# Modus Tollendo Tollens

Jump to navigation
Jump to search

## 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:

\(\ds p\) | \(\implies\) | \(\ds q\) | ||||||||||||

\(\ds \neg q\) | \(\) | \(\ds \) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds \neg p\) | \(\) | \(\ds \) |

## Explanation

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**.

## Sources

- 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**modus tollens** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**modus tollens**