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| {{Transformation rules}}
| | The name of the writer is Luther. Interviewing is what she does but soon she'll be on her personal. Alabama has always been his house. The preferred hobby for my kids and me is taking part in crochet and now I'm trying to earn cash with it.<br><br>Also visit my weblog :: [http://www.evilsharks.de/index.php?mod=users&action=view&id=7634 car warranty] |
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| '''Destructive 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> is the 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 ''Q'' is false or ''S'' is false, then either ''P'' or ''R'' must be false. In sum, if two [[material conditional|conditionals]] are true, but one of their [[consequent]]s is false, then one of their [[Antecedent (logic)|antecedent]]s has to be false. ''Destructive dilemma'' is the [[Logical disjunction|disjunctive]] version of ''[[modus tollens]]''. The disjunctive version of ''[[modus ponens]]'' is the [[constructive dilemma]]. The rule can be stated:
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| :<math>\frac{P \to Q, R \to S, \neg Q \or \neg S}{\therefore \neg P \or \neg R}</math>
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| where the rule is that wherever instances of "<math>P \to Q</math>", "<math>R \to S</math>", and "<math>\neg Q \or \neg S</math>" appear on lines of a proof, "<math>\neg P \or \neg R</math>" can be placed on a subsequent line.
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| ==Formal notation==
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| The ''destructive dilemma'' rule may be written in [[sequent]] notation:
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| : <math>(P \to Q), (R \to S), (\neg Q \or \neg S) \vdash (\neg P \or \neg R)</math>
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| where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>\neg P \or \neg R</math> is a [[logical consequence|syntactic consequence]] of <math>P \to Q</math>, <math>R \to S</math>, and <math>\neg Q \or \neg 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 (\neg Q \or \neg S)) \to (\neg P \or \neg R)</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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| ==Natural language example==
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| :If it rains, we will stay inside.
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| :If it is sunny, we will go for a walk.
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| :Either we will not stay inside, or we will not go for a walk.
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| :Therefore, either it will not rain, or it will not be sunny.
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| ==Proof==
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| {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
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| |+ ''' '''
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| |- style="background:paleturquoise"
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| ! style="width:15%" | ''Proposition''
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| ! style="width:15%" | ''Derivation''
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| |-
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| | <math>(A\rightarrow B)\and (C\rightarrow D)</math> || Given
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| | <math>\neg B\or\neg D</math> || Given
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| | <math>B\rightarrow\neg D</math> || [[Material implication (rule of inference)|Material implication]]
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| |-
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| | <math>\neg D\rightarrow\neg C</math> || [[Transposition (logic)|Transposition]]
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| | <math>B\rightarrow\neg C</math> || [[Hypothetical syllogism]]
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| |-
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| | <math>A\rightarrow B</math> || [[Simplification]]
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| |-
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| | <math>A\rightarrow\neg C</math> || Hypothetical syllogism
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| |-
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| | <math>\neg A\or\neg C</math> || Material implication
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| |}
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| |}
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| ==Example proof==
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| The validity of this argument structure can be shown by using both [[conditional proof]] (CP) and [[reductio ad absurdum]] (RAA) in the following way:
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| {|
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| |align=right| 1. || <math> ((P \rightarrow Q) \And (R \rightarrow S)) \And (\neg Q \vee \neg S) </math>||(CP assumption)
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| |align=right| 2. || <math> (P \rightarrow Q) \And (R \rightarrow S) </math>||(1: Simplification)
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| |-
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| |align=right| 3. || <math> (P \rightarrow Q) </math>||(2: simplification)
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| |-
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| |align=right| 4. || <math> (R \rightarrow S) </math>||(2: simplification)
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| |-
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| |align=right| 5. || <math> (\neg Q \vee \neg S) </math>||(1: simplification)
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| |-
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| |align=right| 6. || <math> \neg (\neg P \vee \neg R) </math>||(RAA assumption)
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| |align=right| 7. || <math> \neg \neg P \And \neg \neg R </math>||(6: [[DeMorgan's Law]])
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| |align=right| 8. || <math> \neg \neg P </math>||(7: simplification)
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| |-
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| |align=right| 9. || <math> \neg \neg R </math>||(7: simplification)
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| |-
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| |align=right| 10. || <math> P </math>||(8: [[double negation]])
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| |-
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| |align=right| 11. || <math> R </math>||(9: double negation)
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| |align=right| 12. || <math> Q </math>||(3,10: modus ponens)
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| |align=right| 13. || <math> S </math>||(4,11: modus ponens)
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| |-
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| |align=right| 14. || <math> \neg \neg Q </math>||(12: double negation)
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| |align=right| 15. || <math> \neg S </math>||(5, 14: [[disjunctive syllogism]])
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| |align=right| 16. || <math> S \And \neg S </math>||(13,15: [[Logical conjunction|conjunction]])
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| |-
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| |align=right| 17. || <math> \neg P \vee \neg R </math>||(6-16: RAA)
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| |-
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| |align=right|
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| |align=right| 18. || <math> (((P \rightarrow Q) \And (R \rightarrow S)) \And (\neg Q \vee \neg S))) \rightarrow \neg P \vee \neg R </math>||(1-17: CP)
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| |}
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| ==References==
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| {{reflist}}
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| * Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan. The Power of Logic (4th ed.). McGraw-Hill, 2009, ISBN 978-0-07-340737-1, p. 414.
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| ==External links==
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| *http://mathworld.wolfram.com/DestructiveDilemma.html
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| {{DEFAULTSORT:Destructive 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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The name of the writer is Luther. Interviewing is what she does but soon she'll be on her personal. Alabama has always been his house. The preferred hobby for my kids and me is taking part in crochet and now I'm trying to earn cash with it.
Also visit my weblog :: car warranty