Modus ponendo tollens

Template:Italic title Template:Transformation rules Modus ponendo tollens (MPT;<ref>Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234.</ref> Latin: "mode that denies by affirming")<ref>Template:Cite book</ref> 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."<ref>Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press, p. 61.</ref>

In logic notation this can be represented as:

1. <math> \neg (A \land B)</math>
2. <math> A</math>
3. <math> \therefore \neg B</math>

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

1. <math> A\,|\,B</math>
2. <math> A</math>
3. <math> \therefore \neg B</math>

Proof

Step	Proposition	Derivation
1	<math>\neg (A \land B) </math>	Given
2	<math>A</math>	Given
3	<math>\neg A \lor \neg B</math>	De Morgan's laws (1)
4	<math>\neg \neg A</math>	Double negation (2)
5	<math>\neg B</math>	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. <math> A \underline\lor B</math>
2. <math> A</math>
3. <math> \therefore \neg B</math>

See also

• Modus tollendo ponens
• Stoic logic