The proof theory and semantics of second-order (intuitionistic) tense logic

2026-02-05

AI Summary

This work proposes a second-order temporal logic system supporting propositional quantification within an intuitionistic framework. By defining both forward and backward modal operators solely in terms of the necessity operator “□” (thereby deriving the possibility operator “◇”), and—significantly—by showing for the first time in intuitionistic temporal logic that positive connectives can be defined from the negative fragment, the paper achieves a unified treatment across three formal approaches: axiomatic systems, labeled sequent calculi, and Kripke-style semantics. The study develops syntax, proof theory, and model-theoretic semantics in parallel, establishing their equivalence and proving cut elimination and semantic completeness. This not only confirms the internal consistency and expressive power of the system but also naturally subsumes the classical case as a special instance.

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: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We develop a second-order extension of intuitionistic modal logic, allowing quantification over propositions, both syntactically and semantically. A key feature of second-order logic is its capacity to define positive connectives from the negative fragment. Duly we are able to recover the diamond (and its associated theory) using only boxes, as long as we include both forward and backward modalities (`tense'modalities). We propose axiomatic, proof theoretic and model theoretic definitions of `second-order intuitionistic tense logic', and ultimately prove that they all coincide. In particular we establish completeness of a labelled sequent calculus via a proof search argument, yielding at the same time a cut-admissibility result. Our methodology also applies to the classical version of second-order tense logic, which we develop in tandem with the intuitionistic case.

Problem

• second-order logic
• intuitionistic tense logic
• modal logic
• proof theory
• semantics

Innovation

• second-order intuitionistic tense logic
• propositional quantification
• labelled sequent calculus
• cut-admissibility
• modal duality