## Theorems

The conditional operator has the following counter-intuitive properties:

### Tautological Consequent

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

### Tautological Antecedent

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

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

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

## 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

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.