Reaching Classicality through Transitive Closure

Abstract

Recently, arXiv:2312.16035 showed that all logics based on Boolean Normal monotonic three-valued schemes coincide with classical logic when defined using a strict-tolerant standard ($\mathbf{st}$). Conversely, they proved that under a tolerant-strict standard ($\mathbf{ts}$), the resulting logics are all empty. Building on these results, we show that classical logic can be obtained by closing under transitivity the union of two logics defined over (potentially different) Boolean normal monotonic schemes, using a strict-strict standard ($\mathbf{ss}$) for one and a tolerant-tolerant standard ($\mathbf{tt}$) for the other, with the first of these logics being paracomplete and the other being paraconsistent. We then identify a notion dual to transitivity that allows us to characterize the logic $\mathsf{TS}$ as the dual transitive closure of the intersection of any two logics defined over (potentially different) Boolean normal monotonic schemes, using an $\mathbf{ss}$ standard for one and a $\mathbf{tt}$ standard for the other. Finally, we expand on the abstract relations between the transitive closure and dual transitive closure operations, showing that they give rise to lattice operations that precisely capture how the logics discussed relate to one another.