Disjunction of Conditional and Converse

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 by Natural deduction
This is proved by the Tableau method.

Proof by Truth Table
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.

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