Monday, March 6, 2023

More steps in the open future and probability dialectics

I’ve often defended a probabilistic objection to open future views on which either future-tensed contingents are all false or are neither true nor false. If T(q) is the proposition that q is true, then:

  1. P(T(q)) = P(q).

But on the open future views, the left-hand-side is zero, since it’s certain that q is not true. So the right-hand-side is zero. But then both q and its negation have zero probability, and we can’t make any predictions about the future.

An open futurist might push the following response. First, deny (1). Then insist that P(q) for a future contingent q is the objective tendency or chance towards q turning true. Thus, P(coin will be heads) is 1/2 for a fair indeterministic coin, since the there is an objective tendence of magnitude 1/2 for the coin to end up heads.

In this post I want to discuss my next step in the dialectics. I think there may be a problem with combining the objective tendency response with epistemic probabilities. Suppose that yesterday a fair coin was flipped. If the coin was heads, then tomorrow two fair indeterministic coins will be flipped, and if the coin is tails, then tomorrow one fair indeterministic coin will be flipped. Let H be the proposition that tomorrow at least one coin will be heads. If yesterday we had heads, then the objective tendency of H is 3/4. If yesterday we had tails, then the objective tendency of H is 1/2. But we need to be able to say:

  1. P(H) = (1/2)(3/4) + (1/2)(1/2) = 5/8.

Now note that we are quite certain that 5/8 is not the objective tendency of H. The objective tendency of H is either 1/2 or 3/4.

So the open futurist needs a more sophisticated story. Here seems the right one. We say that P(q) is the average of the objective tendencies towards q weighted by the subjective probabilities of these tendencies. This is basically causal probability. The story requires that there be a present fact about all the objective tendencies.

On the technical side, this works. But here is a philosophical worry. If P(H) = 5/8 neither represents the objective tendency of H (which is either 1/2 or 3/4) nor one’s credence that H is true (which is zero on open-futurism), why is it that we should be making our decisions about the future in the light of P(H)?


IanS said...

Proponents of the neither-true-nor-false view might run this line:

They could simply reject (1). Alternatively, they could say that credences should be assigned only to ‘ordinary’ propositions, not to ‘meta-’ propositions (i.e., propositions about the truth or falsity or probability of ordinary propositions). This (I think – not sure) would be adequate to all the usual uses of probability theory and decision theory.

Credences have two sorts of requirements: internal consistency (i.e., they should follow the rules of probability theory), and consistency with what you know or believe. So (Principal Principle) if you think that there are relevant objective tendencies, your credences should reflect them. If you believe a proposition to be true or false, your credence in it should reflect this. Note that this does not prescribe credences for neither-true-nor-false propositions. So you are free to calculate them as in the example.

This would not work for the all-false view. Proponents of that view could reject (1) in the same way. But they would also have to say that future-tensed contingents, which they would take to be false, need not be given credence zero. They would have to distinguish systematically between propositions like London is the capital of France (which is false because London isn’t the capital of France) and propositions like It will rain tomorrow, which is taken to be false not because it won’t rain tomorrow, but because it may not rain tomorrow. I’m not sure this can be made to work. But it does not seem like an impossible project.

IanS said...

I’m not seeing how this works as an argument against open futurism.

An open futurist could:

(a) reject (1);

(b) require that our credences be consistent (according standard probability calculus);

(c) require that our (possibly conditional) credences should match any relevant objective tendencies that we know of.

Answering the last question (… why is it that we should be making our decisions about the future in the light of P(H)?), the open futurist would say: Because P(H) = 5/8 is what probability calculus demands (see (b)) given our credence in ‘one flip tomorrow’, the objective tendency to H given one flip tomorrow (see (c)), our credence in two flips tomorrow, and the objective tendency to H given two flips tomorrow.

Note that on this subjectivist view of probability, there is no objection to ‘Frankenstein’ sets of credences, some of which are simply judgements and some of which reflect our beliefs about objective tendencies.