Nested sequents for intermediate logics: the case of Gödel-Dummett logics

Abstract

We present nested sequent systems for propositional Gödel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these Gödel-Dummett logics, we introduce a new structural rule, called the linearity rule, which (bottom-up) operates by linearising branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking if data exists along certain paths within a nested sequent. Such rules require us to generalise our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalisation of the usual nested sequent formalism. Our calculi exhibit favourable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems.