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In this paper, we show that it is their relational product<\/jats:italic>. Similarly, we prove that the set of $$\\textsf{TS}$$<\/jats:tex-math>\n TS<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-valid inferences can be identified using a dual notion, namely as the relational sum<\/jats:italic> of the sets of $$\\textsf{LP}$$<\/jats:tex-math>\n LP<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-valid and $$\\textsf{K}_3$$<\/jats:tex-math>\n \n K<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-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 $$\\textsf{ST}$$<\/jats:tex-math>\n ST<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\textsf{TS}$$<\/jats:tex-math>\n TS<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We present a notion of duality on which $$\\textsf{ST}$$<\/jats:tex-math>\n ST<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\textsf{TS}$$<\/jats:tex-math>\n TS<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are dual in exactly the same sense in which $$\\textsf{LP}$$<\/jats:tex-math>\n LP<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\textsf{K}_3$$<\/jats:tex-math>\n \n K<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are dual to each other.<\/jats:p>","DOI":"10.1007\/s10992-024-09778-z","type":"journal-article","created":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T00:44:22Z","timestamp":1731545062000},"page":"1673-1700","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["$$\\textsf{ST}$$ and $$\\textsf{TS}$$ as Product and Sum"],"prefix":"10.1007","volume":"53","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-2971-9509","authenticated-orcid":false,"given":"Quentin","family":"Blomet","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul","family":"\u00c9gr\u00e9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,11,14]]},"reference":[{"key":"9778_CR1","doi-asserted-by":"publisher","first-page":"1051","DOI":"10.1007\/s11225-017-9739-7","volume":"105","author":"H Albuquerque","year":"2017","unstructured":"Albuquerque, H., P\u0159enosil, A., & Rivieccio, U. 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