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

TL;DR

This paper introduces a second-order extension of intuitionistic modal logic with tense modalities, providing axiomatic, proof theoretic, and model theoretic frameworks, and proving their equivalence and completeness.

Contribution

It develops a comprehensive second-order intuitionistic tense logic, including classical variants, with unified semantics and proof systems, and establishes key proof-theoretic properties.

Findings

Proof of equivalence of axiomatic, proof theoretic, and model theoretic definitions.

Completeness of a labelled sequent calculus with cut-admissibility.

Extension of methodology to classical second-order tense logic.

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.