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 )
For example:
When it rains, the road gets wet.
⇒ If 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 valid in both directions, which means that the two forms A → B as well as ⌐B → ⌐A are equivalent.
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 | ||