Choice & Inference

I just finished reading this interesting paper recently written (and presented at the last PSA) by the philosopher of mathematics Chris Pincock. In this work, Pincock highlights the following puzzle for Bayesians. Paraphrasing,

The rules of the probability calculus mandate that logically equivalent propositions be assigned the same probability; however, insofar as probabilities are interpreted as degrees of belief, this mandate is deeply unrealistic. This is most evident when it comes to mathematical truths. As typically understood, such truths are all logically equivalent to one another. However, nobody is able to separate all of the mathematical truths from the non-truths. In other words, no one is mathematically omniscient. Thus, no one knows exactly which of these propositions are logically equivalent; consequently, it becomes impossible to follow this rule of the probability calculus.

Pincock takes this point to be especially damning to the objective Bayesian position; however, I take it that the objective Bayesian actually gets around this problem more easily than the subjectivist. Here is why: Read the rest of this entry »

Ok sorry about the lull in activity on here. Marking and conference organisation duties haven’t really helped…

Here’s something that has been at the back of my mind for a few months. This  is almost certainly old hat, but given that this is not my area of research and that I do not know this literature well, I would be grateful for any comments.

It has occurred to  me that there is a striking, if perhaps superficial, similarity between what might be called the Sceptical Paradox (henceforth SP) in epistemic logic and the Good Samaritan Paradox (henceforth GSP) in deontic logic.

The puzzles:

SP

1. I know that there is a zebra in the pen. ( )
2. If there is a zebra in the pen there isn’t a cleverly painted mule in there. ()
3. K-Closure: if   and , then . (Okay, this is very strong, but is probably acceptable if K stands in for what is sometimes known as ‘implicit knowledge’)
4. I don’t know that there isn’t a cleverly painted mule in the pen.  ()

GSP:

1. The homeless ought to be helped. ()
2. If the homeless are helped, then there are homeless people. ()
3. O-Closure: if   and , then .
4. It isn’t the case that there ought to be homeless people. ()

What I have called K-closure and O-closure are theorems of any normal epistemic/deontic logic, and are obviously immediate consequences of necessitation and distribution. They also seem to me, on the face of it, fairly intuitive (their roles in the present puzzles aside). 1, 2 and 4 also appear prima facie unobjectionable.

Read the rest of this entry »

A friend and colleague of mine recently sent me some thoughts and questions pertaining to Peirce, probabilities, and single cases. I’d be very interested in getting others’ thoughts on this. Thus, here (with his permission) is an excerpt from my friend’s email:

I was thinking about the problem of the single case today, and I was wondering what you make of it. I’m taking the problem from Peirce, where you will recall, he characterizes it somewhat as follows:

Suppose you stand before God and he presents you with two decks of cards.  The first has 25 white cards and 1 black card; the second has 25 black cards and 1 white card.  He tells you which deck is which, and says that you may choose which deck to draw a single card from.  He also tells you that if you draw a white card, you will be welcomed into heaven but that if you draw a black card, you will be condemned to hell.  I think it would be hard to dispute that one ought to draw from the first deck.  But, as Peirce says, what consolation can we give to the man who chooses from the first deck but draws the black card?

Now, as I understand the literature following the positivists (and maybe Feller), the problem of the single case has been made out to be about the meaningfulness of probability statements with respect to singular events, like drawing once from such a deck.  But it seems to me the problem is about rationality.  Specifically, it is about the rationality of following or being guided by probabilities in the single case, even when you know what the probabilities are.

So, what do you think?  What arguments do frequentists (or propensity theorists) and Bayesians (both subjective and objective) give to connect probability with rational action in cases like this?  Is there a standard move in the literature?

So indeed, what do you think? More specifically, I wonder what you think about my colleague’s distinction between Peirce’s version of the problem and the problem that typically goes by that name these days. And I wonder how you would answer the questions concluding his email.

I have recently written up a very short paper, in which I suggest that an old puzzle from the Martin Gardner books, known as the ‘Second Ace’ puzzle, could be used as a counterexample to van Fraassen’s reflection principles and their more recent descendants. The puzzle doesn’t seem to have been widely discussed in the philosophical literature yet, although some people in the AI community, including Jo Halpern, apparently have something to say about it, albeit not (explicitly) in connection with Reflection.

Reflection

van Fraassen famously suggested the following (where Cr is a credence function an t, t* are times):

Special Reflection (SR): it ought to be the case that where .

This says that one’s current credence in A, conditional on having a future credence of x in A ought to be x. It follows from this that it ought to be the case that your credence at t in A is the average of your future possible credences in A, weighted by the probability that you give at t to having them.

Read the rest of this entry »