{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T00:47:20Z","timestamp":1703033240248},"reference-count":35,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T00:00:00Z","timestamp":1630368000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper is dedicated to extending and adapting to modal logic the approach of <jats:italic>fractional semantics<\/jats:italic> to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020321000411_inline1.png\" \/><jats:tex-math>\n$[0,1]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Focusing on the modal logic <jats:italic>K<\/jats:italic>, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of <jats:italic>K<\/jats:italic> affords an informational refinement with respect to the standard Kripkean semantics (a new proof of Dugundji\u2019s theorem is a case in point) and it raises the prospect of a proof-theoretic semantics for modal logic.<\/jats:p>","DOI":"10.1017\/s1755020321000411","type":"journal-article","created":{"date-parts":[[2021,8,31]],"date-time":"2021-08-31T08:36:02Z","timestamp":1630398962000},"page":"1033-1052","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["FRACTIONAL-VALUED MODAL LOGIC"],"prefix":"10.1017","volume":"16","author":[{"given":"MARIO","family":"PIAZZA","sequence":"first","affiliation":[]},{"given":"GABRIELE","family":"PULCINI","sequence":"additional","affiliation":[]},{"given":"MATTEO","family":"TESI","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,8,31]]},"reference":[{"key":"S1755020321000411_r20","first-page":"221","volume-title":"Selected Papers in Proof Theory","author":"Mints","year":"1992a"},{"key":"S1755020321000411_r10","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-012-9261-3"},{"key":"S1755020321000411_r29","doi-asserted-by":"publisher","DOI":"10.1093\/mind\/109.436.781"},{"key":"S1755020321000411_r6","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-009-0137-3"},{"key":"S1755020321000411_r9","doi-asserted-by":"publisher","DOI":"10.2307\/2268175"},{"key":"S1755020321000411_r5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107050884"},{"key":"S1755020321000411_r14","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-15-2221-5_9"},{"key":"S1755020321000411_r1","doi-asserted-by":"publisher","DOI":"10.2307\/2273828"},{"key":"S1755020321000411_r18","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-013-9310-6"},{"key":"S1755020321000411_r21","volume-title":"A Short Introduction to Modal Logic","volume":"30","author":"Mints","year":"1992b"},{"key":"S1755020321000411_r23","doi-asserted-by":"publisher","DOI":"10.1007\/s10670-020-00264-x"},{"key":"S1755020321000411_r26","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-008-9425-4"},{"key":"S1755020321000411_r13","doi-asserted-by":"publisher","DOI":"10.3390\/axioms8040118"},{"key":"S1755020321000411_r31","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3208-5"},{"key":"S1755020321000411_r33","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-010-0387-2_2"},{"key":"S1755020321000411_r34","volume-title":"Displaying Modal Logic","volume":"3","author":"Wansing","year":"2013"},{"key":"S1755020321000411_r3","first-page":"1","volume-title":"Logic: From Foundations to Applications","author":"Avron","year":"1996"},{"key":"S1755020321000411_r27","first-page":"900","article-title":"Uniform, cut-free formulations of T, S4 and S5","volume":"48","author":"Pottinger","year":"1983","journal-title":"Journal of Symbolic Logic"},{"key":"S1755020321000411_r32","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005217827758"},{"key":"S1755020321000411_r15","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-020-09917-y"},{"key":"S1755020321000411_r8","doi-asserted-by":"publisher","DOI":"10.1007\/s11787-014-0106-4"},{"key":"S1755020321000411_r35","doi-asserted-by":"publisher","DOI":"10.1016\/j.jal.2017.01.002"},{"key":"S1755020321000411_r7","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537793.001.0001","volume-title":"Modal Logic","volume":"35","author":"Chagrov","year":"1997"},{"key":"S1755020321000411_r4","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093882402"},{"key":"S1755020321000411_r22","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-005-2267-3"},{"key":"S1755020321000411_r25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-22686-6"},{"key":"S1755020321000411_r24","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020319000431"},{"key":"S1755020321000411_r30","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139168717"},{"key":"S1755020321000411_r19","first-page":"259","article-title":"Socratic proofs for some normal modal propositional logics","volume":"47","author":"Leszcczy\u0144ska","year":"2004","journal-title":"Logique et Analyse"},{"key":"S1755020321000411_r11","volume-title":"Proof-Theoretic Semantics","author":"Francez","year":"2015"},{"key":"S1755020321000411_r12","first-page":"65","article-title":"Zum intuitionistischen Aussagenkalk\u00fcl","volume":"69","author":"G\u00f6del","year":"1932","journal-title":"Anzeiger der Akademie der Wissenschaften Wien, mathematisch-naturwissenschaftliche Klasse"},{"key":"S1755020321000411_r16","first-page":"1","article-title":"Untersuchungen zum Pr\u00e4dikatenkalk\u00fcl","volume":"I","author":"Ketonen","year":"1944","journal-title":"Annales Academiae Scientiarum Fennicae. 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Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}