\(\textsf{ST}\) and \(\textsf{TS}\) as Product and Sum

The set of \(\textsf{ST}\) $\mathsf{ST}$ -valid inferences is neither the intersection, nor the union of the sets of \(\textsf{K}_3\) ${\mathsf{K}}_3$ -valid and \(\textsf{LP}\) $\mathsf{LP}$ -valid inferences, but despite the proximity to both systems, an extensional characterization of \(\textsf{ST}\) $\mathsf{ST}$ in terms of a natural set-theoretic operation on the sets of \(\textsf{K}_3\) ${\mathsf{K}}_3$ -valid and \(\textsf{LP}\) $\mathsf{LP}$ -valid inferences is still wanting. In this paper, we show that it is their relational product. Similarly, we prove that the set of \(\textsf{TS}\) $\mathsf{TS}$ -valid inferences can be identified using a dual notion, namely as the relational sum of the sets of \(\textsf{LP}\) $\mathsf{LP}$ -valid and \(\textsf{K}_3\) ${\mathsf{K}}_3$ -valid inferences. We discuss links between these results and the interpolation property of classical logic. We also use those results to revisit the duality between \(\textsf{ST}\) $\mathsf{ST}$ and \(\textsf{TS}\) $\mathsf{TS}$ . We present a notion of duality on which \(\textsf{ST}\) $\mathsf{ST}$ and \(\textsf{TS}\) $\mathsf{TS}$ are dual in exactly the same sense in which \(\textsf{LP}\) $\mathsf{LP}$ and \(\textsf{K}_3\) ${\mathsf{K}}_3$ are dual to each other.