Thread:Jacky720/@comment-32182236-20200429164606/@comment-32182236-20200501130915

..Actually.. It seems the proof DOES hold. If we did assume P2 to be false from the start, we'd ALREADY reach the conclusion that the statement is neither true nor false.

And the rest follows.

No need for Set Theory.

But.. That would deny the Law of Excluded Middle.

..

Alright. Here's Part 2 of the proof that the Liar's Paradox isn't a valid proposition, PERIOD, and therefore has no truth value. Now if only I was able to construct Part 1..


 * Cx:Therefore, there exists a set where all propositions lie within (??, ??, ???)
 * Py:Any statement referring to another statement must refer to that specific statement itself. (For instance, a statement referring to the 2nd statement would be known to refer to statement TWO, not statement 4, 1, etc..)
 * Cx+1:Therefore, the Liar's Paradox would take the form ¬PN, while also being statement N on the set. (Cx, Py)
 * Py+2:If there is a statement ¬PN, then the statement ¬PN-1 must occur earlier in the set, as that too, would be a valid proposition, and referring to an earlier statement, thus appearing first.
 * Cx+2:Therefore, the Liar's Paradox must be statement Z on the set, where Z is N plus at least the number of statements that come before ¬P1 (Cx+1, Py+2)
 * Py+3:PN comes before ¬PN
 * Cx+3:Therefore, there exists at least one statement that comes before ¬PN (Py+3)
 * Cx+4:Therefore, the Liar's Paradox must be at least statement N+N, or 2N on the set (Cx+2, Cx+3)
 * Cx+5:The Liar's Paradox must both be statement N and statement 2N, which is a CONTRADICTION (Cx+1, Cx+4)
 * Cx+6:Therefore, the Liar's Paradox cannot be a valid proposition.