Probably Possible

Have been reading an article by Timothy Williamson that I find quite interesting:
http://www.philosophy.ox.ac.uk/faculty/members/docs/infiniteheads.pdf

Suppose that a fair coin is tossed infinitely many times at one second intervals starting from t1, and the tosses are all independent. Let (H1…) be the event that every toss of the coin lands heads. Let (H1) be the event that the first toss at t1 lands heads, and (H2…) be the event that every toss from t2 (separated from t1 by one second) onwards lands heads.

There’s an intuition that (H1…), although extremely unlikely, is possible, and hence, we should not be certain that ~(H1…) - we should be more confident that (H1…) than we are that A & ~A. Williamson shares the intuition, but argues that with a few other plausible assumptions, it leads to contradiction.

Let “>” be read as “is more probable than” and “≥” be read as “is as least as probable as”. The following, Williamson thinks, is a “promising principle” for comparative probability:

(!) If X and Y are each incompatible with Z, then
(a) X > Y iff X v Z > Y v Z
(b) X ≥ Y iff X v Z ≥ Y v Z

But now, let X be (H1…), Y be A&~A, and Z be ~(H1) & (H2…). Since (H1…) and A&~A are both incompatible with ~(H1) & (H2…), we get:

(H1…) > A&~A iff (H1…) v (~H(1) & (H2…)) > A&~A v (~H(1) & (H2…))

(H1…) v (~H(1) & (H2…)) is equivalent to (H2…), and A&~A v (~H(1) & (H2…)) is equivalent to (~H(1) & (H2…)). Hence we get:

(H1…) > A&~A iff (H2…) > (~H(1) & (H2…))

But ~H(1) & (H2…) and (H1…) differ only on the outcome of the first toss; hence,

(H1…) > A&~A iff (H2…) > (H1…)

Now Williamson thinks that (H1…) ≥ (H2…), which means, given the equivalence above, that it’s not the case that (H1…) > A&~A. But this contradicts what intuition purportedly tells us. What is Williamson’s reason for thinking that (H1…) ≥ (H2…)? He writes:

“H(1…) and H(2…) are isomorphic events. More precisely, we can map the constituent single-toss events of (H1…) one-one onto the constituent single-toss events of H(2…) in a natural way that preserves the physical structure of the set-up just by mapping each toss to its successor. H(1…) and H(2…) are events of exactly the same qualitative type; they differ only in the inconsequential respect that H(2…) starts one second after H(1…). That H(2…) is preceded by another toss is irrelevant, given the independence of the tosses. Thus H(1…) and H(2…) should have the same probability.”

I have to admit that my mathematically naive intuition is that (H2…) > (H1…). Suppose that this is not correct, that (H1…) ≥ (H2…). Since (H1) & (H2…) ≥ (H1…), transitivity yields (H1) & (H2…) ≥ (H2…). But it seems to me that (H1) & (H2…) < (H2...). Imagine there's an evil demon who can choose to exercise his ability to tell, before we start to toss a coin infinitely, whether it'll land heads on every toss or not. You've previously been condemned to be tortured for eternity, but the evil demon offers you a way out. He gives you two tickets, one which says (H2...) and the other which says (H1). The coin has yet to be tossed, and the demon, who has thus far refrained from exercising his ability, tells you that you'll be spared from eternal torture if every ticket you hold says something true. Before the demon exercises his unique ability, he gives you the choice to throw away the ticket which says (H1) if you so desire. Should you throw it away?

It seems to me that you should. If (H1) is true, throwing the ticket away does not make it less likely that you'll be spared. If (H1) is false, then throwing it away is better than keeping it, since keeping it means that there's an additional possibility, and hence, additional danger, that you won't be spared. But this means that keeping the ticket decreases the probability that you'll be spared, i.e., (H1) & (H2...) < (H2...).

For all I know, defending the intuition that (H2...) > (H1…) might be fraught with insuperable mathematical problems. In any case, I find the intuition about as strong as the intuition that (H1…) > A&~A. If, by keeping the ticket which says (H1), the additional possibility that I won’t be spared doesn’t mean a lower probability that I’ll be spared, then it’s hard to see why the mere possibility that (H1…) means that I shouldn’t be certain that ~(H1). If it does not make any difference whether I keep or throw away the ticket saying (H1), then I’m inclined to think that a ticket saying (H1…) is worth exactly the same as one saying A&~A.