Date and Time: Thursday, November 12th 2020, 16:30-18:30, Amsterdam time.
Title: Neighborhood products of modal logics
The product of modal logics is often considered in the context of multidimensional modal logics. It is defined as the logic of two-dimensional structures, usually, the product of Kripke frames. But there are other semantics for modal logic. Van Benthem with coauthors (2005, “Multimodal Logics of Products of Topologies”) defined the product of topological spaces and topological product of modal logics for extensions of S4. Note that this product of topological spaces is a bitopological space, it has horizontal and vertical topologies that are different from the product topology. These definitions were later extended to monotone modal logics by Sano (2011, “Axiomatizing Hybrid Products of Monotone Neighborhood Frames”) by defining the product of monotone neighborhood frames. For extension of S4 the neighborhood frames are equivalent to topological spaces.
It turned out that, in general, the neighborhood product of modal logics is weaker than Kripke products. For example the neighborhood (topological) product of S4 with S4 is the fusion of S4 with S4. Later in 2012 and in 2016 the speaker proved that for many serial normal modal logics the neighborhood product coincides with the fusion. And for non-serial logics the neighborhood product of modal logics should include some list of variable-free formulas.
The way we prove completeness for neighborhood products of modal logics is based on Kripke completeness. For a given Kripke frame we construct a neighborhood frame of pseudo infinite paths in the Kripke frame. The details will be given in the talk.
I think that this construction has a nice epistemic intuition which I would like to discuss with the participants of the seminar during and/or after my talk.
In the talk I will formulate several completeness results that were proved by the speaker, including some results about derivational topological semantics.