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:

$(P\wedge Q)⊢(Q\wedge P)$

and

$(Q\wedge P)⊢(P\wedge Q)$

where $⊢$ is a metalogical symbol meaning that $(Q\wedge P)$ is a syntactic consequence of $(P\wedge Q)$, in the one case, and $(P\wedge Q)$ is a syntactic consequence of $(Q\wedge P)$ in the other, in some logical system;

or in rule form:

$\frac{P\wedge Q}{\therefore Q\wedge P}$

and

$\frac{Q\wedge P}{\therefore P\wedge Q}$

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

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

$(P\wedge Q)\to (Q\wedge P)$

and

$(Q\wedge P)\to (P\wedge Q)$

where $P$ and $Q$ are propositions expressed in some formal system.

Generalized principle

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

H₁ $\wedge$ H₂ $\wedge$ ... $\wedge$ H_n

is equivalent to

H_σ(1) $\wedge$ H_σ(2) $\wedge$ H_σ(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.