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.

Lemma: A proposition cannot be both, true and false

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

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