{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T12:38:41Z","timestamp":1740141521741,"version":"3.37.3"},"reference-count":20,"publisher":"Oxford University Press (OUP)","issue":"7","license":[{"start":{"date-parts":[[2023,8,7]],"date-time":"2023-08-07T00:00:00Z","timestamp":1691366400000},"content-version":"vor","delay-in-days":2,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"crossref","award":["DE220100544"],"award-info":[{"award-number":["DE220100544"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,10,23]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>By limiting the range of the predicate variables in a second-order language, one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note, we provide an infinitary strongly complete axiomatization for several systems of this kind having the range of the predicate variables as a parameter. The completeness argument uses simple techniques from the theory of Boolean algebras.<\/jats:p>\n               <jats:p>This article is dedicated to our friend John N. Crossley on the occasion of his 86th birthday.<\/jats:p>","DOI":"10.1093\/logcom\/exad050","type":"journal-article","created":{"date-parts":[[2023,8,7]],"date-time":"2023-08-07T05:25:57Z","timestamp":1691385957000},"page":"1295-1304","source":"Crossref","is-referenced-by-count":0,"title":["A parametrized axiomatization for a large number of restricted second-order logics"],"prefix":"10.1093","volume":"34","author":[{"given":"Guillermo","family":"Badia","sequence":"first","affiliation":[{"name":"School of Historial and Philosophical Inquiry , University of Queensland, Brisbane, , g.badia@uq.edu.au","place":["Australia"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John L","family":"Bell","sequence":"additional","affiliation":[{"name":"Department of Philosophy , University of Western Ontario, London, , jbell@uwo.ca","place":["Canada"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2023,8,5]]},"reference":[{"journal-title":"Journal of Symbolic Logic","article-title":"Maximality of logic without identity","author":"Badia","key":"2024102321255420100_ref1"},{"volume-title":"Model-Theoretic Axiomatization Results for Certain Restricted Second-Order Languages","year":"1969","author":"Bell","key":"2024102321255420100_ref2"},{"volume-title":"Models and Ultraproducts: An Introduction","year":"1969","author":"Bell","key":"2024102321255420100_ref3"},{"volume-title":"A Course in Mathematical Logic","year":"1977","author":"Bell","key":"2024102321255420100_ref4"},{"key":"2024102321255420100_ref5","doi-asserted-by":"crossref","DOI":"10.1017\/9781108981804","volume-title":"Higher-Order Logic and Type Theory","author":"Bell","year":"2022"},{"key":"2024102321255420100_ref6","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1305\/ndjfl\/1039886524","article-title":"On elementary equivalence for identity-free logic","volume":"37","author":"Casanovas","year":"1996","journal-title":"Notre Dame Journal of Formal Logic"},{"volume-title":"Logic and Implication: An Introduction to the General Algebraic Study of Non-Classical Logics","year":"2022","author":"Cintula","key":"2024102321255420100_ref7"},{"key":"2024102321255420100_ref8","first-page":"167","article-title":"Boolean-valued second-order logic","volume":"56","author":"Daisuke","year":"2015","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"2024102321255420100_ref9","doi-asserted-by":"crossref","first-page":"81","DOI":"10.2307\/2266967","article-title":"Completeness in the theory of types","volume":"15","author":"Henkin","year":"1950","journal-title":"The Journal of Symbolic Logic"},{"key":"2024102321255420100_ref10","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511551574","volume-title":"Model Theory","author":"Hodges","year":"1993"},{"key":"2024102321255420100_ref11","first-page":"221","article-title":"A note on weak second order logic with variables for elementarily definable relations","volume-title":"Proceedings of the Bertrand Russell Memorial Conference (Uldum, 1971)","author":"Lindstr\u00f6m","year":"1973"},{"key":"2024102321255420100_ref12","doi-asserted-by":"crossref","first-page":"93","DOI":"10.4064\/fm-61-1-93-103","article-title":"A complete, infinitary axiomatization of weak second-order logic","volume":"61","author":"L\u00f3pez-Escobar","year":"1967","journal-title":"Fundamenta Mathematicae"},{"volume-title":"Introduction to Mathematical Logic","year":"1964","author":"Mendelson","key":"2024102321255420100_ref13"},{"key":"2024102321255420100_ref14","first-page":"45","article-title":"Semantical closure and non-finite axiomatizability I","volume-title":"Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2\u20139 September 1959","author":"Montague","year":"1961"},{"key":"2024102321255420100_ref15","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9452-5","volume-title":"Mathematical Logic","author":"Monk","year":"1976"},{"key":"2024102321255420100_ref16","doi-asserted-by":"crossref","first-page":"246","DOI":"10.2307\/2269096","article-title":"On $\\omega $-consistency and related properties","volume":"21","author":"Orey","year":"1956","journal-title":"Journal of Symbolic Logic"},{"key":"2024102321255420100_ref17","first-page":"265","article-title":"Notes on the Rasiowa\u2013Sikorski lemma","volume":"34","author":"Rauszer","year":"1975","journal-title":"Bulletin of the Section of Logic"},{"key":"2024102321255420100_ref18","doi-asserted-by":"crossref","first-page":"193","DOI":"10.4064\/fm-37-1-193-200","article-title":"A proof of the completeness theorem of G\u00f6del","volume":"37","author":"Rasiowa","year":"1951","journal-title":"Fundamenta Mathematicae"},{"key":"2024102321255420100_ref19","first-page":"673","article-title":"Some model-theoretical results concerning weak second-order logic. 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