Contradiction is Negation of Tautology/Proof 3

Proof
Let $p$ be a propositional formula.

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

Then $v \left({p}\right) = F \iff v \left({\neg p}\right) = T$ by the definition of the logical not.

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