{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T05:15:18Z","timestamp":1735881318009,"version":"3.32.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"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":[[2024,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Gri\u0161in and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and compare it with Gri\u0161in set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifiers. After interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we establish that the logic for these multiplicative quantifiers (but without disquotational truth) is consistent, by proving that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents.<\/jats:p>","DOI":"10.1017\/s1755020323000138","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T09:40:37Z","timestamp":1685958037000},"page":"996-1017","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["NON-CONTRACTIVE LOGICS, PARADOXES, AND MULTIPLICATIVE QUANTIFIERS"],"prefix":"10.1017","volume":"17","author":[{"given":"CARLO","family":"NICOLAI","sequence":"first","affiliation":[]},{"given":"MARIO","family":"PIAZZA","sequence":"additional","affiliation":[]},{"given":"MATTEO","family":"TESI","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"S1755020323000138_r9","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-013-9268-4"},{"key":"S1755020323000138_r16","unstructured":"[16] Takeuti, G. (1987). Proof theory (second edition). Amsterdam: Elsevier ."},{"key":"S1755020323000138_r17","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-022-09998-x"},{"key":"S1755020323000138_r6","doi-asserted-by":"publisher","DOI":"10.2307\/2269029"},{"key":"S1755020323000138_r10","unstructured":"[10] Minari, P. (2014). Lecture notes for a course in structural proof-theory. Unpublished manuscript."},{"key":"S1755020323000138_r4","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020319000571"},{"key":"S1755020323000138_r5","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-2021-0012"},{"key":"S1755020323000138_r13","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005293713265"},{"key":"S1755020323000138_r11","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/14.2.299"},{"key":"S1755020323000138_r12","doi-asserted-by":"publisher","DOI":"10.1007\/s11098-005-7777-x"},{"key":"S1755020323000138_r7","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90045-4"},{"key":"S1755020323000138_r3","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-017-9441-2"},{"key":"S1755020323000138_r15","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71124-8"},{"key":"S1755020323000138_r18","volume-title":"Basic Proof Theory","volume":"43","author":"Troelstra","year":"2003"},{"key":"S1755020323000138_r2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025159016268"},{"key":"S1755020323000138_r8","doi-asserted-by":"publisher","DOI":"10.1070\/IM1982v018n01ABEH001382"},{"key":"S1755020323000138_r19","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020311000177"},{"key":"S1755020323000138_r1","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(92)90073-9"},{"key":"S1755020323000138_r14","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020322000284"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020323000138","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T10:53:41Z","timestamp":1735815221000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020323000138\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,5]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["S1755020323000138"],"URL":"https:\/\/doi.org\/10.1017\/s1755020323000138","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"type":"print","value":"1755-0203"},{"type":"electronic","value":"1755-0211"}],"subject":[],"published":{"date-parts":[[2023,6,5]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. 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"}]}}