Editing Destructive dilemma (section)

{{Short description|Rule of inference of propositional logic}}
{{Infobox mathematical statement
| name = Destructive dilemma
| type = [[Rule of inference]]
| field = [[Propositional calculus]]
| statement = If <math>P</math> implies <math>Q</math> and <math>R</math> implies <math>S</math> and either <math>Q</math> is false or <math>S</math> is false, then either <math>P</math> or <math>R</math> must be false.
| symbolic statement = <math>\frac{P \to Q, R \to S, \neg Q \lor \neg S}{\therefore \neg P \lor \neg R}</math>
}}
{{Transformation rules}}

'''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 (logic)|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 destructive dilemma rule can be stated:

:<math>\frac{P \to Q, R \to S, \neg Q \lor \neg S}{\therefore \neg P \lor \neg R}</math>

where the rule is that wherever instances of "<math>P \to Q</math>", "<math>R \to S</math>", and "<math>\neg Q \lor \neg S</math>" appear on lines of a proof, "<math>\neg P \lor \neg R</math>" can be placed on a subsequent line.