# Modus tollens

 Part of the series on Key articles General logic Bad logic v - t - e

Modus tollens ("mode of talking") is a logical argument, or rule of inference.

## As an argument

A modus tollens argument has the following form:

If P, then Q.
Not Q.
Therefore, not P.

For example:

If it is raining, the ground is wet.
The ground is not wet.
Therefore, it is not raining.

## As a rule of inference

In propositional logic:

$\left\{P\rightarrow Q,\neg Q\right\} \models \neg P$

In first-order logic:

$\models_{\mathfrak{A}}\forall x.\left(P(x)\rightarrow Q(x)\right) \wedge \models_{\mathfrak{A}}\exists x.\left(\neg Q(x)\right) \implies \models_{\mathfrak{A}}\exists x.\left(\neg P(x)\right)$