Modus ponendo tollens

Modus ponendo tollens (MPT;

modus tollendo ponens

.

Overview

MPT is usually described as having the form:

Not both A and B
A
Therefore, not B

For example:

Ann and Bill cannot both win the race.
Ann won the race.
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:

$\neg (A\land B)$
$A$
$\therefore \neg B$

Based on the

Sheffer Stroke

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

$A\,|\,B$
$A$
$\therefore \neg B$

Proof

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

Strong form

Modus ponendo tollens can be made stronger by using

exclusive disjunction

instead of non-conjunction as a premise:

$A{\underline {\lor }}B$
$A$
$\therefore \neg B$

References

^ Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234.
ISBN 0-415-91775-1
.

Taylor and Francis
/CRC Press, p. 61.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Modus_ponendo_tollens&oldid=1211544628"

[1] Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234.

[2] ISBN 0-415-91775-1
.

[3] Taylor and Francis
/CRC Press, p. 61.

[3]

Overview

Proof

Strong form

See also

References