# 6.2 Subtype relation

## Definition

To explain what the subtype relation means, let us offer this definition:

If A ≤ B (read as "A is a subtype of B"), then either:

• A is B, or
• A is a specialization of B.

For example, the type "Bird" is a subtype of the type "Animal". This is because "Bird" is a specialization of "Animal".

## Specialization

A specialization A ("Bird") of another type B ("Animal") includes all of the properties of the original type B ("Animal"), but adds some more restrictions. For example, a Bird not only has all of the characteristics of "Animal", it also has specialized characteristics such as

• It has wings,
• It lays eggs,
• It has a beak, and
• It has feathers.

Thus, "Bird" is a specialization of "Animal", since it has all of the properties of "Animal", but adds further restrictions.

## Identity

But A can also be identical to B. For example,

Animal ≤ Animal

.

## Proper subtype

If A is not B, but A ≤ B, then we are entitled to write

A < B

, which means "A is a proper subtype of B". This is a way of saying that while A is not B; instead, A is a specialization of B.

## Supertype

The converse of the '≤' relation is the '≥' relation, called the "supertype relation". The two formulas

A ≤ B    and     B ≥ A

are equivalent.

## Proper supertype

If A is a supertype of B, i.e., A ≥ B, and A is not B, then we are entitled to write

A > B

, which means "A is a proper supertype of B". The two formulas

A < B     and     B > A

are equivalent.

## Transitivity

It is also true that since

SeaGull ≤ Bird    and    Bird ≤ Animal

, then it follows that

SeaGull ≤ Animal

. (Note to the interested: This is because ≤ is a partial order and partial orders are transitive.)

## Immediate subtype

An immediate subtype A of another type B is a subtype that has no other types in between A and B.

Because of the transitivity of the subtype relation, if A ≤ B, it does not follow that A is an immediate subtype of B: There may be other subtypes in between. It is useful to be able to speak about immediate subtypes, however, even if we have no notation to use for it.

## Next

Next, we discuss two special types, the universal type and the absurd type.

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Up: 6 Core ontological ideas
Next: 6.3 Entity and Absurdity