The Category of Natural Kinds

This is part 3 of the series on natural kinds. The previous post is here.

Having established what natural kinds are, we’re ready to examine maps between them and lift our discussion to the categorical setting. This short post establishes the categorical nature of natural kinds and maps between them.

Onward

In the context of natural kinds, implication (remember, there’s a logic here!) is taken to be the species-genus relation: for natural kinds K₁, K₂, to say K₁ ⇒ K₂ is to say K₁ is a species of K₂.

This is formalized as follows:

For kinds K₁ = (A₁,X₁) and K₂ = (A₂,X₂), saying that K₁ is a species of K₂ amounts to:

• (A₁⊆A₂)
• (X₂⊆X₁)

That is,

• every K₁ individual is also a K₂ individual
• every K₂ trait is also a K₁ trait.

We have the following theorem (T1), for natural kinds K₁ = (A₁,X₁) and K₂ = (A₂,X₂):

K₁ ⇒ K₂ iff A₁ ⊆ A₂ and X₂ ⊆ X₁.

Upward

Natural kinds, together with the species-genus relation, form a category:

Let K = (A,X) be any natural kind.

Identity Arrow

By reflexivity of the subset relation, A ⊆ A and X ⊆ X, so X ⇒ X.

Arrow Composition

Now let K₁ = (A₁,X₁), K₂ = (A₂,X₂), and K₃ = (A₃,X₃) be natural kinds and suppose K₁ ⇒ K₂ and K₂ ⇒ K₃.

By the theorem (T1) in the previous section, K₁ ⇒ K₂ iff A₁ ⊆ A₂ and X₂ ⊆ X₁, and K₂ ⇒ K₃ iff A₂ ⊆ A₃ and X₃ ⊆ X₂.

If A₁ ⊆ A₂ and A₂ ⊆ A₃, then A₁ ⊆ A₃, and if X₃ ⊆ X₂ and X₂ ⊆ X₁, then X₃ ⊆ X₁, so K₁ ⇒ K₃ on the assumption that K₁ ⇒ K₂ and K₂ ⇒ K₃.

The subset relation is associative, so evidently this composition is too.

These facts tell us that natural kinds with the species-genus implication form a category.

In the next post, we’ll examine some of the categorical features of natural kinds, with an eye to building bridges to other parts of logic and mathematics.