Contradiction is Negation of Tautology/Proof 3

Proof
Let $p$ be a propositional formula.

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

Then $\map v p = F \iff \map v {\neg p} = T$ by the definition of the logical not.

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

Hence:
 * $\bot \dashv \vdash \neg \top$

That is, the proposition:
 * If it's not true, it must be false

is valid only in the context where there are only two truth values.