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. arXiv admin note: substantial text overlap with arXiv:2401.07055

@article{paperbot3978,
  title = {The calculus of neo-Peircean relations},
  author = {Filippo Bonchi and Alessandro Di Giorgio and Nathan Haydon and Pawel Sobocinski},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 22, Issue 2},
  year = {2026},
  doi = {10.46298/lmcs-22(2:29)2026}
}