A conjunction joins statements with "and". Its denial, "not both P and Q", says the two cannot be true together. It powers a syllogism much like the disjunctive syllogism, but in the opposite direction.
Key idea
Conjunctive syllogism: from "Not both P and Q" and "P", conclude "not Q". But from "Not both P and Q" and "not P", nothing follows: "not both" allows neither.
Why it works
"Not both" rules out exactly one case: both true. If P is true, Q would make that case happen, so Q must be false. But if P is false, the forbidden case is already avoided, and Q is free to be true or false. "Ben is not both at the game and at home" fits Ben being at the library, where he is at neither.
| Premises | Conclusion | Verdict |
|---|---|---|
| Not both P and Q; P | not Q | valid |
| Not both P and Q; Q | not P | valid |
| Not both P and Q; not P | Q | invalid |
| Not both P and Q; not Q | P | invalid |
Worked examples
Worked example: Affirming one part
"It is not the case that both the traffic light is red and the traffic light is green. The light is red." What follows?
- "Not both" plus one part true.
- The other part must be false.
Answer: The light is not green.
Worked example: Denying one part
"Kim is not both the team captain and the team treasurer. Kim is not the captain." What follows about the treasurer?
- Denying one part avoids the forbidden case already.
- Kim might be the treasurer, or might hold neither job.
Answer: nothing follows
Common mistake
Denying one conjunct and concluding the other is true. "Not both" allows neither to be true. Check: is there a situation where both parts are false? If so, the conclusion can fail.
The Stoic logicians of ancient Greece listed this form, together with modus ponens, modus tollens and two disjunctive forms, among their five basic arguments.