Illicit commutation (Logic)

Various logical fallacies arising from a failure to observe the rules of commutativity for logical operations.

If an animal is a dog, then it has four legs.
Our cat has four legs.
It follows that: Our cat is a dog.

The material conditional (“if A, then B”) is not a commutative operation: if A ⟶ B is true, it does not follow that B ⟶ A must also be true. As the example above shows, failing to observe this rule can lead to absurd results.

• Illicit conversion
• Invalid commutation
• Non-commutativity violation

As in mathematics, the law of commutativity also applies in logic; it determines which operations may be reversed. Just as, e.g. multiplication allows commutation (a × b is the same as b × a), whereas the same is not permitted for division (a ÷ b is not the same as b ÷ a), there are also rules of commutation (though in this context often called “conversion”) for logical operations. For example, swapping the operands is permitted for adjunction, whereas this is not permitted for the (material) conditional.

The most common form of this fallacy is therefore an erroneous reversal of a conditional statement (“if A, then B”) – as in the example above. This form of an illicit commutation is also known as “affirming the consequent”.

This article is still a work in progress.

• Confusion of cause and effect
• Commutativity
• Formal fallacies