Commutativity of conjunction

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Rule of inference

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.

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Formal notation

Commutativity of conjunction can be expressed in sequent notation as:

( P ∧ Q ) ⊢ ( Q ∧ P ) {\displaystyle (P\land Q)\vdash (Q\land P)}

and

( Q ∧ P ) ⊢ ( P ∧ Q ) {\displaystyle (Q\land P)\vdash (P\land Q)}

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

or in rule form:

P ∧ Q ∴ Q ∧ P {\displaystyle {\frac {P\land Q}{\therefore Q\land P}}}

and

Q ∧ P ∴ P ∧ Q {\displaystyle {\frac {Q\land P}{\therefore P\land Q}}}

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

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

( P ∧ Q ) → ( Q ∧ P ) {\displaystyle (P\land Q)\to (Q\land P)}

and

( Q ∧ P ) → ( P ∧ Q ) {\displaystyle (Q\land P)\to (P\land Q)}

where P {\displaystyle P} and Q {\displaystyle 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 ∧ {\displaystyle \land } H2 ∧ {\displaystyle \land } ... ∧ {\displaystyle \land } Hn

is equivalent to

Hσ(1) ∧ {\displaystyle \land } Hσ(2) ∧ {\displaystyle \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.

References

Elliott Mendelson (1997). Introduction to Mathematical Logic. CRC Press. ISBN 0-412-80830-7. 0-412-80830-7 ↩