Bi-intermediate logics of trees and co-trees

A bi-Heyting algebra validates the G\"odel-Dummett axiom $(p\to q)\vee (q\to p)$ 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 kind are called bi-G\"odel algebras and form a variety that algebraizes the extension $\mathsf{bi}$-$\mathsf{LC}$ of bi-intuitionistic logic axiomatized by the G\"odel-Dummett axiom. In this paper we initiate the study of the lattice $\Lambda(\mathsf{bi}$-$\mathsf{LC})$ of extensions of $\mathsf{bi}$-$\mathsf{LC}$. We develop the methods of Jankov-style formulas for bi-G\"odel algebras and use them to prove that there are exactly continuum many extensions of $\mathsf{bi}$-$\mathsf{LC}$. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of $\mathsf{bi}$-$\mathsf{LC}$. We introduce a sequence of co-trees, called the finite combs, and show that a logic in $\mathsf{bi}$-$\mathsf{LC}$ is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest non-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$ and consequently, a unique pre-locally tabular extension of $\mathsf{bi}$-$\mathsf{LC}$. These results contrast with the case of the intermediate logic axiomatized by the G\"odel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.