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:

:$\frac$

and

:$\frac$

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, ... H_''n'', and permutation σ(n) of the numbers 1 through n, it is the case that:

:H₁ $\land$ H₂ $\land$ ... $\land$ H_n

is equivalent to

:H_σ(1) $\land$ H_σ(2) $\land$ H_σ(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.

References

{{logic-stub
Classical logic
Rules of inference
Theorems in propositional logic