// laws and rules · Rule of inference
Modus tollens_
Modus tollens (“the mode that denies”) is the rule of inference that, from an implication p ⇒ q and the negation of its consequent ¬q, concludes the negation of the antecedent ¬p. It is the reasoning behind every refutation: if a theory predicts something that does not happen, the theory is false. The formula ((p ⇒ q) ∧ ¬q) ⇒ ¬p is a tautology.
Example
((p ⇒ q) ∧ ¬q) ⇒ ¬p
What the variables mean
- ▸ p: “The engine gets fuel”
- ▸ q: “The engine starts”
In plain words
If the engine gets fuel, it starts. The engine does not start. Therefore, the engine is not getting fuel.
Truth table
| p | q | p ⇒ q | ¬q | (p ⇒ q) ∧ ¬q | ¬p | ((p ⇒ q) ∧ ¬q) ⇒ ¬p★ |
|---|---|---|---|---|---|---|
| T | T | T | F | F | F | T |
| T | F | F | T | F | F | T |
| F | T | T | F | F | T | T |
| F | F | T | T | T | T | T |
Classification: Tautology · 4 rows
Statement of the rule
Premise 1: p ⇒ q. Premise 2: ¬q. Conclusion: ¬p. As a single formula: ((p ⇒ q) ∧ ¬q) ⇒ ¬p.
The rule rests on a simple idea: a true implication cannot have a true antecedent and a false consequent. If the consequent is false, the antecedent must be false too.
Why it is valid: reading the table
We look for a row with true premises and a false conclusion. For ¬p to be F we need p = T; for ¬q to be T we need q = F. But with p = T and q = F the implication p ⇒ q is F, so the conjunction of premises is F. The problematic row does not exist.
In the row p = F, q = F both premises are true (p ⇒ q is T because its antecedent is false, and ¬q is T) and the conclusion ¬p is T as well. It is the only row where the argument “fires”, and there it works.
How it is used in proofs
It underlies proof by refutation and the scientific method: derive an observable consequence q from a hypothesis p, check that q is false, conclude that p is false.
In natural deduction it is usually derived from modus ponens plus the contrapositive: from p ⇒ q obtain ¬q ⇒ ¬p, then apply MP with ¬q to reach ¬p.
Examples
Everyday: “If there were an exam today, the classroom would be full. The classroom is empty. So there is no exam today.”
Debugging: “If the request reached the server, it appears in the log. It does not appear in the log. Therefore the request never reached the server.” Programmers apply modus tollens every time they rule out a cause.
Mathematics: “If a number is divisible by 4, it is even. 15 is not even. Therefore 15 is not divisible by 4.”
Relation to other laws
It is the same as applying modus ponens to the contrapositive of the implication, which is why the two rules are interchangeable given the law (p ⇒ q) ⇔ (¬q ⇒ ¬p).
Its twin fallacy is denying the antecedent, ((p ⇒ q) ∧ ¬p) ⇒ ¬q: denying p does not let you deny q, because q may be true for other reasons.
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related operators
Frequently asked questions
What does “modus tollens” mean? ▼
From the Latin modus tollendo tollens, “the mode that by denying denies”: it denies the consequent of an implication in order to deny its antecedent.
Is it the same as the contrapositive? ▼
Not quite. The contrapositive is an equivalence between two implications; modus tollens is a rule of inference that is justified by that equivalence.
Why is the row p = F, q = F the important one? ▼
It is the only row where both premises are true. There the conclusion ¬p is true as well, which shows the argument never leads from true premises to a false conclusion.