Monotonicity (of entailment)

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

• Upward or downward entailment

• Distribution

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 (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 of type A, i.e., an affirmative 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:

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:

• All cats are marsupials.
• All cats are artiodactyls.
• All cats are dogs.
• etc.

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

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):

• All vertebrates are mammals.
• All living beings are mammals.

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

• Upward entailment: ⇑ or ↑.
• Downward entailment: ⇓ or ↓.

• Distribution

• Monotonicity of entailment on Wikipedia