Destructive dilemma

Destructive dilemma

Type	Rule of inference
Field	Propositional calculus
Statement	If ${\displaystyle P}$ implies ${\displaystyle Q}$ and ${\displaystyle R}$ implies ${\displaystyle S}$ and either ${\displaystyle Q}$ is false or ${\displaystyle S}$ is false, then either ${\displaystyle P}$ or ${\displaystyle R}$ must be false.
Symbolic statement	${\displaystyle {\frac {P\to Q,R\to S,\neg Q\lor \neg S}{\therefore \neg P\lor \neg R}}}$

Destructive dilemma^[1][2] 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:

${\displaystyle {\frac {P\to Q,R\to S,\neg Q\lor \neg S}{\therefore \neg P\lor \neg R}}}$

where the rule is that wherever instances of "${\displaystyle P\to Q}$", "${\displaystyle R\to S}$", and "${\displaystyle \neg Q\lor \neg S}$" appear on lines of a proof, "${\displaystyle \neg P\lor \neg R}$" can be placed on a subsequent line.