Disjunction of Conditional and Converse

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


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

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
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.

First we put this in the format so we can use a truth table on it:
 * $$\top \dashv \vdash \left({p \implies q}\right) \or \left({q \implies p}\right)$$

Then: