A proof of the logical equivalence of inverse and converse
published on December 30, 2022
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