Closure and Credence
A dilemma, or apparent dilemma, looms for anyone who subscribes to the following three theses:
(1) There are certain closure principles that hold for rational beliefs; for example, rational beliefs are closed under conjunction and modus ponens:
(a) If we’re rational to believe that p and we’re rational to believe that q, then we’re rational to believe that p&q.
(b) If we’re rational to believe that p and we’re rational to believe that p ⊃ q, then we’re rational to believe that q.
(You might worry that the above principles are false as they stand, because of cases in which we believe that p and believe that q (for example), but have not put one and one together. If you like, add the relevant bells and whistles to get your preferred versions of the principles.)
(2) To be rational to believe a proposition, we need to be rational in assigning it a sufficiently high credence.
(3) There are many propositions we are rational to believe.
Let’s not question (2). Now suppose that to be rational to believe a proposition, we need to assign it a credence of 1. Then (3) seems false, for it seems that there are not many propositions to which we would be rational in assigning a credence of 1. But suppose that to be rational to believe a proposition, we just need to be rational in assigning it a sufficiently high credence that may well fall short of 1. Then it seems that (1a) and (1b) are false. Let’s say that we are rational to believe that p and also that q, and that our credences in p and in q are each 0.95. Let’s say that 0.95 is the minimum credence we need to assign to a proposition to be rational to believe it. Then, for all the probability axioms say, our rational credence in p&q may well be less than 0.95, in which case we are not rational to believe it. Similarly, let’s say that we are rational to believe that p and also that p ⊃ q, and that our credences in p and in p ⊃ q are both 0.95. Then, for all the probability axioms say, our rational credence in q may well assume a value in the interval [0.9, 1]. Suppose our credence in q is 0.91. Then, if 0.95 is the minimum credence we need to assign to a proposition to be rational to believe it, we are not rational to believe that q.
How might we resolve this dilemma? On Tuesday, at my midterm review, I argued that there is a way of thinking about the relation between belief and credence so that we can preserve closure without having to hold that rational belief requires certainty. (The talk drew heavily from one of wo’s blog posts, but I’m solely responsible for anything stupid I said!) Think of rational belief in terms of the ruling out of possible worlds. To be rational to believe that p is to be rational to rule out all worlds in which not-p. And to be rational to rule out worlds in which not-p is to be rational to assign those worlds a credence of 0. Now this would mean that rational belief requires certainty, unless we restrict the space of worlds within which ruling out takes place. Let’s hold that one is rational to believe that p iff one is rational to rule out all not-p worlds within the relevant set of worlds. If m is the proposition that is the relevant set of worlds, then one is rational to believe that p iff one is rational to assign a conditional credence of 1 to p given m, and a sufficiently high credence to m. Let’s hold that what is relevant and what counts as sufficiently high vary with context.
For the time being, let’s set aside the following questions:
(i) What exactly is m?
(ii) What determines what counts as sufficiently high?
(iii) Whose context restricts the relevant space of possible worlds?
Assuming that something like the above account of rational belief is correct, we can see how we can preserve closure, and yet not hold that rational belief requires certainty. Suppose that we are in a context in which m is relevant, and that the minimum credence we need to assign to m to be rational to believe that a proposition is true is 0.95. Suppose that we are rational to believe that p and rational to believe that q. Then on the above account, Cr(p|m) = Cr(p ⊃ q|m) = 1. Then Cr(q|m) = 1. Then Cr(q&m) = Cr(m). But Cr(q) ≥ Cr(q&m). So Cr(q) ≥ Cr(m) = 0.95. If we’re rational to believe that p and also that p ⊃ q, the credence we are rational to assign to m sets the lower bound for the credence we are rational to assign to q
One might worry that on this account of rational belief, we would still have very few rational beliefs—if Cr(p|m)=1, then we are rational in being disposed to place limb, life, and soul on the conditional bet that p is true given m. But there are hardly any such bets, or so one might think. To mitigate the worry, however, note that we may well be rational to make such bets when either m entails p, or m, together with anything else that we are certain of, entails p. Suppose I’m certain that there appears to be a table in front of me. Suppose also that m is something like `Appearance reflects reality on this particular occasion’. Then it seems rational for me to assign a conditional credence of 1 to there being a table in front of me, given that appearance reflects reality on this particular occasion.
Another worry with the account is whether it merely shifts the bump in the rug. Someone who subscribes to the Lockean Thesis, and holds that one is rational to believe that p iff one is rational to assign a credence to p that meets a sufficiently high threshold, is at risk of being impaled on the horns of the dilemma. On the current account of rational belief, there is also a threshold to be met: one needs to assign a certain minimum credence to m for one to be rational to believe that p. Does the account face similar problems to those faced by someone who subscribes to the Lockean Thesis? The worry that it does cropped up a number of times during my talk’s Q&A, and I’m afraid I botched up my reply to it. Here’s another attempt to assuage the worry.
Subscribing to the Lockean Thesis seems to be bad for someone who likes (1a) and (1b). But holding that one needs to assign a sufficiently high credence to m to be rational to believe that p need not lead to a similar problem, depending on what m is supposed to be exactly. Let’s say that when one is rational to believe that p, m is what one is rational to presuppose, or rational to take for granted (whatever it may be to presuppose something or to take it for granted). Then we might think that if what is rational for one to presuppose is closed under conjunction or modus ponens, we are led to similar worries that someone who subscribes to the Lockean Thesis faces, but at the level of what is rational to presuppose, rather than at the level of what is rational to believe.
But for the current account of rational belief to work, we don’t need to hold that what is rational for one to presuppose is closed under conjunction and modus ponens. Suppose that you’re in a context in which you’re rational to believe that p, and Cr(p|r&s) = Cr(p|r&t) = Cr(p|r&s&t) = 1. Suppose that you are rational to presuppose that r&s, and you are also rational to presuppose that r&t, but you might not be rational to presuppose that r&s&t, because your credence in r&s&t might be below the minimum required for you to be rational to believe that p. Nonetheless, so long as you presuppose that r&s or you presuppose that r&t (but not both at the same time), you are rational to believe that p.
One might think that on an intuitive understanding of what it is to presuppose something, if one is rational to presuppose that r, and one is rational to presuppose that s, and one is rational to presuppose that t, then one is rational to presuppose that r&s&t. But it also seems intuitive that one shouldn’t presuppose too much! While one may be rational to presuppose any one of a number of things, one shouldn’t presuppose all of them.
In any case, I don’t want to put too much weight on intuitions about how the term `presupposition’ is used in daily life, since it’s really meant to be a technical notion. To be rational to believe that p, there is some special proposition, m, to which one has to assign a sufficiently high credence, and which carves out the not-p possibilities to which one needs to assign a credence of 0. While I’ve been speaking of whether one is rational to presuppose that m, or to take m for granted, it remains an open question whether to be rational to believe that p, one needs to assume any propositional attitude (e.g. that of presupposition, taking for granted) towards m, over and above assigning it a sufficiently high credence. If the answer is `No’, then the above worry dissipates. Of course, more will have to be said about this m that I’ve been talking about, but I’ll try to answer questions (i), (ii), and (iii) in more detail when I’ve got the energy to do so.