# A proof of the logical equivalence of inverse and converse

### 2022-12-30

The inverse of `p → q`

is `¬p → ¬q`

, and its converse is `q → p`

.

This logical proof shows that the converse of a conditional entails its inverse (i.e., that `q → p ⊢ ¬p → ¬q`

):

```
[1] q → p // premise
[2] ¬q ∨ p // disjunction from conditional (1)
[3] p ∨ ¬q // commutativity of disjunction (2)
[4] ¬¬p ∨ ¬q // double negation (3)
[5] ¬p → ¬q // conditional from disjunction (4)
```

Its reversal proves that inverse entails converse (`¬p → ¬q ⊢ q → p`

):

```
[1] ¬p → ¬q // premise
[2] ¬¬p ∨ ¬q // disjunction from conditional (1)
[3] p ∨ ¬q // double negation (2)
[4] ¬q ∨ p // commutativity of disjunction (3)
[5] q → p // conditional from disjunction (4)
```

Because inverse and converse entail each other (`q → p ⊣⊢ ¬p → ¬q`

), they are logically equivalent: `q → p ≡ ¬p → ¬q`