- Destructive dilemma
Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B)
Modus tollens (A→B, ¬B ⊢ ¬A)
Modus ponendo tollens (¬(A∧B), A ⊢ ¬B)
Conjunction introduction (A, B ⊢ A∧B)
Simplification (A∧B ⊢ A)
Disjunction introduction (A ⊢ A∨B)
Disjunction elimination (A∨B, A→C, B→C ⊢ C)
Disjunctive syllogism (A∨B, ¬A ⊢ B)
Hypothetical syllogism (A→B, B→C ⊢ A→C)
Constructive dilemma (A→P, B→Q, A∨B ⊢ P∨Q)
Destructive dilemma (A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B)
Biconditional introduction (A→B, B→A ⊢ A↔B)
Biconditional elimination (A↔B ⊢ A→B)
Predicate calculus Universal generalization
In logic, a destructive dilemma is any logical argument of the following form:
where represents the logical assertion.
The argument can be read in this way:
- If P, then Q
- If R, then S
- Not Q or not S
- Therefore, not P or not R
And to once again restate the argument, one can turn this argument into a conditional, where if the first three premises, then not P or R:
- 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.
- Therefore, either it will not rain, or it will not be sunny.
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. P (8: double negation) 11. R (9: double negation) 12. Q (3,10: modus ponens) 13. S (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)
- Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan. The Power of Logic (4th ed.). McGraw-Hill, 2009, p. 414.
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