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.^[1]

Formal notation

Commutativity of conjunction can be expressed in sequent notation as:

${\displaystyle (P\land Q)\vdash (Q\land P)}$

and

${\displaystyle (Q\land P)\vdash (P\land Q)}$

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

or in rule form:

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

and

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

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

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

${\displaystyle (P\land Q)\to (Q\land P)}$

and

${\displaystyle (Q\land P)\to (P\land Q)}$

where ${\displaystyle P}$ and ${\displaystyle Q}$ 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 ${\displaystyle \land }$ H₂ ${\displaystyle \land }$ ... ${\displaystyle \land }$ H_n

is equivalent to

H_σ(1) ${\displaystyle \land }$ H_σ(2) ${\displaystyle \land }$ 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

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