## Thursday, January 19, 2012

### Presentist counting

In a posthumous paper, David Lewis shows that one can find a presentist paraphrase of sentences like "There have ever been, are or ever will be n Fs" for any finite n. But his method doesn't work for infinite counting.

It turns out that there is a solution that works for finite and infinite counts, using a bit of set theory. For any set S of times, say that an object x exactly occupies S provided that at every time in S it was, is or will be the case that x exists and at no time outside of S it was, is or will be the case that x exists. For any non-empty set S of times, let nF(S) be a cardinality such that at every time t in S it was, is or will be the case that there are exactly nF(S) objects exactly occupying S. This is a presentist-friendly definition. Let N be any set of abstracta with cardinality nF(S) (e.g., if we have the Axiom of Choice, we should have an initial ordinal of that cardinality) and let eF(S) be the set of ordered pairs { <S,x> : xN }. We can think of the members of eF(S) as the ersatz Fs exactly occupying S. Let eF be the union of all the eF(S) as S ranges over all subsets of times. (It's quite possible that I'm using the Axiom of Choice in the above constructions.) Then "There have ever been, are or ever will be n Fs" can be given the truth condition |eF|=n.

This ersatzist construction suggests a general way in which presentists can talk of ersatz past, present or future objects. For instance, "There were, are or ever will be more Fs than Gs" gets the truth condition: |eG|≤|eF|. "Most Fs that have ever been, are or will be were, are or will be Gs" gets the truth condition |eFG|>(1/2)|eF|, where FG is the conjunction of F with G. I don't know just how much can be paraphrased in such ways, but I think quite a lot. Consequently, just as I think the B-theory can't be rejected on linguistic grounds, it's going to be hard to reject presentism on linguistic grounds.

Heath White said...

Can the presentist help herself to a set of times? I might have thought that, given presentism, there was only one time, the present, and necessarily so.

Alexander R Pruss said...

Sure, the times are abstracta.

One standard construction is to identify a time t with a maximal set of compossible propositions whose conjunction was, is or will be true. Then "p was, is or will be true at t" just means "p is a member of t".

Another construction is just to identify a time t with a real number. Then "p was, is or will be true at t" means: "if t<0, then p was true |t| seconds ago; if t=0, then p is true; if t>0, then p will be true in t seconds."

This is why we shouldn't say that presentism holds that only the present time is real. Times might be abstracta (a point independent of presentism) and hence past and future times will exist presently, though of course many of their "contents" won't.

Heath White said...

A general objection to the first construction is that something like Nietzsche's eternal recurrence might have been true. I.e. it is a contingent fact that it is not the case that there are multiple times defined by the same set of propositions. Whereas it is a necessary fact that it is not the case that there are multiple sets defined by the same set of propositions.

Alexander R Pruss said...

Maybe.

But here's one worry. Let's say that there is eternal recurrence. Let IC be Ice Capades, considered as a temporally extended event from 1940 to 1995. Then clones of IC will recur eternally. (The character played by Woody Allen in Hannah and Her Sisters is so horrified at the recurrence of IC that he considers this sufficient to reject eternal recurrence.) But these clones will be numerically distinct from IC.

So, in 1950, the proposition that IC is occurring is true. But if things recur on a million year schedule, then in 1001950, the proposition that IC is occurring isn't true. What is true is that an event just like IC is occurring.

One might wonder what distinguishes IC from its clones if it's not times? But that's a line of thought that will only be impressive to those who accept some version of the identity of indiscernibles, while those who do accept the identity of indiscernibles may simply reject the possibility of eternal recurrence.