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