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
where is a metalogical symbol meaning that is a syntactic consequence of , in the one case, and is a syntactic consequence of in the other, in some logical system;
or in rule form:
and
where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "" and wherever an instance of "" appears on a line of a proof, it can be replaced with "";
or as the statement of a truth-functional tautology or theorem of propositional logic:
and
where 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:
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.
References