Denying a conjunct

The textbook case

“You can’t go to both the concert and the dinner on Saturday. You’re not going to the concert, so you’re going to the dinner.”

Both premises can be true while you stay home. “Not both” ruled out doing both; it never required doing either. Skipping the concert frees up the evening, and that’s all it does.

Two theories that can’t both be right

“Your explanation and mine can’t both be right: they predict opposite results. The data rule yours out. So mine is right.”

This one sounds rigorous, and the first premise may well be true. But two incompatible explanations can both be wrong. Showing that a rival fails doesn’t show that yours succeeds, because a third explanation is still possible. The argument has quietly turned “not both” into “exactly one”.

A rule that forbids a combination

A store’s policy says a coupon “can’t be combined with sale prices.” A shopper says, “This shirt isn’t on sale, so my coupon applies.”

The policy forbids using both; it doesn’t promise that the coupon works whenever there’s no sale. The coupon might have expired, or cover a different department. A rule that forbids a combination is easy to misread as a rule that grants each part on its own.

“Not both P and Q” rules out exactly one situation: P and Q both true. The valid form, which some logic textbooks (Pospesel’s, and Kiersky and Caste’s) call the conjunctive argument, uses that: if P is true, Q can’t be, since both true is the one situation excluded. It works the same way from Q: not both P and Q; Q; so not P. Denying a conjunct doesn’t work, because once P is false the “not both” statement has nothing left to say about Q.

Affirming a disjunct in disguise. “Not both P and Q” means the same as “not P or not Q”: at least one of them is false. Rewrite the fallacy with that premise and it becomes not P or not Q; not P; so Q, which confirms one option of an “or” (“not P”) and concludes the other option (“not Q”) is false. That’s Affirming a disjunct. Rewrite the valid conjunctive argument the same way and you get not P or not Q; P; so not Q, which denies one option and concludes the other: a Disjunctive syllogism.

Contraries, not contradictories. Two statements that can’t both be true but can both be false are called contraries; two that can’t both be true and can’t both be false are contradictories (see False dilemma). “Not both P and Q” says only that P and Q can’t both be true, so on its own it treats them as contraries. Denying a conjunct treats them as contradictories, where ruling one out settles the other. It’s the same slide that many false dilemmas make.

• When at least one of them must be true as well. If “P or Q” also holds, then exactly one of them is true, and ruling one out does establish the other. It helps to say that part out loud, since “not both” doesn’t carry it.
• When the conclusion denies rather than affirms. Not both P and Q; P; so not Q is the valid conjunctive argument.

The test: could both be false? If so, ruling one out doesn’t establish the other.

A switch with two positions

“The breaker switch can’t be both up and down. It isn’t up. So it’s down.”

This has the fallacy’s shape, but a two-position switch is always in one position or the other. The unstated premise “up or down” is true, which makes the pair contradictories, so the conclusion follows. That’s the first condition under “When it isn’t an error”.

Affirming one, denying the other

“Our museum pass can’t be used for both the morning tour and the evening tour. We used it for the morning tour. So it won’t get us into the evening tour.”

This starts from “not both” too, which makes it easy to mistake for the fallacy. But it affirms one conjunct (the pass was used for the morning tour) and concludes the other is ruled out. That’s the valid conjunctive argument, the second condition under “When it isn’t an error”.

“You can’t have both” is often said when a choice between the two is expected: offered cake or pie at a table where dessert is definitely coming, “not both” arrives with an unspoken “but one of them”, and in that setting the listener is right to fill it in. The fallacy is filling it in where nothing supplies it, when “neither” is a live possibility.

It’s also easy to slide from “these are incompatible” to “these are the options”. Once two theories, plans or explanations are set against each other, rejecting one can feel like choosing the other, the same pull that makes a False dilemma persuasive.

• Ask whether both could be false, and name the case where neither holds.
• Ask what shows at least one must be true. If something does, say it; the argument then works.
• Don’t treat the conclusion as refuted. Q may still be true. The argument just hasn’t shown it.