Destructive dilemma
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:
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 is a metalogical symbol meaning that is a syntactic consequence of,, and in some logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
where,, and are propositions expressed in some formal system.Proof
Example proof
The validity of this argument structure can be shown by using both conditional proof and reductio ad absurdum in the following way:
