{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T00:43:34Z","timestamp":1711500214456},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2022,7,8]],"date-time":"2022-07-08T00:00:00Z","timestamp":1657238400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000247_inline1.png\" \/><jats:tex-math>\n$\\mathtt {ML}^{\\infty }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000247_inline2.png\" \/><jats:tex-math>\n$\\mathtt {GML}^{\\infty }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, respectively. We then characterise these logics as the bisimulation-invariant fragments of the naturally corresponding predicate logic, viz., the extension of first-order logic with the infinity quantifier. Furthermore, for both <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000247_inline3.png\" \/><jats:tex-math>\n$\\mathtt {ML}^{\\infty }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000247_inline4.png\" \/><jats:tex-math>\n$\\mathtt {GML}^{\\infty }$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> we provide a sound and complete axiomatisation for the set of formulas that are valid in every Kripke frame, we prove a small model property with respect to a widened class of weighted models, and we establish decidability of the satisfiability problem.<\/jats:p>","DOI":"10.1017\/s1755020322000247","type":"journal-article","created":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T23:59:15Z","timestamp":1657238355000},"page":"1-35","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["COUNTING TO INFINITY: GRADED MODAL LOGIC WITH AN INFINITY DIAMOND"],"prefix":"10.1017","volume":"17","author":[{"given":"IGNACIO","family":"BELLAS ACOSTA","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"YDE","family":"VENEMA","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,7,8]]},"reference":[{"key":"S1755020322000247_r23","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/11.1.85"},{"key":"S1755020322000247_r2","doi-asserted-by":"publisher","DOI":"10.1017\/9781316717158"},{"key":"S1755020322000247_r19","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71188-1"},{"key":"S1755020322000247_r17","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1111\/j.1755-2567.1966.tb00600.x","article-title":"First order predicate logic with generalized quantifiers","volume":"32","author":"Lindstr\u00f6m","year":"1966","journal-title":"Theoria"},{"key":"S1755020322000247_r1","unstructured":"[2] Bellas Acosta, I. (2020). Studies in the extension of standard modal logic with an infinity modality. Master\u2019s Thesis, Institute for Logic, Language and Computation, Universiteit van Amsterdam."},{"key":"S1755020322000247_r29","volume-title":"The Stanford Encyclopedia of Philosophy (Winter 2016 Edition)","author":"Westerst\u00e5hl","year":"2016"},{"key":"S1755020322000247_r12","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19950410410"},{"key":"S1755020322000247_r3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107050884"},{"key":"S1755020322000247_r13","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093890715"},{"key":"S1755020322000247_r15","doi-asserted-by":"publisher","DOI":"10.1016\/S1570-2464(07)80008-5"},{"key":"S1755020322000247_r7","doi-asserted-by":"publisher","DOI":"10.1007\/BF00374047"},{"key":"S1755020322000247_r8","doi-asserted-by":"publisher","DOI":"10.1023\/A:1005245900406"},{"key":"S1755020322000247_r21","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008275906015"},{"key":"S1755020322000247_r24","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511974885"},{"key":"S1755020322000247_r18","doi-asserted-by":"publisher","DOI":"10.4064\/fm-44-1-12-36"},{"key":"S1755020322000247_r26","first-page":"280","volume-title":"Proceedings of the 22 nd IEEE Symposium on Logic in Computer Science (LICS 2007)","author":"van Benthem","year":"2007"},{"key":"S1755020322000247_r5","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/bxp004"},{"key":"S1755020322000247_r6","doi-asserted-by":"publisher","DOI":"10.1007\/s001530100110"},{"key":"S1755020322000247_r16","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0096899"},{"key":"S1755020322000247_r14","first-page":"323","article-title":"Grades of modality","volume":"13","author":"Goble","year":"1970","journal-title":"Logique et Analyse"},{"key":"S1755020322000247_r27","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/5.3.325"},{"key":"S1755020322000247_r20","unstructured":"[23] Otto, M. (2019). Graded modal logic and counting bisimulation. Preprint, arXiv:1910.00039."},{"key":"S1755020322000247_r10","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19990450406"},{"key":"S1755020322000247_r11","doi-asserted-by":"publisher","DOI":"10.1007\/BF00379767"},{"key":"S1755020322000247_r9","doi-asserted-by":"publisher","DOI":"10.1145\/4904.4999"},{"key":"S1755020322000247_r25","unstructured":"[3] van Benthem, J. (1976). Modal correspondence theory. Ph.D. Thesis, Universiteit van Amsterdam."},{"key":"S1755020322000247_r4","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-021-00797-0"},{"key":"S1755020322000247_r28","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0023902"},{"key":"S1755020322000247_r22","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(00)00056-6"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020322000247","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,26]],"date-time":"2024-03-26T09:48:41Z","timestamp":1711446521000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020322000247\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,8]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["S1755020322000247"],"URL":"https:\/\/doi.org\/10.1017\/s1755020322000247","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,8]]},"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"}}]}}