Disjunction of Conditional and Converse/Proof by Truth Table

Theorem
Given any two statements, one of them implies the other.


 * $\vdash \left({p \implies q}\right) \lor \left({q \implies p}\right)$

That is, given any conditional, either it is true or its converse is.

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

As can be seen by inspection, the truth values under the main connective is true for all models, proving a tautology.

$\begin{array}{|ccccccc|} \hline (p & \implies & q) & \lor & (q & \implies & p) \\ \hline F & T & F & T & F & T & F \\ F & T & T & T & T & F & F \\ T & F & F & T & F & T & T \\ T & T & T & T & T & T & T \\ \hline \end{array}$