Disproof of law of excluded middle

Case 1: Self referential statements have no truth value and are not true

Let S be the statement that every statement is either true or false. Then S says that S is either true or false. Therefore, S refers to itself and S is not true

Case 2: Self referential statements have a truth value

Given the statement 'S': ¬S, by substituting S into ¬S we get ¬¬S which is S. So S implies S and not S.

Assuming the law of excluded middle, this is false, so the law of excluded middle says ¬S. However, S is defined as ¬S, so the conclusion is S and ¬S. The axiom of law of excluded middle results in a contradiction, so the law of excluded middle is false

integralcurveinc@gmail.com