A Puzzle about Evidence

(I suspect that I may have heard about this puzzle before, but I don't know where. If anyone has seen something similar in the literature, I would appreciate the reference.)

The following two principles seem true:
(1) For all x, y, and z, if y and z are propositions and x has (some degree of) evidence for y, then x has (some, perhaps lower, degree of) evidence for the conjunction of y and z.
(2) For all x, y, and z, if x has (some degree of) evidence for the conjunction of y and z, then x has (some degree of) evidence for y and x has (some degree of) evidence for z.

(1), I think, will appeal to many. (It seems pretty plausible to me. Also, though I'm by no means an expert in this area, it seems to me that (1) will likely be accepted by those who are attracted to a close connection between evidence and evidential probability.) (2), on the other hand, seems undeniable. If you have evidence for a conjunction then of course you have evidence for each of its conjuncts!

The puzzle, of course, is that, given that any two propositions have a conjunction, (3) follows from (1) and (2):
(3) For all x, if there is a proposition y such that x has (some degree of) evidence for y, then: for all propositions z, x has evidence for z.

Suppose otherwise. Then there is an x such that (i) there is a proposition y such that x has evidence for y, and (ii) there is a proposition z such that it is not the case that x has evidence for z. But consider the conjunction of y and z. Since x has evidence for y (by (i)), x has evidence for the conjunction of y and z (by (1)). But if x has evidence for the conjunction of y and z, then x has evidence for y and x has evidence for z (by (2)). So, x has evidence for z. But this contradicts (ii), and our reductio is complete. (3) follows from (1) and (2).

It is important to keep in mind what the is puzzling about this result. It is not puzzling for the reason that it entails that everyone is justified in believing some proposition is justified in believing every proposition. It is not puzzling for this reason because the entailment simply doesn't hold. Presumably despite the fact that you have evidence for every proposition, there are some propositions such that your evidence for them is not sufficient to justify you in believing them.

What is puzzling about the result is simply that it entails that you have evidence for certain propositions that you might have thought you have no evidence for. For instance, you might have responded to someone's claim that some proposition p was the case by saying that you have no evidence for p. If the result discussed in this post holds, then (assuming that you have evidence for some proposition or other), your response is false. You do have evidence for p. If the result discussed in this post holds, then, people are often mistaken about what propositions they have evidence for.