Rule of Commutation

This page has been identified as a candidate for refactoring of medium complexity. In particular: extension Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

Conjunction

Conjunction is commutative:

Formulation 1

$p \land q \dashv \vdash q \land p$

Formulation 2

$\vdash \paren {p \land q} \iff \paren {q \land p}$

Disjunction

Disjunction is commutative:

Formulation 1

$p \lor q \dashv \vdash q \lor p$

Formulation 2

$\vdash \paren {p \lor q} \iff \paren {q \lor p}$

Its abbreviation in a tableau proof is $\text{Comm}$.

Also known as

The rule of commutation is also known as the commutative law.

Note that this term is also used throughout mathematics in the context of addition and multiplication of numbers:

the Commutative Law of Addition

the Commutative Law of Multiplication

so it is wise to be aware of the context.

Also see

• Rule of Association
• Principle of Commutation

Technical Note

When invoking the Rule of Commutation in a tableau proof, use the {{Commutation}} template:

{{Commutation|line|pool|statement|depends|type}}

where:

line is the number of the line on the tableau proof where Rule of Commutation is to be invoked

pool is the pool of assumptions (comma-separated list)

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

depends is the line (or lines) of the tableau proof upon which this line directly depends

type is the type of Rule of Commutation: specifically Disjunction or Conjunction, whose link will be displayed in the Notes column.