A modular approach to completeness, Beth-definability and Craig interpolation
Ivan Di Liberti, FLoV
We provide an algebraic definition of “fragment of geometric logic”. Those are understood as families of locales enjoying algebraic properties. Meet-semilattices, Distributive lattices, and other relevant classes of posets corresponding to established propositional logics arise as fragments of geometric logic in this sense. Thanks for the modularity of our approach we can tell what family of fragments of geometric logic verify a completeness theorem (with respect to booleans), Craig-interpolation, and Beth definability. In its current form, the framework cannot encompass modal logics, as modal operators are structure (as opposed to property). This is a report of joint work with Lingyuan Ye.