In most logics disjunction and conjunction distribute over each other, meaning that a ∧ (b ∨ c) is logically equivalent to (a ∧ b) ∨ (a ∧ c). However, in some cases it is useful to let go of distributivity. A recent paper called "Positive Modal Logic Beyond Distributivity" describes a new frame semantics, so-called semilattice semantics, for the logic whose algebraic semantics is given by lattices. This logic is a positive logic in the sense it only contains disjunction, conjunction, true and false with no other connectives.

This talk will introduce a way to add implication to this logic, inspired by conditional logic. Doing so, we gain sound and complete semantics, a duality result, and a finite model property via (algebraic) filtrations. We conclude by discussing extensions of this logic, showing that it captures subintuitionistic and superintuitionistic logics, along with pointing out directions for further research.

This talk is part of a second year mathematics research project.