''Modus ponendo tollens'' (MPT;
Latin
Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
: "mode that denies by affirming") is a
valid rule of inference
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a ...
for
propositional logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
. It is closely related to ''
modus ponens
In propositional logic, (; MP), also known as (), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P'' implies ''Q.'' ''P'' is true. Therefore, ''Q'' must ...
'' 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 the United Kingdom that publishes books and academic journals. Its parts include Taylor & Francis, CRC Press, Routledge, F1000 Research and Dovepress. It is a division of ...
/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 negation of the conjunction operation, expressed in ordinary language as "not both". It is also called non-conjunction, ...
(alternative denial), ", ", the inference can also be formalized in this way:
#
#
#
Proof
Strong form
''Modus ponendo tollens'' can be made stronger by using
exclusive disjunction
Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or Logical_equality#Inequality, logical inequality is a Logical connective, logical operator whose negation is the logical biconditional. With two inputs, X ...
instead of non-conjunction as a premise:
#
#
#
See also
* ''
Modus tollendo ponens''
*
Stoic logic
Stoicism is a school of Hellenistic philosophy that flourished in ancient Greece and Rome. The Stoics believed that the universe operated according to reason, ''i.e.'' by a God which is immersed in nature itself. Of all the schools of ancient p ...
References
{{Reflist
Latin logical phrases
Rules of inference
Theorems in propositional logic
nl:Modus tollens#Modus ponendo tollens