Modus ponendo tollens

Transformation rules
Propositional calculus
Rules of inference
Implication introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive / hypothetical syllogism Constructive / destructive dilemma Absorption / modus tollens / modus ponendo tollens
Rules of replacement
Associativity Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology Negation introduction
Predicate logic
Universal generalization / instantiation Existential generalization / instantiation

Modus ponendo tollens (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to modus ponens and modus tollens.

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:

1. $\neg (A\land B)$
2. $A$
3. $\therefore \neg B$

Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:

1. $A\,|\,B$
2. $A$
3. $\therefore \neg B$

See also

• Modus tollendo ponens
• Stoic logic