Modus Tollendo Tollens

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Proof Rule

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

Proof Rule

If we can conclude $\phi \implies \psi$, and we can also conclude $\neg \psi$, then we may infer $\neg \phi$.

Sequent Form

The Modus Tollendo Tollens can be symbolised by the sequent:

\(\ds p\) \(\implies\) \(\ds q\)
\(\ds \neg q\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds \neg p\) \(\) \(\ds \)


The Modus Tollendo Tollens can be expressed in natural language as:

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.

Also known as

Modus Tollendo Tollens is also known as:

  • Modus tollens, abbreviated M.T.
  • Denying the consequent.

Also see

The following are related argument forms:

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:

Linguistic Note

Modus Tollendo Tollens is Latin for mode that by denying, denies.

The shorter form Modus Tollens means mode that denies.