Of course, the following lemma follows immediately from the law of excluded middle which is used in the definition of propositions. But now we will formally prove that a proposition cannot be both - true and false - by showing that this would be a contradiction.
If \(x\) is a proposition, then the conjunction \(x\wedge \neg x\) is a contradiction (i.e. always invalid).
\(\text{"Socrates is a philosopher and Socrates is not a philosopher"}\) is invalid.
\(a= 2\wedge a\neq 2\) is invalid.
Proofs: 1
Proofs: 1