Why is the Two Envelopes paradox paradoxical? After some thought I've decided that I don't really know (so the following is less clear than I'd hoped it would be) but maybe the reason is related to our difficulties with such concepts as deliberation and probability. Perhaps we overlook (for some reason) something that we intuit easily enough in the case of, for example, Pascal’s Wager:
......Even were you totally unsure, whether or not there was a God, it would clearly be silly to act as though God probably existed just because (even if you did know that) the net rewards for doing so were very much greater than those for not doing so. Intuitively, such rewards are irrelevant if there is no genuine chance of whichever is not already the case (if it is not as if someone will toss a fair coin fairly, and give you those rewards on heads)—and that is, I think, reminiscent of the calculation (see my previous post) of the expected gain on exchanging your envelope (when only one of the ‘X’s in the equation for expected gain named the definite amount in your envelope). Such a distinction, between genuine and other chances, is obscure but apposite (cf. Denyer's 'proof,' below) because whenever we are thinking rationally we have already presupposed that our decisions might make a genuine difference.
......After all, with your original choice of envelope (which gave you the 50% chance of X = 1 and 50% chance of X = 2) there was no temptation to wonder how it could be that, whilst X = 3/2 is impossible, 50% of X (= 1) + 50% of X (= 2) = 3/2. And those chances were genuine (so to speak; and even if, counterfactually, your envelope had been given to you, via some deterministic mechanism, your epistemic uncertainty would have yielded a similarly quasi-genuine, if then entirely imaginary, chanciness). By contrast, the chances that seemed paradoxical, when focusing upon just part of what you know (when ignoring the parenthetical identities in the equation below for expected gain on exchange), could have measured nothing more than your ignorance about what was already the case.
......Regarding that possibility (that ‘could’), it seems (as below) that simply to be rational is to have (rather obscurely) related presuppositions about what is possible—beliefs so obscure (so analytically dubious) that we might well overlook their import when a scenario encourages us to (and plausibly the way in which that happens within the two envelopes scenario is a clue as to how to analyse such beliefs). Similarly, we might presuppose that (certainly) 2 + 2 = 4, even those of us who think that a (Cartesian) demon, fooling us about such things, is not just possible (despite its inconsistency with 2 + 2 = 4) but is not necessarily unlikely. And if determinism is like that, if its nonexistence is a necessary presupposition of the practice of our rationality (for all that various determinisms are prima facie possible), then the fact that possible worlds are, in themselves, rather deterministic objects (whereas in no possible world is it false that 2 + 2 = 4) might help to account for why philosophers (who have tended to favour possible worlds analyses) have tended to find the Two Envelopes so paradoxical (cf. Pascal being a deterministic Jansenist, which might explain his view of his Wager).
......So maybe the answer (to why we find the two envelopes so paradoxical) has something to do with the following argument (inspired by Denyer's "Time, Action & Necessity") for free will (and against Stephen Law's most recent argument). Many Naturalists believe that there are, in reality, such relativistic particles and fields as are the subject-matter of physics, and that what the other subjects (biochemistry, psychology, politics and so forth) concern themselves with are really just relatively large chunks of that. Many Naturalists also believe that, given how much we know, from such sciences, it would be irrational to believe in such things as souls and revelations; and they think that it would, of course, be wrong to be irrational.
......Thinking is important (we can all agree, at least here). Coming to the right decision is important—it should not just be a random event, choosing what to think about the real world, of real people (many Naturalists are humanists too). One could be irrational, but that would be wrong. And thinking rationally (making a responsible decision on the basis of the evidence) cannot be something that you are bound to do, in a totally determined way, if it is something that you might neglect to do, could be at fault for not doing. Naturalists arguing against the possibility of religious knowledge are, in particular, very aware of the ‘ought’ in ‘thought;’ and ought implies can—there would be little point in arguing against irrational beliefs if believers are bound to believe as they do, as would be the case were the physics of their physicalistic minds deterministic.
......But the only alternative to determinism, in this physical universe (of which we know a vast amount, thanks to modern physics), is randomness; if there is another alternative, and if mind is reducible to physics, then it’s odd that we don't know about it in modern physics (which has even revealed to us the relativity of space and time). And how could mere randomness be the cause of the difference between a moral act (thinking properly) and an immoral act (thinking improperly); how could it be that we should try to get result A rather than B, if it is just random whether A or B? So (to end with my reply to Law) the very rationality that Naturalists are fond of opposing to religious knowledge presupposes something that cannot be either of the options (determinism or randomness) available to them, as physicalistic atheists; so that, according to their own scientific standards, all the evidence available to us, at any time, must be evidence between the major alternatives (such as some modern monotheisms).
Young Peregrine Falcons, Treecreepers and Butterflies galore! - It's mid-Summer and my usual lack of mojo in photography at this time of year has reared it's head again, although my recent trip to amazing Skomer Island ...
8 hours ago