Bi-intermediate logics of trees and co-trees
Nick Bezhanishvili, Miguel Martins, Tommaso Moraschini

Abstract:
A bi-Heyting algebra validates the Go ̈del-Dummett axiom (p → q) ∨ (q → p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this form are called bi-Go ̈del algebras and form a variety that algebraizes the extension bi-LC of bi-intuitionistic logic axiomatized by the Go ̈del-Dummett axiom. In this paper we initiate the study of the lattice Λ(bi-LC) of extensions of bi-LC. We develop the method of Jankov formulas for bi-Go ̈del algebras and use them to prove that Λ(bi-LC) has the size of the continuum. We also show that bi-LC is not locally tabular and give a criterion of locall tabularity in Λ(bi-LC).