Denying the antecedent

The textbook case

If it rained last night, the street is wet. It didn’t rain last night. So the street isn’t wet.

Both premises can be true and the conclusion still false. A street cleaner, a burst water main or a neighbor’s sprinkler would also leave the street wet. Rain is one way to get a wet street, and ruling it out doesn’t rule out the others.

A reassurance that doesn’t follow

“If an email is full of spelling mistakes, it’s a scam. This one is perfectly written, so it’s not a scam.”

Spelling mistakes are one sign of a scam, not the only one. A careful scammer can write cleanly. The premise says what typos tell you; it says nothing about what their absence tells you. This version is easy to miss because the conclusion is comforting and the first premise sounds like sensible advice.

Hidden inside “all”

“All members get free shipping. Dana isn’t a member. So Dana won’t get free shipping.”

“All members get free shipping” means if someone is a member, they get free shipping. The speaker denies the antecedent (Dana isn’t a member) and concludes the consequent is false. But the store might also ship free on large orders or during a sale. Phrased as a generalization about a group, the conditional is harder to spot, and so is the fallacy.

Both forms deny one half of the conditional. The whole difference is which half.

“If P, then Q” rules out exactly one situation: P true and Q false. Modus tollens works because of that: if Q is false, P can’t be true, since P with no Q is the one situation the conditional excludes. Denying the antecedent doesn’t work because the conditional excludes nothing when P is false. Not-P with Q and not-P without Q are both still open.

“If not P, then not Q” is called the inverse of “if P, then Q” (hence the alias “inverse error”), and it doesn’t follow from it. What does follow is the contrapositive, “if not Q, then not P”, which is what modus tollens uses.

Its sibling fallacy, Affirming the consequent (“If P, then Q. Q. So P.”), fails for the same reason: Q might happen without P.

• When the conditional really means “if and only if”. “Q if and only if P” says P is the only way to get Q, so not P does rule out Q. So does “Q only if P”, which means if Q, then P; with that premise, going from not P to not Q is modus tollens. The error is treating a one-way “if” as one of these.
• As evidence rather than proof. If P is one of the main ways Q comes about, learning that P is false can reasonably make Q less likely. That’s legitimate when the conclusion is stated as less likely, not ruled out, and how much it should move you depends on how many other ways Q could happen.

The test: could Q be true without P? If so, not P alone can’t establish not Q.

Denying the consequent

“If the package had been delivered, the doorbell camera would have recorded it. The camera recorded nothing all day. So the package wasn’t delivered.”

This has a “not” in it and runs from one missing thing to another, which can look like denying the antecedent. But what’s denied is the consequent (the camera recording it), and the conclusion denies the antecedent. That’s modus tollens, which is valid. Whether the premise is true, say if the camera was offline, is a separate question about soundness, not form; see Valid vs. true.

“Only if”, or a promise that runs both ways

A parent says, “You get dessert only if you finish your dinner.” Sam leaves half his dinner. His sister says, “So you’re not getting dessert.”

“Dessert only if dinner” means if you get dessert, you finished your dinner. Sam didn’t finish, which denies the consequent, so “no dessert” follows by modus tollens. Even if the parent had said “If you finish your dinner, you’ll get dessert”, most families would take it to run both ways, and the sister would be drawing the conclusion the parent meant. That’s the first condition under “When it isn’t an error”: the conditional really is “if and only if”.

In everyday speech, “if” often carries an unstated “and if not, then not”. Linguists Michael Geis and Arnold Zwicky noted in 1971 that a statement of the form “if P, then Q” invites the inference “if not P, then not Q”. Their example: “If you mow the lawn, I’ll give you five dollars” is normally heard as a promise that you won’t get the money otherwise. That invited inference, called conditional perfection, is exactly the step denying the antecedent takes. It’s usually harmless in conversation, where the speaker meant it. Carried into an argument where the conditional is really one-way, it becomes this fallacy.

Content matters too. In one experiment, Denise Cummins and colleagues gave students arguments of this form built on cause-and-effect statements. Students were more willing to accept the conclusion when the effect had few other possible causes and less willing when it had many. The authors illustrate the idea with two arguments of the same form: “If my finger is cut, it bleeds. My finger isn’t cut. So it isn’t bleeding” (few other ways to bleed) is more tempting than “If I eat candy often, I have cavities. I don’t eat candy often. So I don’t have cavities” (many other causes of cavities). The fallacy is most tempting when no other way to get Q comes to mind, which is not the same as there being none.

Stripped of content, the form still gets accepted often. A meta-analysis of studies using abstract materials found that participants accepted denying the antecedent about half the time.