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:

and

is a metalogical is a syntactic consequence in the other, in some logical system;

or in rule form:

and

";

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

and

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:

1. H1 H2 Hn

is equivalent to

1. Hσ(1) Hσ(2) Hσ(n).

For example, if H1 is

1. It is raining

H2 is

1. Socrates is mortal

and H3 is

1. 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.

References