Each valid form from the last two lessons has an invalid twin that looks almost the same.
| Form | Pattern | Valid? |
|---|---|---|
| Modus ponens | If P then Q; P; so Q | valid |
| Modus tollens | If P then Q; not Q; so not P | valid |
| Affirming the consequent | If P then Q; Q; so P | invalid |
| Denying the antecedent | If P then Q; not P; so not Q | invalid |
Key idea
Affirming the consequent (Q, so P) and denying the antecedent (not P, so not Q) are invalid: the consequent can come about some other way.
Why it works
"If P, then Q" says P is one way to get Q. It does not say it is the only way. So learning that Q happened does not tell you P happened (affirming the consequent), and learning that P did not happen does not tell you Q failed (denying the antecedent). The truth table shows a counterexample row for each: P false and Q true makes both premises true and the conclusion false.
Worked examples
Worked example: Affirming the consequent
"If the dog hears the doorbell, it barks. The dog is barking. So it heard the doorbell." Valid?
- The second premise affirms the consequent; the conclusion affirms the antecedent.
- Counterexample: a squirrel ran past the window. The dog barks, though no doorbell rang.
Answer: invalid: affirming the consequent
Worked example: Denying the antecedent
"If the number is divisible by 4, it is even. This number is not divisible by 4. So it is not even." Valid?
- The second premise denies the antecedent.
- Counterexample: 6 is not divisible by 4, but it is even.
Answer: invalid: denying the antecedent
Common mistake
Affirming the consequent is the most common formal fallacy ("If it rained, the street is wet; the street is wet; so it rained"). Show it with a counterexample where Q is true for a different reason: a street cleaner went by.