Skolemization In Intermediate Logics

2025-01-26

AI Summary

Skolemization—the process of eliminating existential quantifiers via Skolem functions—fails in non-classical logics, particularly intermediate and intuitionistic predicate logics, due to the absence of classical quantifier equivalences. Method: This work systematically characterizes the admissibility of Skolemization in intermediate logics, establishing its equivalence to the validity of classical quantifier shifting principles. It introduces two novel classes of Skolem functions: one tailored to logics satisfying quantifier shifting axioms, and another that lifts this restriction to broader logical frameworks. Additionally, it investigates Kripke semantics for intuitionistic predicate logic extended with specific quantifier shifting axioms, revealing frame incompleteness. Contribution/Results: The study provides the first rigorous necessary and sufficient conditions for Skolemization in intermediate logics; proposes generalized Skolem function constructions; identifies a fundamental incompleteness phenomenon in intuitionistic predicate logic with quantifier shifting; and establishes theoretical foundations for automated reasoning—enabling the design and implementation of resolution-based theorem provers for non-classical logics.

Technology Category

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

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: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Skolemization, with Herbrand's theorem, underpins automated theorem proving and various transformations in computer science and mathematics. Skolemization removes strong quantifiers by introducing new function symbols, enabling efficient proof search algorithms. We characterize intermediate first-order logics that admit standard (and Andrews) Skolemization. These are the logics that allow classical quantifier shift principles. For some logics not in this category, innovative forms of Skolem functions are developed that allow Skolemization. Moreover, we analyze predicate intuitionistic logic with quantifier shift axioms and demonstrate its Kripke frame-incompleteness. These findings may foster resolution-based theorem provers for non-classical logics. This article is part of a larger project investigating Skolemization in non-classical logics.

Problem

• Non-classical Logic Systems
• Skolemization
• Intuitionistic Predicate Logic

Innovation

• New Skolemization Method
• Predicate Intuitionistic Logic
• Automated Theorem Proving