Affirming a disjunct

The textbook case

“Either the printer is out of paper or it’s jammed. It’s out of paper. So it isn’t jammed.”

A printer can be out of paper and jammed. The “or” said at least one of the two problems was there, and you’ve found one. That leaves the other exactly as open as it was before you looked.

The first fault found

A developer’s app crashes on launch. “It’s either the config file or the new login code. There’s a typo in the config file, so the login code is fine.” After the typo is fixed, the app still crashes.

Troubleshooting often narrows a problem to a short list, and it’s natural to stop at the first fault that turns up. But two things can be broken at once. The typo confirms one disjunct; it doesn’t clear the other. This version is common because finding a cause feels like finding the cause.

An “or” in a rule

A job posting asks for “a bachelor’s degree or five years of experience.” A recruiter skims a résumé and says, “She has the degree, so she doesn’t have the five years.”

An “or” in a requirement lists ways to qualify, and an applicant can meet both. Nobody would read the posting as “a degree or experience, but not both”, yet the recruiter’s conclusion only follows on that reading. The fallacy is harder to spot here because no one states it as an argument; the “so” slips in while reading.

“P or Q” rules out exactly one situation: P and Q both false. Disjunctive syllogism uses that: if Q is false, P can’t be false too, so P is true. Affirming a disjunct gets no help from it, because “P or Q” doesn’t rule out the situation where both are true.

Inclusive and exclusive “or”. Logic textbooks distinguish two senses of “or”:

• Inclusive: “P or Q, or both”. This is how standard logic reads a plain “or”.
• Exclusive: “P or Q, but not both”, as in “the flight lands at 6 or at 7”.

With an exclusive “or”, the argument is valid: P or Q, but not both. P. So not Q. Traditional logic called that valid version modus ponendo tollens (“the way that denies by affirming”). The fallacy is running the same step on an inclusive “or”.

David Kelley’s textbook explains why logic treats a plain “or” as inclusive. When an argument that affirms a disjunct seems valid, it’s because of some fact about the subject, such as that one person can’t be in two rooms at once, and not because of the argument’s form. Logicians call a step like that a material rather than a formal consequence: it depends on what the statements are about, not only on their structure. Stating the fact as a premise (“not both”) makes the argument valid by its form.

Denying a conjunct is the same mistake in other words. “Not both P and Q” means “not P or not Q”, and denying a conjunct affirms one of those disjuncts.

• The “or” is exclusive by its own terms. “One or the other, but not both”, “exactly one of”. With that stated, confirming one option does rule out the other.
• The options can’t both be true, and that’s plainly a fact. One flight leaves from one gate; a single die roll shows one number. The conclusion then follows from the argument plus that fact. It’s clearer to say the fact out loud, because a listener can’t always tell which “or” you meant.
• The options are contradictories. “The door is locked or it isn’t.” Exactly one is true, so confirming one denies the other.

The test: could both be true? If so, confirming one doesn’t rule out the other.

One flight, one gate

“The board has switched our flight between gate 12 and gate 14 all morning, so it’s leaving from one of those. The agent just announced gate 14. So it’s not leaving from gate 12.”

This affirms a disjunct, but a flight boards from one gate. The unstated premise “not both” is true, and with it the conclusion follows. That’s the second condition under “When it isn’t an error”: the options can’t both be true, as a fact about flights rather than about the form.

Denying instead of affirming

“My keys are either in my coat or in my bag. I’ve emptied the coat pockets and they aren’t there. So they’re in my bag.”

Same kind of “or” premise, and easy to lump in with the fallacy. But this denies a disjunct and concludes the other, which is Disjunctive syllogism, a valid form. Whether the keys could be somewhere else entirely is a question about whether the first premise is true, not about the form; see False dilemma.

In conversation, “or” often suggests “not both”. “You can take the bus or walk” sounds like a choice of one. The philosopher Paul Grice argued that this “not both” isn’t part of what “or” means but something listeners infer: if the speaker knew both were true, they would have said “and”. The inference is easy to cancel (“she invited John or Bill, or both”), which is part of Grice’s case that it isn’t built into the word. Carried into an argument where both options really can hold, that everyday inference becomes this fallacy.

It also travels with a False dilemma about causes. When two explanations are framed as rivals, evidence for one feels like evidence against the other, even when both could be at work.

• Ask whether both could be true, and describe the case where they are.
• Ask what rules out “both”. If it’s a real fact about the situation, the conclusion may be fine; if it’s only how the sentence sounded, it isn’t.
• Check the other option directly rather than inferring it from the first.
• Don’t swing the other way. Spotting the fallacy doesn’t show the other option is true. It may well be false; the argument just hasn’t shown it.