constructive dilemma

{{Short description|Rule of inference of propositional logic}}

{{Infobox mathematical statement

| name = Constructive dilemma

| type = Rule of inference

| field = Propositional calculus

| statement = If $P$ implies $Q$ and $R$ implies $S$, and either $P$ or $R$ is true, then either $Q$ or $S$ has to be true.

| symbolic statement = $\backslash frac\{(P\; \backslash to\; Q),\; (R\; \backslash to\; S),\; P\; \backslash lor\; R\}\{\backslash therefore\; Q\; \backslash lor\; S\}$

}}

{{Transformation rules}}

Constructive dilemmaHurley, Patrick. A Concise Introduction to Logic With Ilrn Printed Access Card. Wadsworth Pub Co, 2008. Page 361Moore and ParkerCopi and Cohen 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:

:$\backslash frac\{(P\; \backslash to\; Q),\; (R\; \backslash to\; S),\; P\; \backslash lor\; R\}\{\backslash therefore\; Q\; \backslash lor\; S\}$

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