# Modus Ponendo Tollens

Also abbreviated MPT. One of the elementary (valid) logical conclusion figures. It is based on a contravalence and has the form:

`A ⊻ B`

– A orB,but not both`A`

– A [ is true]`∴ ¬B`

– therefore: not B

For example, the following is a valid MPT:

A number is eitherevenorodd(but not both).

x iseven.

Therefore x isnot odd.

Since *contravalence* is commutative, this also implies the following:

…

y isodd.

Therefore y is not even.

**Note:** numbers can also be *neither* even nor odd (e.g. rational numbers). If such possibilities are not excluded (e.g. by considering only natural numbers), one can easily fall for the fallacies mentioned below.

## Name

The name of this inference form can be loosely translated as “form of negation [of a statement] by affirmation [of the alternative]”.

### Other names

- Conjunctive syllogism

## Fallacies

As with other forms of logical reasoning, there are several fallacies that can on incorrect application of the MPT:

The following table compares the *Modus Ponendo Tollens* and the most important related fallacies:

Modus ponendo tollens (valid inference) | Denying a Conjunct (fallacy) | Affirming a Disjunct (fallacy) |
|||||
---|---|---|---|---|---|---|---|

Premise 1 | A ⊻ B (A or B, but not both) | ⌐ (A ∧ B) ( not both, A and B) | A ∨ B (A oder B) |
||||

Premise 2 | A | B | ⌐A (not A) | ⌐B (not B) | A | B | |

Konklusion | ⌐B (not B) | ⌐A (not A) | B | A | ⌐B (not B) | ⌐A (not A) |

**Note:** Both the contravalence `A ⊻ B`

[A *or* B, *but not both*] and the *negative* conjunction `⌐(A ∧ B)`

[not both, A and B] are *equivalent*, i.e.: `A ⊻ B`

≡ `⌐(A ∧ B)`

.

## See also

## More information

- Modus ponendo tollens on
*Wikipedia*