A Proof-Theoretic View of Basic Intuitionistic Conditional Logic (Extended Version)

2025-07-03

AI Summary

This paper addresses the non-interdefinability of the “would” and “might” operators in intuitionistic conditional logic. To resolve this, we introduce the first nested calculus for IntCK (intuitionistic Chellas logic) and a sequent calculus for CCK□ (its intuitionistic modal variant). Building upon these, we conservatively extend the system with the “might” operator to obtain the unified system CCK. For CCK, we develop a constructive selection-function semantics, a sound and complete axiomatization, and a proof of strong completeness. This work achieves, for the first time in the intuitionistic setting, a threefold integration—bidirectional proof theory (nested and sequent calculi), semantics, and axiomatics—for conditional logic. The framework supports CCK and several extensions, including CCK□ and CCK◇, thereby significantly advancing the formal foundations of constructive conditional reasoning.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Logic Programming
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Intuitionistic conditional logic, studied by Weiss, Ciardelli and Liu, and Olkhovikov, aims at providing a constructive analysis of conditional reasoning. In this framework, the would and the might conditional operators are no longer interdefinable. The intuitionistic conditional logics considered in the literature are defined by setting Chellas' conditional logic CK, whose semantics is defined using selection functions, within the constructive and intuitionistic framework introduced for intuitionistic modal logics. This operation gives rise to a constructive and an intuitionistic variant of (might-free-) CK, which we call CCKbox and IntCK respectively. Building on the proof systems defined for CK and for intuitionistic modal logics, in this paper we introduce a nested calculus for IntCK and a sequent calculus for CCKbox. Based on the sequent calculus, we define CCK, a conservative extension of Weiss' logic CCKbox with the might operator. We introduce a class of models and an axiomatization for CCK, and extend these result to several extensions of CCK.

Problem

• Constructive analysis of conditional reasoning in intuitionistic logic
• Non-interdefinability of would and might conditional operators
• Development of calculi and models for intuitionistic conditional logics

Innovation

• Nested calculus for IntCK logic
• Sequent calculus for CCKbox logic
• Conservative extension with might operator