I’ve been thinking again about the issues about knowledge justified belief and practical interests that I explored a bit in this old paper. In that paper I have a rather complicated example that’s meant to show that a principle Jeremy Fantl and Matthew McGrath endorse, namely (PC) is false. Here is the principle.

(PC) S is justified in believing that *p* only if S is rational to prefer as if *p*.

The rough outline of why (PC) is wrong is that whether one is rational to prefer as if *p* might depend not only on whether one has justified attitudes towards *p*, but on whether one’s other attitudes are justified. Here is one example in which that distinction matters.

S justifiably has credence 0.99 in *p*. She unjustifiably has credence 0.9999 in *q*. (She properly regards *p* and *q* are probabilistically independent.) In fact, given her evidence, her credence in *q* should be 0.5.

S is offered a bet that pays $1 if _p_v_q_ is true, and loses $1000 otherwise. Assume S has a constant marginal utility for money. It is irrational for S to prefer to take the bet. Given her evidence, it has a negative expected value. Given her (irrational) beliefs, it has a positive expected value, but if she properly judged the evidence for *q*, then she would not take the bet.

Of course, given *p* the bet is just a free grant of $1, so she should take it.

So this is a case where it is not rational to prefer as if *p*. She should prefer to decline the bet, but to accept the bet given *p*.

If we accept (PC), it follows that S is not justified in believing *p*. But this conclusion seems wrong. S’s credence in *p* is perfectly justified. And on any theory of belief that seems viable around here, S’s credence in *p* counts as a belief. (On my preferred view, S believes *p* iff she prefers as if *p*. And she does. The main rival to this view is the “threshold view”, where belief requires a credence above the threshold. And the usual values for the threshold are lower than 0.99.)

So this is a counterexample to (PC). In a recent paper, Fantl and McGrath defend a weaker principle, namely (KA).

(KA) S knows that *p* only if S is rational to act as if *p*.

Is this case a counterexample to (KA) as well? (Assume that *p* is true, so the agent could possibly know it.) I don’t believe that it is a counterexample. I think the things that an agent knows are the things she can use to frame a decision problem. If the agent knows *p*, then the choice between taking or declining the bet just is the choice between taking a dollar and refusing it. So she should take the bet. This would be irrational, so that must be the wrong way to frame the bet. Hence she doesn’t know that *p*.

The upshot of this is that these practical cases give us a new kind of counterexample to K = JTB. In the case I’ve described, the agent has a justified true belief that *p*, but does not know *p*.