Cian had a lot to say about this paper. Most of our discussion stemmed from certain kinds of counter-example that he suggested. I hope I’m not misrepresenting him in what follows:
The first kind of case is meant as a counterexample to the left-right direction of principle (1), which says roughly that you believe P if and only if conditionalizing on P doesn’t change any conditional preferences over things that matter. Weatherson says that the L-R direction ‘seems trivial’, but what about the following simple case:
I’m considering whether to buy insurance against meteor strikes. I believe my house won’t be struck by a meteor tomorrow; but I still buy the insurance just in case. Conditonal on my house not being struck by a meteor tomorrow, however, I prefer not to buy the insurance.
I guess what Weatherson would have to say is that your choosing to buy the insurance is incompatible with your believing that your house won’t be struck. Is this really that plausible? Perhaps it could be motivated by the following line of thought: a) if I don’t believe that it might be struck, I won’t buy the insurance. But b) if I believe that it won’t be struck, I don’t believe it might be struck. So c) if I believe it won’t be struck, I won’t buy the insurance.
The weak point here seems to be b). If we can’t believe that something won’t happen unless we believe that it’s not true that it might happen, then it’s hard to maintain we have many beliefs at all.
(It’s not essential to the case that the striking is in the future. Consider instead a kind of insurance which pays out if, unbeknownst to you, the house has already been struck.)
The second kind of case is also directed against the left-right direction of 1). Let q be a conspiracy theory which says that the real president of the US is chosen randomly by aliens, and the election is just a sham. Let p be that Obama is president. It seems coherent that we could prefer armed rebellion to acquiescence conditional on q, but prefer acquiescience conditional on p & q (reasoning that in this case, since the aliens by chance picked the right man for the job, we don’t need to bother rebelling). Then the left-right direction of 1) tells us that we don’t believe p. But this seems silly – of course we believe that Obama is president.
Can Weatherson appeal to his restriction to live and salient options or to relevant and salient propositions to deflect these counterexamples? It seems not – presumably there is no problem with describing cases where armed rebellion and insurance-buying are possible for us, and we’re seriously considering them while still retaining our beliefs that the house won’t be struck by a meteor and that Obama is president. Similarly, presumably we can retain these beliefs even while taking seriously the conspiracy theory and meteor-striking possibilities, and/or currently considering them.
One way out would be to deny that, if we take the conspiracy theory or the meteor-striking possibility seriously, then we can’t believe that Obama is president or that the house won’t be struck. But what justifies this restriction? We worried that any sense of ‘taking seriously’ that would do the job would end up being parasitic on belief and therefore unavailable for use in characterising it. For example, the proposal that a necessary condition for taking something seriously consists in not believing that it’s false seems to fall prey to circularity.
The third kind of case Cian suggested was meant as a counterexample to the set of principles on page 10. In particular, he didn’t like the consequence that believing that p entails that conditionalizing on it can’t move you from believing q to believing ¬q. Suppose you have credence 0.5 in a coin landing heads, and credence 0.5 in it landing tails. Let q be ‘the coin lands heads’, and p be ‘there is no goblin that will make the coin land tails’. Now conditional on there being no goblin which will make the coin land tails, your credence in q should go up a tiny bit (after all, you’re still leaving the option open that there’s a goblin to make it land heads.) But if the threshold for belief is 1/2 as these principles suggest, then conditionalizing on p is enough to shift you from not believing q to believing q. So in normal cases of coin tossing we don’t believe that there is no goblin which will make the coin land tails. This isn’t good.
We had plenty of other thoughts, but that’s probably enough to be going on with!