An Equational Presentation of Boolean Hyperdoctrines: Reconciling Lawvere's Logic with Peirce's Calculus of Relations

Core Concepts

There exists an adjunction between the category of first-order bicategories and the category of boolean hyperdoctrines, revealing a formal correspondence between Lawvere's algebraic approach to logic and Peirce's calculus of relations.

Abstract

The paper presents a formal correspondence between first-order bicategories and boolean hyperdoctrines, two algebraic frameworks for studying first-order logic.
First, the authors introduce the notion of peircean bicategories, which provide a more concise characterization of first-order bicategories. They show that peircean bicategories are equivalent to first-order bicategories.
Next, the authors establish an adjunction between the category of first-order bicategories and the category of boolean hyperdoctrines. This adjunction reveals a formal connection between Lawvere's algebraic approach to logic, captured by hyperdoctrines, and Peirce's calculus of relations, captured by first-order bicategories.
The authors further show that this adjunction becomes an equivalence when restricted to well-behaved hyperdoctrines. They also demonstrate that functionally complete first-order bicategories are equivalent to boolean categories.
The key insights are:

Peircean bicategories provide a more concise presentation of first-order bicategories, while preserving the essential properties.
There is an adjunction between first-order bicategories and boolean hyperdoctrines, linking Lawvere's and Peirce's perspectives on logic.
This adjunction becomes an equivalence for a well-behaved class of hyperdoctrines.
Functionally complete first-order bicategories are equivalent to boolean categories.

The results reconcile the algebraic approaches to logic developed by Lawvere and Peirce, providing a deeper understanding of the connections between these two influential perspectives.

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Quotes

"Logic in his adolescent phase was algebraic. There was Boole's algebra of classes and Peirce's algebra of relations. But in 1879 logic come of age, with Frege's quantification theory. Here the bound variables, so characteristic of analysis rather than of algebra, became central to logic."

Quine

Key Insights Distilled From

When Lawvere meets Peirce: an equational presentation of boolean hyperdoctrines

by Filippo Bonc... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18795.pdf

Deeper Inquiries

How can the insights from this work be applied to improve the design and implementation of programming languages and proof assistants that leverage relational reasoning

The insights from this work can be applied to improve the design and implementation of programming languages and proof assistants that leverage relational reasoning in several ways. Firstly, by understanding the correspondence between first-order bicategories and boolean hyperdoctrines, developers can create more efficient and powerful tools for reasoning about relationships and dependencies in code. This can lead to the development of programming languages with enhanced capabilities for handling complex data structures and relationships between variables.
Additionally, the formal link established in this work can be utilized to enhance the design of proof assistants that rely on relational algebra for verification and validation of logical statements. By leveraging the insights from peircean bicategories and boolean hyperdoctrines, proof assistants can offer more robust and comprehensive support for formal reasoning and theorem proving. This can result in more reliable and accurate verification of complex logical statements and proofs.
Overall, the application of the concepts and structures introduced in this work can lead to advancements in the development of programming languages and proof assistants that are better equipped to handle relational reasoning, leading to more efficient and effective software development and formal verification processes.

Are there other logical frameworks or mathematical structures that could be fruitfully connected to the correspondence between first-order bicategories and boolean hyperdoctrines

The correspondence between first-order bicategories and boolean hyperdoctrines opens up possibilities for connecting with other logical frameworks and mathematical structures to further enhance the understanding and application of relational reasoning. One potential area of exploration could be the integration of these concepts with modal logic, which deals with the notions of necessity and possibility in logical statements. By incorporating modal logic principles into the framework of first-order bicategories and boolean hyperdoctrines, it may be possible to develop more sophisticated reasoning systems that can handle complex modalities and temporal dependencies in logical reasoning.
Furthermore, the connection between these structures and category theory could provide insights into the relationship between relational reasoning and categorical semantics. By exploring the connections between first-order bicategories, boolean hyperdoctrines, and other categorical structures, researchers and developers can gain a deeper understanding of the underlying principles governing logical reasoning and computation. This could lead to the development of more advanced computational models and formal systems that leverage the power of category theory for reasoning about relationships and dependencies in various domains.

What are the potential implications of this work for the development of novel approaches to automated reasoning and theorem proving

The implications of this work for the development of novel approaches to automated reasoning and theorem proving are significant. By establishing the formal correspondence between first-order bicategories and boolean hyperdoctrines, researchers and practitioners can enhance the capabilities of automated reasoning systems and theorem provers. This can lead to the development of more efficient and accurate tools for automated theorem proving, formal verification, and logical reasoning.
One potential application of this work is in the field of artificial intelligence, where automated reasoning systems play a crucial role in decision-making and problem-solving tasks. By incorporating the insights from this research, AI systems can be equipped with more advanced reasoning capabilities that enable them to handle complex logical statements and relationships with greater accuracy and efficiency. This can lead to improvements in AI applications such as natural language processing, knowledge representation, and automated planning.
Additionally, the development of novel approaches to automated reasoning based on the principles of first-order bicategories and boolean hyperdoctrines can have implications for various domains such as cybersecurity, formal methods, and computational logic. By leveraging the formal correspondence established in this work, researchers can explore new avenues for enhancing the efficiency and effectiveness of automated reasoning systems, ultimately leading to advancements in the field of automated reasoning and theorem proving.