Modus ponendo tollens

Modus ponendo tollens (MPT;^[1] Latin: "mode that denies by affirming")^[2] 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."^[3]

In logic notation this can be represented as:

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

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

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

Proof

Step	Proposition	Derivation
1	${\displaystyle \neg (A\land B)}$	Given
2	${\displaystyle A}$	Given
3	${\displaystyle \neg A\lor \neg B}$	De Morgan's laws (1)
4	${\displaystyle \neg \neg A}$	Double negation (2)
5	${\displaystyle \neg B}$	Disjunctive syllogism (3,4)

Strong form

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

1. ${\displaystyle A{\underline {\lor }}B}$
2. ${\displaystyle A}$
3. ${\displaystyle \therefore \neg B}$

See also

• Modus tollendo ponens
• Stoic logic