Bicategorical Models of Classical Propositional Logic

TL;DR

This paper develops bicategorical models of classical propositional logic that are symmetric and non-degenerate, extending previous categorical models to higher categorical levels with examples like Rel, Span, and Prof.

Contribution

It introduces bicategorical models of classical propositional logic that are symmetric and non-degenerate, expanding the modeling framework to 2-cells and negations.

Findings

Models include Rel, Span, and Prof, demonstrating non-degeneracy at multiple categorical levels.

Bicategorical models preserve symmetry and non-degeneracy, avoiding reduction strategy choices.

Extends categorical logic models to bicategories with richer structure.

Abstract

F\"uhrmann and Pym constructed models of classical propositional logic in an order-enriched categorical setting, whose typical example is the category $\mathbf{Rel}$ of sets and relations. It is remarkable in that they are both non-degenerate and symmetric, i.e., free from the choices of the reduction strategy. As a furter categorification of this direction, we give bicategorical models of classical propositional logic that is also symmetric and non-degenerate. Primal examples of our models include $\mathbf{Rel}$, $\mathbf{Span}$, and $\mathbf{Prof}$, which shows that we can construct models that are non-degenerate not only for $1$-cells but also for $2$-cells and the logical negations.