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(Journal of Philosophical Logic 49:93\u2013120, 2020) and others. The calculus allows one to establish and generalize in a very natural manner several recent results, such as the coincidence of some of these notions with their classical counterparts, and the possibility of expressing some notions of satisfaction for higher-level inferences using notions of satisfaction for inferences of lower level. We also show that at each level all notions of satisfaction considered are pairwise distinct and we address some remarks on the possible significance of this (huge) number of notions of consequence.<\/jats:p>","DOI":"10.1007\/s10992-021-09639-z","type":"journal-article","created":{"date-parts":[[2021,11,20]],"date-time":"2021-11-20T11:02:28Z","timestamp":1637406148000},"page":"1417-1452","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Higher-level Inferences in the Strong-Kleene Setting: A Proof-theoretic Approach"],"prefix":"10.1007","volume":"51","author":[{"given":"Pablo","family":"Cobreros","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elio","family":"La Rosa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2844-129X","authenticated-orcid":false,"given":"Luca","family":"Tranchini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,11,20]]},"reference":[{"key":"9639_CR1","doi-asserted-by":"crossref","unstructured":"Baaz, M., Ferm\u00fcller, C.G., & Salzer, G. 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