====== Monotonicity (of entailment) ======

Generalized conceptual qualities that reflect the hierarchical order in a category system. 

===== Other names =====

  * Upward or downward entailment

  * Distribution

===== Description =====

Terms that describe an object in a hierarchical category system may also imply the levels above or below it in this hierarchyt. This quality may change depending on the context.

For example, in the statement:

> All //cats// are //mammals//.

Both “cats” and “mammals” are terms that refer to the biological classification ([[wp>Taxonomy (biology)|taxonomy]]) of living organisms. In this context, “cats” are defined as a subgroup of “mammals,” which means that this statement is necessarily //true//.

In this statement (a [[categorical_statement|categorical statement]] of type A, i.e., an affirmative [[universal_quantification|universal quantification]]), both the //subject// (here: “cats”) and the //predicate// (here: “mammals”) are entailing in different ways with regard to the other elements of this hierarchy:

==== Downward entailment ====

In the statement “All //cats// are //mammals//,” the subject (here: “cats”) is //downward// entailing. This means that the statement also implies all subgroups of cats. For example, we can deduce:

  * All //Siamese cats// are mammals.
  * All //long-haired cats// are mammals.
  * //Our house cat// is a mammal.
  * etc.

 However, this does not apply to the //predicate//. We cannot deduce:

  * <s invalid "invalid conclusion">All cats are //marsupials//.</s>
  * <s invalid "invalid conclusion">All cats are //artiodactyls//.</s>
  * <s invalid "invalid conclusion">All cats are //dogs//.</s>
  * etc.

In this case, //downwards entailment// is synonymous with the concept of [[distribution|distribution]] in syllogistic logic.

==== Upwards entailment ====

Conversely, however, we can replace the predicate of the above statement with broader categories (i.e. upwards in the hierarchy tree):

  * All cats are //feliformia//.
  * All cats are //vertebrates//.
  * All cats are //living beings//.
  * etc.

Similarly, upwards extension of the subject term is not possible. The following examples are all invalid (and also incorrect):

  * <s invalid "invalid conclusion">All //vertebrates// are mammals.</s>
  * <s invalid "invalid conclusion">All living beings are mammals.</s>

===== Symbols =====

Analogous to the logical symbols for [[conditional|conditionals]], upward and downward arrow symbols are used to indicate the direction of entailment of a term:

  * Upward entailment: ''⇑'' or ''↑''.
  * Downward entailment: ''⇓'' or ''↓''.

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

  * [[distribution|Distribution]]

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

  * [[wp>Monotonicity of entailment|Monotonicity of entailment]] on //Wikipedia//