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{{Transformation rules}} | |||
'''Disjunction introduction''' or '''addition'''<ref>Hurley</ref><ref>Moore and Parker</ref><ref>Copi and Cohen</ref> is a simple [[validity|valid]] [[argument form]], an [[immediate inference]] and a [[rule of inference]] of [[propositional calculus|propositional logic]]. The rule makes it possible to introduce [[logical disjunction|disjunctions]] to [[formal proof|logical proofs]]. It is the [[inference]] that if ''P'' is true, then ''P or Q'' must be true. | |||
:Socrates is a man. | |||
:Therefore, either Socrates is a man or pigs are flying in formation over the English Channel. | |||
The rule can be expressed as: | |||
:<math>\frac{P}{\therefore P \or Q}</math> | |||
where the rule is that whenever instances of "<math>P</math>" appear on lines of a proof, "<math>P \or Q</math>" can be placed on a subsequent line. | |||
Disjunction introduction is controversial in [[paraconsistent logic]] because in combination with other rules of logic, it leads to [[Principle of explosion|explosion]] (i.e. everything becomes provable). See [[Paraconsistency#Tradeoff|Tradeoffs in Paraconsistent logic]]. | |||
== Formal notation == | |||
The ''disjunction introduction'' rule may be written in [[sequent]] notation: | |||
: <math>P \vdash (P \or Q)</math> | |||
where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>P \or Q</math> is a [[logical consequence|syntactic consequence]] of <math>P</math> in some [[formal system|logical system]]; | |||
and expressed as a truth-functional [[tautology (logic)|tautology]] or [[theorem]] of propositional logic: | |||
:<math>P \to (P \or Q)</math> | |||
where <math>P</math> and <math>Q</math> are propositions expressed in some formal system. | |||
== References == | |||
{{reflist}} | |||
{{DEFAULTSORT:Disjunction Introduction}} | |||
[[Category:Rules of inference]] | |||
[[Category:Paraconsistent logic]] | |||
[[Category:Theorems in propositional logic]] | |||
Revision as of 12:06, 6 July 2013
Disjunction introduction or addition[1][2][3] is a simple valid argument form, an immediate inference and a rule of inference of propositional logic. The rule makes it possible to introduce disjunctions to logical proofs. It is the inference that if P is true, then P or Q must be true.
- Socrates is a man.
- Therefore, either Socrates is a man or pigs are flying in formation over the English Channel.
The rule can be expressed as:
where the rule is that whenever instances of "" appear on lines of a proof, "" can be placed on a subsequent line.
Disjunction introduction is controversial in paraconsistent logic because in combination with other rules of logic, it leads to explosion (i.e. everything becomes provable). See Tradeoffs in Paraconsistent logic.
Formal notation
The disjunction introduction rule may be written in sequent notation:
where is a metalogical symbol meaning that is a syntactic consequence of in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
where and are propositions expressed in some formal system.
References
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