Tautology is Negation of Contradiction

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.

Comment
Note that the proof of:
 * $\neg \bot \vdash \top$

relies indirectly, via Reductio Ad Absurdum, upon Law of Excluded Middle, and it can be seen that this is just another way of stating that truth.

The proposition:
 * If it's not false, it must be true

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

From the intuitionistic perspective, this result does not hold.

Proof by Truth Table
We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values in the appropriate columns match.

$\begin{array}{|c||cc|} \hline \bot & \neg & \top \\ \hline F & F & T \\ \hline \end{array}$

Proof by Boolean Interpretation
Let $p$ be a propositional formula.

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

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

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

Hence $\top \dashv \vdash \neg \bot$.