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

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.

💡 Research Summary

The paper develops a comprehensive theory of second‑order (quantification over propositions) intuitionistic tense logic, denoted IKt₂, together with its classical counterpart Kt₂. The authors begin by fixing a language that contains propositional symbols, second‑order variables, implication, two “white” modalities (□ for forward, ■ for backward) and second‑order universal quantification. They show that, thanks to the expressive power of second‑order intuitionistic propositional logic, all usual propositional connectives (∧, ∨, ⊥, ∃) can be defined via impredicative encodings, and, crucially, the diamond modalities (◇ and its backward analogue) can be defined solely in terms of the box modalities and second‑order quantification: \