Paradoxes of Material Implication

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Theorems

The conditional operator has the following counter-intuitive properties:

Tautological Consequent

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


Tautological Antecedent

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


Contradictory Antecedent

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


Contradictory Consequent

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


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

\(\ds p\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds q \implies p\) \(\) \(\ds \)

Formulation 2

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


False Statement implies Every Statement

If something is false, then it implies anything.

Formulation 1

\(\ds \neg p\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds p \implies q\) \(\) \(\ds \)

Formulation 2

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


Also note this counterintuitive result:

Disjunction of Conditional and Converse

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


Examples

Red Grass and Green Moon

The compound statement:

If grass is red then the moon is made of green cheese

is true, despite being semantically meaningless.


Historical Note

The Paradoxes of Material Implication 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.


Sources

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