(note: a previous version of this post was deleted by me. The reply above was a response to that post)
There's an obvious problem: if we assign semantics in a way that every statement implicit asserts its own truth, then we cannot prove anything true! To wit, we can consistently assign the truth value "false" to every
For clarity: if we assign the truth value "false", then the statement's implicit assertion of its own truth is false, and therefore the statement is false.
Your version still runs afoul of the liar's paradox. If we interpret "this sentence is false" as a proposition satisfying
P = (P --> not P)
- Case 1: P is true. In this case, we have True --> False, which is false, and therefore P is false.
- Case 2: P is false. This can only happen if the hypothesis is true and the conclusion is false: that is, we can conclude P is true and not P is false. Therefore, P is true.