Abstract The set of ST -valid inferences is neither the intersection, nor the union of the sets of K 3 -valid and LP -valid inferences, but despite the proximity to both systems, an extensional characterization of ST in terms of a natural set-theoretic operation on the sets of K 3 -valid and LP -valid inferences is still wanting. In this paper, we show that it is their relational product . Similarly, we prove that the set of TS -valid inferences can be identified using a dual notion, namely as the relational sum of the sets of LP -valid and 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 ST and TS . We present a notion of duality on which ST and TS are dual in exactly the same sense in which LP and K 3 are dual to each other.