# 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)