# Modus tollens

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Modus tollens ("mode of taking") is a logical argument, or rule of inference. (Compare with modus ponens, or "mode of putting.")

## As an argument

A modus tollens argument has the following form:

P1: If X, then Y. (i. e. Either not X or Y)
P2: Not Y.
C1: Therefore, not X.

For example:

P1: If it is raining, the ground is wet. (i. e. It is not raining or the ground is wet.)
P2: The ground is not wet.
C1: Therefore, it is not raining.

## As a rule of inference

In propositional logic:

$\left\{X\rightarrow Y,\neg Y\right\} \models \neg X$

In first-order logic:

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