Modus Tollendo Tollens

Proof Rule
The modus tollendo tollens is a valid deduction sequent in propositional logic: If we can conclude $p \implies q$, and we can also conclude $\neg q$, then we may infer $\neg p$.

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:
 * $\displaystyle {p \implies q \quad \neg q \over \neg p} \text{MTT}$

Tableau Form
In a tableau proof, the modus tollendo tollens can be invoked in the following manner:


 * Abbreviation: $\mathrm{MTT}$
 * Deduced from: The pooled assumptions of each of $p \implies q$ and $\neg q$.
 * Depends on: Both of the lines containing $p \implies q$ and $\neg q$.

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 the Rule of Implication.

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.