Beliefs and thresholds
Following up on Weng-Hong (1, 2, 3), here are a few thoughts on thresholds for belief.
If beliefs come in different degrees or strength, what do we mean when we say not that Fred believes that P with strength x, but simply that Fred believes that P? Perhaps we mean that Fred believes that P with sufficient strength, where context may help determining what counts as sufficient. However, on this account, the following principles should be obviously invalid (both descriptively and normatively):
1) If one believes that P, and one believes that Q, then one believes that P and Q.
2) If one believes that P, and one believes that if P then Q, then one believes that Q.
3) If one believes that P or Q, and one believes that not P, then one believes that Q.
4) If one believes that P, and one believes that P is equivalent to Q, then one believes that Q.
5) If one believes that P is equivalent to Q, then one has equal confidence in P and Q.
For instance, one can easily believe P and Q each to degree 0.8, while one believes their conjunction only to degree 0.6. Still, something is odd about saying "I believe P, and I believe Q, but I don't believe P and Q". Likewise for the other cases: if you believe P and you believe that if P then Q, then you ought to believe Q; ...; if you believe that P is equivalent to Q, you shouldn't say that P is is more probable than Q. Why is that?
Some have concluded that our non-graded notion of belief is not reducible to the graded notion: to believe something in the all-or-nothing sense is not to assign it sufficiently high credence, but, say, to have it in one's belief box, or to take it for granted in action.
To get around the problems, taking for granted in action etc. must not themselves come in degrees. But that is doubtful: aren't some things taken for granted more robustly than others? Moreover, I'm not convinced there really is such an all-or-nothing sense of "belief". For lots of everyday decisions -- from betting and arguing to searching friends and waiting in cafes -- what matters are credences, not simple yes-no judgments. So I don't think there is a special attitude towards propositions we are ready to act upon. Finally, the present suggestion fails to explain the intuitive oddity of violating (5). If belief and credence are completely different beasts, it should be okay to say "I'm more confident in P than in Q, and I believe that the two are equivalent".
Here is an alternative proposal, modeled on contextualism about knowledge. The threshold for belief, on this account, is always 1. What varies from context to context is the relevant space of possibilities.
Suppose for simplicity that there are only finitely many ways things could be for an individual. (This is to avoid technical problems with infinitesimals, and actually not a serious simplification if one assumes that for a finite mind there can be only finitely many equivalence classes of doxastically indistinguishable possibilities.) A credence function assigns values to each individual possibility. To believe P, your credence function must assign 0 to every possibility in which not-P. So "Fred believes" functions as a universal quantifier over all possibilities that Fred doesn't rule out by assigning them credence 0.
This type of quantifier is normally restricted. Assume Fred is a skeptic and assigns credence 0.5 to the hypothesis that he is a brain in a vat. Still, when I ask you why Fred is angry at Mary, you may truly answer "because he believes that she threw away all his wine"; when I ask why he voted republican, you may truly say "because he believes their propaganda about terrorism" -- even though Fred assigns credence below 0.5 to the relevant propositions. We simply ignore the vat possibilities to which the rest of his credence goes.
I don't know the exact rules for which possibilities can be ignored in a given context. In the Fred case, what is important is not so much that the vat scenarios are in some sense remote, but that they are irrelevant for his behaviour under discussion. When the relative expected utility of the available options on a given occasion are invariant under conditionalizing credences on P, we can ignore all possibilities in which not-P. And so it is no surprise that we sometimes do.
Anyway, holding fixed a context, (1)-(4) above now come out valid. For instance, if one assigns credence 0 to all relevant possibilities where not-P and to all relevant possibilities where not-Q, then one ought to assign credence 0 to all relevant possibilities where not-(P and Q). And we can explain the oddity of violating (5): when I say that I'm more confident in P than in Q, I raise attention to open possibilities in which P is true and Q false; but when I continue "I believe that P and Q are equivalent", all such possibilities must be ignored.
(We can also explain why "I believe that P; but I'm not sure: perhaps not-P" sounds better than "I'm not sure that P: perhaps not-P; but I believe that P". They are like "everyone has left; but Fred is still here" versus "Fred is still here; but everyone has left".)
I hate to be self-promoting but the view I defend in "Can We Do Without Pragmatic Encroachment?" is meant to respect 1 through 5 without being forced to say belief is degree of belief 1.
Having said that, I don't quite end up *endorsing* 1 through 5, though I do endorse restricted versions of them. (You can believe p, believe q and not believe p & q on my view, but you can't do all three things consciously.)
The advantage my view was meant to have over the degree of belief 1 view (which isn't crazy I think) is that it explains why certain changes in what is relevant change the space of possibilities, while other changes do not. Or, to put it another way, it can let our degrees of belief determine not only what we believe, but also our dispositions to gain and lose beliefs when the circumstances change. I think the belief = degree of belief 1 view has to take those dispositions to be given by a separate part of the model.