Tuesday, January 18, 2011

A-theory and worlds

According to the best A-theories (i.e., those that accept an Aristotelian view of propositions as changing in truth value), there is an objective and non-relational fact as to what time it is, a fact that won't obtain tomorrow. Assume this. Here is one way to think about this. Let w be the actual world. The actual world holds all the actually true facts, including presumably the fact that it is January 18, since it is indeed Tuesday. Moreover, there will be a world w* at which everything happens just as at w except that at w* it is some time t on January 19. World w* will be a world that we will inhabit at t, tomorrow.

On this view, at every time, we are in a different world. We will then have an earlier-than relation between worlds defined as follows: w is earlier than w* if and only if at w* it is true that w was actual. Assume the earlier-than relation is transitive. Say that two worlds are directly temporally related if and only if either they are identical or one is earlier than the other. We then get:

  1. The future is closed if and only if direct temporal relatedness is transitive.
Suppose the future is closed—the best A-theory will say that (or so I claim). Then direct temporal relatedness is an equivalence relation, and for any world w, we can form the equivalence class T(w) of all the worlds directly temporally related to w.

We need one more thing in the formalism. We need a way to compare times between worlds that aren't directly temporally related. Thus, there is a simultaneity relation between worlds. Worlds w and w* are simultaneous provided that at both worlds it is the same time. This relation is also an equivalence relation, and we can let S(w) be the equivalence class of all worlds simultaneous with w.

Each world w is then a member of two orthogonal equivalence classes: T(w) which contains all the directly past and future worlds, and S(w) which contains all the simultaneous worlds. This provides resources for the formation of new modal operators, using one or the other of the equivalence relations as an accessibility relation.

Enough formalism. Maybe in a future post I will try to criticize the view.


Heath White said...

I'm not following the argument for (1). If time is branching into the future, the earlier-than relation is still transitive (so is identity, obviously) but the future is not closed. What am I missing?

Alexander R Pruss said...

Suppose w1 has two futures at t2: w2a and w2b. Then, w2a is directly temporally related to w1 and w1 is directly temporally related to w2b, but w2a and w2b are not directly temporally related.

The reason is that A and B directly temporally related iff either they are identical or one is earlier than the other, i.e., iff A=B or A<B or B<A.

Alexander R Pruss said...

The argument for (1) also assumes that there can't be any branching into the past.

Heath White said...

OK, gotcha: w2b is DTR to w1, w1 is DTR to w2a, but w2b is not DTR to w2a.

Next question: why would the best A-theories endorse (1)? I can't see any particular connection between believing an A-theory and believing in a closed future.

Alexander R Pruss said...

Because everyone should accept a closed future. :-)

Why? Well, the main reason to believe in an open future is a certain fatalism-based argument. The argument has some logical premises and some metaphysical premises. The metaphysical premises include:
1. the past is fixed
2. if p is fixed and p entails q, then q is fixed
3. there will be free actions in the future
4. an action in the future is not free if the proposition that it will take place is fixed
5. if t is a past time and p is a true proposition and the proposition p existed at t, then <p is true at t> is a fact about the past.

The logical premises include:
6. temporalized bivalence: for all p and t, if t is a time then either p is true at t or p is not true at t

There are also some additional metaphysical premises about the nature of propositions, and there are some less controversial logical premises. But the methodological point is this. Given a choice between revising metaphysics and revising logic, ceteris paribus we should revise metaphysics. Now, the onus of argument is on someone who thinks that the ceteris paribus clause applies here. There are cases where the metaphysics is so important to our lives that we should rather revise logic than metaphysics. But of the metaphysical premises above, only premise 3 has that property. And there are cases where a metaphysical premise is utterly uncontroversial. But none of the metaphysical premises have that property. So, barring a really strong argument, we should on methodological grounds deny the disjunction of 1, 2, 4 and 5 rather than denying 6. But open futurists insist on denying 6.

Alexander R Pruss said...

Ooops. 6 should read:
6. temporalized bivalence: for all p and t, if t is a time then either p is true at t or not-p is true at t

Heath White said...

OK this helps. I was confused about where you were going.

enigMan said...

Coming from a physics background, I've always found the best argument for an open future to be the plausibility of collapse interpretations of quantum mechanics. Many physicists still like those, because the world does seem to behave as though possible but non-actual futures exist enough to interact. My problem with the alternative (Many Worlds) interpretations is that according to them we are all always about to do everything that it's physically possible for us to do at that time.

The problem here is not fatalism, but is, similarly, irresponsibility. I take us to be responsible agents (while many physicists have no such qualms). Here there are scientific observations, metaphysical premises (e.g. we are responsible agents) and logical ones. But while (6) is logical in form, the intuition against revising logic is not against rejecting propositions of logical form (such as that there are true contradictions). And surely the fact that so many thinkers, at the time of Aristotle and ever since, have taken it to be the case that it's neither true nor false that there will be X tomorrow (where X is some not unlikely event) shows that (6) is not part of our background logic.

Rob K said...

Couldn't an open-futurist deny (5) instead? That's the idea behind Ockham's soft/hard fact distinction.

Also, if Aristotelians believe that propositions can change their truth value, then (5) seems ill-formed, since it refers to 'a true proposition' without specifying when it's true.

Alexander R Pruss said...


But if you deny (5), then you don't have an argument for an open future view any more, right? As far as I can see, all the arguments for an open future go through something like (5).

I may be missing something, being in bed with a flu.

alyssa said...
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Drew said...

Quick question: Is there anything in Catholic Dogma that demands a B-Theory of time?

Alexander R Pruss said...

I don't think Catholic dogma requires anything either way.

But: If the A-theory holds, then what God knows changes with time. Yesterday he knew that it was objectively Saturday, and today he knows that it is objectively Sunday. I do not think this change of knowledge actually implies a change in God--I am an externalist about God's beliefs. But if you think this change of knowledge implies a change in God, then you have to reject the A-theory, since it is dogma that God is unchanging and all-knowing.