I'd never heard of \\(\textbf{Belnap4}\\), but my knowledge of multi-valued logics is poor. Here's what this little poset looks like:

In this picture, taken from the _Stanford Encylopedia of Philosophy_, the two intermediate truth values are called \\(\varnothing\\), where we are _ignorant_ of whether something is true or false, and \\( \\{\bot,\top\\} \\), where we have _contradictory information_ saying that something is both true and false! \\(\top\\) is true and \\(\bot\\) is false.

So, in a \\(\textbf{Belnap4}\\)-category, we can say

* yes, \\(x \leq y\\)

* no, \\( x \nleq y \\)

* I don't know whether \\(x \leq y\\) or \\( x \nleq y \\)

* I've got contradictory information suggesting both \\(x \leq y\\) and \\( x \nleq y \\).

Cool!

There's a nice monoidal monotone \\(f: \textbf{Bool} \to \textbf{Belnap4} \\) embedding ordinary Boolean logic in this 4-valued logic, so using "base change" (as explained in [comment #54](https://forum.azimuthproject.org/discussion/comment/18470/#Comment_18470)) we can turn any preorder into a \\(\textbf{Belnap4}\\)-category.

But this is more interesting: are there any monoidal monotones \\(g: \textbf{Bool} \to \textbf{Belnap4} \\)? If so, we can use them to "crush down" \\(\textbf{Belnap4}\\)-categories into preorders.

In general it should be lots of fun to combine multi-valued logic with enriched categories as we are doing here.

In this picture, taken from the _Stanford Encylopedia of Philosophy_, the two intermediate truth values are called \\(\varnothing\\), where we are _ignorant_ of whether something is true or false, and \\( \\{\bot,\top\\} \\), where we have _contradictory information_ saying that something is both true and false! \\(\top\\) is true and \\(\bot\\) is false.

So, in a \\(\textbf{Belnap4}\\)-category, we can say

* yes, \\(x \leq y\\)

* no, \\( x \nleq y \\)

* I don't know whether \\(x \leq y\\) or \\( x \nleq y \\)

* I've got contradictory information suggesting both \\(x \leq y\\) and \\( x \nleq y \\).

Cool!

There's a nice monoidal monotone \\(f: \textbf{Bool} \to \textbf{Belnap4} \\) embedding ordinary Boolean logic in this 4-valued logic, so using "base change" (as explained in [comment #54](https://forum.azimuthproject.org/discussion/comment/18470/#Comment_18470)) we can turn any preorder into a \\(\textbf{Belnap4}\\)-category.

But this is more interesting: are there any monoidal monotones \\(g: \textbf{Bool} \to \textbf{Belnap4} \\)? If so, we can use them to "crush down" \\(\textbf{Belnap4}\\)-categories into preorders.

In general it should be lots of fun to combine multi-valued logic with enriched categories as we are doing here.