Bicategorical Models of Classical Propositional Logic

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.