====== Contraposition ======
A [[logic:inferences:index|valid inference]] in which the [[glossary:antecedent|antecedent]] and [[glossary:consequent|consequent]] of a [[glossary:subjunction|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”.
===== Description =====
The contraposition is a logical transformation rule that follows from the [[logic:inferences:modus_tollens|Modus Tollens]]. It is a [[glossary:commutativity|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 | | B → A \\ (if B, then A) | ⌐A → ⌐B \\ (if not‑A, then not‑B) |
^See also: | [[logic:inferences:modus_tollens|Modus Tollens]] || | [[logic:formal_fallacies:affirming_the_consequent|Affirming the consequent]] | [[logic:formal_fallacies:denying_the_antecedent|Denying the antecedent]] |
===== See also =====
* [[logic:inferences:modus_tollens|Modus Tollens]]
* [[glossary:subjunction|Subjunction]]
===== More Information =====
* [[wp>Contraposition]] on //Wikipedia//