Contradiction is Negation of Tautology/Proof by Truth Table
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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
We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, the truth values in the appropriate columns match.
$\begin{array}{|c||cc|} \hline \top & \neg & \bot \\ \hline \T & \T & \F \\ \hline \end{array}$
$\blacksquare$