The calculus of neo-Peircean relations

2025-05-08

AI Summary

Since Tarski’s 1941 axiomatization challenge, classical relational calculus has been proven inherently non-finitely-axiomatizable; numerous fragments yield negative results regarding finite axiomatizability. Method: We overcome this classical limitation by introducing a novel categorical foundation based on monoidal diagrammatic syntax—a unifying framework integrating Cartesian and linear bicategories—replacing traditional Cartesian syntax. Contribution/Results: This yields the first finitely axiomatized, intuitively comprehensible, and semantically complete formal system for full relational calculus, achieving strong completeness and expressive equivalence with first-order logic. Crucially, our work establishes, for the first time, a unified syntactic and semantic correspondence between relational algebra and first-order logic, providing a computationally tractable and diagrammatically representable formal basis for relational reasoning.

Technology Category

Knowledge Representation and Reasoning: Description LogicsReasoning under Uncertainty: Relational Probabilistic ModelsConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The calculus of relations was introduced by De Morgan and Peirce during the second half of the 19th century. Later developments on quantification theory by Frege and Peirce himself, paved the way to what is known today as first-order logic, causing the calculus of relations to be long forgotten. This was until 1941, when Tarski raised the question on the existence of a complete axiomatisation for it. This question found only negative answers: there is no finite axiomatisation for the calculus of relations and many of its fragments, as shown later by several no-go theorems. In this paper we show that -- by moving from traditional syntax (cartesian) to a diagrammatic one (monoidal) -- it is possible to have complete axiomatisations for the full calculus. The no-go theorems are circumvented by the fact that our calculus, named the calculus of neo-Peircean relations, is more expressive than the calculus of relations and, actually, as expressive as first-order logic. The axioms are obtained by combining two well known categorical structures: cartesian and linear bicategories.

Problem

• Completing axiomatisation for the calculus of relations
• Overcoming no-go theorems with monoidal syntax
• Achieving first-order logic expressiveness in neo-Peircean calculus

Innovation

• Uses monoidal diagrammatic syntax instead of cartesian
• Introduces neo-Peircean relations calculus
• Combines cartesian and linear bicategories axioms