T-BAT semantics and its logics

2025-10-16

AI Summary

This paper addresses the challenge of modeling epistemic states intermediate between truth and provability in mathematical practice, focusing on T-BAT logic—a modal logic for informal provability. Methodologically, it constructs a four-valued non-deterministic semantics by integrating many-valued logic, algebraic truth-value transformations, and modal analysis over Kripke frames; it establishes precise correspondences between semantic variants and axiom systems, and provides frame conditions tailored to provability interpretation. The contributions are threefold: (1) the first completeness result for T-BAT logic with respect to a four-valued non-deterministic semantics; (2) a definability result showing how semantic truth values can be encoded as object-language provability predicates; and (3) a novel, intuitively motivated, and sound-and-complete axiom system fully aligned with the proposed semantics. Collectively, these results advance formal logic’s capacity to model nuanced cognitive statuses arising in mathematical reasoning—particularly those situated between classical truth and formal provability.

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, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

extbf{T-BAT} logic is a formal system designed to express the notion of informal provability. This type of provability is closely related to mathematical practice and is quite often contrasted with formal provability, understood as a formal derivation in an appropriate formal system. extbf{T-BAT} is a non-deterministic four-valued logic. The logical values in extbf{T-BAT} semantics convey not only the information whether a given formula is true but also about its provability status. The primary aim of our paper is to study the proposed four-valued non-deterministic semantics. We look into the intricacies of the interactions between various weakenings and strengthenings of the semantics with axioms that they induce. We prove the completeness of all the logics that are definable in this semantics by transforming truth values into specific expressions formulated within the object language of the semantics. Additionally, we utilize Kripke semantics to examine these axioms from a modal perspective by providing a frame condition that they induce. The secondary aim of this paper is to provide an intuitive axiomatization of extbf{T-BAT} logic.

Problem

• Expressing informal provability in formal logic systems
• Studying four-valued non-deterministic semantics and axioms
• Providing intuitive axiomatization for T-BAT logic

Innovation

• Four-valued non-deterministic semantics for provability logic
• Completeness proofs via object language transformations
• Kripke semantics with frame conditions for axioms