Nested Sequents for Intuitionistic Logics

Abstract

Relatively recently nested sequent systems for modal logics have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. In an earlier paper I showed there was a strong connection between modal nested sequents and modal prefixed tableaux. In this paper I show the connection continues to intuitionistic logic, both standard and constant domain, relating nested intuitionistic sequent calculi to intuitionistic prefixed tableaux. Modal nested sequent machinery generalizes one sided sequent calculi-intuitionistic nested sequents similarly generalize two sided sequents. It is noteworthy that the resulting system for constant domain intuitionistic logic is particularly simple. It amounts to a combination of intuitionistic propositional rules and classical quantifier rules, a combination that is known to be inadequate when conventional intuitionistic sequent systems are used.

Cut Elimination in Nested Sequents for Intuitionistic Modal Logics

“…Furthermore, note that our treatment of the "intuitionistic" in nested sequents is different from the one in [9] (which is two-sided inside each nesting and does not treat modalities), and the one in [13], (which focuses on variants of biintuitionistic tense logics, and does not cover all 15 logics in the IS5-cube).…”

Inducing Syntactic Cut-Elimination for Indexed Nested Sequents

“…Nested sequent calculi have been presented for intuitionistic logic Ip [8] and also for extensions of intuitionistic logic with modalities [22,15]. Logics between classical and intuitionistic logic are called intermediate logics.…”

Modal interpolation via nested sequents

“…Further, our method heavily relies on the classical nature of the underlying logics because Brünnler's nested sequent calculus we use is not suitable, for example, for intuitionistic-based logics. There are several recently developed versions of nested sequents adapted for intuitionistic logic of various flavors, notably by Fitting [5], by Goré et al [9], and by Straßburger [13]. Thus, it is natural to see whether our method can be extended to such intuitionistic nested sequents.…”