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.