''Modus ponendo tollens'' (MPT;
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
: "mode that denies by affirming") is a
valid rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
for
propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
. It is closely related to ''
modus ponens
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
'' and ''
modus tollendo ponens''.
Overview
MPT is usually described as having the form:
#Not both A and B
#A
#Therefore, not B
For example:
# Ann and Bill cannot both win the race.
# Ann won the race.
# Therefore, Bill cannot have won the race.
As
E. J. Lemmon describes it:"''Modus ponendo tollens'' is the principle that, if the negation of a conjunction holds and also one of its conjuncts, then the negation of its other conjunct holds."
[ Lemmon, Edward John. 2001. ''Beginning Logic''. ]Taylor and Francis
Taylor & Francis Group is an international company originating in England that publishes books and academic journals. Its parts include Taylor & Francis, Routledge, F1000 Research or Dovepress. It is a division of Informa plc, a United Ki ...
/CRC Press, p. 61.
In
logic notation this can be represented as:
#
#
#
Based on the
Sheffer Stroke
In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the logical negation, negation of the logical conjunction, conjunction operation, expressed in ordinary language as "not both". ...
(alternative denial), ", ", the inference can also be formalized in this way:
#
#
#
Proof
See also
* ''
Modus tollendo ponens''
*
Stoic logic
Stoic logic is the system of propositional logic developed by the Stoic philosophers in ancient Greece.
It was one of the two great systems of logic in the classical world. It was largely built and shaped by Chrysippus, the third head of the Stoi ...
References
{{Reflist
Latin logical phrases
Rules of inference
Theorems in propositional logic
nl:Modus tollens#Modus ponendo tollens