Modus ponendo tollens

${\displaystyle \neg (A\land B)}$ ${\displaystyle A}$ ${\displaystyle \neg B}$

modus ponendo tollens

The modus ponendo tollens (also conjunctive syllogism) is a final figure of classical propositional logic and a final rule of many logical calculi that allows two sentences with the forms Not (A and B). and A. , the premises of inferring a sentence of the form Not B. as a conclusion .

In terms of content, it is therefore concluded from the knowledge that two specific facts cannot exist at the same time, but that one of the two facts does exist, that the other of the two does not exist.

History and naming

The Latin name modus ponendo tollens , freely: "In conclusion ( modus ), which by setting ( ponendo ) [a statement] rejects [another] statement ( tollens ), is explained by the fact that given the first premise, ¬ (A ∧ B), by setting a second, positive (non-negated) premise, A, a statement, B, is "rejected" (negated).

It corresponds to one of the five types of hypothetical syllogism according Chrysippus : 'Either the first or the second; but the first; so not the second '.

Scheme and example

Scheme	example
${\displaystyle \neg (A\land B)}$ ${\displaystyle A}$ modus ponendo tollens ${\displaystyle \neg B}$	It cannot be that it is raining and the road is dry. It's raining. modus ponendo tollens The road is not dry.

proof

The logical equivalence of the statements ¬ (A ∧ B) and A → ¬B follows from the definitions of conjunction , subjunction and negation .

A. B. f f f w w f w w

A ∧ B ¬ (A ∧ B) f w f w f w w f

¬B A → ¬B w w f w w w f f

See also

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

Individual evidence

1. ↑ Thomas Zoglauer: Introduction to formal logic for philosophers . ( limited preview in Google Book search).
2. ↑ Modus ponens and modus tollens | logic . In: Encyclopedia Britannica . ( britannica.com [accessed October 18, 2018]).
3. ↑ See Peter Thom : Syllogismus; Syllogistics. in: Historical Dictionary of Philosophy Vol. 10, p. 695