In my last set of sketchy thoughts, I proceeded in the indexing of the premises of the argument, as presented in the SEP, as far as premise (4):
(4') K{∃x|∃c}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone, to be true.
Now, the way the argument moves from here requires that a reductio be made, and a contradiction be derived from (4). Here's the original set of premises:
(4) K(p ∧ ¬Kp) It is known that both the particular claim is true and that it is not known to be true.
(5) Kp ∧ K¬Kp The particular claim is known to be true and it is known that it is not known to be true.
(6) Kp ∧ ¬Kp The particular claim is known to be true and it is not known to be true.
(5) comes from (4) by the standard rule that a conjunction is known only if its conjuncts are known. So (5) has to be indexed as:
(5') K{∃x|∃c}p ∧ K{∃x|∃c}¬K{∃y|∃c}p It is known by someone or other that the particular claim is true and it is known by that same someone or other that it is not known, by someone or other, who may or may not be the same someone, to be true.
And here, of course, we are stymied a bit. For the only thing that can be derived from this is:
(6') K{∃x|∃c}p ∧ ¬K{∃y|∃c}p It is known by someone or other that the particular claim is true, and it is not known by someone or other (who may or may not be the same someone or other) that it is true.
And that's obviously not a contradiction.
However, let's assume that x=y here. That is, let's assume that the same someone or other is involved all the way through. Even this, as it turns out, is not quite enough to give us a contradiction. The astute reader will have noticed that I have been ignoring the |c side of the index, and pretending that we can assume that all the relevant circumstances or conditions of knowing are the same throughout. This, of course, is not quite legitimate, and a more accurate indexing will take this into account. I would suggest the following so far:
(1') p ∧ ¬K{∃y|∃c}p A particular claim is true, and it is not known to be true by someone or other (given some set of circumstnaces).
(2') (p ∧ ¬K{∃y|∃c}p) → ◊K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) If a particular claim is true and is not known to be true by someone or other (given the original set of circumstances), it follows that it is possible that "a particular claim is true and is not known to be true by someone or other" is known by someone or other (who may or may not be the same person and who may or may not be in the same circumstances).
(3') ◊K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) It is possible that it is known by someone or other that both the particular claim is true and that it is not known, by someone or other who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(4') K{∃x|∃d}(p ∧ ¬K{∃y|∃c}p) It is known by someone or other in some circumstances that both the particular claim is true and that it is not known, by someone or other, who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(5') K{∃x|∃d}p ∧ K{∃x|∃c}¬K{∃y|∃c}p It is known by someone or other in some circumstances that the particular claim is true and it is known by that same someone or other that it is not known, by someone or other, who may or may not be the same someone and may or may not be in the same circumstances, to be true.
(6') K{∃x|∃d}p ∧ ¬K{∃y|∃c}p It is known by someone or other that the particular claim is true, and it is not known by someone or other (who may or may not be the same someone or other and may or may not be in the same circumstances) that it is true.
It can easily be seen that, taking this into account, the assumption that x and y are the same person(s) is not enough to yield a contradiction and make the reductio viable. Suppose that x and y are both Tom, and suppose that p is the claim that the sky is blue. And suppose the set of conditions c is describable as 'being a baby who has never seen the sky and cannot understand any testimony about it' and the set of conditions d is describable as 'being a young man who has seen the sky'. The result then is:
It is known by Tom, when he was a young man who had seen the sky, that the sky is blue; and it is not known by Tom, when he was a baby who had never seen the sky and could not understand any testimony about it, that the sky is blue.
Which is not a contradiction in the slightest; it's exactly what you'd expect if Tom originally (as a baby) didn't know that the sky was blue, and came to know at some point (by the time he was a young man) that the sky was blue. In other words, it's exactly what you'd expect if we can come to know things through learning about them.
So to have a genuine contradiction here, not only must we assume that x is y; we must assume that d is c, as well. So what has been shown to be contradictory is the following:
(4'*) K{∃x|∃c}(p ∧ ¬K{∃c|∃c}p) It is known by someone or other in some circumstances that both the particular claim is true and that it is not known by the same someone or other in those same circumstances to be true.
(4'*) is necessarily false. This leads us into the rest of the argument.
(7') ¬K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) It is not known by the same person under a given set of conditions both that the particular claim is true and that it is not known by the same person under the same set of conditions to be true.
(8') ¬K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) (7) is necessary.
(9') ¬◊K{∃x|∃c}(p ∧ ¬K{∃x|∃c}p) It is not possible that this is known by the same person under a given set of conditions: both that p is true and that it is not known by that same person under that same set of conditions to be true.
(10') ¬∃p(p ∧ ¬K{∃x|∃c}p) There is no particular claim for which it is true both that the claim is true and that it is not known by someone under some set of conditions to be true.
(11') ∀p(p → Kp) For any particular claim, if it is true, it is known to be true by someone under some set of conditions.
(7') we get from the reductio of (4'*); (8') we know because (4'*) was not just false but impossible. (8') tells us that it is impossible for anyone to know both a given claim and that they don't know it (setting aside all differences of conditions, e.g., different times of life). So (9') follows from (8'). It still contradicts (3') only if we assume that x is y and that c is d. But it does do that, given that assumption.
Can we get (10') from (9'), though? We can get the unindexed (10) from (9) by using the negation of (3) to deny the consequent of (2), which by modus tollens gives us the negation of (1). We've been assuming as well that x is y and that c is d, that is, that there is only one knower involved, and only one set of conditions for knowing. But anyone who has gone this far, of course, can, instead of following the trail back to the negation of (1), simply hold that the argument up to (9') shows that it is impossible for us to make this assumption, at least legitimately, for non-omniscient knowers. Nothing precludes such a response; and it is certainly more plausible than the argument itself.
Thus KP doesn't seem to result in the paradox. What creates the paradox is the assumption that KP requires that a given claim is true only if it can possibly be known by any particular knower under any set of conditions. But, of course, this assumption is manifestly false. Not only is the alleged requirement absurd, it's not what people mean when they say that every truth is such that it is possible for it to be known.
Which raises the question of what people really do mean, or, at least, what they can consistently mean, when they use principles like KP. Another post, although I can't promise that it will necessarily be the next one.
And keep in mind that these are all sketchy thoughts, and that none of them should be taken as any sign that I know what I'm talking about.
