Rule of Commutation

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Theorem

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 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 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 : specifically Disjunction or Conjunction, whose link will be displayed in the Notes column.