Mode tollendo ponens

The modus tollendo ponens or disjunctive syllogism is a final figure in classical propositional logic or a rule of many logical calculi that allows one to infer a sentence of form B from a sentence of the form A or B and a sentence of the form not A. In terms of content, it is therefore concluded from the knowledge that at least one of two facts must exist, but that one of the two does not exist, that the other of the two must exist.

The Latin name modus tollendo ponens , freely: "Inference ( modus ), which by rejecting [denying] ( tollendo ) [a statement] sets [derives] ( ponens ) a [other] statement ", is explained by the fact that with a given first premise, A ∨ B, by negating (¬A) a statement another statement, B, is “set” (derived).

Since a sentence A ∨ B is also called a disjunction , the modus tollendo ponens is sometimes referred to as a "disjunctive syllogism ".

Formulation and example

The conclusion follows from the premises of the form and . ${\displaystyle A\lor B}$ ${\displaystyle \neg A}$ ${\displaystyle B}$

Scheme	example
${\displaystyle A\lor B}$ ${\displaystyle \neg A}$ modus tollendo ponens ${\displaystyle B}$	It's day or night. It is not day. modus tollendo ponens It's night.

proof

The logical equivalence of the statements A ∨ B and ¬A → B follows from the definitions of disjunction , subjunction and negation .

A. B. ¬A A∨B ${\displaystyle \Leftrightarrow }$ ¬A → B f f w f f f w w w w w f f w w w w f w w

See also

• Modus ponens , actually Modus ponendo ponens
• Mode tollens , actually mode tollendo tollens
• Modus ponendo tollens