The Knower Paradox is as follows:

(K) Somebody knows that (K) is false.

Suppose that (K) is true. Then somebody knows that (K) is false. Since knowledge is factive, if somebody knows that (K) is false, (K) must be false. Contradiction.

So we seem to have proven that (K) is false. But if we have proven that (K) is false, then it seems that we know that (K) is false. But is is exactly what (K) says! So (K) is true. Contradiction.

I don't know if the following point has been made in the literature on the Knower or not, but it seems like an interesting result. Suppose that we specify the knower in (K) and adjust the reference accordingly. Thus:

(Kd) Dan Korman knows that (Kd) is false.

Clearly, (Kd) cannot be true. For, once again, this would imply that (Kd) is false.

It does seem, however, that (Kd) can be false. For it appears that if Dan never considers (Kd) and, in particular, if he never considers the preceding proof that (Kd) cannot be true, then we can consistently say that (Kd) is false. However, the moment that Dan does consider these facts, he will see (smart fellow that he is!) that (Kd) cannot be true and so he will know that it is false. But that, of course, is just what (Kd) claims. So, (Kd) is true.

Now what is interesting about the subject-specific knower is that the inconsistency is contingent (cf. Kripke). [Note: time indexing the propositions won't avoid the contingency result. For in that case we just consider worlds in which Dan first considers (Kd) at different times.]

But perhaps the solution is this. Prior to considering (Kd), Dan fails to know that (Kd) is false (even though it is) because he doesn't believe that it is false or because he isn't justified in believing that it is false. The preceding argument turns on the following assumption: if Dan can provide a proof that (Kd) is false, then he will have a justified belief that (Kd) is false. And, if (Kd) is in fact false, then he will know that (Kd) is false. But we know that JTB doesn't necessarily yeild knowledge. So why not just say that (Kd) is intrinsically "Gettierized" (to use a term of art) for Dan. (Of course, we can know that (Kd) is false; it is only Dan who can't).

But if that is so, then perhaps we can say the same thing about (K). Perhaps we can all have a justified, true belief that (K) is false, it is just that this proposition is unknowable.

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