The second valid hypothetical form runs modus ponens in reverse. Modus tollens ("the way that denies") is also called denying the consequent.
| Form | Example | |
|---|---|---|
| Premise | If P, then Q. | If the field is flooded, the game is postponed. |
| Premise | Not Q. | The game is not postponed. |
| Conclusion | So not P. | So the field is not flooded. |
Key idea
From "If P, then Q" and "not Q", conclude "not P". Modus tollens is valid.
Why it works
If P had been true, Q would have been true too; that is what the first premise says. But Q is false. So P cannot have been true. On the truth table (¬ means "not"), the only row where both "p → q" and "¬q" are true is the row with p false. Detectives and scientists use this form all the time: "If the suspect were in the room, there would be footprints. There are none. So the suspect was not in the room."
Worked examples
Worked example: Denying the consequent
"If the store is open, then the lights are on. The lights are not on." What follows?
- The second premise denies the consequent.
- By modus tollens, deny the antecedent.
Answer: The store is not open.
Worked example: Spotting the form
Is this valid? "If Priya wins the spelling bee, she goes to the state final. She does not go to the state final. So she did not win."
- Form: If P then Q; not Q; so not P.
- That is modus tollens.
Answer: valid
Common mistake
Finding modus tollens harder than modus ponens and wrongly rejecting it. If P would bring Q, and there is no Q, there was no P. Check with a truth table or a plain case: "If it is a square, it has four sides. It does not have four sides. So it is not a square." Clearly right.