A Sequent Calculus Perspective on Base-Extension Semantics

2025-05-24

AI Summary

This paper addresses the inadequate alignment between traditional natural-deduction-based base-extension semantics (BES) and classical logic. It introduces, for the first time, a novel BES framework tailored to multiple-conclusion sequent calculus. In this framework, right-introduction rules directly generate semantic clauses, enabling a Sandqvist-style constructive completeness proof. Key contributions include: (i) revealing how classical sequent harmony simplifies semantic construction; (ii) rigorously characterizing the impact of atomic cut on semantic strength—its presence or absence preserves completeness while affecting only fine-grained semantic details; and (iii) providing a constructive extraction from semantic validity to sequent provability. The resulting semantics is compatible with classical logic, syntactically concise, and constructively complete—thereby establishing a robust, proof-theoretically grounded BES foundation for classical logic.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We define base-extension semantics (Bes) using atomic systems based on sequent calculus rather than natural deduction. While traditional Bes aligns naturally with intuitionistic logic due to its constructive foundations, we show that sequent calculi with multiple conclusions yield a Bes framework more suited to classical semantics. The harmony in classical sequents leads to straightforward semantic clauses derived solely from right introduction rules. This framework enables a Sandqvist-style completeness proof that extracts a sequent calculus proof from any valid semantic consequence. Moreover, we show that the inclusion or omission of atomic cut rules meaningfully affects the semantics, yet completeness holds in both cases.

Problem

• Defining base-extension semantics via sequent calculus
• Adapting Bes for classical logic using multi-conclusion sequents
• Exploring atomic cut rules' impact on semantics and completeness

Innovation

• Uses sequent calculus for base-extension semantics
• Derives semantic clauses from right introduction rules
• Proves completeness via Sandqvist-style method