====== 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.
> <span conclusio>It follows that: <s invalid "Illicit commutation">Our cat is a dog.</s></span>

The [[glossary:conditional|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.

===== Other names =====

  * Illicit conversion
  * Invalid commutation
  * Non-commutativity violation

===== Description =====

As in mathematics, the law of commutativity also applies in logic; it determines which operations may be reversed. Just as, e.g. [[wp>Multiplication|multiplication]] allows commutation (''a × b'' is the same as ''b × a''), whereas the same is not permitted for [[wp>Division (mathematics)|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 [[glossary:adjunction|adjunction]], whereas this is not permitted for the [[glossary:conditional|(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 “[[formal_fallacies:affirming_the_consequent|affirming the consequent]]”.

#TODO **This article is still a work in progress.**

===== See also =====

  * [[causality:cause-effect|Confusion of cause and effect]]
  * [[glossary:commutativity|Commutativity]]
  * [[logic:formal_fallacies:index|Formal fallacies]]