Affirming the consequent

The textbook case

If it rained last night, the street is wet. The street is wet. So it rained last night.

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. The wet street fits rain; it doesn’t prove it.

A diagnosis stated as certain

“If you have the flu, you’ll run a fever. You’re running a fever. So you have the flu.”

Fever has many causes: infections of every kind, heat exhaustion, some medications. The fever makes flu more likely than it was, but the argument claims it settles the question. The error is the “so”, which presents a possibility as a conclusion.

Hidden inside “all”

“All successful startups use agile development. We use agile development. So we’ll be successful.”

“All successful startups use agile” means if a startup is successful, it uses agile. The speaker has the consequent (we use agile) and concludes the antecedent (we’ll be successful). Unsuccessful startups use agile too, so sharing the trait can’t show which group you’re in. The fallacy is harder to see when the conditional is phrased as a generalization.

In “if P, then Q”, P is the antecedent (the “if” part) and Q is the consequent (the “then” part). Modus ponens affirms the antecedent and moves forward to the consequent. Affirming the consequent affirms the consequent and tries to move back.

Its sibling fallacy, Denying the antecedent (“If P, then Q. Not P. So not Q.”), fails for the same reason: Q might happen without P.

• As evidence rather than proof. If P makes Q more likely, then seeing Q can reasonably raise your confidence in P. Reasoning from a result to its likely cause is legitimate when the conclusion is stated as probable, and the other ways Q could have come about have been considered and ruled out or weighed.
• When the conditional actually runs the other way, or both ways. “Only if P, Q” and “Q if and only if P” both let you get from Q to P. The error is treating a one-way “if” as if it were one of these.

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

“Only if” instead of “if”

“The alarm sounds only if someone opens the door. The alarm is sounding. So someone opened the door.”

This looks backwards, but “only if” reverses the direction. “The alarm sounds only if the door is opened” means if the alarm sounds, the door was opened. With that premise, concluding that the door was opened is modus ponens, which is valid. (Whether the premise is true, and whether the alarm can misfire, is a separate question about soundness, not form.)

A controlled test, with a hedged conclusion

“If the new fertilizer works, the treated plots will yield more than the untreated ones. Across forty randomly assigned plots, the treated ones yielded far more than chance would explain. So the fertilizer very likely works.”

The shape resembles the fallacy, but two things change it. The conclusion claims likely, not certain. And the random assignment and comparison with untreated plots were designed to rule out the other ways Q could happen (soil, weather, luck). That’s evidence rather than proof, handled properly.

In everyday speech, “if” often does suggest “only if”. “If you mow the lawn, I’ll give you five dollars” is normally understood to mean you won’t get the money otherwise. Linguists call this tendency to hear a one-way conditional as two-way conditional perfection. Carried into an argument where it doesn’t hold, it produces this fallacy.

It also feels natural because results are usually some evidence for their causes. The step from “this fits” to “this proves” is small, and easy to take without noticing.