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| {{Transformation rules}}
| | My name: Wilhelmina Hutchins<br>Age: 39<br>Country: Austria<br>City: Gries <br>Post code: 4871<br>Address: Leobnerstrasse 49<br><br>Here is my weblog :: [http://hemorrhoidtreatmentfix.com/hemorrhoid-relief hemorrhoid relief] |
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| :''For the [[theorem]] of [[propositional calculus|propositional logic]] which expresses Disjunction elimination, see [[Case analysis]]''. | |
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| In [[propositional logic]], '''disjunction elimination'''<ref>http://www.wordiq.com/definition/Disjunction_elimination</ref><ref>http://www.lawrence.edu/dept/philosophy/research/ryckmant/Disjunction%20Elimination.htm</ref><ref>http://www.cs.gsu.edu/~cscskp/Automata/proofs/node6.html</ref> (sometimes named '''proof by cases''' or '''case analysis'''), is the [[validity|valid]] [[argument form]] and [[rule of inference]] that allows one to eliminate a [[logical disjunction|disjunctive statement]] from a [[formal proof|logical proof]]. It is the [[inference]] that if a statement <math>P</math> implies a statement <math>Q</math> and a statement <math>R</math> also implies <math>Q</math>, then if either <math>P</math> or <math>R</math> is true, then <math>Q</math> has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true.
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| :If I'm inside, I have my wallet on me.
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| :If I'm outside, I have my wallet on me. | |
| :It is true that either I'm inside or I'm outside. | |
| :Therefore, I have my wallet on me.
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| It is the rule can be stated as:
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| :<math>\frac{P \to Q, R \to Q, P \or R}{\therefore Q}</math>
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| where the rule is that whenever instances of "<math>P \to Q</math>", and "<math>R \to Q</math>" and "<math>P \or R</math>" appear on lines of a proof, "<math>Q</math>" can be placed on a subsequent line.
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| == Formal notation ==
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| The ''disjunction elimination'' rule may be written in [[sequent]] notation:
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| : <math>(P \to Q), (R \to Q), (P \or R) \vdash Q</math>
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| where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>Q</math> is a [[logical consequence|syntactic consequence]] of <math>P \to Q</math>, and <math>R \to Q</math> and <math>P \or R</math> in some logical system;
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| and expressed as a truth-functional [[tautology (logic)|tautology]] or theorem of propositional logic:
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| :<math>(((P \to Q) \and (R \to Q)) \and (P \or R)) \to Q</math>
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| where <math>P</math>, <math>Q</math>, and <math>R</math> are propositions expressed in some [[formal system]].
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| ==See also==
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| * [[Disjunction]]
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| * [[Argument in the alternative]]
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| * [[Disjunct normal form]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Disjunction Elimination}}
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| [[Category:Rules of inference]]
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My name: Wilhelmina Hutchins
Age: 39
Country: Austria
City: Gries
Post code: 4871
Address: Leobnerstrasse 49
Here is my weblog :: hemorrhoid relief