Unifying the field of modal logics

Nowadays there is an enormous and at times bewildering variety of modal and other non-classical logics. Nevertheless, it still makes sense to speak of `modal logic' as such since there is a well-developed mathematical theory which emphasizes the unity of the field. This theory is supported by a number of general results that apply to all modal logics alike.

But next to such a process of abstraction and generalization there is also a unifying research direction which seeks to compare various logics. An important way to do so is to use interpretation functions for the formulas, and simulation functions linking logics of the one type to that of the other. The relation between intermediate logics and certain modal logics provides a well-known case.

Results obtained over the last ten years have provided simulations linking most well-known families of modal logics: normal, monadic, uni-modal logics (i.e., the standard ones) can simulate normal, polymodal logics (i.e., in a language with several unary modalities). And the latter logics can simulate classical and monotone modal logics, as well as normal modal logics with modalities of higher arity.

In the talk I will briefly introduce and survey the area, and then focus on some of my own contributions. My last topic will concern ongoing research concerning modal logics with a propositional basis that is weaker than the usual Boolean one.  

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