On the Correspondence between Nested Calculi and Semantic Systems for Intuitionistic Logics

TL;DR

This paper explores the connection between labelled and nested proof calculi for various forms of intuitionistic logic, showing how nested calculi can be derived from labelled ones through rule elimination, inheriting key proof properties.

Contribution

It demonstrates that nested calculi for intuitionistic logic naturally originate from labelled calculi via structural rule elimination, enabling derivation from semantics.

Findings

Nested calculi inherit properties like completeness and cut admissibility from labelled calculi.

The method allows deriving nested calculi directly from a logic's semantics.

Refined labelled calculi can be used to obtain nested calculi with desirable proof-theoretic features.

Abstract

This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting's nested calculi naturally arise from their corresponding labelled calculi--for each of the aforementioned logics--via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic's semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic's semantics.