Frames and Theories I

Philosophy-wise, there are a few things that need to be motivated. The first is frames; the second is frame morphisms. Once the motivations for these are in place, much of the rest of the philosophical picture will follow along.

Regarding the first bit, recall first that frames have theories in them and that theories are related to each other in a few ways. The natural question to ask is then "what are these theories theories of?"

Whenever I get to this point in doing the philosophy bit, I start to regret my vocabulary choices. Calling the elements of a frame 'theories' is fantastic for motivating the technical shenaniganry we've been engaged in -- it makes sense to close one theory under another, to talk about one theory containing another, etc. The problem, though, is that theories as paradigmatic syntactic entities. They're sets of sentences. And what we're trying to do is build a semantics. If we were building a proof-theoretic semantics, it would be totally fine to ground out our semantics in something syntactic. But we're giving a model-theoretic semantics. So we need to tell some story about what theories are that doesn't amount to their being, well, theories.

To tell that story, it helps to first remember that a frame is only a part of a model. Models also have, in addition to their frame-bits, a domain-bit and a valuation-bit. What these additional pieces do is tell us what objects there are and what features those objects have at each theory in the frame. So theories can be thought of as states of affairs -- they're places where things can be and can have properties. Corresponding to any such state of affairs (semantic entity) is a natural syntactic entity: the set containing p whenever p is true in that state of affairs and 'not p' whenever p is false in that state of affairs.

So that's what we have to say about theories. The natural thing to say about frames, given this, is that if we’re investigating subject matter S, the correct frame to use is the one that contains all the theories compatible with our information about S. But what we want to think about, really, is logic. And I think it’s not at all obvious what frame/class of frames this leaves us with. Next time around, I want to talk about two plausible answers to this question. I also think there is an interesting philosophical debate to be had about which of these is right, and that this debate will have overtones that should be recognizable to anyone familiar with the debates about whether Tarski’s account of logical consequence was domain relative. (This Greg Ray paper, together with the citations it contains, makes a great intro to the area if you’re unfamiliar.)

That's enough for one day. Next up I'll actually give the two answers. Then I'll turn to giving philosophical motivation for frame morphisms. After that we'll return to our categorification quest.