The conference is going well, but for now I just have time now to write up a few quick thoughts on one of yesterday's talks. Lauren Ashwell (MIT/Auckland) was discussing the following widely-accepted principle:
(NBD) It is a necessary condition for S to be justified in believing that p that S not believe that her belief that p is unjustified.
This can be formalized as: "B~JBp -> ~JBp". That is, if you believe that another belief of yours is unjustified, then the latter belief really is unjustified. Self-doubt thus justifies itself. If you start to doubt whether you're justified in believing that you have two hands, then this doubt suffices to remove ("defeat") your justification for the common-sense belief. That seems silly to me, and I'm not really sure why anyone (let alone everyone) would believe NBD. To offer something more than mere intuition, in response to Lauren's talk I thought of an argument which seems to show that NBD leads to a contradiction.
I will assume that knowledge is closed under known entailment:
(Kp & K(p -> q)) -> Kq
If you know p, and you know p implies q, then you (are in a position to) know q.
I will also assume that our beliefs are accessible to us through introspection:
Bp -> KBp
If you believe that p, then you (can) know that you believe that p.
Finally, I assume that it is possible for someone to simultaneously believe that another belief is unjustified, and yet also believe that this former skeptical belief is itself unjustified. So the following is possible:
B~JBp & B~JB~JBp
Now, if someone (S) fitting the above assumptions could know NBD, then we get a contradiction. Here's how:
1. S knows: (i) B~JBp (starting belief, introspection)
(ii) B~JBp -> ~JBp (NBD)
Thus, by closure, S also knows: (iii) ~JBp
2. JB~JBp (knowledge entails justified belief; apply to (iii).)
3. B~JB~JBp (starting belief)
4. B~JB~JBp -> ~JB~JBp (NBD for belief that ~JBp)
5. ~JB~JBp (3,4 modus ponens)
Which contradicts (2)!
So given the other assumptions, NBD cannot be known. So you shouldn't believe it.
[Update: 8 Dec 05] I came up with a parallel argument that's slightly simpler, though with slightly less plausible starting conditions: Bp and B~JBBp. It would be rather odd, but surely possible, to believe p whilst thinking you're not justified in believing that you have the former belief. So consider this:
1) Bp (starting belief)
2) KBp (from 1, introspection)
3) JBBp (from 2, knowledge entails justified belief)
4) B~JBBp (starting belief)
5) B~JBBp -> ~JBBp (NBD for belief that Bp)
6) ~JBBp (4,5 modus ponens)
Contradiction: 3, 6.
Actually, this argument is stronger than the previous one. It doesn't merely show NBD to be unknowable, but indeed straight out false. Neat. (Though I suppose the supporter of NBD would instead reject my introspection principle, and thus (2).)
One final point of interest: it would seem that if NBD were true, then we should try not to doubt our existing beliefs (well, unless we go on to give them up, I suppose). After all, we presumably want to have justified beliefs. But a necessary condition for your belief being justified is that you not doubt it (in the strong sense of believing it to be unjustified). If you don't doubt it, it might be justified or it might not. If you do doubt it, then it's guaranteed to be unjustified. So it seems preferable to take the option where you at least have *some* chance of being justified.