◀ ▲ ▶Branches / Logic / Lemma: A proposition cannot be both, true and false

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).

Examples:

\(\text{"Socrates is a philosopher and Socrates is not a philosopher"}\) is invalid.

\(a= 2\wedge a\neq 2\) is invalid.

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