Paradoxes of Material Implication

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Theorems

The conditional operator has the following counter-intuitive properties:

$\top \dashv \vdash p \implies \top$, 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

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 boolean interpretations:


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

$\blacksquare$


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

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


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

$\blacksquare$


Also presented as

These results are also presented in the following forms:

True Statement is implied by Every Statement

If something is true, then anything implies it.

Formulation 1

$p \vdash q \implies p$

Formulation 2

$\vdash q \implies \paren {p \implies q}$


False Statement implies Every Statement

If something is false, then it implies anything.

Formulation 1

$\neg p \vdash p \implies q$

Formulation 2

$\vdash \neg p \implies \paren {p \implies q}$


Also note this counterintuitive result:

Disjunction of Conditional and Converse

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


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.)


Sources