Destructive dilemma is the name of a valid rule of inference of 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 conditionals are true, but one of their consequents is false, then one of their antecedents has to be false. Destructive dilemma is the disjunctive version of modus tollens. The disjunctive version of modus ponens is the constructive dilemma. The destructive dilemma rule can be stated:
|Rules of inference|
|Rules of replacement|
where the rule is that wherever instances of "", "", and "" appear on lines of a proof, "" can be placed on a subsequent line.
The destructive dilemma rule may be written in sequent notation:
where , , and are propositions expressed in some formal system.
Natural language example
- If it rains, we will stay inside.
- If it is sunny, we will go for a walk.
- Either we will not stay inside, or we will not go for a walk, or both.
- Therefore, either it will not rain, or it will not be sunny, or both.
|3||Material implication (2)|
|4||Conjunction elimination (1)|
|6||Hypothetical syllogism (3,5)|
|7||Conjunction elimination (1)|
|8||Hypothetical syllogism (6,7)|
|7.||(6: De Morgan's Law)|
|10.||(8: double negation)|
|11.||(9: double negation)|
|12.||(3,10: modus ponens)|
|13.||(4,11: modus ponens)|
|14.||(12: double negation)|
|15.||(5, 14: disjunctive syllogism)|
- Hurley, Patrick. A Concise Introduction to Logic With Ilrn Printed Access Card. Wadsworth Pub Co, 2008. Page 361
- Moore and Parker
- 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.