# Commutativity of conjunction

In propositional logic, the **commutativity of conjunction** is a valid argument form and truth-functional tautology. It is considered to be a law of classical logic. It is the principle that the conjuncts of a logical conjunction may switch places with each other, while preserving the truth-value of the resulting proposition.^{[1]}

## Formal notation[edit]

*Commutativity of conjunction* can be expressed in sequent notation as:

and

where is a metalogical symbol meaning that is a syntactic consequence of , in the one case, and is a syntactic consequence of in the other, in some logical system;

or in rule form:

and

where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "" and wherever an instance of "" appears on a line of a proof, it can be replaced with "";

or as the statement of a truth-functional tautology or theorem of propositional logic:

and

where and are propositions expressed in some formal system.

## Generalized principle[edit]

For any propositions H_{1}, H_{2}, ... H_{n}, and permutation σ(n) of the numbers 1 through n, it is the case that:

- H
_{1}H_{2}... H_{n}

is equivalent to

- H
_{σ(1)}H_{σ(2)}H_{σ(n)}.

For example, if H_{1} is

*It is raining*

H_{2} is

*Socrates is mortal*

and H_{3} is

*2+2=4*

then

*It is raining and Socrates is mortal and 2+2=4*

is equivalent to

*Socrates is mortal and 2+2=4 and it is raining*

and the other orderings of the predicates.

## References[edit]

**^**Elliott Mendelson (1997).*Introduction to Mathematical Logic*. CRC Press. ISBN 0-412-80830-7.