Clearly indexicals pose a problem for classical logic, else there would be no interest in constructing alternative logics to deal with them
This does not follow. There’s lots of interest in things (such as constructing alternative logics) that are driven by pure curiosity, status-seeking, and other reasons which have nothing to do with solving a problem. I support many of these reasons, and it’s fun, but that doesn’t mean I have to reject simpler tools when they work. There is no problem with indexicals in classical logic—you can construct clear propositions for any indexical case.
Your thirder argument (I lose twice for tails, but only win once for heads) sounds right for me, the halfer argument is even simpler (I lose for tails, win for heads, just do the math). The key is that they are _different_ propositions. The thirder argument is correct if the bet is resolved once on heads and resolved twice on tails. The halfer argument is correct if the bet is resolved once, no matter how many times the subject is awoken.
When we say “The dog found a bone”, we are assuming that the person we’re speaking to will be able to figure out which dog we’re referring to. The proposition is about that dog. It’s not about the process the speaker goes through to infer which dog we’re talking about, which could involve all sorts of background knowledge regarding the speaker, the listener, the dogs in the neighborhood, laws restricting when dogs are allowed off leash, etc. The same for “I” or “today”. It’s a convenience in ordinary speech. There’s no need to introduce this issue in the Sleeping Beauty problem, since the problem is about inferring the result of a single coin flip, not about inferring what day of the week today is (whatever that means).
I’m not sure I understand what you’re saying about bets. Surely you’re not saying that Beauty’s belief about whether the coin flip was Heads depends on what betting scheme (if any?) has been set up? For the “only one” bet case, you might want to look at my reply to Part 1 of this post.
I think I agree completely with your first paragraph. There’s no need to introduce indexical complexity, and classical probability copes with this just fine.
In all cases, the probability is 50% (or, once known, it’s 1 or 0). The casual discussion of betting methodology conflates probability and payouts. If there’s a 50% that you’ll lose twice and a 50% chance that you’ll win once, you should require a 2:1 payout on the bet, which leads people to say 33% probability when they combine the two. If you’ll only lose once EVEN WHEN ASKED TWICE, then 1:1 odds are fine and 50% is clear.
I don’t know what you mean by “in all cases, the probability is 50%”. What situation are you referring to? I’m arguing that Beauty’s probability of Heads should be 1⁄3 when woken on Monday or Tuesday, and I believe this is quite consistent with obtaining good results in any betting scenario, when the decision is made properly based on this probability. If you think Beauty’s probability for Heads should be 1⁄2, you need to do more than just assert this.
This does not follow. There’s lots of interest in things (such as constructing alternative logics) that are driven by pure curiosity, status-seeking, and other reasons which have nothing to do with solving a problem. I support many of these reasons, and it’s fun, but that doesn’t mean I have to reject simpler tools when they work. There is no problem with indexicals in classical logic—you can construct clear propositions for any indexical case.
Your thirder argument (I lose twice for tails, but only win once for heads) sounds right for me, the halfer argument is even simpler (I lose for tails, win for heads, just do the math). The key is that they are _different_ propositions. The thirder argument is correct if the bet is resolved once on heads and resolved twice on tails. The halfer argument is correct if the bet is resolved once, no matter how many times the subject is awoken.
When we say “The dog found a bone”, we are assuming that the person we’re speaking to will be able to figure out which dog we’re referring to. The proposition is about that dog. It’s not about the process the speaker goes through to infer which dog we’re talking about, which could involve all sorts of background knowledge regarding the speaker, the listener, the dogs in the neighborhood, laws restricting when dogs are allowed off leash, etc. The same for “I” or “today”. It’s a convenience in ordinary speech. There’s no need to introduce this issue in the Sleeping Beauty problem, since the problem is about inferring the result of a single coin flip, not about inferring what day of the week today is (whatever that means).
I’m not sure I understand what you’re saying about bets. Surely you’re not saying that Beauty’s belief about whether the coin flip was Heads depends on what betting scheme (if any?) has been set up? For the “only one” bet case, you might want to look at my reply to Part 1 of this post.
I think I agree completely with your first paragraph. There’s no need to introduce indexical complexity, and classical probability copes with this just fine.
In all cases, the probability is 50% (or, once known, it’s 1 or 0). The casual discussion of betting methodology conflates probability and payouts. If there’s a 50% that you’ll lose twice and a 50% chance that you’ll win once, you should require a 2:1 payout on the bet, which leads people to say 33% probability when they combine the two. If you’ll only lose once EVEN WHEN ASKED TWICE, then 1:1 odds are fine and 50% is clear.
I don’t know what you mean by “in all cases, the probability is 50%”. What situation are you referring to? I’m arguing that Beauty’s probability of Heads should be 1⁄3 when woken on Monday or Tuesday, and I believe this is quite consistent with obtaining good results in any betting scenario, when the decision is made properly based on this probability. If you think Beauty’s probability for Heads should be 1⁄2, you need to do more than just assert this.