Commutativity of conjunction

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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 \and Q) \vdash (Q \and P)


(Q \and P) \vdash (P \and Q)

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

or in rule form:

\frac{P \and Q}{\therefore Q \and P}


\frac{Q \and P}{\therefore P \and Q}

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

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

(P \and Q) \to (Q \and P)


(Q \and P) \to (P \and Q)

where P and 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 \land H2 \land ... \land Hn

is equivalent to

Hσ(1) \land Hσ(2) \land Hσ(n).

For example, if H1 is

It is raining

H2 is

Socrates is mortal

and H3 is



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.


  1. Elliott Mendelson (1997). Introduction to Mathematical Logic. CRC Press. ISBN 0-412-80830-7.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>