The calculus of neo-Peircean relations

TL;DR

This paper introduces the calculus of neo-Peircean relations, a diagrammatic, monoidal approach that achieves complete axiomatisation and expressiveness comparable to first-order logic, overcoming previous no-go theorems.

Contribution

It presents a novel diagrammatic calculus that extends the traditional calculus of relations, enabling complete axiomatisation and high expressiveness.

Findings

The calculus is more expressive than traditional relation calculus.

It achieves complete axiomatisation using categorical structures.

It matches the expressiveness of first-order logic.

Abstract

The calculus of relations was introduced by De Morgan and Peirce during the second half of the 19th century, as an extension of Boole's algebra of classes. 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.