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Definability and Interpolation in Philosophy

This paper provides a historical overview of the applications of the Craig interpolation and Beth definability theorems in philosophy since the 1950s, identifying "dependence" as a central theme while contributing new technical findings on logical translations and generalized definability.

Original authors: Johan van Benthem

Published 2026-02-10

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CC BY 4.0

Original authors: Johan van Benthem

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery. You have two different sets of clues, and you’re trying to figure out how they connect. This paper, written by a prominent logician, explores how two powerful mathematical "rules of connection"—called Beth’s Theorem and Craig’s Interpolation Theorem—act like the ultimate detective tools for understanding how different ideas, scientific theories, and even human beliefs are linked.

Here is the breakdown of the paper using everyday metaphors.

1. The Two Master Tools

To understand the paper, you first need to meet the two "superheroes" of logic:

The "Hidden Identity" Tool (Beth’s Theorem):
Imagine you are watching a play. You don't know the actor's real name, but you see that every time they enter the stage, they always wear a red hat, carry a cane, and walk with a limp. Even though you haven't been told their name, the "role" they play is so specific that the name is effectively locked in.

In logic: If a concept is "implicitly" defined (meaning its behavior is so consistent that it can't be anything else), Beth’s Theorem proves it can be "explicitly" defined (meaning we can write down a clear, direct formula for it).

The "Common Ground" Tool (Craig’s Interpolation):
Imagine two people, Alice and Bob, are having a heated argument. Alice is talking about cooking, and Bob is talking about car engines. They seem to be speaking different languages. But suddenly, they both start talking about heat.

In logic: If one idea leads to another, even if they use totally different vocabularies, there must be a "middleman" concept—a bridge made only of the words they both share—that explains the connection.

2. How These Tools Apply to Real Life

The author shows that these aren't just math puzzles; they are the "glue" that holds various parts of human thought together.

Supervenience: The "Shadow" Effect

Think of a shadow on the ground. The shadow (the "mental" or "moral" state) depends entirely on the object casting it (the "physical" state). If you move the object, the shadow moves. The paper discusses how we can use these logical tools to ask: Is the shadow just a direct result of the object, or is there something else going on?

Determinism: The "Movie Script"

Imagine watching a movie. If you know exactly where every character is at minute 10, can you predict exactly where they will be at minute 60? That is determinism. The paper uses these logical tools to look at scientific theories (like physics) to see if the "rules of the universe" allow us to write a clear "script" that connects the present to the future.

Questions and Information: The "Missing Puzzle Piece"

When you ask a question, you are looking for a specific piece of information to complete a picture. The paper explains that the logic of questions works just like the logic of statements. If knowing the answer to Question A and Question B automatically gives you the answer to Question C, there is a logical "bridge" connecting them.

Scientific Theories: The "Invisible Scaffolding"

In science, we have things we can see (like a thermometer reading) and things we can't see (like "electron spin"). The paper explores how the "unseen" parts of a theory are tied to the "seen" parts. It asks: How much of our scientific "truth" is just a way of organizing the things we can actually observe?

Knowledge and Belief: The "Mental Filing Cabinet"

Our brains don't store everything in one giant pile; we organize knowledge into topics (food, family, work). The paper uses the "Common Ground" tool to show how our beliefs are "modular." If you learn something new about "food," it shouldn't necessarily rewrite your entire "work" file, unless there is a logical bridge connecting the two.

3. The Big Picture: Why Does This Matter?

The author concludes that language and vocabulary are everything.

Just as a painter uses different brushes to create different textures, scientists and philosophers use different "vocabularies" to describe the world. These two logical theorems are the rules that tell us when those different "brushes" are actually painting the same picture.

By studying these connections, we aren't just doing math; we are learning the fundamental architecture of how information, science, and human thought are woven together.

Technical Summary: Definability and Interpolation in Philosophy

Author: Johan van Benthem
Context: Draft chapter for Theory and Applications of Craig Interpolation (Ubiquity Press, 2026).

1. Problem Statement

The paper explores the intersection between formal logic and various branches of philosophy (philosophy of logic, science, language, epistemology, and metaphysics). The central problem is to identify how two fundamental meta-logical properties—Beth’s Definability Theorem and Craig’s Interpolation Theorem—manifest as underlying structural themes in philosophical discourse.

While these theorems are often viewed as technical properties of logical systems (specifically first-order logic), the author investigates whether they serve as formal analogues for philosophical concepts such as supervenience, determinism, dependence, information flow, and modularity. The paper seeks to move beyond mere coincidence to show that these theorems capture the essence of how different "realms" or "vocabularies" relate to one another.

2. Methodology

The author employs a comparative and conceptual mapping methodology. Rather than presenting a new mathematical proof, the paper performs a "light tour" of existing philosophical literature, re-interpreting classical philosophical problems through the lens of model theory and algebraic logic.

The methodology follows a specific logical trajectory:

Semantic Invariance: Mapping philosophical notions of "similarity" or "equivalence" (e.g., supervenience) to model-theoretic notions like isomorphism or potential isomorphism.
Implicit vs. Explicit Dependence: Using Beth’s Theorem to bridge the gap between implicit relations (how one property behaves in relation to another across models) and explicit relations (how one property can be defined via a formula in a specific vocabulary).
Cross-Model Transfer: Utilizing the "surplus" of Craig Interpolation to explain how information or truth-values transfer between different structures or "situations" (as seen in Situation Theory).

3. Key Contributions and Results

The paper categorizes the application of these theorems into several thematic domains:

Supervenience and Determinism (Implicit vs. Explicit):
- Supervenience: The author identifies supervenience as a form of implicit semantic definability. He notes that the philosophical debate over whether supervenience implies "reduction" is essentially a debate over whether Beth’s Theorem applies to the specific logic used to describe the realms in question.
- Determinism: Using Montague’s analysis of physical theories, the paper shows that "Laplacian determinism" (where the state at $t_0$ determines the state at $t$ ) can be formulated as implicit definability. Craig Interpolation then guarantees that this dependence can be expressed as an explicit functional definition.
Questions and Dependence Logics:
- The paper demonstrates that the logic of questions (where consequence is defined by the refinement of partitions) relies on the Projective Beth Theorem. This provides a formal basis for the "expressive completeness" of logical constants in answering questions.
- It links these to Dependence Logics, where functional dependence between variables is the central mechanism.
Information Flow and Situation Theory:
- A significant result is the characterization of Interpolation as Entailment along Potential Isomorphism. The author shows that the existence of an interpolant $\alpha$ in vocabulary $L$ is equivalent to the ability to transfer truth from model $M$ to model $N$ whenever $M$ and $N$ are $L$ -potentially isomorphic.
Scientific Theories and Modularity:
- Theoretical Terms: The paper discusses how the "empirical content" of a theory is related to the expansion of observational structures to include theoretical ones, touching upon the concept of "Ramsey-eliminability."
- Modularity: Using the Robinson Joint Consistency Theorem and Interpolation, the author shows that bodies of knowledge (beliefs) can be formally "split" into disjoint modules, and that interpolation governs the information flow between these modules.

4. Significance

The significance of this work is twofold:

For Philosophy: It provides a rigorous mathematical framework to refine "fuzzy" philosophical concepts. It suggests that when philosophers argue about reductionism or causality, they are often implicitly making assumptions about the meta-logical properties (like the Beth property) of the underlying logic they are using.
For Logic: It highlights the "expressive completeness" and "information-theoretic" value of interpolation. It suggests that interpolation is not just a technical curiosity but a fundamental requirement for any logic intended to model "transitions," "definitions," or "information transfer" between different conceptual domains.

Conclusion: The paper concludes that while the "linguistic turn" in philosophy has brought language to the forefront, the formal study of how vocabularies interact—via interpolation and definability—remains a vital, unifying bridge between the mathematical rigor of logic and the conceptual depth of philosophy.