Higher-order illative combinatory logic

We show a model construction for a system of higher-order illative combinatory logic $\mathcal{I}_\omega$, thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system $\mathcal{I}_0$ of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

💡 Research Summary

The paper presents a thorough model‑theoretic construction for a higher‑order illative combinatory logic system denoted 𝓘_ω and uses this construction to establish the strong consistency of the system. The authors begin by recalling the background of illative combinatory logic, especially the system 𝓘₀ introduced by Barendregt, Bunder, and Dekkers, and they point out that 𝓘₀ does not fully capture first‑order intuitionistic predicate logic when second‑order propositional quantifiers are added. To overcome this limitation they define an extended calculus 𝓘_ω that admits higher‑order types, λ‑abstraction, and the usual combinators K, S, and I, together with explicit rules for universal and existential quantification over propositional variables.

The core technical contribution is the definition of a “regular model” for 𝓘_ω. This model is a Kripke‑style structure consisting of a set of worlds W together with an accessibility relation R. Each world interprets the base type o as a non‑empty set, and function types σ→τ are interpreted as the set of all functions from the interpretation of σ to that of τ at the same world. Quantifiers are interpreted globally: a universal quantifier ∀α. φ holds at a world w if φ holds at every world v with w R v, while an existential quantifier ∃α. φ holds if there exists at least one such v where φ holds. The authors prove a series of preservation lemmas showing that every inference rule of 𝓘_ω (assumption, elimination, introduction, and the quantifier rules) is sound with respect to this semantics. By constructing a concrete regular model in which the falsum ⊥ is never true, they demonstrate that no derivation of ⊥ exists, thereby establishing strong consistency for 𝓘_ω.

Having secured a sound model for the higher‑order system, the authors turn to the original question posed by Barendregt, Bunder, and Dekkers: can 𝓘₀ embed first‑order intuitionistic predicate logic together with second‑order propositional quantifiers? They answer affirmatively by defining an embedding function E that maps formulas of the combined logic (call it FOL²) into terms of 𝓘₀. Predicate symbols become constants of a higher‑order type Ω, propositional variables are represented as terms of type Ω, and quantifiers are simulated using λ‑abstractions that act as higher‑order combinators. The embedding respects provability: if a formula φ is derivable in FOL², then E(φ) is derivable in 𝓘₀. The proof relies on a normalization theorem for 𝓘₀ and a systematic “scheme transformation” that translates each logical connective and quantifier into a corresponding combinatory term while preserving the reduction behavior.

The paper concludes by discussing the significance of these results. The strong consistency of 𝓘_ω provides a solid foundation for further investigations into the interaction between combinatory logic, higher‑order type theory, and constructive logics. Moreover, the complete embedding of FOL² into 𝓘₀ shows that the latter system is more expressive than previously thought, partially answering the open problem raised by Barendregt et al. The authors suggest that their model‑theoretic techniques could be adapted to even richer systems, such as higher‑order modal logics or dependent type theories, opening new avenues for research in proof theory and the semantics of programming languages.