A Question about Quantified Modal Logic

In propositional modal logic, we have a clear distinction between models and frames. A frame is a pair of a set W and a binary relation R on W. A model is a frame plus a valuation function that maps atomic sentence letters to subsets of W. Given this distinction, we can define truth at a point in a model, truth on a model (i.e. truth at all points on a model) and truth on a frame (i.e. truth at all points on all models on the frame.)

In the previous post I was considering logics which, as well as modal operators, had actuality operators and quantifiers. Given that extra expressive capacity, where should we draw the frame/model distinction? There might well be a well established precedent for this, but I wasn’t sure what it was. And it might be that different distinctions are useful for different purposes.

Here are two more specific questions.

First, if we have an actuality operator, we need a way of designating one world as actual in each model. Is this something we do at the level of the frame, or is it something done by something like the valuation function?

Second, is the number of individuals in each world set by the frame, or do different models on the same frame have different individuals in the worlds?

As I said, I imagine there are conventional answers to these questions, and the conventions are probably well motivated. But I wasn’t sure what they were.

10 Replies to “A Question about Quantified Modal Logic”

  1. We can think of a designated world as designated by the frame and, thus, fixed across all models built on that frame. In this sense it is similar to designated semantic values which are fixed independent of a particular model.

    I’m not well-versed in quantified modal logic, but I think there are a couple of intuitive ways to answer the second question. It depends on whether you want a “fixed domain” or “variable domain” semantics. Let’s define QML frames as triples with a set of worlds, an access relation, and set of objects O. If you want a fixed domain semantics, then for any model built on a frame, the set of objects O supplied by the frame can serve as the domain for interpretation of names and predicates at any world. If you want a variable domain semantics, then you need a function f mapping worlds to subsets of O where f(w) will then serve as the domain for interpretation of names and predicates at world w. This is off the top of my head, so let me know if anything about this seems wrong.

  2. The second issue is handled on frames, but the first (perhaps) could go either way.

    The standard move (and the only one that I’m aware of) to associate individuals with worlds is to construct an augmented frame, which is a Krikpe frame plus a domain function that assigns individuals from a first-order domain to each world (for variable domain frames) and which assigns the same individuals to every world (for constant domain frames). The valuation function for first-order augmented models is modified to handle the quantifiers, but otherwise works the way you’d expect. Note that you still have a frame/model distinction which does some work for you. For instance, the Barcan and Converse Barcan formulas are valid w.r.t. the class of anti-monotone frames and w.r.t. the class of monotone frames, respectively.

    The first question depends more on the application you have in mind. One move is to view questions as strictly satisfiability problems for the designated world and to treat it as a issue handled by the model. But, on the other hand, I suppose that you could decorate the actual world in W, treat W as sorted, and work with a multi-modal frame. I guess I’d have to see what kind of lifting the logic needed to do in order to recommend one route over the other. Conceptually, treating the first question as a question for model theory is probably easier to start with.

    You might want to consider the temporal logic literature, too.

  3. Thanks Gregory and Colin, those are helpful. Especially Gregory’s point that we can still make a useful frame/model distinction even with these somewhat richer frames.

    So it looks like I was wrong to think that we could characterise frames where the Limit Assumption fails. My principle (L) from the previous post holds on all frames with every world finite. So that proposal of mine fails…

  4. It seems like a funny terminological way to be wrong about something.

    I always thought the choice to call something a frame was pragmatic, determined by what you were trying to do. If you’re interested in which properties of a certain kind of structure you can define in a given language, call those structures the frames. (So, e.g., if you want to know what digraphs you can define stick to the ordinary kripke frames be, or if you’re interested in what sizes you can force the set of worlds to be, let frames just be the worlds, etc…) Is there something more to it than that?

  5. Hi Andrew,

    The distinction between Kripke frames and Kripke models is more the terminological. While the distinction naturally comes for students interested in computation, I think it should be part of the list of basic concepts that a contemporary modal logic course should include. You might be interested in a recent discussion at Certain Doubts, here, which I suspect traces back to mixing up models and frames.

    Now, in some cases you can do creative things to transform what is essentially a satisfiability problems into a validity problem for a weird class of frames. But, the transformation might not preserve the behavior that you set out to model.

  6. Thanks Gregory,

    In propositional modal logic, I see that you shouldn’t confuse frames and models, where the latter are concieved as including interpretations of the non-logical non-modal vocabulary (since, for example, sets of formulae validated by classes of models needn’t be closed under substitution.)

    But when we’re concerned with defineability in QML, is there a principled reason to include, say, the domain of each world as part of the frame or allow it to vary? I can’t see here what rests on it, but I’d like to know for sure.

  7. Hi Andrew,

    Think of frames as rules of the road. Some roads have traffic circles. Others are one-way. For example, you can look at the Google Map of Lisbon to get a sense of how traffic squirrels around, where the bus stops are located, —but you can do all of this from your home, without imagining a single vehicle on the streets, and without booking a flight to LIS, hiring a car, and determining whether or not a trip across town avoids entering a traffic circle.

    The metaphor is a bit flimsy, but if it helps install in your mind the idea that frames and models explore different types of questions, that’s probably the right/better starting point.

    So, on propositional Kripke frames you get the rules of the road for how worlds are related. If I’m in a world and you’re in a world and I can see you, can you see me? If so, can we be in different worlds or do we have to be in the same one? Or does it not matter which worlds we inhabit? These are frame questions, since they have to do directly with properties (if any at all) imposed on the accessibility relation.

    For QML, you augment W and R of a Kripke frame with a function that associates individuals with worlds. There are analogous structural questions you might ask about that function. Are worlds in my frame arranged so that wherever I might travel, from world to world, I might lose an individual during the trip that I had in the world where anti-monotonic frames are behave this way. Might I visit a world in my trip and meet someone new? Monotonic augmented frames behave this way. Is it impossible for me to lose or gain anythone during my travels from world to world? Then the frame is a constant domain augmented frame.

    QML models differ from standard Kripke models in two respects. The former are based on augmented frames, which we just discussed, and the latter are not. But the valuation functions are different, too. If you look at your QML textbook, this is where the business with x-variant variable assignments comes in.

  8. Remedial question: if I understand the previous comments, the set of individuals at a world in QML is assigned by the frame, and the set of true propositions in PML is assigned by the valuation function. But then there’s a difference of status between objects and states of affairs, which I would not have expected given the usual analogy (names:objects::propositions:states of affairs). Is this correct? If so, is it something to be concerned about?

  9. Ah, good question owlbear. I am assuming here that we’re looking at basic QML (i.e., with no constant symbols, and no function symbols). We’re just looking at what happens when you mix together (normal) box modality and its dual with the universal quantifier and its dual.

    My roadmap metaphor of me being a proposition right before talking about QML is not helping, either.

    Or maybe it is not so unhelpful: At bottom, both the box and \forall are types of quantifiers. The former ranges over worlds, the latter over individuals in a domain. A perfect kind of frame question is to ask whether there are interesting properties for how these two combine, whatever the truth assignments. That’s what the discussion of the Barcan / Converse Barcan schemas is getting at. (My typos are even worse than usual…sorry about that.)

  10. Hi Greg,

    That wasn’t quite the question I asked. There is clearly a non-terminological issue about whether frames should include information about the non-modal non-logical vocabulary (sentences letters in PML, and predicate/constants/etc in QML.) Obviously they shouldn’t (see the point about modal logics being closed under substitution) – I took that to be common knowledge which was why I didn’t explicitly exclude it.

    But I wanted to know to what extent including or excluding other information in the frame was terminological, or serving a particular purpose.

    That said, I think you’ve over simplified even the propositional case. I PML there are all kinds of notions of frame which include more or less information than Kripke frames without infringing on the frame/model distinction. Kripke frames are just a subclass of descriptive frames, which in turn are just a subclass of general frame (and there are yet orthogonal ways you can generalise, such using neighborhood frames.)

    In the above cases we are building in information about the algebra of propositions into the frame (as well as information about the “road map”.) Each kind of frame has its own nice properties, and I assume the choice of which to use just depends on what you’re trying to do.

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