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Proof

(related to Lemma: A proposition cannot be both, true and false)

Let $x$ be a proposition.
Using the semantics of propositional logic and the definition of conjunction we get the following truth table:

$[[x]]_I$	$[[\neg x]]_I$	$[[x \wedge \neg x]]_I$
\(1\)	\(0\)	\(0\)
\(0\)	\(1\)	\(0\)

It follows that \(x\wedge \neg x\) is a contradiction.
Thus, $x$ cannot be both, true and false.
∎

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References

Bibliography

Mendelson Elliott: "Theory and Problems of Boolean Algebra and Switching Circuits", McGraw-Hill Book Company, 1982