A valid inference in which the antecedent and consequent of a subjunction are exchanged and both negated.
As a formula, this looks as follows:
( A ⟶ B ) ≡ ( ⌐B ⟶ ⌐A )– “when A, then B” is logically equivalent to “when not-B, then not-A”
For example:
“When it rains, the road gets wet”
is logically equivalent to
“When the road does not get wet, it does not rain”.
The contraposition is a logical transformation rule that follows from the Modus Tollens. It is a commutative operation, which means that the operation is valid in both directions.
In the table below the valid transformations and a selection of invalid ones are compared:
| Contraposition (valid forms) | Invalid forms | ||||
|---|---|---|---|---|---|
| Origin | A → B (if A, then B | ⌐B → ⌐A (if not‑B, then not‑A) | A → B (if A, then B) | A → B (if A, then B) |
|
| Transformation | ⌐B → ⌐A (if not‑B, then not‑A) | A → B (if A, then B | (if B, then A) | (if not‑A, then not‑B) |
|
| See also: | Modus Tollens | Affirming the consequent | Denying the antecedent | ||