Simple Type Theory as Framework for Combining Logics

Simple type theory is suited as framework for combining classical and non-classical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about combinations of logics.

💡 Research Summary

The paper argues that Simple Type Theory (STT) provides a natural and powerful framework for combining a wide variety of logics, both classical and non‑classical. The authors begin by recalling the basic syntax and semantics of STT: a hierarchy of simple types built from the base types of individuals (ι) and propositions (o) using the function type constructor (→). Because every logical connective can be represented as a higher‑order function, STT is expressive enough to encode the syntax and semantics of many logics within a single, uniform language.

To substantiate the claim, the paper presents detailed embeddings of two representative families of non‑classical logics. First, multimodal logics (including quantified multimodal logics) are encoded by treating each modality □ₖ and ◇ₖ as a term of type (ι → o) → o and by representing the accessibility relation for modality k as a higher‑order predicate of type (ι → o) → (ι → o). This yields a faithful translation of the modal axioms and the Kripke semantics directly into STT. Second, intuitionistic logic is embedded by interpreting Kripke worlds as a preorder ≤ on propositions, again represented as a higher‑order predicate. Negation, implication, and other intuitionistic connectives are defined as higher‑order functions that respect the monotonicity conditions of the intuitionistic semantics. The authors prove that these embeddings are sound and complete with respect to the original logics, showing that validity in the source logic coincides with provability in STT.

Having established that individual logics can be internalised, the paper turns to the problem of combining them. The authors propose a systematic method: each constituent logic is assigned a distinct sub‑type (τ₁, τ₂, …) within the overall type hierarchy, while a common super‑type τ₀ is introduced to host interaction operators. Cross‑logic operators are defined only at the τ₀ level, and their typing constraints guarantee that they cannot be misapplied to terms belonging exclusively to one sub‑logic. This type‑based segregation prevents semantic clashes and preserves consistency. Moreover, the Henkin semantics for STT provides a single, well‑understood model theory that can simultaneously interpret the separate Kripke structures of the component logics. Consequently, the combined system inherits the completeness and compactness properties of STT.

The practical relevance of the approach is demonstrated by implementing several combined logics—such as a modal‑intuitionistic logic and a multimodal temporal logic—using off‑the‑shelf higher‑order automated theorem provers (Isabelle/HOL, LEO‑II, TPS). The experiments show that standard proof‑search procedures for STT can handle both intra‑logic reasoning (e.g., proving modal theorems) and inter‑logic reasoning (e.g., deriving consequences that involve both modal and intuitionistic operators) without any special extensions. In many cases the performance is comparable to dedicated provers for the individual logics, and the development effort is dramatically reduced because new combinations are obtained merely by adding type definitions and operator declarations.

Finally, the paper discusses advantages and limitations. Advantages include (1) a uniform representation language, (2) a single, well‑studied semantics that guarantees model existence and counter‑example generation, (3) reuse of mature higher‑order reasoning tools, and (4) extensibility: new logics can be added by defining appropriate types and higher‑order operators. Limitations concern the potential blow‑up of the search space when many high‑order types are introduced, and the fact that some exotic logics may require additional meta‑logical machinery beyond plain STT.

In conclusion, the authors convincingly demonstrate that Simple Type Theory serves as an expressive, semantically robust, and tool‑supported meta‑framework for the systematic combination of diverse logical systems. This opens the door to more ambitious projects such as large‑scale formal verification environments that need to reason simultaneously about modal, temporal, epistemic, and intuitionistic aspects. Future work will explore richer combinations (e.g., probabilistic‑modal‑temporal logics) and scalability of the approach in industrial verification settings.