Supposing the truth

Here is a coin. What would have happened if I had just tossed it? It might have landed heads, and it might have landed tails. If the coin is biased towards tails, it is more likely that it would have landed heads. If it's a fair coin, both outcomes are equally likely. That is, they are equally likely on the supposition that the coin had been tossed. Let's write this as P(Heads // Toss) = 1/2, where the double slash indicates that the supposition in question is "subjunctive" rather than "indicative".

How should we analyze subjunctive supposition? Some say that P(B // A) measures the expected chance of B given A: P(B // A) = \sum_x P(Ch(B/A)=x) x. If chance is time-dependent, the relevant chance should arguably be indexed to a time "shortly before A", in cases where A stands for a possible event at a particular time. In the coin toss example, the conditional chance of heads given toss at the time just before I decided whether to toss the coin was plausibly 1/2.

But now suppose I actually decided to toss the coin, and it landed Heads. And suppose you saw all that. What, then, is your credence in Heads on the subjunctive supposition that coin was tossed? That is, if the coin had been tossed, how confident are you that it would have landed heads?

The answer is not immediately obvious because our knowledge that the coin was tossed makes the question pragmatically infelicitous. Nonetheless, I think the answer should be 1. In general, if you know that A and B are true, then your credence in B on the subjunctive supposition that A should be 1.

Why? Mainly because I'd like to think of supposing as an instance of constrained probability revision, and a minimum condition on such revisions is that if the original probability P already satisfies the constraint (here, P(A)=1), then the "revision" should leave it unchanged.

But then the chance-based analysis can't be right. For the conditional chance of Heads given Toss is not affected by our knowledge of Heads and Toss. On the chance-based analysis, P(Heads // Toss) should still be 1/2.

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