Truth Accessibility Relations among Times

In my two previous posts, I have explained the notion of 'truth at a time' and argued that for a (contingent) proposition to be true at T it must supervene upon what exists at T. In short, truth at T requires a truthmaker at T.

A question that remains concerns accessibility relations among times. If a proposition is true at some earlier time, is it necessary that it be true at all later times that it was true at that earlier time? And if a proposition is true at some later time, is it necessary that it be true at all earlier times that it will be true at that later time? In common tense logical notation, these questions concern the theses:

1. p → FPp
   
2. p → PFp

(1) says that if p is true at some time, say T, then it will be true at all times subsequent to T that p was true at T. (2) says that if p is true at T, then it has been true at all times prior to T that p will be true at T.

I believe that (1) is true and (2) is false. In other words, I believe that earlier times are accessible to later times, but I don't think that later times are accessible to earlier times. This antisymmetry means that I take S4 and not S5 to be the right system of modal logic as far as times are concerned.

S4 ensures reflexivity and transitivity among times. Reflexivity means that each time is accessible to itself. Hence, if it is true at T that p, then it is true at T that it is true at T that p, etc. Transitivity means that if T1 is accessible to T2, and T2 is accessible to T3, then T1 is accessible to T3. Hence, if it is true at T1 that p, and if it is true at T2 that it is true at T1 that p, then it is true at T3 that it is true at T1 that p.

To reflexivity and transitivity, S5 adds symmetry. This means that if T1 is accessible to T2, then T2 is accessible to T1. Hence, if it is true at T1 that p, then it is true at T2 that it is true at T1 that p, and it is true at T1 that it is true at T2 that it is true at T1 that p, etc.

Now, if S5 is right, then both (1) and (2) have to be accepted - what's true at any time is mirrored at all times. If S4 is right, however, then we cannot accept both (1) and (2), for that would violate antisymmetry. The questions, then, are whether we should accept antisymmetry or not and, if so, whether we should go with (1) or (2).

My argument turns on the relation between time and causality and on the falsity of determinism. Earlier events can cause later events, but not vice-versa (or so most of us think). That motivates antisymmetry, but to establish it we have to appeal to the idea that what's true at a time depends upon what exists at that time. Suppose event E1 at T1 indeterministically causes event E2 at T2. It is true, therefore, at T2 that E2 is occurring. Given symmetric accessibility between T1 and T2, it is true at T1 that it is true at T2 that E2 is occurring. But this is false if the causal relation between E1 and E2 is indeterministic, for in that case we do not have a truthmaker at T1 for "E2 is occurring at T2".

If determinism is true, then a unique sequence of later times is implied by earlier times. It can be true at T1, therefore, that "E2 is occurring" will be true at T2. Hence, T2 is implicitly accessible from T1. But if determinism is false, then no unique sequence of later times is implied by earlier times. Let p represent the occurrence of some event that is causally contingent from the perspective of T1 (e.g., "E2 occurs at T2"). From that perspective, it may be that p is true at T2, and it may be that p is false at T2. Moreover, there is nothing at T1 sufficient for making it true either that p will be the case at T2 or that p will not be the case at T2. Since what's true at T1 depends on what exists at T1, it cannot be true at T1 either that p will be true at T2 or that p will not be true at T2. Given indeterminism, therefore, it follows that (2) is false. If, for example, there being a sea battle at T2 is causally contingent at T1, then even if a sea battle does occur at T2, it does not make "A sea battle will occur at T2" true at T1, contrary to (2).

But what about (1)? It would be very counterintuitive if (1) were false. It's falsity would allow, for example, that "Caesar crossed the Rubicon" might be true in, say, 1000 A.D., but not true in 2000 A.D. So I want to insist that (1) is true, but I emphasize that it's truth is not a mere matter of logic. If we accept that truth supervenes on being, then (1) commits us to a substantive metaphysical position. Since what's (contingently) true at T2 depends upon what exists at T2, for it to be true at T2 that some event E occurred at T1, there must be something at T2 sufficient for making that true. E's occurring at T1 is not enough. There must also be something that carries E's occurring at T1 forward in time and that constitutes at T2 E's having occurred at T1. As to what that "something" is, I refer the reader to this forthcoming paper of mine.

In summary, if determinism is false, as I believe, then (2) is false. (1), however, is true - it would be intuitively quite odd (to say the least) if it weren't - though it comes at what some will regard as an unacceptable metaphysical price. Those folks either have to deny that truth supervenes on being or reject (1). I think either is a much bigger price to pay.