Context, Judgement, Deduction: A Categorical Framework for Deductive Systems

Core Concepts

This research paper introduces a novel categorical framework called "judgemental theories" to unify and analyze various deductive systems, including dependent type theory and natural deduction.

Abstract

• Bibliographic Information: Coraglia, G., & Di Liberti, I. (2024). Context, Judgement, Deduction. arXiv:2111.09438v3 [math.LO].
• Research Objective: This paper aims to develop a unified mathematical framework, grounded in category theory, to represent and analyze diverse deductive systems.
• Methodology: The authors utilize concepts from category theory, particularly fibrations and 2-categorical constructions, to formalize the notions of context, judgement, and deduction rules. They demonstrate the framework's expressiveness by applying it to dependent type theory and natural deduction.
• Key Findings: The proposed "judgemental theories" framework provides an algebraic approach to defining deduction rules and analyzing proof theory. It offers a clear definition of extensional type constructors in type theory and elucidates the role of structural rules in proof theory. The framework also highlights the computational aspect of proofs due to its inherent structure.
• Main Conclusions: Judgemental theories offer a powerful and versatile tool for studying deductive systems. The framework's ability to unify diverse systems like dependent type theory and natural deduction underscores its potential for broader applications in logic and computer science.
• Significance: This research contributes significantly to the field of categorical logic by providing a novel framework for understanding and analyzing deductive systems. The framework's computational nature also holds promise for applications in proof assistants and formal verification.
• Limitations and Future Research: The paper primarily focuses on dependent type theory and natural deduction. Exploring the application of judgemental theories to other logical systems, such as modal logic or linear logic, could further demonstrate the framework's versatility and lead to new insights.

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Quotes

"Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly." - [Wit22, 4.116]

Key Insights Distilled From

Context, Judgement, Deduction

by Greta Coragl... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2111.09438.pdf

Deeper Inquiries

How might the framework of judgemental theories be extended to encompass non-classical logics, such as modal or linear logic?

Extending judgemental theories to encompass non-classical logics like modal or linear logic requires carefully adapting the core components of contexts, judgements, rules, and policies to reflect the specific features of these logics. Here's a breakdown of potential approaches:
Modal Logic:

Contexts: Introduce a notion of "possible worlds" into the contexts. This could be achieved by:

Adding a new judgement classifer for worlds, allowing judgements like "Γ ⊢ w World".
Enriching the category of contexts with a functorial action of a suitable category representing the accessibility relation between worlds.

Judgements: Modify the interpretation of judgements to be relative to worlds. For example, "Γ, w ⊢ φ" could mean "φ holds in world w under context Γ".
Rules: Introduce rules for modal operators that reflect their semantics. For instance, a rule for the necessity operator (□) might look like:Γ, w ⊢ φ
----------- (□I)
Γ, w ⊢ □φ if for all w' accessible from w, Γ, w' ⊢ φ

Policies: Adapt policies to account for the modal structure. This might involve considering the interaction of modal operators with substitution and other structural rules.
Linear Logic:

Contexts: Represent contexts as multisets instead of sets to reflect the resource-sensitive nature of linear logic. This change affects how context extension and substitution are handled.
Judgements: Introduce different judgement forms to distinguish between linear and non-linear assumptions. For example:

"Γ ⊸ A" for linear assumptions.
"Γ ⊢ A" for non-linear assumptions.

Rules: Adapt rules to manage resources appropriately. For instance, the introduction rule for the multiplicative conjunction (⊗) should consume its premises:Γ1 ⊸ A Γ2 ⊸ B
----------------- (⊗I)
Γ1, Γ2 ⊸ A ⊗ B

Policies: Revise policies to ensure the soundness of resource manipulation within the logic.
General Considerations:

Fibred Categories: Employing fibred categories over a suitable base category (e.g., a category of worlds for modal logic) can provide a natural framework for handling the additional structure introduced by non-classical logics.
Internal Logic: Investigating the internal logic of appropriate categorical structures (e.g., presheaf toposes for modal logic) can offer insights into how to represent these logics within the judgemental theories framework.

Could the inherent rigidity of the judgemental theories framework limit its applicability to less structured or more informal deductive systems?

The inherent rigidity of the judgemental theories framework, while advantageous for its clarity and computational nature, could pose challenges when applied to less structured or more informal deductive systems. Here's a nuanced look at the potential limitations:
Challenges with Less Structured Systems:

Informal Rules: Systems with informal or ambiguous rules might be difficult to capture directly within the strict categorical framework. The precise definition of rules as functors and policies as natural transformations requires a level of formality that might not be present.
Dynamic or Evolving Systems: Deductive systems that evolve over time or allow for dynamic modifications of rules or judgements might not fit neatly into the static structure of judgemental theories.
Challenges with Informal Systems:

Implicit Assumptions: Informal systems often rely on implicit assumptions or contextual understanding that is not explicitly formalized. Representing such implicit knowledge within the explicit framework of judgemental theories could be difficult.
Lack of Clear Syntax/Semantics: Systems lacking a well-defined syntax or semantics might pose challenges for translation into the formal language of categories and functors.
Potential Mitigations:

Abstraction and Generalization: The framework might be adaptable by abstracting away from concrete details and focusing on higher-level structural properties shared by a broader class of deductive systems.
Hybrid Approaches: Combining judgemental theories with other formalisms or techniques might offer a more flexible approach. For instance, integrating aspects of rewriting systems or logical frameworks could provide tools for handling less structured aspects.
Overall Perspective:
While the rigidity of judgemental theories might limit its direct applicability to certain less structured or informal systems, it's important to view this rigidity not as an insurmountable obstacle but rather as a design choice that prioritizes clarity, precision, and computational meaning. Exploring extensions, generalizations, or hybrid approaches could potentially broaden its scope while preserving its core strengths.

In what ways could the computational nature of judgemental theories be leveraged to develop more efficient proof assistants or automated reasoning tools?

The computational nature of judgemental theories, stemming from its foundation in category theory and its emphasis on explicit constructions, offers several promising avenues for developing more efficient proof assistants and automated reasoning tools:
1. Proof Construction and Verification:

Proof Objects as Programs: The correspondence between proofs and well-typed programs in type theory, known as the Curry-Howard isomorphism, can be directly leveraged. Judgemental theories provide a natural framework for representing proof objects as categorical structures, enabling their manipulation and verification within the system.
Proof Search Strategies: The categorical structure can guide the development of efficient proof search strategies. For example, properties like pullbacks and fibrations can inform tactics for decomposing goals and searching for applicable rules.
2. Efficient Representation and Manipulation:

Structure Sharing: The use of categorical constructs like pullbacks and limits allows for efficient representation of proofs by sharing common substructures. This can significantly reduce the size and complexity of proof objects.
Compositionality: The compositional nature of categorical constructions enables modular reasoning and proof reuse. Proofs can be built from smaller, independently verified components, improving efficiency and maintainability.
3. Automation and Decision Procedures:

Term Rewriting: The explicit representation of computation rules within judgemental theories facilitates the implementation of term rewriting systems for automated simplification and normalization of proofs.
Decision Procedures: For specific fragments of logic or type theory, the categorical structure might enable the development of specialized decision procedures. For example, identifying decidable subcategories within a judgemental theory could lead to efficient algorithms for certain proof obligations.
4. Formalization and Metaprogramming:

Meta-Level Reasoning: The abstract nature of judgemental theories allows for meta-level reasoning about proofs and deductive systems. This enables the development of powerful tactics, proof transformations, and automated proof assistants.
Extensible Systems: The modularity and extensibility of the framework facilitate the development of proof assistants that can be easily extended with new logics, theories, and decision procedures.
5. Integration with Existing Tools:

Bridging the Gap: Judgemental theories can serve as a unifying framework for integrating different proof assistants and automated reasoning tools. By providing a common language and foundation, it can facilitate communication and interoperability between systems.
Overall Impact:
By leveraging the computational nature of judgemental theories, we can strive towards proof assistants and automated reasoning tools that are not only sound and expressive but also efficient, scalable, and user-friendly. This approach holds the potential to significantly advance the field of formal verification and automated reasoning, with applications in software engineering, hardware design, and other domains where formal guarantees are crucial.