Commutativity of conjunction

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:


 * $$(P \land Q) \vdash (Q \land P)$$

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:


 * $$\frac{P \land Q}{\therefore Q \land P}$$

and


 * $$\frac{Q \land P}{\therefore P \land Q}$$

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, ... 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
 * 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.