On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

Abstract

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus.

On the correspondence between nested calculi and semantic systems for intuitionistic logics

“…This refinement process allows us to capture the best of both worlds: we invoke the general results of the labelled setting to obtain satisfactory labelled calculi for a class of logics, and via refinement, transform the systems into nested calculi better suited for applications. Similar ideas and relationships have been discussed in the literature [13,17,16,20,21], where refined calculi (which can be considered nested calculi) were derived from labelled calculi for modal, intuitionistic, and related logics. (NB.…”

“…8 are entirely new. Beyond this, the paper was written with an increased focus and more detailed exposition on how the labelled calculi are refined in order to extract each nested calculus-this provides the paper with more explanatory force than [16]. Also, it should be noted that this paper corrects an error that occurs in the conference version.…”

“…Also, it should be noted that this paper corrects an error that occurs in the conference version. In [16], the labelled calculus G3IntQC is lacking a structural rule (called (ihd) in this paper) corresponding to the condition that domains in first-order intuitionistic models are non-empty, or inhabited. Without the inclusion of this rule, G3IntQC is incomplete as modus ponens cannot be simulated in the calculus (see Thm.…”

“…The last section, Sect. 9,concludes. This paper serves as an enhanced and revised version of the conference paper [16]. Most significantly, the nested-to-labelled translation of Sect.…”

“…3 and Appendix A for details). We have included this rule here and adjusted the content of [16] regarding G3IntQC accordingly. Furthermore, to increase the flow and readability of the paper, some results outside the scope of, or auxiliary to, our main focus (i.e.…”