Inference-Behaviour Semantics for All$^\ast$ Connectives in Two-Dimensional Sequent Calculi

2026-07-19

AI Summary

This study addresses the precise semantic definition of logical connectives grounded in inferential behavior and systematically identifies all semantically meaningful connectives. Within a framework of inferential semantics, the authors conduct the first comprehensive definability analysis of all 10,816 possible pairs of connective rules by integrating two-dimensional sequent calculus with minimal substructural consequence relations. The investigation successfully isolates 21 semantically significant connectives—including intuitionistic and dual-intuitionistic negation, conjunction, disjunction, implication, and their converses—and provides rigorous semantic clauses for each. Furthermore, the work demonstrates that classical logical connectives decompose into two distinct semi-semantic components, thereby elucidating their semantic interrelations across different logical systems.

Abstract

Inference-behaviour semantics (I-bS) is a new approach to proof-theoretic semantics, grounded in two inferentialist principles: (1) the use of an expression in reasoning determines its meaning, and (2) a connective is defined by its operational rules. I-bS operationalises these ideas by measuring the syntactic use of a connective in the proof of its definability, against a substructurally minimal derivability relation. I-bS thereby gives the meaning of a connective in terms of its semantic clause, i.e. minimal substructural rule pair. This paper validates and verifies I-bS by applying it to all $10,816$ connective rule pairs that can be formulated in two-dimensional sequent calculi using at most two premiss sequents and at most two active formulae. As a result, we find semantic clauses for exactly $21$ meaningful connectives, namely bottom, top, two negations (intuitionistic and dual-intuitionistic), group and lattice conjunction, disjunction and implication, as well as their converses and inverses. We use these results to precisely map the semantic interrelations among the connectives, across linear, classical, intuitionistic, dual-intuitionistic, minimal, and lattice logic. Most notably, we find that intuitionistic negation, disjunction and implication each capture half of the meaning of their classical counterparts.

Problem

• inference-behaviour semantics
• connectives
• sequent calculi
• proof-theoretic semantics
• semantic clause

Innovation

• Inference-Behaviour Semantics
• two-dimensional sequent calculi
• substructural logic
• semantic clause
• proof-theoretic semantics