(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.∎

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