Just a quick post on a familiar topic. I've been working on a paper where I say that one of the problems with E=K is that, intuitively, it does not seem that inference is a process for gathering or acquiring new evidence. Inference is a way of extending our knowledge by using old evidence to settle new questions. If I know p and deduce p v q and then deduce (p v q) v r, I don't think I've just added some new evidence to my stock of evidence but I know things I mightn't have not known before. And, if I deduce ~ p --> q, that's not more evidence but it's something I know that I've not known before.
One objection to this is that I'm just working with a very narrow conception of evidence, W is working with a broader conception of evidence, and there's little here that gives us reason to revise E=K. Fair enough. The problem cannot be that there's redundant evidence on E=K, there's going to be redundant evidence on views that restrict our evidence to things we know non-inferentially because we can know the same thing non-inferentially in different ways. Moreover, there is a perfectly good sense in which things we know via competent deduction are reasons for belief. It's because I knew p v q that I could knowingly deduce (p v q) v r and deduce ~p --> q. There's no real difference between bits of evidence and reasons for belief, so there's really nothing to the objection.
Okay, but I still think we ought to distinguish evidence (or ultimate evidence) from derivative reasons for belief. Suppose I see that there's been a fox in the garden and suppose I then see that the fox in the garden ate blueberries off of the bushes. I then read that there's never been an observed case of male foxes that eat blueberries, but there's solid evidence that suggests that female foxes just love the things. So, let's say I have good inductive evidence, e, for the belief:
(1) There's been a female fox in the garden.
Now, suppose that my inductive evidence is strong enough that I can be said to know (1). Because I know that it is analytic that female foxes are vixens, I can knowingly deduce from (1):
(2) A vixen has been in the garden.
Now, I deduce that since (1) is deducible from (2), (1) is at least as well supported by what I know as (2) is.
Here's what I'd like to say. I'd like to say that the evidential probability of (2) is neither less nor greater than (1). I'd like to say that the evidential probability of (1) prior to believing (2) is less than 1. I don't think I can say these things if E=K is true. If among the propositions included in my evidence is (1), then (2) has the evidential probability of 1. So, what about (1)? If I know full well where (1) came from, the evidential probability for that should be lower than 1 and remain that way, even when I realize that (1) is a deductive consequence of (2) and so realize that (1) is deducible from (2) and so as well supported by what I know as (2) is.
One of the advantages of the view that limits evidence to what you know non-inferentially is that you can say what I'd like to say. If our evidential probabilities are determined by conditionalizing on our evidence and our bodies of evidence include everything E=K says, you get the unfortunate result that the negation of (1) is inconsistent with your evidence even if your evidence for (1) consists of a circular argument that cites (1) itself and inductive grounds that are consistent with (~1).