# Constructive dilemma

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

**Constructive dilemma**^{[1]}^{[2]}^{[3]} is a 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 *P* or *R* is true, then *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 rule can be stated:

where the rule is that whenever instances of "", "", and "" appear on lines of a proof, "" can be placed on a subsequent line.

## Formal notation

The *constructive 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.

## Variable English

If P then Q. If R then S. P or R. Therefore, Q or S.

## Natural language example

- If I win a million dollars, I will donate it to an orphanage.
- If my friend wins a million dollars, he will donate it to a wildlife fund.
- I win a million dollars or my friend wins a million dollars.
- Therefore, either an orphanage will get a million dollars, or a wildlife fund will get a million dollars.

The dilemma derives its name because of the transfer of disjunctive operator.