Editing Associative property (section)

== Propositional logic ==
{{Transformation rules}}

=== Rule of replacement ===
In standard truth-functional propositional logic, ''association'',<ref>{{cite book |last1=Moore |first1=Brooke Noel |last2=Parker |first2=Richard |date=2017 |title=Critical Thinking |location=New York |publisher=McGraw-Hill Education |page=321 |isbn=9781259690877|edition=12th }}</ref><ref>{{cite book |last1=Copi |first1=Irving M. |last2=Cohen |first2=Carl |last3=McMahon |first3=Kenneth |date=2014 |title=Introduction to Logic |location=Essex |publisher=Pearson Education |page=387 |isbn=9781292024820|edition=14th }}</ref> or ''associativity''<ref>{{cite book |last1=Hurley |first1=Patrick J. |last2=Watson |first2=Lori |date=2016 |title=A Concise Introduction to Logic |location=Boston |publisher=Cengage Learning |page=427 |isbn=9781305958098|edition=13th }}</ref> are two [[Validity (logic)|valid]] [[rule of replacement|rules of replacement]]. The rules allow one to move parentheses in [[well-formed formula|logical expressions]] in [[formal proof|logical proofs]]. The rules (using [[Logical connective#In language|logical connectives]]  notation) are:

<math display="block">(P \lor (Q \lor R)) \Leftrightarrow ((P \lor Q) \lor R)</math>

and

<math display="block">(P \land (Q \land R)) \Leftrightarrow ((P \land Q) \land R),</math>

where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a [[Formal proof|proof]] with".

=== Truth functional connectives ===
''Associativity'' is a property of some [[logical connective]]s of truth-functional [[propositional logic]]. The following [[logical equivalence]]s demonstrate that associativity is a property of particular connectives. The following (and their converses, since {{math|↔}} is commutative) are truth-functional [[tautology (logic)|tautologies]].{{citation needed|reason=Stack Exchange is not a reliable source?|date=June 2022}}

;Associativity of disjunction
:<math>((P \lor Q) \lor R) \leftrightarrow (P \lor (Q \lor R))</math>
;Associativity of conjunction
:<math>((P \land Q) \land R) \leftrightarrow (P \land (Q \land R))</math>
;Associativity of equivalence
:<math>((P \leftrightarrow Q) \leftrightarrow R) \leftrightarrow (P \leftrightarrow (Q \leftrightarrow R))</math>

[[Logical NOR|Joint denial]] is an example of a truth functional connective that is ''not'' associative.