Paradoxes of Material Implication

Theorems

 * $$q \vdash p \implies q$$

"If something is true, then anything implies it."


 * $$\neg p \vdash p \implies q$$

"If something is false, then it implies anything."

These results can be formalized alternatively as part of the following set:


 * $$\top \dashv \vdash p \implies \top$$
 * $$p \dashv \vdash \top \implies p$$


 * $$\top \dashv \vdash \bot \implies p$$
 * $$\neg p \dashv \vdash p \implies \bot$$

Proof by Natural deduction
These are proved by the Tableau method.

Proof by Truth Table
The second set of assertions will be proved by truth tables.

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

Comment
These counter-intuitive results have caused debate and confusion among philosophers for millennia.

In particular, the result $$\neg p \vdash p \implies q$$ is known as a vacuous truth. It is exemplified by the (rhetorical) argument:

"If England win the Ashes this year, then I'm a monkey's uncle."