Paradoxes of Material Implication

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


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


 * "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:


 * $$\top \dashv \vdash p \implies \top$$, 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 Natural deduction
These are proved by the Tableau method.

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


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

As can be seen for all models by inspection, where the truth value in the relevant column on the LHS is $$T$$, that under the one on the RHS is also $$T$$:

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

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


 * $$\top \dashv \vdash p \implies \top$$ 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 father's brother was indeed a simian.)