From talking to Alex Pruss, again, a worry about pragmatic encroachment combined with closure. Suppose you know that p entails q, where p is something with serious risks concerning error but q isn’t. Suppose your underlying evidence puts you barely above the the threshold for rational belief when nothing is at stake. You believe p and you believe q because you’ve deduced it from p. But p is at risk, in the sense required to raise the threshold needed for rational belief. In the case as imagined, your underlying evidential basis is the same for both beliefs. But the PE’er shouldn’t be happy saying you don’t know or rationally believe p, but you do know or rationally believe q.
Don’t say the adequate closure principles don’t license the belief that q. Those principles offer sufficient conditions only (e.g., “knowledge is closed under known deduction”), and whatever one says about belief, the evidential support for q can be the same as that for p. If you choose to appeal to some condition on proper basing, let the person note the entailment, and then base the belief directly on the evidence for p rather than on the entailment: on any adequate account of basing, one can realize that q follows from p, deduce it from p, and yet know it on the basis of something other than this entailment. That’s a claim that needs an argument, but I won’t go into it here.
Think of it this way. You are strict purist about evidential support. You always honor an invariant threshold of epistemic support. The PE’ers claim that your behavior sometimes yields rational belief and sometimes not. But whatever they want to say, they should not say that your belief that p is not rational and your belief that q is. The question is how to secure this result.