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.") It is also known as indirect proof or proof by contrapositive, and is a valid form of argument in formal logic.[1]

As an argument[edit]

A modus tollens argument has the following form:

P1: If X, then Y. (i. e. Either not X or Y)
P2: Not Y.
C: 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.
C: Therefore, it is not raining.

The contrapositive of "if X then Y" is "if not Y then not X"; if a proposition is true, then so is its contrapositive.[2]

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)

Denying the antecedent[edit]

See the main article on this topic: Denying the antecedent

It can be contrasted with the fallacy of denying the antecedent, for instance (using the above example) "it is not raining, therefore the ground is not wet" (obviously untrue if you're standing in a lake). Denying the antecedent asserts not-X rather than not-Y.

See also[edit]

References[edit]

  1. Modus Tollens, Jordan Bell, Wolfram Mathworld
  2. Converse and Contrapositive, CS381 Discrete Structures/Discrete Mathematics Web Course Material, Old Dominion University