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In [[propositional calculus|propositional logic]], the '''commutativity of conjunction''' is a [[validity|valid]] [[argument form]] and truth-functional [[tautology (logic)|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.<ref>{{cite book|title=Introduction to Mathematical Logic|author=Elliott Mendelson|year=1997|publisher=CRC Press|isbn=0-412-80830-7}}</ref> | |||
== Formal notation == | |||
''Commutativity of conjunction'' can be expressed in [[sequent]] notation as: | |||
: <math>(P \and Q) \vdash (Q \and P)</math> | |||
and | |||
: <math>(Q \and P) \vdash (P \and Q)</math> | |||
where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>(Q \and P)</math> is a [[logical consequence|syntactic consequence]] of <math>(P \and Q)</math>, in the one case, and <math>(P \and Q)</math> is a syntactic consequence of <math>(Q \and P)</math> in the other, in some [[formal system|logical system]]; | |||
or in [[rule of inference|rule form]]: | |||
:<math>\frac{P \and Q}{\therefore Q \and P}</math> | |||
and | |||
:<math>\frac{Q \and P}{\therefore P \and Q}</math> | |||
where the rule is that wherever an instance of "<math>(P \and Q)</math>" appears on a line of a proof, it can be replaced with "<math>(Q \and P)</math>" and wherever an instance of "<math>(Q \and P)</math>" appears on a line of a proof, it can be replaced with "<math>(P \and Q)</math>"; | |||
or as the statement of a truth-functional tautology or [[theorem]] of propositional logic: | |||
:<math>(P \and Q) \to (Q \and P)</math> | |||
and | |||
:<math>(Q \and P) \to (P \and Q)</math> | |||
where <math>P</math> and <math>Q</math> are [[proposition]]s expressed in some formal system. | |||
== Generalized principle == | |||
For any propositions H<sub>1</sub>, H<sub>2</sub>, ... H<sub>''n''</sub>, and permutation σ(n) of the numbers 1 through n, it is the case that: | |||
:H<sub>1</sub> <math>\land</math> H<sub>2</sub> <math>\land</math> ... <math>\land</math> H<sub>n</sub> | |||
is equivalent to | |||
:H<sub>σ(1)</sub> <math>\land</math> H<sub>σ(2)</sub> <math>\land</math> H<sub>σ(n)</sub>. | |||
For example, if H<sub>1</sub> is | |||
:''It is raining'' | |||
H<sub>2</sub> is | |||
:''[[Socrates]] is mortal'' | |||
and H<sub>3</sub> 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== | |||
{{reflist}} | |||
[[Category:Classical logic]] | |||
[[Category:Rules of inference]] | |||
[[Category:Theorems in propositional logic]] |
Revision as of 17:29, 11 April 2013
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:
and
where is a metalogical symbol meaning that is a syntactic consequence of , in the one case, and is a syntactic consequence of in the other, in some logical system;
or in rule form:
and
where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "" and wherever an instance of "" appears on a line of a proof, it can be replaced with "";
or as the statement of a truth-functional tautology or theorem of propositional logic:
and
where and 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:
is equivalent to
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
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