In model theory for standard first-order logic, one constructs a single model, a reduced product, from a collection of first-order models, together with an index set and a filter on the index set.

In model theory for modal first-order logic using Kripke frames, one constructs a single model containing substructures ("possible worlds") that are similar to individual classical models.

It appears to me that for certain systems of modal logic such as S5 and perhaps S4, one might construct modal models as reduced products of classical models. One would need to make an appropriate choice of a filter on the index set, and also of a topology on the index set, so that the modal operators might be mapped to the corresponding topological operations.

I have not located any research along these lines. I would be interested to hear about anything that has been done, or from anyone who might be interested in pursuing this topic.


I am not sure whether this is what you are driving at, but every normal modal logic can be described by algebraic models. An algebraic model is a modal algebra (Boolean algebra with additional modal operations) together with an ultrafilter and an assignment of elements of the modal algebra to the propositional variables. A modal formula is true with respect to the model if its Boolean truth value is in the ultrafilter.

Algebraic models correspond to generalized Kripke frames via an extension of Stone duality. The worlds are the ultrafilters of the modal algebra, the topology on the set of worlds is the Stone space topology and the visibility relations can be reconstructed from the modal operations on the algebra. So, no ultraproducts, but plenty of ultrafilters.

  • $\begingroup$ Thanks for your concise survey of what's been done in this area. It agrees with my own impression that nothing's been published about constructing modal models as products. Since I'm a MathOverflow newbie and the FAQ says "MathOverflow is not a discussion forum" I'll leave it at that. Please contact me via email (address available via my "Website" link) if you like. $\endgroup$ – MikeC Aug 9 '10 at 21:43

Ultraproducts of Kripke models have been studied a lot since the 70s (for a start, see §6.3–6.6 of [1]). I don’t know if more general reduced products have been considered as well.

[1] Robert Goldblatt, Mathematical modal logic: a view of its evolution, in: D. Gabbay and J. Woods (eds.), Handbook of the History of Logic, vol. 6, Elsevier, 2006.


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