For an argument to be sound, all its premises must be true at once. We see all the Rockettes at once. So of course it seems to us that (1a)-(3a) are all true at once. For that matter, we tend to think all identity-statements true omnitemporally if true at all (or at least true for all time after their subjects begin to exist48), and so again true at once (once their subjects exist). But when Jane has gone home, nothing satisfies the description "the leftmost Rockette'' (though Jane of course satisfies "the person who was the leftmost Rockette''). If nothing satisfies "the leftmost Rockette,'' the description does not refer. And if the description does not refer, (1a) is not true. If we see (1a) as omnitemporally true (or as true for all time after Jane starts to exist), this is because we treat "the leftmost Rockette'' as temporally rigid, picking out Jane at all times if it picks her out at any (or all future times once it picks her out at any). But we needn't. And read as involving a temporally non-rigid description, (1a) can cease to be true. This might suggest that (1a) is not "really'' an identity-statement, that at some deep level it is "really'' the predication that Jane is the leftmost Rockette. I take no stand on this. What's clear is that the predication can cease to be true, and on the non-rigid reading, the identity-statement is true only as long as the predication is.
With the descriptions rigid, (1a)—(4a) is sound but irrelevant: we took the descriptions as rigid in the last section. Now let us read (1a) and (2a) non-rigidly. If we do, an ambiguity emerges. (1a)—(3a) are all true at the same point in our lives. So for us, the argument is sound (though still irrelevant). But on Jane's timeline, things differ. When Jane is the leftmost Rockette, she has not yet lived through dancing in the rightmost Rockette's spot—even if she has a perfect memory, she has no memory of this. Jane then shares the stage with the rightmost. So Jane is then living through existing simultaneously in public time with her dancing in the rightmost Rockette's spot. But still, on Jane's timeline, (2a) is not yet true, because Jane has not yet done what she must to satisfy the description "the rightmost Rockette,'' even though (1a) and (2a) are true at once on our timeline. So too, when (2a) is true on Jane's personal timeline, (1a) has ceased to be true. Jane recalls dancing in the leftmost spot, and so qualifying for the title "the leftmost Rockette,'' and one can only remember what has happened in one's past. Dancing as leftmost is something she once did but is no longer doing—on her timeline. This is so even though Jane is then living through existing simultaneously in public time with her dancing in the leftmost spot. So on Jane's personal timeline, with its premises read non-rigidly, (1a)—(4a) is never sound.
48 But even here, one has to wonder. Is it really still true that Lincoln = Lincoln? "Lincoln = Lincoln'' is after all a more precise rendering of''Lincoln is Lincoln.'' The latter is present-tensed. Perhaps if Lincoln no longer exists, nobody has any longer the property of identity with Lincoln.
This suggests that there is actually a tense involved in (1a) and (2a), at least when we treat the descriptions as non-rigid. The ordinary-language sentences (1a) renders are after all "Jane is the leftmost'' or "Jane is identical with the leftmost'' (and so for (2a)). In these, "is" is present-tensed. Whether non-rigid (1a) and (2a) are true at once depends on whether the present the tense brings in is Jane's own or that of the public timeline. So here is the ambiguity: the non-rigid-description reading of (1a)—(4a) is sound (though as irrelevant as the rigid reading) if the tenses invoke the public timeline, but never sound if they invoke Jane's. If pastward time-travel can occur, it's Jane's timeline that counts: that (1a)—(4a) is sound along the public timeline is irrelevant. To show why, I now sketch another problem about time-travel.
In the same period of public time, Jane does (as rightmost) and does not (as leftmost) remember exiting the stage after the leftmost Rockette danced her number. So it seems that time-travel entails a flat-out contradiction. To avoid this, the defender of time-travel must relativize Jane's remembering somehow. One option would be to relativize to different places: Jane recalls this in the rightmost but not the leftmost spot, and it is no contradiction to recall-this-in-the-rightmost-spot at t but not recall-this-in-the-left-most-spot at t, even though it is one to recall this and not recall this at t. But this is unintuitive, it's hard to avoid the feeling that if Jane remembers in one spot then Jane remembers simpliciter (which re-instates the contradiction), and it isn't sufficiently general, as it would not handle time-travelers not located in space (angels, perhaps). Another option would be to relativize to different temporal parts of Jane—the rightmost part does recall and the leftmost does not—and reject the move from Jane's part's remembering then to Jane's remembering then or else parse "Jane remembers then'' strictly in terms of her temporal parts. But that Jane has temporal parts so ordered implies that Jane has a personal timeline distinct from the public timeline.49 So if relativizing to Jane's timeline will solve the problem by itself, it's a cheaper solution: it doesn't commit one to temporal parts. It's better, then, to relativize to Jane's time-line, i.e. to say that there's no contradiction involved in Jane's time-travel because Jane recalls and does not recall at different points in her life. But this dissolves the problem only if Jane's personal timeline takes precedence over the public time-line: that is, only if that Jane recalls P at a point in her life coinciding with public time t does not imply that Jane recalls P at t simpliciter. For of course, if this did follow, the original contradiction would be re-instated. Pastward time-travel is possible only if it does not involve the contradiction above. The best way to block the contradiction is to relativize Jane's recalling to her timeline. But this works
49 If Jane had just one temporal part at public time t, she would be like the rest of us. Her personal timeline would not diverge from public time at t. If Jane has distinct temporal parts at public t, then one of them is in the other's past along Jane's timeline. But along the public timeline, whatever is at t when t is present is present. So if Jane has distinct temporal parts at public t, her timeline diverges from that of public time.
only if facts about the order of segments of Jane's own life supersede facts about the public timeline—only if, as it were, Jane while time-travelling is really in her own time even though every instant of her life coincides with some instant of public time. This last is exactly what one would expect if there can be pastward time-travel, in which individual timelines break free of the public timeline.
If time-travel can occur, then that P at a point in one's life which coincides with public time t does not entail that P at t. Nor does P's being so at public t entail that P is so at every point in one's life which coincides with public t. Jane exits the stage from her leftmost position at public time t + 1. So at public time t, she has not yet made the exit, and does not remember it. As Jane exits the stage after her first dancing, her timeline has not yet diverged from public time. So if her not recalling this exit at public t, before she made it, entailed that she did not recall it at every point in her life which coincided with t, then she would both recall it and not recall it as rightmost. If time-travel can occur, then, that Jane does not at public time t recall her exit does not entail that Jane does not recall this at every point in her life which coincides with t. Not all facts about the public timeline impose themselves on the time-traveler's timeline.
There are stages of our lives at which Jane is presently both the leftmost and the rightmost Rockette. During any such stage, (1a)—(4a) is sound. But Jane's life has no such stage. For again, as we watch the Rockettes, we see all at once events that for Jane are successive, i.e. not all co-present. On her timeline, when she is presently the rightmost Rockette, being the leftmost Rockette is in her past.50 Now if time-travel can occur, facts about time-travelers' personal timelines supersede facts about public time. That (1a)-(3a) are all true at once in public time does not entail that they are all true at once during Jane's life, even though all segments of Jane's life coincide with public times. So if time-travel can occur, the fact that (1a)—(4a) is sound in public time does not matter. Along Jane's timeline it is wrong to treat (1a)—(4a) as a proof, even though we would be right to do so. Even if the argument were able (as it were) to shorten the chorus line, it would do so only if it were sound along Jane's timeline.
It would strengthen my overall case if I could show that what must be so if time-travel is possible is so- i.e. that for quite general reasons, personal timelines supersede public time. For the nonce I can only suggest something weaker. There is not (say I) some single substantival entity, Time, which passes at Newton's "uniform and equable rate.'' What we call time's passage is just a function of what events occur, and how they occur. Events compose all timelines, public and personal. So at the very least, there is no a priori reason that we should treat public time as having priority over personal, and able (as it were) to impose itself upon it. For both are wholes composed equally of the same basic parts. They
50 Another angle on the same fact: on Jane's timeline, there are times between the leftmost and the rightmost Rockettes' arrivals onstage. There are none in our timeline—for us, all Rockettes step onstage from backstage at once.
simply arrange those parts differently, if they differ. Now in some cases, parts compose wholes only by composing larger parts of those wholes: players compose baseball leagues only by composing those leagues' teams, atoms compose walls only by composing bricks which compose walls. It is at least arguable that public time is really just the fusion of personal times—that events compose public time only by composing personal times which compose public time. If this is not just arguable but true, of course, then traits of personal times take precedence.
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