{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T08:13:12Z","timestamp":1780560792671,"version":"3.54.1"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,2,13]],"date-time":"2025-02-13T00:00:00Z","timestamp":1739404800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,2,13]],"date-time":"2025-02-13T00:00:00Z","timestamp":1739404800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100009093","name":"Scuola Normale Superiore","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100009093","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stud Logica"],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Fractional semantics provides a multi-valued interpretation of a variety of logics, governed by purely proof-theoretic principles. This approach employs a method of systematic decomposition of formulas through a well-disciplined sequent calculus, assigning a fractional value that measures the \u201cquantity of identity\u201d (intuitively, \u201cquantity of truth\u201d) within a sequent. A key consequence of this framework is the breakdown of the traditional symmetry between truth and contradiction. In this paper, we explore the ramifications of this novel perspective on classical logic. Specifically, we (i) introduce an alternative\n                    <jats:italic>paraconsistent<\/jats:italic>\n                    consequence relation, and (ii) show how the gradual character of contradictions induces a corresponding characterization of tautologies, thereby obtaining a full-fledged informational refinement of classical logic.\n                  <\/jats:p>","DOI":"10.1007\/s11225-025-10168-y","type":"journal-article","created":{"date-parts":[[2025,2,13]],"date-time":"2025-02-13T03:16:57Z","timestamp":1739416617000},"page":"673-702","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Inside Classical Logic: Truth, Contradictions, Fractionality"],"prefix":"10.1007","volume":"114","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9545-3912","authenticated-orcid":false,"given":"Mario","family":"Piazza","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2810-5835","authenticated-orcid":false,"given":"Matteo","family":"Tesi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,2,13]]},"reference":[{"issue":"2","key":"10168_CR1","doi-asserted-by":"publisher","first-page":"707","DOI":"10.2307\/2274660","volume":"55","author":"A Avron","year":"1990","unstructured":"Avron, A., Relevance and paraconsistency\u2013a new approach, The Journal of Symbolic Logic, 55(2):707\u2013732, 1990.","journal-title":"The Journal of Symbolic Logic"},{"issue":"1","key":"10168_CR2","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1016\/j.jal.2004.07.015","volume":"3","author":"A Avron","year":"2005","unstructured":"Avron, A., Combining classical logic, paraconsistency and relevance, Journal of Applied Logic 3(1):133\u2013160, 2005.","journal-title":"Journal of Applied Logic"},{"key":"10168_CR3","volume-title":"Meaning and Necessity","author":"R Carnap","year":"1947","unstructured":"Carnap, R., Meaning and Necessity, The University of Chicago Press, 1947."},{"key":"10168_CR4","unstructured":"Cozzo, C., Meaning and argument. A theory of meaning centred on immediate argumental role, Almqvist & Wiksell International, Stockholm, 1994."},{"key":"10168_CR5","unstructured":"De Castro Alves, T., Synonymy and Identity of Proofs: A Philosophical Essay, PhD Thesis, 2018."},{"issue":"4","key":"10168_CR6","doi-asserted-by":"publisher","first-page":"477","DOI":"10.2178\/bsl\/1067620091","volume":"9","author":"K Do\u0161en","year":"2003","unstructured":"Do\u0161en, K., Identity of proofs based on normalization and generality, Bulletin of Symbolic Logic 9(4):477\u2013503, 2003.","journal-title":"Bulletin of Symbolic Logic"},{"issue":"1\u20132","key":"10168_CR7","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1002\/malq.201020020","volume":"58","author":"K Do\u0161en","year":"2012","unstructured":"Do\u0161en, K., and Z. Petri\u0107, Isomorphic formulae in classical propositional logic, Mathematical Logic Quarterly 58(1\u20132):5\u201317, 2012.","journal-title":"Mathematical Logic Quarterly"},{"key":"10168_CR8","unstructured":"G\u00f6del, K., Zur intuitionistischen arithmetik und zahlentheorie, in Ergebnisse eines mathematischen Kolloquiums, 1933, pp. 34\u201338."},{"issue":"1","key":"10168_CR9","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1093\/logcom\/exv051","volume":"27","author":"M Piazza","year":"2017","unstructured":"Piazza, M., and G. Pulcini, Unifying logics via context-sensitiveness, Journal of Logic and Computation 27(1):21\u201340, 2017.","journal-title":"Journal of Logic and Computation"},{"issue":"4","key":"10168_CR10","doi-asserted-by":"publisher","first-page":"810","DOI":"10.1017\/S1755020319000431","volume":"13","author":"M Piazza","year":"2020","unstructured":"Piazza, M., and G. Pulcini, Fractional semantics for classical logic, The Review of Symbolic Logic 13(4):810\u2013828, 2020.","journal-title":"The Review of Symbolic Logic"},{"key":"10168_CR11","doi-asserted-by":"crossref","unstructured":"Piazza, M., G. Pulcini, and M. Tesi, Fractional-valued modal logic, The Review of Symbolic Logic 16(4):1033\u20131052, 2021.","DOI":"10.1017\/S1755020321000411"},{"issue":"3","key":"10168_CR12","doi-asserted-by":"publisher","first-page":"275","DOI":"10.18778\/0138-0680.2023.17","volume":"52","author":"M Piazza","year":"2023","unstructured":"Piazza, M., G. Pulcini, and M. Tesi, Fractional-valued modal logic and soft bilateralism, Bulletin of the Section of Logic 52\/3:275\u2013299, 2023.","journal-title":"Bulletin of the Section of Logic"},{"key":"10168_CR13","doi-asserted-by":"crossref","unstructured":"Piazza, M., G. Pulcini, and M. 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C., Complementary sentential logics, Bulletin of the Section of Logic 19(4):112\u2013116, 1990.","journal-title":"Bulletin of the Section of Logic"}],"container-title":["Studia Logica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11225-025-10168-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11225-025-10168-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11225-025-10168-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T07:53:08Z","timestamp":1780559588000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11225-025-10168-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,13]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["10168"],"URL":"https:\/\/doi.org\/10.1007\/s11225-025-10168-y","relation":{},"ISSN":["0039-3215","1572-8730"],"issn-type":[{"value":"0039-3215","type":"print"},{"value":"1572-8730","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,13]]},"assertion":[{"value":"15 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2025","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}