A dilemma combines two conditionals with a disjunction. It is the argument of "either way, you lose" (or "either way, you win").
| Constructive dilemma | Destructive dilemma |
|---|---|
| If P then Q, and if R then S. | If P then Q, and if R then S. |
| P or R. | Not Q or not S. |
| So Q or S. | So not P or not R. |
Key idea
A constructive dilemma affirms the antecedents (P or R) and concludes the consequents (Q or S): modus ponens on each horn. A destructive dilemma denies the consequents and concludes the denial of the antecedents: modus tollens on each horn. Both are valid.
Why it works
Take the constructive dilemma. At least one of P and R is true. If it is P, the first conditional gives Q; if it is R, the second gives S. Either way, Q or S. The destructive dilemma is the same reasoning with modus tollens. The two options in the "or" premise are called the horns of the dilemma.
Worked examples
Worked example: Constructive
"If Jo studies tonight, Jo misses the concert; if Jo goes to the concert, Jo does poorly on the quiz. Jo studies tonight or goes to the concert." What follows?
- The "or" premise affirms the two antecedents.
- Constructive dilemma: one consequent or the other.
Answer: Jo misses the concert or does poorly on the quiz.
Worked example: A broken dilemma
"If the family drives, the trip takes two days; if they fly, the trip costs more. The trip takes two days or costs more. So the family drives or flies." Valid?
- The "or" premise affirms the consequents, and the conclusion affirms the antecedents.
- That is affirming the consequent on each horn: the trip could take two days because of a storm while they take the train.
Answer: invalid
Common mistake
Mixing the two patterns: affirming consequents in a destructive dilemma or denying antecedents in a constructive one, which makes it invalid. Check: a constructive dilemma moves forward along the arrows (antecedents to consequents); a destructive one moves backward with "not" (denied consequents to denied antecedents).