Positive modal logic beyond distributivity: duality, preservation and completeness
Anna Dmitrieva
Abstract:
In this thesis, we study positive (non-distributive) logics and their modal extensions by means of duality theory. Our work is inspired by topological dualities for semilattices and lattices established by Jipsen and Moshier (2014). First we construct a choice-free version of this duality using methods of Bezhanishvili and Holliday (2020). Then we establish a Priestley-like duality based on Jipsen & Moshier duality for arbitrary lattices. We call it Principal upset Priestley (PUP) duality. We define a filter completion of a lattice and, using PUP duality, prove by a Sahlqvist style argument that filter completions preserve all inequalities. That allows us to obtain a purely dual proof of a classical result by Baker and Hales (1974).
We also extend PUP duality by adding modal operators and prove preservation under filter completions for it, thus obtaining a modal version of the Baker and Hales theorem. Furthermore, we show that Sahlqvist-like inequalities correspond to first-order sentences, just as in standard modal logic. We also consider a PUP duality with a non-standard modality nabla, which can be seen as a generalization of the orthocomplementation operation on ortholattices. Therefore, our duality specializes to a duality for ortholattices that turns out to be equivalent to the one constructed by Goldblatt (1975) and BimboĢ (2007). Finally, we develop deductive systems reflecting the PUP dualities. We introduce general team semantics for these deductive systems and demonstrate how preservation by filter completions implies completeness for this type of semantics.