Proof-theoretic Semantics for Second-order Logic

2025-08-11

AI Summary

Traditional model-theoretic semantics for second-order logic—namely full and Henkin models—face foundational limitations in semantic clarity and philosophical burden, particularly due to their reliance on set-theoretic ontology. Method: This paper proposes a proof-theoretic alternative: logical meaning is defined via inference roles within atomic systems; set-theoretic models are replaced by “base-extension semantics”; second-order quantifiers are reconstrued as systematic rule substitutions; and Hilbert-style calculi are integrated with inference-role theory. Contribution/Results: The framework avoids ontological commitments to sets, accommodates both classical and intuitionistic logic, and achieves modal-style soundness and completeness for each. Its semantics is equivalent to Henkin semantics yet grounded exclusively in use-theoretic principles—yielding a philosophically lightweight, expressively adequate, and non-substantial semantic foundation for higher-order logic.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

We develop a proof-theoretic semantics (P-tS) for second-order logic (S-oL), providing an inferentialist alternative to both full and Henkin model-theoretic interpretations. Our approach is grounded in base-extension semantics (B-eS), a framework in which meaning is determined by inferential roles relative to atomic systems -- collections of rules that encode an agent's pre-logical inferential commitments. We show how both classical and intuitionistic versions of S-oL emerge from this set-up by varying the class of atomic systems. These systems yield modular soundness and completeness results for corresponding Hilbert-style calculi, which we prove equivalent to Henkin's account of S-oL. In doing so, we reframe second-order quantification as systematic substitution rather than set-theoretic commitment, thereby offering a philosophically lightweight yet expressive semantics for higher-order logic. This work contributes to the broader programme of grounding logical meaning in use rather than reference and offers a new lens on the foundations of logic and mathematics.

Problem

• Develops proof-theoretic semantics for second-order logic
• Offers alternative to model-theoretic interpretations
• Reframes second-order quantification as systematic substitution

Innovation

• Base-extension semantics for second-order logic
• Modular soundness and completeness results
• Reframing quantification as systematic substitution