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:

[math]\displaystyle{ (P \land Q) \vdash (Q \land P) }[/math]

and

[math]\displaystyle{ (Q \land P) \vdash (P \land Q) }[/math]

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

or in rule form:

[math]\displaystyle{ \frac{P \land Q}{\therefore Q \land P} }[/math]

and

[math]\displaystyle{ \frac{Q \land P}{\therefore P \land Q} }[/math]

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

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

[math]\displaystyle{ (P \land Q) \to (Q \land P) }[/math]

and

[math]\displaystyle{ (Q \land P) \to (P \land Q) }[/math]

where [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] 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 [math]\displaystyle{ \land }[/math] H₂ [math]\displaystyle{ \land }[/math] ... [math]\displaystyle{ \land }[/math] H_n

is equivalent to

H_σ(1) [math]\displaystyle{ \land }[/math] H_σ(2) [math]\displaystyle{ \land }[/math] 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.