{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:43:27Z","timestamp":1762508607561},"reference-count":64,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2021,10,20]],"date-time":"2021-10-20T00:00:00Z","timestamp":1634688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,1,25]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Weak filters were introduced by K. Schlechta in the \u201990s with the aim of interpreting defaults via a generalized \u2018most\u2019 quantifier in first-order logic. They arguably represent the largest class of structures that qualify as a \u2018collection of large subsets\u2019 of a given index set $I$, in the sense that it is difficult to think of a weaker, but still plausible, definition of the concept. The notion of weak ultrafilter naturally emerges and has been used in epistemic logic and other knowledge representation (KR) applications. We provide a comprehensive exposition of weak filters and ultrafilters, comparing them with their classical counterparts that have found very important applications in logic, set theory and topology. Weak (ultra)filters capture the \u2018majorities\u2019 of social choice theory (which coincide with the commonsense understanding of a \u2018large subset\u2019) and in that respect, they outperform classical (ultra)filters in their role as \u2018collections of large subsets\u2019. Yet, they lack some of the elegant properties of classical (ultra)filters that make them an almost perfect match for logical theories. We investigate the extent to which some important classical results carry through in this new setting, and we focus on genuinely weak filters and genuinely weak ultrafilters. For weak ultrafilters, we proceed to provide concrete examples, answer questions of existence and characterize their construction. The class of weak (ultra)filters represents a genuine contribution of KR to set theory that might be of some interest to set theorists too and we initiate this study by providing a glimpse on natural set-theoretic questions.<\/jats:p>","DOI":"10.1093\/jigpal\/jzab030","type":"journal-article","created":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T19:15:16Z","timestamp":1632165316000},"page":"68-95","source":"Crossref","is-referenced-by-count":1,"title":["On weak filters and ultrafilters: Set theory from (and for) knowledge representation"],"prefix":"10.1093","volume":"31","author":[{"given":"Costas D","family":"Koutras","sequence":"first","affiliation":[{"name":"College of Engineering and Technology, American University of the Middle East , Egaila 54200, Kuwait"}]},{"given":"Christos","family":"Moyzes","sequence":"additional","affiliation":[{"name":"Knowledge Representation Group, Department of Computer Science, University of Liverpool , Liverpool L69 7ZF, UK"}]},{"given":"Christos","family":"Nomikos","sequence":"additional","affiliation":[{"name":"Department of Computer Science & Engineering, University of Ioannina , Ioannina 45110, Greece"}]},{"given":"Konstantinos","family":"Tsaprounis","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Aegean , Karlovassi, Samos 83200, Greece"}]},{"given":"Yorgos","family":"Zikos","sequence":"additional","affiliation":[{"name":"Graduate Programme in Algorithms, Logic, & Discrete Mathematics , Department of Informatics & Telecommunications, National and Kapodistrian University of Athens, Athens 15784, Greece"}]}],"member":"286","published-online":{"date-parts":[[2021,10,20]]},"reference":[{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/11663081.2016.1214804","article-title":"Knowledge means \u2018all\u2019, belief means \u2018most\u2019","volume":"26","author":"Askounis","year":"2016","journal-title":"Journal of Applied Non-Classical Logics"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/S0004-3702(99)00081-8","article-title":"A modal logic for subjective default reasoning","volume":"116","author":"Ben-David","year":"2000","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107050884","volume-title":"Modal Logic","author":"Blackburn","year":"2001"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1090\/conm\/530\/10440","article-title":"Ultrafilters and set theory","volume-title":"Ultrafilters Accross Mathematics","author":"Blass","year":"2010"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-34514-5","volume-title":"Measure Theory","author":"Bogachev","year":"2007"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0004-3702(94)90096-5","article-title":"Conditional logics of normality: a modal approach","volume":"68","author":"Boutilier","year":"1994","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","article-title":"Dolciani Mathematical Expositions","volume-title":"Proofs That Really Count: The Art of Combinatorial Proof","author":"Benjamin","year":"2003"},{"key":"2023022307091069300_","volume-title":"Models and Ultraproducts","author":"Bell","year":"1969"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.18653\/v1\/2021.nuse-1.10","article-title":"Towards a formal model of narratives","volume-title":"3rd Workshop on Narrative Understanding @NAACL-HLT","author":"Castricato","year":"2021"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/BF00693270","article-title":"Basic conditional logic","volume":"4","author":"Chellas","year":"1975","journal-title":"Journal of Philosophical Logic"},{"key":"2023022307091069300_","article-title":"Studies in Logic and the Foundations of Mathematics","volume-title":"Model Theory","author":"Chang","year":"1990"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-65780-1","volume-title":"The Theory of Ultrafilters","author":"Comfort","year":"1974"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/0004-3702(94)90031-0","article-title":"From ordering-based nonmonotonic reasoning to conditional logics","volume":"66","author":"del Cerro","year":"1994","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/0004-3702(87)90053-1","article-title":"A first-order conditional logic for prototypical properties","volume":"33","author":"Delgrande","year":"1987","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/0004-3702(88)90079-3","article-title":"An approach to default reasoning based on a first-order conditional logic: revised report","volume":"36","author":"Delgrande","year":"1988","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","first-page":"89","article-title":"What\u2019s in a default? Thoughts on the nature and role of defaults in nonmonotonic reasoning","volume-title":"NonMonotonic Reasoning, Essays Celebrating Its 30th Anniversary","author":"Delgrande","year":"2012"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511809088","volume-title":"Introduction to Lattices and Order","author":"Davey","year":"2002"},{"key":"2023022307091069300_","volume-title":"A Mathematical Introduction to Logic","author":"Enderton","year":"2001"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1145\/502090.502092","article-title":"Plausibility measures and default reasoning","volume":"48","author":"Friedman","year":"2001","journal-title":"Journal of the ACM"},{"key":"2023022307091069300_","volume-title":"Real Analysis: Modern Techniques and Their Applications","author":"Folland","year":"1999"},{"key":"2023022307091069300_","volume-title":"Measure Theory","author":"Fremlin","year":"2008"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"102831","DOI":"10.1016\/j.apal.2020.102831","article-title":"Ultrafilters, finite coproducts and locally connected classifying toposes","volume":"171","author":"Garner","year":"2020","journal-title":"Annals of Pure and Applied Logic"},{"key":"2023022307091069300_","article-title":"Number 7 in CSLI Lecture Notes","volume-title":"Logics of Time and Computation","author":"Goldblatt","year":"1992"},{"key":"2023022307091069300_","first-page":"45","article-title":"Knowledge representation: modalities, conditionals, and nonmonotonic reasoning","volume-title":"Knowledge Representation, Reasoning and Learning","author":"Herzig","year":"2020"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1090\/surv\/059","volume-title":"Consequences of the axiom of choice","author":"Howard","year":"1998"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-0346-0330-0","volume-title":"Introduction to Cardinal Arithmetic","author":"Holz","year":"1999"},{"key":"2023022307091069300_","volume-title":"An Introduction to Real Analysis","author":"Hunter","year":"2014"},{"key":"2023022307091069300_","volume-title":"Modalities, Conditionals and Nonmonotonic Reasoning","author":"Jauregui","year":"2008"},{"key":"2023022307091069300_","article-title":"Springer Monographs in Mathematics","volume-title":"Set Theory","author":"Jech","year":"2002"},{"key":"2023022307091069300_","volume-title":"The Axiom of Choice","author":"Jech","year":"2008"},{"key":"2023022307091069300_","volume-title":"Discovering Modern Set Theory I: The Basics","author":"Just","year":"1996"},{"key":"2023022307091069300_","volume-title":"Discovering Modern Set Theory II: Set-Theoretic Tools for Every Mathematician","author":"Just","year":"1997"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/0004-3702(90)90101-5","article-title":"Nonmonotonic reasoning, preferential models and cumulative logics","volume":"44","author":"Kraus","year":"1990","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","first-page":"267","article-title":"Default reasoning via topology and mathematical analysis: a preliminary report","volume-title":"Principles of Knowledge Representation and Reasoning: Proceedings of the Sixteenth International Conference, KR 2018","author":"Koutras","year":"2018"},{"key":"2023022307091069300_","article-title":"Default consequence relations from topology and measure theory","author":"Koutras","year":"2021"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"2303","DOI":"10.1093\/logcom\/exx016","article-title":"A modal logic of knowledge, belief and estimation","volume":"27","author":"Koutras","year":"2017","journal-title":"Journal of Logic and Computation"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/3-540-68804-8_2","article-title":"On the expressive power of logics on finite models","volume-title":"Finite Model Theory and Its Applications","author":"Kolaitis","year":"2007"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s10849-017-9251-5","article-title":"In all but finitely many possible worlds: model-theoretic investigations on \u2018overwhelming majority\u2019 default conditionals","volume":"26","author":"Koutras","year":"2017","journal-title":"Journal of Logic Language and Information"},{"key":"2023022307091069300_","article-title":"Problem Books in Mathematics","volume-title":"Problems and Theorems in Classical Set Theory","author":"Komj\u00e1th","year":"2006"},{"key":"2023022307091069300_","first-page":"357","article-title":"S4 as the conditional logic of nonmonotonicity","volume-title":"Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning (KR\u201991)","author":"Lamarre","year":"1991"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1093\/logcom\/11.2.229","article-title":"Nonmonotonic logics and semantics","volume":"11","author":"Lehmann","year":"2001","journal-title":"Journal of Logic and Computation"},{"key":"2023022307091069300_","first-page":"332","article-title":"Codensity and the ultrafilter monad","volume":"28","author":"Leinster","year":"2013","journal-title":"Theory and Applications of Categories"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0004-3702(92)90041-U","article-title":"What does a conditional knowledge base entail","volume":"55","author":"Lehmann","year":"1992","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","volume-title":"Bridges from Classical to Nonmonotonic Logic","author":"Makinson","year":"2005"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/134","volume-title":"Analytic Number Theory: Exploring the Anatomy of Integers","author":"de Koninck","year":"2012"},{"key":"2023022307091069300_","volume-title":"Elementary Induction on Abstract Structures","author":"Moschovakis","year":"1974"},{"key":"2023022307091069300_","article-title":"Undergraduate Texts in Mathematics","volume-title":"Notes on Set Theory","author":"Moschovakis","year":"2005"},{"key":"2023022307091069300_","article-title":"Lecture Notes in Logic","author":"Moschovakis","year":"2014"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-8966-5","volume-title":"Topics in Conditional Logic","author":"Nute","year":"1980"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-67149-9","volume-title":"Neighborhood Semantics for Modal Logic","author":"Pacuit","year":"2017"},{"key":"2023022307091069300_","article-title":"Frontiers in Artificial Intelligence and Applications\u2014Dissertations in AI","volume-title":"Conditional and Preferential Logics: Proof Methods and Theorem Proving","author":"Pozzato","year":"2010"},{"key":"2023022307091069300_","first-page":"598","article-title":"Majority logic","volume-title":"Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR2004)","author":"Pacuit","year":"2004"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0004-3702(80)90014-4","article-title":"A logic for default reasoning","volume":"13","author":"Reiter","year":"1980","journal-title":"Artificial Intelligence"},{"key":"2023022307091069300_","volume-title":"Introduction to Modern Set Theory","author":"Roitman","year":"1990"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1093\/logcom\/5.4.473","article-title":"Defaults as generalized quantifiers","volume":"5","author":"Schlechta","year":"1995","journal-title":"Journal of Logic and Computation"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1093\/jigpal\/5.5.753","article-title":"Filters and partial orders","volume":"5","author":"Schlechta","year":"1997","journal-title":"Logic Journal of the IGPL"},{"key":"2023022307091069300_","volume-title":"Coherent Systems","author":"Schlechta","year":"2004"},{"key":"2023022307091069300_","volume-title":"Inquiry","author":"Stalnaker","year":"1984"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s11098-005-4062-y","article-title":"On logics of knowledge and belief","volume":"128","author":"Stalnaker","year":"2006","journal-title":"Philosophical Studies"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511614316","volume-title":"Social Choice and the Mathematics of Manipulation","author":"Taylor","year":"2005"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/163","volume-title":"Introduction to Analytic and Probabilistic Number Theory","author":"Tenenbaum","year":"2015"},{"key":"2023022307091069300_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511974885","volume-title":"Model and Games","author":"V\u00e4\u00e4n\u00e4nen","year":"2011"},{"key":"2023022307091069300_","first-page":"487","article-title":"Logics for qualitative reasoning","volume-title":"Logic, Epistemology, and the Unity of Science","author":"Paulo","year":"2004"},{"key":"2023022307091069300_","volume-title":"Modal Epistemic Logics Without Negative Introspection: Epistemic Structures and Extensions With Estimation and Information","author":"Zikos","year":"2012"}],"container-title":["Logic Journal of the IGPL"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/31\/1\/68\/49301730\/jzab030.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/31\/1\/68\/49301730\/jzab030.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,9]],"date-time":"2023-11-09T02:35:50Z","timestamp":1699497350000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jigpal\/article\/31\/1\/68\/6399094"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,20]]},"references-count":64,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,10,20]]},"published-print":{"date-parts":[[2023,1,25]]}},"URL":"https:\/\/doi.org\/10.1093\/jigpal\/jzab030","relation":{},"ISSN":["1367-0751","1368-9894"],"issn-type":[{"value":"1367-0751","type":"print"},{"value":"1368-9894","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,2]]},"published":{"date-parts":[[2021,10,20]]}}}