The counterexample method shows a form is invalid. Here we use it on the most common formal fallacies, which start from an "if… then…" premise.
Key idea
From "If P, then Q": you may infer Q from P (modus ponens), and not-P from not-Q (modus tollens). You may not infer P from Q (affirming the consequent) or not-Q from not-P (denying the antecedent).
Why it works
"If it rained, the street is wet" says rain is one way to get a wet street. So rain guarantees a wet street, and a dry street rules out rain. But a wet street does not tell you it rained (a sprinkler could have done it), and no rain does not tell you the street is dry, for the same reason.
| Form | Second premise | Conclusion | Verdict |
|---|---|---|---|
| Modus ponens | P (it rained) | Q (the street is wet) | Valid |
| Modus tollens | not Q (the street is dry) | not P (it did not rain) | Valid |
| Affirming the consequent | Q (the street is wet) | P (it rained) | Invalid |
| Denying the antecedent | not P (it did not rain) | not Q (the street is dry) | Invalid |
Worked example: Name the form
"If the dog is hungry, it barks at the door. It is barking at the door. So it is hungry."
- The second premise says the "then" part happened (it is barking).
- The conclusion is the "if" part (it is hungry).
- It might bark because it wants to go out.
Answer: Affirming the consequent: invalid.
Worked example: A valid one
"If Ben won the race, he got a medal. Ben did not get a medal. So Ben did not win."
- The second premise denies the "then" part.
- If Ben had won, he would have a medal; he doesn't, so he didn't win.
Answer: Modus tollens: valid.
Common mistake
Affirming the consequent feels natural: "If it rained, the street is wet. The street is wet, so it rained." Before accepting, ask: what else could make the "then" part true? A counterexample with the same form: "If it is a dog, it is a mammal. It is a mammal. So it is a dog" (it could be a cat).