Modal logics which have an algebraic semantics but not a Kripke semantics

Question

A colleague told me that there are modal logics which have an algebraic semantics of some kind but which do not have a Kripke semantics and in which both $\Box$ is not monotonic with respect to $\to$, in the sense that

$$\frac{\phi \to \psi}{\Box\, \phi \to \Box \,\psi} $$

fails and in which the rule of necessitation fails. My colleague could not recall which logics he had in mind.

Can anyone tell me of any interesting logics which have an algebraic semantics but not a Kripke semantics? Are any of these logics in which $\Box$ is not monotonic with respect to $\to$ and in which necessitation fails?

Answer 1

There is a semantics based on neighborhood frames. A logic of a class of neighborhood frames is not necessarily normal and even not necessarily monotone (i.e. closed under the ϕ→ψ / □ϕ→□ψ rule). On the other hand, any neighborhood frame (W,N) defines an algebra on the set of subsets of W; see E.Pacuit. Neighborhood Semantics for Modal Logic, p.32.

Answer 2

I guess the logic your colleague mentioned is quasi-normal modal logic. The book Modal Logic by Alexander Chagrov and Michael Zakharyaschev contains this. But it indeed can have a non-standard Krikpe semantics by distinguishing some "actual worlds" from a standard Kripke frame (see Chap 5.6), and its algebraic semantics is constructed via modal matrices (see Chap 7.5). This logic has to do with provability logic and Solovay's second theorem, and you can find a concise intro in the last part of Chap 3.8 where there's a subsection called Logic $\mathbf{S}$.