{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,20]],"date-time":"2024-09-20T16:52:21Z","timestamp":1726851141478},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T00:00:00Z","timestamp":1644278400000},"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":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In \u201cSome Remarks on Extending and Interpreting Theories with a Partial Truth Predicate\u201d, Reinhardt [21] famously proposed an instrumentalist interpretation of the truth theory Kripke\u2013Feferman (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline3.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>) in analogy to Hilbert\u2019s program. Reinhardt suggested to view <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline4.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> as a tool for generating \u201cthe significant part of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline5.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>\u201d, that is, as a tool for deriving sentences of the form <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline6.png\" \/><jats:tex-math>\n$\\mathrm{Tr}\\ulcorner {\\varphi }\\urcorner $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. The constitutive question of <jats:italic>Reinhardt\u2019s program<\/jats:italic> was whether it was possible \u201cto justify the use of nonsignificant sentences entirely within the framework of significant sentences\u201d. This question was answered negatively by Halbach &amp; Horsten [10] but we argue that under a more careful interpretation the question may receive a positive answer. To this end, we propose to shift attention from <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline7.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-provably true <jats:italic>sentences<\/jats:italic> to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline8.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-provably true <jats:italic>inferences<\/jats:italic>, that is, we shall identify the significant part of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline9.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with the set of pairs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline10.png\" \/><jats:tex-math>\n$\\langle {\\Gamma , \\Delta }\\rangle $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, such that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline11.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> proves that if all members of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline12.png\" \/><jats:tex-math>\n$\\Gamma $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> are true, at least one member of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline13.png\" \/><jats:tex-math>\n$\\Delta $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is true. In way of addressing Reinhardt\u2019s question we show that the provably true inferences of suitable <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline14.png\" \/><jats:tex-math>\n$\\mathrm {KF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-like theories coincide with the provable sequents of matching versions of the theory Partial Kripke\u2013Feferman (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000089_inline15.png\" \/><jats:tex-math>\n$\\mathrm {PKF}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>).<\/jats:p>","DOI":"10.1017\/s1755020322000089","type":"journal-article","created":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T04:29:45Z","timestamp":1644294585000},"page":"33-58","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":5,"title":[", , AND REINHARDT\u2019S PROGRAM"],"prefix":"10.1017","volume":"16","author":[{"given":"LUCA","family":"CASTALDO","sequence":"first","affiliation":[]},{"given":"JOHANNES","family":"STERN","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,2,8]]},"reference":[{"key":"S1755020322000089_r11","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-017-9424-3"},{"key":"S1755020322000089_r6","doi-asserted-by":"publisher","DOI":"10.2307\/2274902"},{"key":"S1755020322000089_r18","volume-title":"Proof Theory: The First Step into Impredicativity","author":"Pohlers","year":"2009"},{"key":"S1755020322000089_r4","doi-asserted-by":"publisher","DOI":"10.2307\/2269866"},{"key":"S1755020322000089_r7","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-017-9751-y"},{"key":"S1755020322000089_r9","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139696586"},{"key":"S1755020322000089_r12","doi-asserted-by":"publisher","DOI":"10.2307\/2024634"},{"key":"S1755020322000089_r14","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-017-9727-y"},{"key":"S1755020322000089_r20","first-page":"227","article-title":"Remarks on significance and meaningful applicability","volume":"94","author":"Reinhardt","year":"1985","journal-title":"Mathematical Logic and Formal Systems"},{"key":"S1755020322000089_r1","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-0458-8_5"},{"key":"S1755020322000089_r10","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1146620166"},{"key":"S1755020322000089_r13","volume-title":"Truth, Vagueness, and Paradox: An Essay on the Logic of Truth","author":"McGee","year":"1990"},{"key":"S1755020322000089_r23","volume-title":"Proof Theory","volume":"81","author":"Takeuti","year":"1987"},{"key":"S1755020322000089_r8","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1309952527"},{"key":"S1755020322000089_r2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19890350202"},{"key":"S1755020322000089_r15","doi-asserted-by":"publisher","DOI":"10.2307\/2272731"},{"key":"S1755020322000089_r5","doi-asserted-by":"publisher","DOI":"10.2307\/2274093"},{"key":"S1755020322000089_r17","doi-asserted-by":"publisher","DOI":"10.1093\/jigpal\/jzy059"},{"key":"S1755020322000089_r19","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801174"},{"key":"S1755020322000089_r22","first-page":"1","volume-title":"International Symposium on Lambda-Calculus and Computer Science Theory","author":"Scott","year":"1975"},{"key":"S1755020322000089_r3","volume-title":"Logical Frameworks for Truth and Abstraction: An Axiomatic Study","volume":"135","author":"Cantini","year":"1996"},{"key":"S1755020322000089_r16","doi-asserted-by":"publisher","DOI":"10.1093\/jigpal\/jzy059"},{"key":"S1755020322000089_r21","doi-asserted-by":"publisher","DOI":"10.1007\/BF00305492"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020322000089","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T10:05:06Z","timestamp":1675850706000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020322000089\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,8]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["S1755020322000089"],"URL":"https:\/\/doi.org\/10.1017\/s1755020322000089","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,8]]},"assertion":[{"value":"\u00a9 The Author(s), 2022. 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"}]}}