On Canonical Modal Logics That Are Not Elementarily Determined
Robert Goldblatt, Ian Hodkinson, and Yde Venema
There exist modal logics that are validated by their canonical frames
but are not sound and complete for any elementary class of frames.
Continuum many such bimodal logics are exhibited, including one of each
degree of unsolvability, and all with the finite model property.
Monomodal examples are also constructed that extend K4 and are related
to the proof of non-canonicity of the McKinsey axiom.
We dedicate this
paper to Max Cresswell, a pioneer in the study of canonicity, on the
occasion of his 65th birthday.