not equally good: under the system of analysis J provides, the past-argument is sound, but the future-argument is not even valid. Accordingly, I think this essay’s approach to the semantics of physical modality lets us clear one of the biggest hurdles that stands in the way of the opponent of Taylor’s claim that we are “forced” into accepting future-fatalism by the reasoning of his paper.
In another famous piece of work, “Time, Truth and Modalities,” cowritten with Keith Lehrer,42 Taylor again exploits the complex relations between time and physical possibility to produce a vexing puzzle. Here the difficulty is said to be that: “It often happens that one does not do what he can do, and thus forfeits, through neglect, some end for which that act would have been the unique means,” and that analyses of such commonplace situations seem “to entail a contradiction, and it is far from evident how such a contradiction is to be resolved.”43
Here I’m simply going to transcribe Taylor’s and Lehrer’s (T and L’s) picture of the relevant case from their paper itself, since their description is concise and important.Smith, who works in the country, has promised his wife to be in the city at four o’clock. It is now shortly before half past three, and Smith is seated at a table in the country airport not far from a plane that is about to depart for the city. At the present time, this plane, which leaves at half past three, is the swiftest possible means of transportation to the city, and the plane arrives non-stop in the city at precisely four o’clock. Finally, let us suppose that although there is nothing to prevent Smith from leaving on the plane at half past three, he in fact does not do so.
In the case that we have imagined the following statements are all true of Smith:(1) If Smith does not leave at 3:30, then he cannot arrive at 4:00;
(2) If Smith does leave at 3:30, then he will arrive at 4:00
(3) Smith can leave at 3:30, BUT
(4) Smith does not leave at 3:30.
These four statements are, thus, seemingly consistent, but unfortunately they also seem to entail a contradiction. For:(2) If Smith does leave at 3:30, then he will arrive at 4:00 and
(3) Smith can leave at 3:30
seem clearly to entail
(5) Smith can arrive at 4:00.
But, on the other hand:(1) If Smith does not leave at 3:30, then he cannot arrive at 4:00
and(4) Smith does not leave at 3:30
seem equally clearly to entail
(6) Smith cannot arrive at 4:00.
Statements (5) and (6) are clearly contradictories.44
The reader who has borne so far through this essay’s work on the semantics of physical modality should be inclined to say that here the apparent “contradiction” depends at some level on ambiguities in the formulated relations between physical possibility and time. I agree that this is true, but I think that, once the time issues are cleaned up, the Smith problem actually hinges on confusions about physical-possibility-at-the-present, a fact which system J gives us the tools to recognize and explain. The following will not be the only valid criticisms of T and L’s analysis of their case, but I think they will be the central ones.
We should note first that the “can” in this case pretty clearly stands in place of a modal operator denoting physical possibility. T and L vaguely acknowledge this by saying that “... ‘can’ as it occurs in our four statements is, by ordinary standards, a rather peculiar modal term,” and that the reason why standard modal-logic analyses of the Smith problem run into such trouble (trouble T and L rub their hands over at great length) is that “... the paradigm modality of the development of modal logic has been the logical modalities such as logical possibility as opposed to non-logical modalities such as causal possibility.” (For “causal” here we can read “physical.”)
Next we should recall that, as I have argued at tiresome length, physical-modal propositions are acutely time-and-situation-sensitive, and are symbolically well-formed and perspicuously analyzable only when formalized with explicit temporal operators designating both the indices over which the modality is asserted to range and the indices with respect to which it is to be evaluated. Thus I have argued that a standard physical-modal proposition must be formally expressed as containing two temporal operators. Again, T and L more or less agree, saying that: “The kind of possibility expressed by ‘can’ is unusual in that it is essential with respect to this kind of possibility to distinguish the time of an event that is possible from the time of the possibility of the event,” that: “Abilities can be won and lost, such as that what one is at one time able to do he may subsequently be able to do no longer, and such facts can be expressed non-misleadingly by the appropriate tenses and time references,” and so finally, that in cases such as that of Smith: “... ‘can’-statements have an implicit double time-reference.”45
Given this, we might first wonder why Taylor has never publicly applied such considerations to his original “Fatalism”-problem, and then be tempted to say that the Smith problem itself can be solved simply by making explicit the times and situations with respect to which the two contradictory physical-modal statements, “Smith can arrive at 4:00” and “Smith cannot arrive at 4:00,” are to be evaluated. Let’s look at the second statement first. Recall that physical possibility is to be understood in terms of physical compatibility among situations-at-times. “Smith cannot arrive at 4:00” is true, then, with respect to what time-and-situation? We look around for some situation-at-a-time that would stand in an appropriate causal relation to Smith’s potential flight to the city—the only way in the case he can arrive at 4:00—with which such a flight, and so such an arrival-at-4: 00, would be incompatible. The relevant modally-limiting situation is easily seen here to be the situation in which Smith did not in fact leave on the plane at 3:30. We can assume that there was an interval greater than one moment when Smith could have boarded the plane, but let’s suppose that the very first moment when such an option was no longer open to him, the moment when the plane’s doors closed and its engines revved, was the moment: 3:30:19.97. It appears reasonable to say that, with respect to any temporal point at or after this moment, as T and L cook the case, it is not physically possible that Smith arrives at 4:00, so that the proposition: (3:30:19.97 (~◊4:00 Smith arrives)) appears to be true.
The analysis of “Smith can arrive at 4:00” looks just as easy. The statement “Smith can arrive at 4:00” is at some time true just in case there stands in the relation R to the actual-world-at-that-time no situation-at-a-time physically incompatible with Smith’s arrival at 4:00. At, for example, 3:30:18.00 no such situation stands in the appropriate mother-relation to the actual world, and the statements: (◊4:00 Smith arrives) looks at this time to be true. The proposition accordingly looks to remain true at all moments up to 3:30:19.97, for it is at this moment that an actual situation causally incompatible w
ith Smith’s timely arrival begins to obtain. I think we may thus reasonably conclude here that:VI-7) (∀tn(tn<3:30:19.97)(tn◊4:00 Smith arrives))
is true, and that:VI-8) (∀tn(tn≥3:30:19.97)(tn~◊4:00 Smith arrives))
is also true. Thus T and L’s original statements (5) and (6) look to be in fact perfectly compatible, as long as the crucial time-intervals with respect to which the modalities are to be evaluated are made explicit.
But T and L have an extremely clever way to avoid this solution, though I am going to make their dodge more perspicuous than they make it. T and L now fasten on the very last instant at which Smith was able to board the plane—say, the instant: 3:30:19.96. (T and L call this instant simply “3:30,” which is, I think confusing in its 60-second breadth of reference.) Here 3:30:19.96 is the critical instant: if Smith does not board before or at this instant, he does not board. Suppose Smith is hovering right outside the door to the plane at 3:30:19.96, able to dart in if he wishes or stay out instead if he wishes. We designate the critical moment, 3:30:19.96, by the word “now.” T and L thus restate their problem:(1′) If Smith does not board at 3:30:19.96, then he cannot now (at 3:30:19.96) arrive at 4:00.
(2′) If Smith boards at 3:30:19.96, then he will arrive at 4:00.
(3′) Smith can now (at 3:30:19.96) board at 3:30:19.96.
(4′) Smith does not board at 3:30:19.96.
By (2′) and (3′) we get:(5′) Smith can now (at 3:30:19.96) arrive at 4:00.
But by (1′) and (4′) we get:(6′) Smith cannot now (at 3:30:19.96) arrive at 4:00.
Thus, by collapsing the relevant moment of the case to the last, vanishingly small, ideal moment when Smith can but does not board the plane, T and L seem to have kept their paradox alive by arguing that both:(VI-9) 3:30:19.96 (◊4:00 Smith arrives)
and(VI-10) 3:30:19.96 (∼◊4:00 Smith arrives)
are validly derivable from the apparently consistent (1′)-(4′). Since the moment 3:30:19.96 is the moment I (and T and L) have designated “now,” the locus of the Smith problem is now fixed at the issue of physical-possibility-at-the-present.
T and L now propose to resolve the problem by denying the validity of the inference from (2′) and (3′) to (5′). First, they say that they understand a proposition like (tn◊p) to mean really that nothing has happened by time tn sufficient to “prevent” p (this is an analysis of (tn◊p) with which I am obviously in complete agreement). T and L accordingly reformulate (2′), (3′) and (5′) as:(2”) If Smith boards at 3:30:19.96, then he will arrive at 4:00.
(3”) Nothing has happened by 3:30:19.96 sufficient to prevent Smith from boarding at 3:30:19.96.
(5”) Nothing has happened by 3:30:19.96 sufficient to prevent Smith from arriving at 4:00.
T and L claim that this inference is no good, that something has happened by 3:30:19:96 sufficient to prevent Smith from arriving at 4:00: namely, his failure to board at 3:30:19.96. They thus claim that the established modal inference:(If p, then q)
(p is possible)
∴ (q is possible)
is in fact not valid with respect to physical modalities. They “solve” their problem by asserting that, if p is the only means to q, then, even though p is “possible,” if p does not in fact occur, q is not possible.
This seems to me quite curious. It has the following consequences: that if p-at-tn is the only means for and is sufficient for q-at-tn+1, then, if p fails to occur at tn, p-at-tn still remains somehow possible, and it is only the consequence of p-at-tn, the consequence q-at-tn+1, which is impossible. This I simply reject. I think T and L are here accepting only an implication of a feature of physical modality while seeking to deny the feature itself. The reader should be able to anticipate that I propose to solve the reformulated Smith problem differently: namely, simply by denying the compatibility of T and L’s “consistent” premises (3′) and (4′). That is, it is a consequence of my understanding of physical modality that, given the truth of “Smith does not board at time tn,” the proposition “At tn it is possible that Smith boards at tn” is simply false.
It’s easy to see why this is so. See that (3′), perspicuously put, amounts to:(VI-11) 3:30:19.96 (◊3:30:19.96 Smith boards).
This, of course, fixes the situation in the context of which the modality is to be evaluated at that situation obtaining in the actual-world-at-3: 30:19.96. The assertion that at 3:30:19.96 it is physically possible that Smith boards at 3:30:19.96 is thus true only if there is no feature of the actual situation that obtains at 3:30:19.96 which is physically incompatible with Smith’s boarding at 3:30:19.96. But see that premise (4′) established just such a feature: the actual situation at 3:30:19.96 is that Smith does not board at 3:30:19.96. I hold that, given that an actual feature of the situation that obtains at time tn is Smith’s failure-to-board-at-tn, there is no situation-at-tn, physically compatible with this actual situation, in which Smith boards at tn, since this would have as a consequence the fact that it is physically possible at tn that Smith both does and does not board at tn, which is clearly absurd.
T and L, however, explicitly reject this J-type analysis of the reformulated Smith problem. They prefer to treat Smith’s boarding-at-t n as an event somehow physically-compatible-at-tn with the actual situation-at-tn in which Smith does not board-at-tn, while treating only the causal consequence of this boarding-event as physically incompatible with the actual situation. This strikes me as both confused and ad hoc. T and L justify their rejection of system J’s solution in the following way: “The crux of the matter is that Smith’s not ... (boarding at 3:30:19.96) ... does not count as a condition that prevents Smith from ... (boarding at 3:30:19.96) ..., because if it did it would follow that no one can ever do anything he does not do, which is surely false.”46
This claim seems very strange when read alongside Taylor’s own position in “Fatalism.” And under this essay’s analysis of physical possibility the claim is in fact not true. Under this essay’s analysis, it is quite possible at t1 that any number of things happen at t2, all but one of which will not actually happen at t2. My analysis yields only the conclusion that, at t2, it is not physically possible for something both to happen-at-t2 and not-happen-at t2. Given the actual situation that obtains at, say, 3:30:19.95, it is at this time both physically possible that Smith boards at 3:30:19.96 and physically possible that Smith does not board at 3:30:19.96, because neither event-at-3:30:19.96 is physically incompatible with the actual situation-at-3:30:19.95. Given, though, that an explicit feature of the actual situation-at -3:30:19.96 is Smith not boarding at 3:30:19.96, I hold that there is no compatible situation-at-3:30:19.96 in which he boards. It is not at all the case that Smith “can never do anything he does not do,” but only that Smith cannot possibly both do and not do something at the same time. This fact seems to me completely and obviously true, and it’s also captured beautifully by this essay’s proposed system for properly understanding terms like “can.” I thus firmly believe that the system J is able to offer a far more satisfactory solution to T and L’s own Smith problem than the authors themselves come up with.
Finally, I’ll quickly note that if system J is regarded as constituting a correct way to understand the semantics of physical modality, the whole “problem of future contingents,” at least as a problem in such semantics, seems capable of being put to rest. A solution to the physical-modal version of the future-contingents problem would seem to be found if we could show that the inference:
is invalid under a coherent and workable system for understanding physical-modal propositions. And under J, the inference is indeed invalid. We can conceive of the actual world-at-t1 as located at an intersection of two causal paths, jx and jy, extending from t1 to t2. Let jx represent the causal path from the actual world-at-t1 to the actual world-at-t2, and let jy represent a path from the actual world-at-t1 to some world-at-t2 whose features are not physically incompatible with the features of the actual world-at-t1, to which the actual world-at-t 1 could possibly have given
rise. The diagram for such a structure-fragment would look like:
Recall that, by system J’s Rule (1), (t1(t2p)) is true in the actual world-at-t1 just in case p is true in the actual world-at-t2, in W2. Recall also that, by Rule (2), (t1◊~(t2p)) is true in the actual world-at-t 1 just in case there is some world-at-t2, which shares a physical-compatibility path with the actual world-at-t1, in which p is false. We can easily imagine, for some p (say, my deciding whether to scratch), a possible world-at-t2, an A2, causally joined in jy to W1, in which not-p obtains, and so in which p is false, while there is also a world-at-t 2 that is actual, and thus shares the actuality-path jx with W1, in which p is true, obviously making t1(t2p) true in W1.
System J, then, conceiving a physical possibility structure as a set of distinct but intersecting causal paths, constitutes a workable rigorous device with which we can justify our quite reasonable belief that the fact that something now will be the case does not make it now physically impossible that it will not be the case. That is, in J the present truth of a future-tensed proposition does not entail that it is at present physically necessary that the state of affairs denoted by the proposition will obtain. (The alethic-modal version of the problem of future contingents luckily does not concern us here.)
Fate, Time, and Language: An Essay on Free Will Page 21