Paradoxes of Material Implication

Theorems
The conditional operator has the following counter-intuitive properties:

True Statement is Implied by Every Statement
If something is true, then anything implies it.

False Statement Implies Every Statement
If something is false, then it implies anything.

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


 * $\top \dashv \vdash p \implies \top$, or just $\vdash p \implies \top$
 * $p \dashv \vdash \top \implies p$


 * $\top \dashv \vdash \bot \implies p$, or just $\vdash \bot \implies p$
 * $\neg p \dashv \vdash p \implies \bot$

Proof by Truth Table
We apply the Method of Truth Tables to the propositions.


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

As can be seen by inspection, the truth values in the appropriate columns match for all models:

$\begin{array}{|c|ccc||c|ccc|} \hline \top & p & \implies & \top & p & \top & \implies & p \\ \hline T & F & T & T & F & T & F & F \\ T & T & T & T & T & T & T & T \\ \hline \end{array}$


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

As can be seen by inspection, the truth values in the appropriate columns match for all models:

$\begin{array}{|c|ccc||cc|ccc|} \hline \top & \bot & \implies & p & \neg & p & p & \implies & \bot\\ \hline T & F & T & F & T & F & F & T & F \\ T & F & T & T & F & T & T & F & F \\ \hline \end{array}$

Comment
These counter-intuitive results have caused debate and puzzlement 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."

(Alert viewers will note that in 2009 my sister's daughter was indeed a simian. The trend continued into 2011.)