Disjunctive syllogism

“The meeting is either in the library or in the lab. I just walked past the lab and it’s empty. So the meeting is in the library.”

P and Q are called the disjuncts (the options joined by “or”). Disjunctive syllogism denies one disjunct and concludes the other. It doesn’t matter which one is denied: P or Q; not P; so Q is the same form.

“P or Q” rules out exactly one situation: P and Q both false. If you know Q is false, the only way P could be false is that ruled-out situation. So P is true. Hurley and Watson’s textbook calls this the “method of elimination”: one premise offers alternatives, the other removes one, and what’s left is the conclusion.

This is the same kind of reasoning as Modus tollens, which denies the “then” part of “if P, then Q” and concludes the “if” part is false. Each premise pair leaves only one possibility standing. In standard logic, “P or Q” also says the same as “if not Q, then P”, so a disjunctive syllogism can be recast as a Modus ponens on that conditional.

A valid argument only guarantees its conclusion if its premises are true, and here the premise doing the work is “P or Q”.

“Either the café closed early today or we have the wrong address. The address is right. So it closed early.”

That’s a correct disjunctive syllogism, and the conclusion may still be false. The café might be closed all day for a holiday, or have moved last month. The “or” premise left options out, so it’s false, and the valid form can’t rescue it.

That is what a False dilemma is: the form is valid, but the either/or premise is false because it leaves out options. It’s why false dilemma counts as an informal fallacy, not a formal one: the fault is in what the premise claims, not in the step. For more on the difference, see Valid vs. true.

Here’s the same kind of premise with a disjunct affirmed instead:

“Either Sam or Priya can unlock the office. Sam can. So Priya can’t.”

This is invalid. They might both have keys. “P or Q” says at least one is true, and confirming one leaves the other open. That’s the fallacy of Affirming a disjunct.

It becomes valid only if the “or” means “one or the other, but not both” and that is stated or plainly true. Logic textbooks read a bare “or” as inclusive (“or both”), so the extra premise has to be supplied.

Disjunctive syllogism is valid in classical logic and in intuitionistic logic. Some relevance logics, developed by Anderson and Belnap in the 1960s and 1970s, reject it in its general form, because together with another simple rule it lets you derive any conclusion at all from a contradiction. For everyday reasoning with consistent premises this doesn’t change anything, but it’s a reminder that “valid” is always valid in a given logic.

The idea is old. A story in Sextus Empiricus has Chrysippus crediting a hunting dog with it: at a fork of three paths, the dog sniffs two, finds no scent, and runs down the third without sniffing.

Five forms that look similar. Three are valid, two are not.

The valid forms eliminate an option by denying it (or, with “not both”, by affirming its partner). The invalid ones assume that confirming or ruling out one option settles the other.