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Lewis\u2019 systems $$\\textsf{S2}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>\u2013$$\\textsf{S5}$$<\/jats:tex-math>\n \n S<\/mml:mi>\n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating $$\\textsf{STn}$$<\/jats:tex-math>\n STn<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$\\textsf{Sn}$$<\/jats:tex-math>\n Sn<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and backsimulating $$\\textsf{Sn}$$<\/jats:tex-math>\n Sn<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$\\textsf{STn}$$<\/jats:tex-math>\n STn<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, respectively (for $${\\textsf{n}} =2, \\ldots , 5$$<\/jats:tex-math>\n \n n<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>). Next, $$\\textsf{G3}$$<\/jats:tex-math>\n \n G<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of $$\\textsf{G3}$$<\/jats:tex-math>\n \n G<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-style calculi, that they are sound and complete, and it is shown that the proof search for $$\\mathsf {G3.ST2}$$<\/jats:tex-math>\n \n G<\/mml:mi>\n 3<\/mml:mn>\n .<\/mml:mo>\n ST<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is terminating and therefore the logic is decidable.<\/jats:p>","DOI":"10.1007\/s11225-023-10048-3","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T14:03:00Z","timestamp":1685973780000},"page":"249-294","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Proof Systems for Super- Strict Implication"],"prefix":"10.1007","volume":"112","author":[{"given":"Guido","family":"Gherardi","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4021-8667","authenticated-orcid":false,"given":"Eugenio","family":"Orlandelli","sequence":"additional","affiliation":[]},{"given":"Eric","family":"Raidl","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"unstructured":"Anderson, A.R., and N. 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