Tautology is Negation of Contradiction/Proof 3

Theorem

A tautology implies and is implied by the negation of a contradiction:

$\top \dashv \vdash \neg \bot$

That is, a truth can not be false, and a non-falsehood must be a truth.

Proof

Let $p$ be a propositional formula.

Let $v$ be an arbitrary boolean interpretation of $p$.

Then:

$\map v p = T \iff \map v {\neg p} = F$

by the definition of the logical not.

Since $v$ is arbitrary, $p$ is true in all interpretations if and only if $\neg p$ is false in all interpretations.

Hence:

$\top \dashv \vdash \neg \bot$

$\blacksquare$