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:

[math]\displaystyle{ (P \land Q) \vdash (Q \land P) }[/math]

and

[math]\displaystyle{ (Q \land P) \vdash (P \land Q) }[/math]

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

or in rule form:

[math]\displaystyle{ \frac{P \land Q}{\therefore Q \land P} }[/math]

and

[math]\displaystyle{ \frac{Q \land P}{\therefore P \land Q} }[/math]

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

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

[math]\displaystyle{ (P \land Q) \to (Q \land P) }[/math]

and

[math]\displaystyle{ (Q \land P) \to (P \land Q) }[/math]

where [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] 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 [math]\displaystyle{ \land }[/math] H2 [math]\displaystyle{ \land }[/math] ... [math]\displaystyle{ \land }[/math] Hn

is equivalent to

Hσ(1) [math]\displaystyle{ \land }[/math] Hσ(2) [math]\displaystyle{ \land }[/math] 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

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

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Classical logic

General|

* Quantifiers
* Predicate
* Connective
* Tautology
* Truth tables
* Truth function
* Truth value
* Well-formed formula
* Monotonicity of entailment
* Idempotency of entailment
* Logicism
* Problem of multiple generality
* Associativity
* Distribution

Classical logics|

* Propositional
* First-order
* Second-order
* Higher-order

Principles|

* Commutativity of conjunction
* Excluded middle
* Bivalence
* Noncontradiction
* Explosion

Rules|

* De Morgan's laws
* Material implication
* Transposition
* modus ponens
* modus tollens
* modus ponendo tollens
* Constructive dilemma
* Destructive dilemma
* Disjunctive syllogism
* Hypothetical syllogism
* Absorption

| Introduction|

* Negation
* Double negation
* Existential
* Universal
* Biconditional
* Conjunction
* Disjunction

|

Elimination|

* Double negation
* Existential
* Universal
* Biconditional
* Conjunction
* Disjunction

People|

* Bernard Bolzano
* George Boole
* Georg Cantor
* Richard Dedekind
* Augustus De Morgan
* Gottlob Frege
* Kurt Gödel
* Hugh MacColl
* Giuseppe Peano
* Charles Sanders Peirce
* Bertrand Russell
* Ernst Schröder
* Henry M. Sheffer
* Alfred Tarski
* Willard Van Orman Quine
* Ludwig Wittgenstein
* Jan Łukasiewicz

Works|

* Begriffsschrift
* Function and Concept
* The Principles of Mathematics
* Principia Mathematica
* Tractatus Logico-Philosophicus

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