Modus Tollendo Tollens

Theorem
The modus tollendo tollens is a valid deduction sequent in propositional logic:


 * $p \implies q, \neg q \vdash \neg p$

That is:
 * If the truth of one statement implies the truth of a second, and the second is shown not to be true, then neither can the first be true.

It can be written:
 * $\displaystyle {p \implies q \quad \neg q \over \neg p} \text{MTT}$

Its abbreviation in a tableau proof is $\mathrm{MTT}$.

Also known as
Modus tollendo tollens is also known as:


 * Modus tollens
 * Denying the consequent.

Also see
The following are related argument forms:
 * Modus Ponendo Ponens
 * Modus Ponendo Tollens
 * Modus Tollendo Ponens

The Rule of Transposition is conceptually similar, and can be derived from the MTT by a simple application of Conditional Proof.

These are classic fallacies:


 * Affirming the Consequent
 * Denying the Antecedent

Linguistic Note
Modus tollendo tollens is Latin for mode that by denying, denies.

Modus tollens means mode that denies.