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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[edit]

A modus tollens argument has the following form:


P1: If X, then Y.
P2: Not Y.
C1: Therefore, not X.

For example:


P1: If it is raining, the ground is wet.
P2: The ground is not wet.
C1: Therefore, it is not raining.

As a rule of inference[edit]

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)

External links[edit]