Tautology is Negation of Contradiction/Proof by Truth Table

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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

We apply the Method of Truth Tables to the proposition.

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

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

$\blacksquare$