Contradiction is Negation of Tautology

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


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

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

Proof by Boolean Interpretation
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.

Also see

 * Tautology is Negation of Contradiction