Disjunction elimination - Revision history

101.167.18.166 at 12:44, 11 November 2014

2014-11-11T12:44:40Z

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en>Alejandrocaro35 at 13:06, 24 February 2014

2014-02-24T13:06:46Z

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	~~:''For~~ the ~~[[theorem]]~~ of ~~[[propositional calculus\|propositional logic]] which expresses Disjunction elimination~~, ~~see [[Case analysis]]'~~'.

	~~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~~.

	~~:If I'm inside, I have my wallet on me~~.
	~~: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~~.

	~~It is the rule can~~ be ~~stated as:~~

	:<~~math~~>~~\frac{P \to Q, R \to Q, P \or R}{\therefore Q}~~<~~/math~~>

	~~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.~~

	~~== Formal notation ==~~
	~~The ''disjunction elimination'' rule~~ may ~~be written in [[sequent]] notation~~:

	~~: <math>~~(~~P \to Q~~)~~, (R \to Q), (P \or R) \vdash Q</math>~~

	~~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;~~

	and ~~expressed as a truth~~-~~functional [[tautology (logic)\|tautology]] or theorem~~ of ~~propositional logic:~~

	:~~<math>(((P \to Q) \and (R \to Q)) \and (P \or R)) \to Q<~~/~~math>~~

	~~where <math>P<~~/~~math>, <math>Q</math>, and <math>R</math> are propositions expressed in some [[formal system]]~~.

	~~==See also==~~
	* [[Disjunction]]
	* [[Argument in the alternative]]
	* [[Disjunct normal form]]

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Disjunction Elimination}}~~
	~~[[Category:Rules of inference]]~~

en>Graham87: 1 revision from :nost:Disjunction elimination: import old edit from the Nostalgia Wikipedia, see User:Graham87/Import

2013-07-06T11:08:03Z

1 revision from nost:Disjunction elimination: import old edit from the Nostalgia Wikipedia, see User:Graham87/Import

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213.22.73.54 at 23:20, 12 March 2012

2012-03-12T23:20:39Z

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