Ian Hodkinson

Canonicity in power

Whether canonicity of a (normal) modal logic depends on the number of ambient propositional atoms has been a basic question in modal logic since Kit Fine raised it in a famous 1973 Uppsala conference. Essentially it asks whether every modal logic that is valid in its canonical frame generated by countably many atoms is actually valid in all its canonical frames. Though this remains open after 50 years, in joint work with Rob Goldblatt we found a positive solution for a number of logics, which are generalisations of Fine's logics of finite width. The solution uses model theory and is based on a kind of Löwenheim-Skolem argument.

References:
K. Fine, Some connections between elementary and modal logic, Proc. 3rd Scandinavian logic symposium, S. Kanger (ed), Uppsala, 1973, North-Holland, 1975, pp.15-31.
R. Goldblatt, I. Hodkinson, Canonicity in power and modal logics of finite achronal width, Rev. Symbolic Logic (2023), 31pp.

Link:
https://doi.org/10.1017/S1755020323000060