Modus Tollendo Tollens

Proof Rule

is a valid argument in types of logic dealing with conditionals $\implies$ and negation $\neg$.

This includes propositional logic and predicate logic, and in particular natural deduction.

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

The can be symbolised by the sequent:

$\ds p$

$\implies$

$\ds q$

$\ds \neg q$

$\ds $

$\ds \vdash \ \ $

$\ds \neg p$

$\ds $

The 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.

is also known as:

The following are related argument forms:

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

These are classic fallacies:

is Latin for mode that by denying, denies.

The shorter form Modus Tollens means mode that denies, or method of denying.

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): modus tollens
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): modus tollens