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| {{Transformation rules}}
| | Friends call her Felicidad and her husband doesn't like it at all. He currently life in Arizona and his mothers and fathers reside close by. What she enjoys doing is to play croquet but she hasn't made a dime with it. Bookkeeping is what he does.<br><br>My blog - [http://Www.sagittarius-Games.com/profile/mamarvin extended auto warranty] |
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| '''Constructive dilemma'''<ref>Hurley, Patrick. A Concise Introduction to Logic With Ilrn Printed Access Card. Wadsworth Pub Co, 2008. Page 361</ref><ref>Moore and Parker</ref><ref>Copi and Cohen</ref> is a name of a [[validity|valid]] [[rule of inference]] of [[propositional calculus|propositional logic]]. It is the [[inference]] that, if ''P'' implies ''Q'' and ''R'' implies ''S'' and either ''P'' or ''R'' is true, then ''Q or S'' has to be true. In sum, if two [[material conditional|conditionals]] are true and at least one of their antecedents is, then at least one of their consequents must be too. ''Constructive dilemma'' is the [[Logical disjunction|disjunctive]] version of [[modus ponens]], whereas, | |
| [[destructive dilemma]] is the disjunctive version of [[modus tollens]]. The rule can be stated:
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| :<math>\frac{P \to Q, R \to S, P \or R}{\therefore Q \or S}</math>
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| where the rule is that whenever instances of "<math>P \to Q</math>", "<math>R \to S</math>", and "<math>P \or R</math>" appear on lines of a proof, "<math>Q \or S</math>" can be placed on a subsequent line.
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| == Formal notation ==
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| The ''constructive dilemma'' rule may be written in [[sequent]] notation:
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| : <math>(P \to Q), (R \to S), (P \or R) \vdash (Q \or S)</math>
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| where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>Q \or S</math> is a [[logical consequence|syntactic consequence]] of <math>P \to Q</math>, <math>R \to S</math>, and <math>Q \or S</math> in some [[formal system|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 S)) \and (P \or R)) \to (Q \or S)</math>
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| where <math>P</math>, <math>Q</math>, <math>R</math> and <math>S</math> are propositions expressed in some formal system.
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| == Variable English ==
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| If P then Q.
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| If R then S.
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| P or R.
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| Therefore, Q or S.
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| == Natural language example ==
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| :If I win a million dollars, I will donate it to an orphanage.
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| :If my friend wins a million dollars, he will donate it to a wildlife fund.
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| :Either I win a million dollars, or my friend wins a million dollars.
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| :Therefore, either an orphanage will get a million dollars, or a wildlife fund will get a million dollars.
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| The dilemma derives its name because of the transfer of disjunctive operants.
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Constructive Dilemma}}
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| [[Category:Rules of inference]]
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| [[Category:Dilemmas]]
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| [[Category:Theorems in propositional logic]]
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Friends call her Felicidad and her husband doesn't like it at all. He currently life in Arizona and his mothers and fathers reside close by. What she enjoys doing is to play croquet but she hasn't made a dime with it. Bookkeeping is what he does.
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