Definability and Interpolation in Philosophy

This paper is a historical tour of occurrences of the Craig interpolation theorem and the Beth definability theorem in philosophy since the 1950s. We identify the notion of dependence as one major red thread behind these, and include some new technical results, in particular, on logical system translations and generalized definability

💡 Research Summary

The paper offers a comprehensive historical and technical survey of the Craig interpolation theorem and the Beth definability theorem as they have appeared in philosophical discourse since the 1950s. Although there is no single research programme uniting these topics, the authors trace a series of intellectual threads that connect the two theorems to a central philosophical notion: dependence. The survey is organized into three major parts.

First, the authors restate Beth’s definability theorem in modern model‑theoretic language. Implicit definability is presented as the condition that any two models of a first‑order theory which agree on the interpretation of a base vocabulary L must also agree on the interpretation of an extra predicate P. Explicit definability is the existence of a formula φ(x) in the language L such that the theory proves ∀x(Px↔φ(x)). The equivalence of these two notions is proved, and the authors extend the theorem by introducing the weaker notion of “potential isomorphism” to capture less rigid invariance conditions.

Second, Craig’s interpolation theorem is recalled in its simplest form: if a formula φ(P,Q) semantically entails ψ(Q,R), then there exists an intermediate formula α(Q) using only the shared predicate Q such that φ entails α and α entails ψ. The paper shows how this theorem immediately yields Beth’s theorem for first‑order logic, but also emphasizes that interpolation is strictly stronger: there are logical fragments (e.g., the guarded fragment) that satisfy interpolation without satisfying Beth’s theorem. This asymmetry is used to illustrate the distinct logical strength of the two results.

The core philosophical discussion then focuses on what counts as a “logical system.” The authors adopt an abstract model‑theoretic view: a logic consists of a set of sentences, a class of structures, and a truth relation. Two logics L₁ and L₂ are considered the same when there exist faithful syntactic translations τ: L₁→L₂ and σ: L₂→L₁ together with semantic back‑translations T, S between their model classes such that for every pair of sentences φ,ψ the entailment φ⊨ψ in L₁ holds iff τ(φ)⊨τ(ψ) in L₂, and similarly in the opposite direction. This bidirectional fidelity is proposed as a minimal criterion for logical identity, and the authors argue that requiring interpolation to be preserved under such translations imposes a non‑trivial constraint on admissible translations.

A novel contribution of the paper is the reinterpretation of translations as “generalized definability.” If a syntactic translation τ provides an explicit definition of the primitive symbols of L₁ in terms of formulas of L₂, then the corresponding semantic transformation T (mapping L₂‑models to L₁‑models) yields an implicit definability condition across possibly different model domains. By formalising this situation in a first‑order theory Σ that talks about two distinguished unary predicates M and N (representing the domains of the source and target logics) and a binary relation Z encoding an L₂‑isomorphism, the authors show how the Projective Beth theorem can be derived from Craig interpolation. The construction uses compactness to turn a set of first‑order consequences into a single interpolant δ(M,L₂,b) that captures the essential property of an object b in the source model. Consequently, an explicit definition of the target predicate P in terms of the source language is obtained.

The philosophical applications are then surveyed. Sections 3‑6 apply Beth’s theorem to topics such as super‑venience, determinism, the nature of questions, and various notions of dependence, showing how implicit versus explicit dependence can be captured by the theorem. Sections 7‑9 turn to Craig interpolation, using it to analyse generalized consequence relations, the architecture of scientific theories, and modularity in belief systems. In each case the shared vocabulary of an interpolation serves as a formal analogue of the “common ground” that allows two theories or belief sets to be linked without loss of content.

Finally, the paper revisits the 19th‑century work of Bernard Bolzano, arguing that his early logical investigations anticipate the modern view of definability and interpolation as tools for relating different logical systems.

In conclusion, the authors argue that both Beth’s definability theorem and Craig’s interpolation theorem provide a unifying meta‑logical framework for many philosophical debates about dependence, modularity, and the identity of logical systems. By formalising translations as generalized definitions and by highlighting the extra strength of interpolation, the paper opens new avenues for research at the interface of logic, philosophy of language, epistemology, and the foundations of science.