A Brief Outline of Fitch's Paradox of Knowability

P1) If (i) all truths are knowable, and (ii) there exists an unknown truth, X, then “X & X is unknown” is a truth.

P2) If “X & X is unknown” is a truth, then, ex hypothesi, it is possible to know that “X & X is unknown.”

C1) Therefore, if (i) all truths are knowable, and (ii) there exists an unknown truth X, then it is possible to know that “X & X is unknown.” [From P1 and P2]

P3) It is not possible to know that “X & X is unknown”—on pain of contradiction.

C2) Therefore, if (i) all truths are knowable, then it is not the case that (ii) there exists an unknown truth—on pain of contradiction. [From C1 and P3]

Defense of P3

P1) If it is possible to know that “X & X is unknown,” then it is possible to know that “X” when it is known that “X is unknown.”

P2) If it is known that “X is unknown”, then “X is unknown” is true.

P3) If “X is unknown” is true, then it is the case that “X” is unknown.

P4) If it is the case that “X” is unknown, then it is not the case that it is known that “X”.

C1) Therefore, if it is possible to know that “X & X is unknown,” then it is possible to know that “X” when it is not the case that it is known that “X”. [From P1—P4]

P5) It is not possible to know that “X” when it is not the case that it is known that “X”.

C2) Therefore, it not is possible to know that “X & X is unknown.” [From C1 and P5]