====== Modus Ponendo Tollens ======

Also abbreviated <abbr :la "Modus ponendo tollens">MPT</abbr>. One of the elementary (valid) logical conclusion figures. It is based on a [[glossary:contravalence|contravalence]] and has the form:

> <span "A or B, but not both">''A ⊻ B''</span>    –    <q><var>A</var> or <var>B</var>, but not both</q>
> <span "A [is true]">''A''</span>    –    <q><var>A</var> [is true]</q>
> <span conclusio><span "therefore not B">''∴ ¬B''</span>    –   <q>therefore: not <var>B</var></q></span>

For example, the following is a valid <abbr>MPT</abbr>:

> A number is either //even// or //odd// (but not both).
> <var>x</var> is //even//.
> <span conclusio>Therefore <var>x</var> is //not odd//.</span>

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

> ...
> <var>y</var> is //odd//.
> <span conclusio>Therefore <var>y</var> is not even.</span>

<aside info>
**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.
</aside>

===== 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 <abbr :la "Modus ponendo tollens">MPT</abbr>:

The following table compares the <i :la>Modus Ponendo Tollens</i> and the most important related fallacies:

<div print-wide">
| ^  //Modus ponendo tollens// \\ (valid inference)  ^^ ^  [[logic:formal_fallacies:denying_a_conjunct|Denying a Conjunct]]  \\ (fallacy)  ^^  [[logic:formal_fallacies:affirming_a_disjunct|Affirming a Disjunct]]  \\ (fallacy)  ^^
^Premise 1 |  <span "A or B, but not both">A ⊻ B</span> \\ <small>(A //or// B, //but not both//)</small>  || |  <span "noth both, A and B">⌐ (A ∧ B)</span> \\ <small>(//not both//, A //and// B)</small>  ||  <span "A or [incl.] B">A ∨ B</span> \\ <small>(A oder B)</small>  ||
^Premise 2 |  A  |  B  | |  <span "not A">⌐A</span> <small>(not A)</small>  |  <span "not B">⌐B</span> <small>(not B)</small>  |  A  |  B  |
^Konklusion |  <span "not B">⌐B</span> <small>(not B)</small>  |  <span "not A">⌐A</span> <small>(not A)</small>  | |  <span "B (invalid)" invalid short>B</span>  |  <span "A (invalid)" invalid short>A</span>  |  <span "not B (invalid)" invalid short2>⌐B</span> <small>(not B)</small>  |  <span "not A (invalid)" invalid short2>⌐A</span> <small>(not A)</small>  |

<aside box>
**Note:** Both the [[glossary:contravalence|contravalence]] <span "A or B but not both">''A ⊻ B''</span> [A //or// B, //but not both//] and the //negative// [[glossary:conjunction|conjunction]] <span "not both, A or B">''⌐(A ∧ B)''</span> [not both, A and B] are //equivalent//, i.e.: ''A ⊻ B'' ≡ ''⌐(A ∧ B)''.
</aside>
</div>

===== See also =====

  * [[glossary:conjunction|Conjunction]]
  * <i :la>[[logic:inferences:modus_tollendo_ponens|Modus Tollendo Ponens]]</i>

===== More information =====

  * [[wp>Modus ponendo tollens]] on //Wikipedia//

{{page>templates:banner#Short-BG-Logic&noheader&nofooter}}