## Saturday, February 26, 2005

The other day I thought of a paradox that arises from admitting fallibility. For each of my beliefs, I believe it to be true (that's just what belief is). So I believe that all my beliefs are true. But I don't believe this at all! Rather, I know I'm fallible - that I have some false beliefs. So we have a contradiction.

This reminds me of the lottery paradox:
The lottery paradox begins by imagining a fair lottery with a thousand tickets in it. Each ticket is so unlikely to win that we are justified in believing that it will lose. So we can infer that no ticket will win. Yet we know that some ticket will win.

(They go on to describe the 'preface paradox', which sounds equivalent to my fallibility paradox, though I had not heard of it before.)

In both cases, the problem involves joining many individual beliefs into one big conjunction. It seems plausible that if you believe that X and you believe that Y, then you believe that X and Y. But I think this general closure principle is mistaken.

The most obvious explanation for this is that belief comes in degrees. Of each individual lottery ticket, I believe with 99.9% sureity that it will lose. Join five of them together, and I believe the conjunction with only 99.5% sureity. Conjoin all 1000, and my 'belief' is zero - I am certain that not all of them will lose.

A similar solution will also explain my fallible beliefs. I will have more confidence in individual beliefs than in the conjunction of several of them. Make the group too large and my confidence level could fall so low that I wouldn't even affirm a 'belief' in the conjunction any more.

So, I've two general questions:
1) Are there any problems with this solution?
2) Are there any alternatives?