Английская Википедия:Commutativity of conjunction

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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:

<math>(P \land Q) \vdash (Q \land P)</math>

and

<math>(Q \land P) \vdash (P \land Q)</math>

where <math>\vdash</math> is a metalogical symbol meaning that <math>(Q \land P)</math> is a syntactic consequence of <math>(P \land Q)</math>, in the one case, and <math>(P \land Q)</math> is a syntactic consequence of <math>(Q \land P)</math> in the other, in some logical system;

or in rule form:

<math>\frac{P \land Q}{\therefore Q \land P}</math>

and

<math>\frac{Q \land P}{\therefore P \land Q}</math>

where the rule is that wherever an instance of "<math>(P \land Q)</math>" appears on a line of a proof, it can be replaced with "<math>(Q \land P)</math>" and wherever an instance of "<math>(Q \land P)</math>" appears on a line of a proof, it can be replaced with "<math>(P \land Q)</math>";

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

<math>(P \land Q) \to (Q \land P)</math>

and

<math>(Q \land P) \to (P \land Q)</math>

where <math>P</math> and <math>Q</math> 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 <math>\land</math> H₂ <math>\land</math> ... <math>\land</math> H_n

is equivalent to

H_σ(1) <math>\land</math> H_σ(2) <math>\land</math> 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.