@Zetherin,
Zetherin;169229 wrote:I think your error is that you believe that for contradictory beliefs to be held, that there must be a contradictory happening (p and ~p). But this isn't true.
For instance, I can believe that the chemical composition of water is H2O, and I can believe that the chemical composition of water is not H2O. But that I believe both, as we know, doesn't mean both propositions are true. We know that one is false.
Contradictory happenings are believable, but that doesn't imply that there are contradictory happenings. What we believe and what is, are two different things.
B(p & ~p) <-> ~K(p v ~p) but, all tautologies are knowable by
1. (p is a tautology) -> Kp.
That is, K(p v ~p) is valid and therefore, ~K(p v ~p) is contradictory.
That is, ~K~(~p & p) is contradictory. Therefore, B(~p & p) is a contradiction.
~B(p & ~p), it is not believable that (p is true and p is false), is a theorem of this logic.