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First, I provide sequent rules and axioms for all the logics in the hierarchy, for all inferential levels, and establish soundness and completeness results. Second, I show how to extend those systems with a corresponding hierarchy of validity predicates, each one of which is meant to capture \u201cvalidity\u201d at a different inferential level. Then, I point out two potential philosophical implications of these results. (i) Since the logics in the hierarchy differ from one another on the rules, I argue that each such logic maintains its own distinct identity (contrary to arguments like the one given by Dicher and Paoli in 2019). (ii) Each validity predicate need not capture \u201cvalidity\u201d at more than one metainferential level. Hence, there are reasons to deny the thesis (put forward in Barrio, E., Rosenblatt, L. &amp; Tajer, D. (<jats:italic>Synthese<\/jats:italic>, 2016)) that the validity predicate introduced in by Beall and Murzi in (<jats:italic>Journal o f Philosophy<\/jats:italic>, <jats:italic>110<\/jats:italic>(3), 143\u2013165, 2013) has to express facts not only about what follows from what, but also about the metarules, etc.<\/jats:p>","DOI":"10.1007\/s10992-021-09616-6","type":"journal-article","created":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T13:02:56Z","timestamp":1627304576000},"page":"1295-1325","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Metainferences from a Proof-Theoretic Perspective, and a Hierarchy of Validity Predicates"],"prefix":"10.1007","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4923-9998","authenticated-orcid":false,"given":"Rea","family":"Golan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,7,26]]},"reference":[{"key":"9616_CR1","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/0890-5401(91)90023-U","volume":"92","author":"A Avron","year":"1991","unstructured":"Avron, A. 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A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49, 93\u2013120. https:\/\/doi.org\/10.1007\/s10992-019-09513-z.","journal-title":"Journal of Philosophical Logic"},{"issue":"2","key":"9616_CR5","doi-asserted-by":"publisher","first-page":"326","DOI":"10.1017\/S1755020311000074","volume":"4","author":"J Beall","year":"2011","unstructured":"Beall, J. (2011). Multiple-conclusion LP and default classicality. Review of Symbolic Logic, 4(2), 326\u2013336. https:\/\/doi.org\/10.1017\/S1755020311000074.","journal-title":"Review of Symbolic Logic"},{"issue":"3","key":"9616_CR6","doi-asserted-by":"publisher","first-page":"143","DOI":"10.5840\/jphil2013110336","volume":"110","author":"J Beall","year":"2013","unstructured":"Beall, J., & Murzi, J. (2013). Two flavors of Curry paradox. 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