Commutativity of conjunction

Formal notation

Commutativity of conjunction can be expressed in sequent notation as:

and

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

or in rule form:

and

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

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

and

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:

is equivalent to

For example, if H₁ is

H₂ is

and H₃ is

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.