Constructive dilemma

Constructive dilemma

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

Constructive dilemma^[1][2][3] is a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either P or R is true, then either Q or S has to be true. In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too. Constructive dilemma is the disjunctive version of modus ponens, whereas, destructive dilemma is the disjunctive version of modus tollens. The constructive dilemma rule can be stated:

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

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