Modus Tollendo Tollens

Theorem
The modus tollendo tollens (or modus 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.

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

Proof 1
By the tableau method of natural deduction:

Proof 2
We apply the Method of Truth Tables to the proposition.

As can be seen for all models by inspection, where the truth value under the main connective on the LHS is $T$, that under the one on the RHS is also $T$:

$\begin{array}{|cccccc||cc|} \hline (p & \implies & q) & \land & \neg & q & \neg & p \\ \hline F & T & F & T & T & F & T & F \\ F & T & T & F & F & T & T & F \\ T & F & F & F & T & F & F & T \\ T & T & T & F & F & T & F & T \\ \hline \end{array}$

Hence the result.

Note that the two formulas are not equivalent, as the relevant columns do not match exactly.

Also known as
This is sometimes known as denying the consequent.

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

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.