Modus ponendo tollens (MPT;Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234. Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to modus ponens 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/CRC Press, p. 61.
In logic notation this can be represented as:
Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:
Proof Step Proposition Derivation 1 Given 2 Given 3 De Morgan's laws (1) 4 Double negation (2) 5 Disjunctive syllogism (3,4)
Strong form Modus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise:
See also Modus tollendo ponens Stoic logic
References
nl:Modus tollens#Modus ponendo tollens