# 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.

## Formal notation

Commutativity of conjunction can be expressed in sequent notation as:

$\displaystyle{ (P \land Q) \vdash (Q \land P) }$

and

$\displaystyle{ (Q \land P) \vdash (P \land Q) }$

where $\displaystyle{ \vdash }$ is a metalogical symbol meaning that $\displaystyle{ (Q \land P) }$ is a syntactic consequence of $\displaystyle{ (P \land Q) }$, in the one case, and $\displaystyle{ (P \land Q) }$ is a syntactic consequence of $\displaystyle{ (Q \land P) }$ in the other, in some logical system;

or in rule form:

$\displaystyle{ \frac{P \land Q}{\therefore Q \land P} }$

and

$\displaystyle{ \frac{Q \land P}{\therefore P \land Q} }$

where the rule is that wherever an instance of "$\displaystyle{ (P \land Q) }$" appears on a line of a proof, it can be replaced with "$\displaystyle{ (Q \land P) }$" and wherever an instance of "$\displaystyle{ (Q \land P) }$" appears on a line of a proof, it can be replaced with "$\displaystyle{ (P \land Q) }$";

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

$\displaystyle{ (P \land Q) \to (Q \land P) }$

and

$\displaystyle{ (Q \land P) \to (P \land Q) }$

where $\displaystyle{ P }$ and $\displaystyle{ Q }$ are propositions expressed in some formal system.

## Generalized principle

For any propositions H1, H2, ... Hn, and permutation σ(n) of the numbers 1 through n, it is the case that:

H1 $\displaystyle{ \land }$ H2 $\displaystyle{ \land }$ ... $\displaystyle{ \land }$ Hn

is equivalent to

Hσ(1) $\displaystyle{ \land }$ Hσ(2) $\displaystyle{ \land }$ Hσ(n).

For example, if H1 is

It is raining

H2 is

Socrates is mortal

and H3 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.