# Destructive dilemma

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

**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 rule can be stated:

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

## Formal notation

The *destructive 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.

## 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.

## Proof

Proposition |
Derivation |
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Given | |

Given | |

Material implication | |

Transposition | |

Hypothetical syllogism | |

Simplification | |

Hypothetical syllogism | |

Material implication |

## 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:

1. | (CP assumption) | |

2. | (1: Simplification) | |

3. | (2: simplification) | |

4. | (2: simplification) | |

5. | (1: simplification) | |

6. | (RAA assumption) | |

7. | (6: DeMorgan's Law) | |

8. | (7: simplification) | |

9. | (7: simplification) | |

10. | (8: double negation) | |

11. | (9: double negation) | |

12. | (3,10: modus ponens) | |

13. | (4,11: modus ponens) | |

14. | (12: double negation) | |

15. | (5, 14: disjunctive syllogism) | |

16. | (13,15: conjunction) | |

17. | (6-16: RAA) | |

18. | (1-17: CP) |

## References

## Bibliography

- Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan. The Power of Logic (4th ed.). McGraw-Hill, 2009, ISBN 978-0-07-340737-1, p. 414.