Arrows! (jk, Arrows are tomorrow)

The first thing to note is that Fine actually gives three different versions of his theory. In the work I’ve done so far on stratified semantics (work that I’m extending with Andrew Tedder) I’ve used Fine’s third option: ternary-relation frames a la Routley and Meyer. But I’ve recently become convinced that Fine was right to focus (both in this paper and in his earlier work) on operational models. So that’s where I’m going to hang out in this series. Also I lied yesterday. I won’t get you all the way to arrows today; all I’ll actually do is do some setup.

As I said in the previous post, the category we’re looking for will have models of the quantifier-free (aka zero-order) fragment of a first-order language as its objects. From the operational perspective, such models have eight pieces: a set-sized domain, a set-sized collection of theories, a designated subset of the theories called the prime theories, a binary closure operation that acts on the theories, a binary containment relation among the theories, a base or ground theory, a complementation function, and a valuation. From top to bottom, here’s what we require of these:

  1. The domain must be nonempty.
  2. The set of theories must be nonempty.
  3. The prime theories must be dense among the theories. This means that if the closure of the theory t under the theory u is contained in the prime theory p, then there must be prime theories pt (extending t) and pu (extending u) so that the closure of pt under u is still contained in p and the closure of t under pu is also still contained in p. The intuition to have here is that prime theories are theories that take a stand of every disjunction they accept. (So if they accept 'A or B', then they also either accept A or accept B.) The density requirement then captures the following intuition: if we close t under u, then take a stand on each of the disjunctions we end up with, the resulting theory is compatible with some way of having taken a stand on all the disjunctions in t and some way of having taken a stand on all the disjunctions in u.
  4. The closure operation must be monotonic: if t is contained in u (so that u is an extension of t) then the closure of t under v should be contained in the closure of u under v.
  5. The containment relation must be a partial ordering.
  6. Closing the base theory under any theory t results in a theory that extends t.
  7. The complementation function must be order reversing (if t extends u, then the complement of u extends the complement of t), decreasing above the base theory (if t extends the base theory, then t extends its own complement as well), and have order two (so the complement of the complement of t is t).
  8. The valuation function maps each predicates to its extension at each theory and maps variables to objects in the usual way.

I think these are fairly natural conditions given the intended interpretation of, e.g. ‘theory’, ‘prime’, etc. Nonetheless, they allow some surprising things to be models. First, say that a frame is what we get when we strip off the domain and the valuation from a model. Then we might build a kind of funny frame by taking the set of ‘theories’ to be the positive rational numbers, the set of ‘prime’ theories to be the set of fractions (that is, the non-whole (unnatural?) numbers), the ‘closure’ operation to be ordinary multiplication, the ‘base’ theory to be the number 1, and complementation to be reciprocation.

Despite the fact that we have unusual examples of frames (and thus of models), for the most part this sort of operational model is pretty well-behaved. The category we need — the one whose arrows I said I’d describe today, but that I will actually describe tomorrow — has these models as its objects. Actually, to make things a bit simpler, I’m going to restrict my attention to models that are a bit simpler. Specifically, I will always take the domain to be the set of variables and suppose that every valuation maps every variable to itself. These assumptions can be done away with by the usual tricks, so are, despite being weird, totally harmless.