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Many Logics, One Methodology: A Plea for Logical Pluralism in Formalised Reasoning (preprint)

This position statement advocates for logical pluralism within a unifying meta-logical framework like LogiKEy, arguing that supporting multiple object logics in proof assistants—rather than enforcing a single foundational logic—better enables interdisciplinary research and large-scale theory development.

Original authors: Christoph Benzmüller, Daniel Kirchner, Luca Pasetto

Published 2026-05-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Christoph Benzmüller, Daniel Kirchner, Luca Pasetto

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

The Big Idea: One Toolbox, Many Rules

Imagine you are an architect. Usually, when you build a house, you pick one set of building codes (the "logic") and stick to them from the foundation to the roof. If you want to build a house with a different style of code, you have to start over with a completely new set of blueprints and tools.

The authors of this paper argue that this is a bad way to do things, especially when you are trying to build complex structures that mix different fields (like math and philosophy). They call the rigid approach "Logical Imperialism" (forcing one rulebook on everything) and propose "Logical Pluralism" instead.

Their solution is a method called LogiKEy. Think of LogiKEy as a universal translation hub. Instead of building a new house for every different rulebook, you build one giant, super-strong "Meta-House" (based on Classical Higher-Order Logic). Inside this Meta-House, you can set up different "rooms." Each room has its own specific rulebook (like a rulebook for time, a rulebook for ethics, or a rulebook for God).

Because all these rooms are inside the same Meta-House, you can use the same powerful tools (like automated proof-checkers) to inspect, compare, and even mix the rules of different rooms without having to rebuild the whole foundation every time.

The Problem with "One Size Fits All"

The paper warns that modern computer systems for math often act like imperialists. They pick one foundational logic (like a specific type of math logic) and say, "This is the only truth."

The authors give a funny example: Division by Zero.

In some computer math libraries, they just decide that $1/0 = 0$ to make the computer calculations easier.
This works fine for engineering, but if you are a philosopher asking deep questions about existence, this rule is weird. It implies that "nothing" is actually "something."
If you build a massive library of math based on this rule, future users (or even AI) might accidentally treat this weird rule as a universal truth of the universe, not just a convenient shortcut.

The authors want a system where you can see these shortcuts clearly and say, "Oh, that's just a rule for this specific room, not the whole building."

The Case Study: Gödel's God Argument

To prove their method works, the authors applied it to a famous philosophical puzzle: Gödel's Modal Ontological Argument. This is a complex mathematical proof trying to show that a "God-like" being must exist based on the definition of "positive properties" (goodness, power, knowledge, etc.).

The Old Way:
Previously, people tried to prove this using standard math logic. But standard math often assumes the world is finite or simple. This led to "trivial" proofs where the argument worked only because the math was too simple (like trying to prove a complex mystery by assuming there are only two people in the world).

The New Way (Using LogiKEy):
The authors used their "Universal Translation Hub" to do something new:

They took Gödel's philosophical argument (which lives in a "Modal Logic" room—logic that deals with possibility and necessity).
They brought in "Mathematical Realism" (the idea that infinite mathematical objects, like numbers, really exist).
They combined them inside the Meta-House.

The Surprise Result:
When they combined Gödel's rules with the existence of infinite mathematical objects, the math changed the philosophy.

They discovered that if you accept that infinite mathematical objects exist, then the set of "positive properties" in Gödel's theory cannot be finite or even countable.
It forces the set of "good things" to be uncountably infinite (like the number of points on a line, rather than just a list of numbers).
This rules out "simple" or "small" versions of God that some previous computer proofs had accidentally allowed.

Why This Matters

The paper isn't just about proving God exists or doesn't exist. It's about how we use computers to think.

Flexibility: It allows researchers to swap out the underlying rules of a theory to see how the results change, without throwing away all their work.
Transparency: It makes sure that hidden assumptions (like "division by zero equals zero") are visible and can be questioned.
Interdisciplinary Work: It lets philosophers and mathematicians work together in the same digital space, even if they usually speak different "logical languages."

Summary Analogy

Imagine a Swiss Army Knife.

Logical Imperialism is like having a knife with only one blade. If you need to saw wood, you are stuck.
Logical Pluralism (LogiKEy) is the full Swiss Army Knife. You have a blade, a screwdriver, a can opener, and a saw all in one handle. You can switch tools instantly to fit the job.
The authors showed that by using this "Swiss Army Knife" approach, they could take a philosophical argument about God, mix it with advanced math about infinity, and discover that the argument requires a much more complex, infinite structure than anyone had realized before.

The paper concludes that this flexible, multi-tool approach is the best way to handle the messy, complex, and interdisciplinary questions of the future.

Technical Summary: Many Logics, One Methodology: A Plea for Logical Pluralism in Formalised Reasoning

Problem Statement
The paper addresses the tension between the need for diverse logical foundations in interdisciplinary research (specifically bridging mathematics, metaphysics, and theology) and the prevailing trend in modern proof assistants toward "logical imperialism." Logical imperialism is defined as the rigid adoption of a single foundational logic (e.g., dependent type theory or classical set theory) as the unquestioned basis for all large-scale theory developments. The authors argue that this monoculture fosters "invisible assumptions" (such as the law of excluded middle, existential import, or specific treatments of undefinedness) that hinder the reuse of libraries across disciplines with conflicting foundational commitments. Furthermore, in specific domains like Gödel's modal ontological argument, standard mathematical assumptions can lead to trivial or inconsistent interpretations if not carefully managed within a flexible logical framework.

Methodology: LogiKEy and Shallow Embeddings
The paper advocates for Logical Pluralism at the object-logic level, implemented within a unifying meta-logical framework called LogiKEy (Logic and Knowledge Engineering Framework and Methodology).

Architecture: LogiKEy employs a layered architecture:
- L0 (Meta-Logic): Classical Higher-Order Logic (HOL), serving as the host environment (e.g., Isabelle/HOL).
- L1 (Object-Logic): Various non-classical logics (e.g., Higher-Order Modal Logic, deontic logic, free logic) are embedded into L0.
- L2 (Domain Theories): Specific theories (e.g., Gödel's ontological theory) are formalized on top of L1.
- L3 (Applications): Applications utilizing the domain theories.
Embedding Technique: The paper primarily utilizes shallow embeddings, where object-logic formulas are identified with HOL terms (via standard translation). This approach sacrifices access to object-logic-specific syntactic tooling but gains access to the full automation stack of the host system (e.g., Sledgehammer, Nitpick, Metis, SMT bridges).
Flexibility: Within this framework, the object-logic is a negotiable parameter. It can be analyzed, revised, exchanged, or combined without rebuilding the entire system. The methodology is not bound to Isabelle/HOL but requires a sufficiently strong host logic.
Deep vs. Shallow: While the current work relies on shallow embeddings, the authors note that deep embeddings (inductive datatypes of formulas) can coexist with shallow ones, with mutual faithfulness proofs being mechanized for propositional and first-order modal logics. Extending this to Higher-Order Modal Logic (HOML) is a priority.

Key Contributions and Results
The paper presents a concrete instantiation of the LogiKEy methodology by extending Gödel's modal ontological argument with modalized mathematical notions.

Formalization of HOML in HOL: The authors utilize a shallow embedding of Higher-Order Modal Logic (HOML) into classical HOL. This includes definitions for world-relativized propositions, actualist vs. possibilist quantifiers, and modal operators (necessity $\Box$ ).
Modalized Mathematical Notions: The paper introduces foundational mathematical concepts (natural numbers, equipollence, cardinality, infinity) as definitions (abbreviations) within the HOL meta-logic, rather than as new axioms. This allows these concepts to interact naturally with the embedded modal logic.
Cardinality of Positive Properties:
- The authors formalize Gödel's notion of "positive properties" as a modal ultrafilter.
- They demonstrate that assuming the existence of mathematical entities (specifically, an infinite domain of entities) within Gödel's theory forces the set of positive properties to be uncountably infinite.
- This result is derived via a modalized variant of Cantor's theorem: there is no surjective mapping from the set of entities to the set of positive properties.
- The proof shows that trivial finite models (often found in literature) are excluded if Gödel's mathematical realism (the independent existence of mathematical objects) is granted.
Verification of Church's Postulates: The paper verifies that core postulates of Church's Type Theory (for HOL) hold within the embedded HOML, with the expected exception of unrestricted Boolean extensionality (which is blocked by the modal operators to preserve intensionality).

Significance and Claims
The paper positions itself as a position statement arguing for the necessity of logical pluralism in formal reasoning, particularly for interdisciplinary applications.

Against Logical Imperialism: The authors claim that rigid foundational commitments risk embedding invisible assumptions that future users (and AI systems) may inherit unreflectively. LogiKEy offers a way to make these commitments explicit and revisable.
Methodological Advantage: The LogiKEy approach allows researchers to "step outside" any given system (such as Zalta's Principia Logico-Metaphysica) to critically examine its foundations and compare it with alternatives. This is contrasted with "principled monism," where a single rich foundation is adopted, potentially limiting the scope of inquiry for those who do not endorse its specific ontological commitments.
Formal Precision: The paper claims that the combination of logical pluralism and shallow embeddings enables the study of complex metaphysical arguments (like Gödel's) with high formal precision and automation, revealing consequences (such as the uncountability of positive properties) that might otherwise remain hidden.
Future Work: The authors identify the integration of deep-and-shallow embeddings for HOML and the further exploration of Gödel's generalized axioms (specifically regarding properties as existing entities) as key future directions. They also note that while the current proofs rely on the HOL meta-layer, constructing pure HOML-level proofs remains an open task.

In summary, the paper argues that a unifying meta-logical framework like LogiKEy, which supports multiple object-logics, is essential for rigorous, interdisciplinary formal reasoning, preventing the stagnation caused by foundational monocultures while enabling the discovery of non-trivial results in metaphysical and mathematical theories.

The Big Idea: One Toolbox, Many Rules

The Problem with "One Size Fits All"

The Case Study: Gödel's God Argument

Why This Matters

Summary Analogy

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