Quentin Blomet, Paul Égré

• The set of ST-valid inferences is neither the intersection, nor the union of the sets of K3-valid and LP-valid inferences, but despite the proximity to both systems, an extensional characterization of STin terms of a natural set-theoretic operation on the sets of K3-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 K3-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 STand TS. We present a notion of duality on which STand TSare dual in exactly the same sense in which LPand K3are dual to each other.