Commutativity of conjunction explained

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

Commutativity of conjunction can be expressed in sequent notation as:

(P \land Q) \vdash (Q \land P)

and

(Q \land P) \vdash (P \land Q)

where

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

or in rule form:

(P \land Q)/(\therefore Q \land P)

and

(Q \land P)/(\therefore P \land Q)

where the rule is that wherever an instance of "

(P \land Q) " appears on a line of a proof, it can be replaced with "(Q \land P) " and wherever an instance of "(Q \land P) " appears on a line of a proof, it can be replaced with "(P \land Q) ";

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

(P \land Q) \to (Q \land P)

and

(Q \land P) \to (P \land 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

\land H₂ \land ... \land H_n

is equivalent to

H_σ(1)

\land H_σ(2) \land 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.

Notes and References

1. Book: Introduction to Mathematical Logic. Elliott Mendelson. 1997. CRC Press. 0-412-80830-7.