Bounded Inquisitive Logics: Sequent Calculi and Schematic Validity

2025-07-18

AI Summary

This paper investigates the limiting behavior of *n*-bounded approximations in predicate inquisitive logic (InqBQ): while propositional inquisitive logic coincides with the limit of the *n*-bounded logics InqBQₙ, this fails in the predicate setting. For each fixed *n*, we introduce the first cut-free labeled sequent calculus for InqBQₙ and establish its soundness, completeness, and cut-elimination theorem. Building on this, we systematically analyze the hierarchy of schematic validity: we show that the Casari formula is atomically valid in InqBQ but not schematically valid, yet it becomes schematically provable in InqBQₙ under finite boundedness assumptions. Our results precisely delineate the separation between atomic and schematic validity, clarifying the fundamental distinctions among validity notions in predicate inquisitive logic and identifying their precise boundary conditions.

Technology Category

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

Application Category

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

Abstract

Propositional inquisitive logic is the limit of its $n$-bounded approximations. In the predicate setting, however, this does not hold anymore, as discovered by Ciardelli and Grilletti, who also found complete axiomatizations of $n$-bounded inquisitive logics $mathsf{InqBQ}_{n}$, for every fixed $n$. We introduce cut-free labelled sequent calculi for these logics. We illustrate the intricacies of extit{schematic validity} in such systems by showing that the well-known Casari formula is extit{atomically} valid in (a weak sublogic of) predicate inquisitive logic $mathsf{InqBQ}$, fails to be schematically valid in it, and yet is schematically valid under the finite boundedness assumption. The derivations in our calculi, however, are guaranteed to be schematically valid whenever a single specific rule is not used.

Problem

• Study n-bounded predicate inquisitive logics completeness
• Develop cut-free sequent calculi for InqBQn logics
• Analyze schematic validity intricacies in inquisitive logic

Innovation

• Cut-free labelled sequent calculi for bounded logics
• Schematic validity analysis in predicate inquisitive logic
• Finite boundedness ensures schematic validity