Modus ponendo tollens


Modus ponendo tollens 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:
  1. Not both A and B
  2. A
  3. Therefore, not B
For example:
  1. Ann and Bill cannot both win the race.
  2. Ann won the race.
  3. 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."
In logic notation this can be represented as:
Based on the Sheffer Stroke, "|", the inference can also be formalized in this way:
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Strong form

Modus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise:
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