But it seems I can very easily get to know. Let P be the property of having once had life. Now, stipulate the predicate "is xyzzy" as follows: if Mars once had life, the predicate expresses P, and otherwise, it expresses ~P. Use "<s>" to abbreviate "the proposition that s". Then, plausibly:
- (Premise) I know the proposition <Mars is xyzzy>. (For I know that "is xyzzy" expresses P if and only if Mars has P, and I can do all the logic needed to yield the claim that Mars is xyzzy.)
- (Premise) The proposition <Mars is xyzzy> is the proposition <Mars once had life> if "is xyzzy" expresses P.
- (Premise) The proposition <Mars is xyzzy> is the proposition <Mars never had life> if "is xyzzy" expresses ~P.
- (Premise) "Is xyzzy" either expresses P or it expresses ~P.
- So, either I know <Mars once had life> or I know <Mars never had life>. (By 1-4)
- So, either I know the proposition that Mars once had life or I know the proposition that Mars never had life. (Expanding abbreviations)
- (Premise schema) If I know the proposition that s, then I know that s.
- So, either I know that Mars once had life or I know that Mars never had life. (By 6 and 7)
And so whichever disjunct is true, I know whether Mars once had life! So, I didn't know it, but once I stipulated "xyzzy" and came to know that Mars is xyzzy, I got to know it.
The argument is valid, but the conclusion is surely false. So what should we deny?
My inclination is either to deny that it is possible to stipulate predicates in the way I stipulated "is xyzzy" (and hence (1) is nonsense, since it uses a bit of nonsense, namely "is xyzzy", as if it were a predicate) or to allow such stipulation but not allow that sentences using the stipulated predicate express precisely the propositions that (2) and (3) claim they do. I also feel some pull to denying (7).
An interesting move would be to deny (1) on the grounds that I once know that it is true that Mars is xyzzy, but I do not know that Mars is xyzzy. That sounds odd.
I think Jon Kvanvig may have once used something in the vicinity of this puzzle, but I could be wrong.