Take the assumptions that seem least convenient for my argument, as given in my previous comment. Say the probability that “arithmetic will never tell us ‘2+2=3’,” does in fact converge to 1.
Let lim{P(x)} mean the limit of x’s probability as we follow a sequence of evidence, such that no truths added to the middle of the sequence could change the limit. If A means “physical causes which duplicate Chalmers’ actions and speech, and duplicate the functional process producing them, would produce qualia,” and B means “arithmetic will never tell us ‘2+2=3’,” and C means all of Chalmers’ premises, I think my analysis of Gettier cases leads in part to this:
lim{P(B)} - lim{P(A|C)} ⇐ | P(B) - lim{P(B)} |
I’d also argue that we should interpret probabilities as rational, logical degrees of belief. For this it helps if you’ve read the Intuitive Explanation of Bayes’ Theorem. Then I can show a flaw in frequentism and perhaps in subjective-Bayesianism just by pointing out that EY’s first problem has a unique answer, whether or not you choose to call it a probability. It has a unique answer even if we call the woman Martha. From there we can go straight to the problem of priors, as I did in my third note.
Going back to the math, then: if you think the right-hand side has a non-zero value, then logically you should adjust P(B). Because you must expect evidence that would lead you to change your mind, and that means you should change your mind now.
Ergo, logic tells us to believe A|C at least as much as we believe B.
Take the assumptions that seem least convenient for my argument, as given in my previous comment. Say the probability that “arithmetic will never tell us ‘2+2=3’,” does in fact converge to 1.
Let lim{P(x)} mean the limit of x’s probability as we follow a sequence of evidence, such that no truths added to the middle of the sequence could change the limit. If A means “physical causes which duplicate Chalmers’ actions and speech, and duplicate the functional process producing them, would produce qualia,” and B means “arithmetic will never tell us ‘2+2=3’,” and C means all of Chalmers’ premises, I think my analysis of Gettier cases leads in part to this:
lim{P(B)} - lim{P(A|C)} ⇐ | P(B) - lim{P(B)} |
I’d also argue that we should interpret probabilities as rational, logical degrees of belief. For this it helps if you’ve read the Intuitive Explanation of Bayes’ Theorem. Then I can show a flaw in frequentism and perhaps in subjective-Bayesianism just by pointing out that EY’s first problem has a unique answer, whether or not you choose to call it a probability. It has a unique answer even if we call the woman Martha. From there we can go straight to the problem of priors, as I did in my third note.
Going back to the math, then: if you think the right-hand side has a non-zero value, then logically you should adjust P(B). Because you must expect evidence that would lead you to change your mind, and that means you should change your mind now.
Ergo, logic tells us to believe A|C at least as much as we believe B.