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 rule can be stated:
Contents
where the rule is that wherever instances of "
Formal notation
The destructive dilemma rule may be written in sequent notation:
where
and expressed as a truth-functional tautology or theorem of propositional logic:
where
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.Example proof
The validity of this argument structure can be shown by using both conditional proof (CP) and reductio ad absurdum (RAA) in the following way: