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

Alternative rendition
These can alternatively be rendered as:


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


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

They can be seen to be logically equivalent to the forms above by application of the Rule of Implication and Modus Ponendo Ponens.

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.)