{"title":"CATEGORICAL QUANTIFICATION","authors":"Constantin C. Brîncuş","doi":"10.1017/bsl.2024.3","DOIUrl":"https://doi.org/10.1017/bsl.2024.3","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139602276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. On the basis of Poincar´e and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which Γ 0 and much bigger ordinals can be defined as von-Neumann ordinals. This refutes the accepted view of Γ 0 as the “limit of predicativity”.
{"title":"POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND","authors":"A. Avron","doi":"10.1017/bsl.2024.2","DOIUrl":"https://doi.org/10.1017/bsl.2024.2","url":null,"abstract":". On the basis of Poincar´e and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which Γ 0 and much bigger ordinals can be defined as von-Neumann ordinals. This refutes the accepted view of Γ 0 as the “limit of predicativity”.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139613497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
KIRSTEN EISENTRÄGER, RUSSELL MILLER, CALEB SPRINGER, LINDA WESTRICK
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{"title":"A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF","authors":"KIRSTEN EISENTRÄGER, RUSSELL MILLER, CALEB SPRINGER, LINDA WESTRICK","doi":"10.1017/bsl.2023.37","DOIUrl":"https://doi.org/10.1017/bsl.2023.37","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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{"title":"BSL volume 29 issue 3 Cover and Front matter","authors":"","doi":"10.1017/bsl.2023.35","DOIUrl":"https://doi.org/10.1017/bsl.2023.35","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135688059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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{"title":"BSL volume 29 issue 3 Cover and Back matter","authors":"","doi":"10.1017/bsl.2023.36","DOIUrl":"https://doi.org/10.1017/bsl.2023.36","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135688179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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{"title":"John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.","authors":"Bruno Bentzen","doi":"10.1017/bsl.2023.28","DOIUrl":"https://doi.org/10.1017/bsl.2023.28","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135687673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
THREE PAPERS ON RECENT WORK ON META-VALIDITY - David Ripley, One step is enough. Journal of Philosophical Logic, vol. 51 (2022), pp. 1233–1259. - Isabella McAllister, Classical logic is not uniquely characterizable. Journal of Philosophical Logic, vol. 51 (2022), pp. 1345–1365. - Rea Golan, There is no tenable notion of global metainferential validity. Analysis, vol. 81 (2021), no. 3, pp. 411–420. - Volume 29 Issue 3
三篇关于元效度的最新研究——大卫·里普利，一步就够了。哲学逻辑杂志，vol. 51 (2022)， pp. 1233-1259。——伊莎贝拉·麦卡利斯特，经典逻辑不是唯一可表征的。哲学逻辑学报，vol. 51 (2022)， pp. 1345-1365。——雷亚·戈兰，没有一个站得住脚的全球元指效度概念。分析，vol. 81 (2021)， no。3，第411-420页。-第29卷第3期
{"title":"THREE PAPERS ON RECENT WORK ON META-VALIDITY - David Ripley, <i>One step is enough</i>. Journal of Philosophical Logic, vol. 51 (2022), pp. 1233–1259. - Isabella McAllister, <i>Classical logic is not uniquely characterizable</i>. Journal of Philosophical Logic, vol. 51 (2022), pp. 1345–1365. - Rea Golan, <i>There is no tenable notion of global metainferential validity</i>. Analysis, vol. 81 (2021), no. 3, pp. 411–420.","authors":"Chris Scambler","doi":"10.1017/bsl.2023.17","DOIUrl":"https://doi.org/10.1017/bsl.2023.17","url":null,"abstract":"THREE PAPERS ON RECENT WORK ON META-VALIDITY - David Ripley, One step is enough. Journal of Philosophical Logic, vol. 51 (2022), pp. 1233–1259. - Isabella McAllister, Classical logic is not uniquely characterizable. Journal of Philosophical Logic, vol. 51 (2022), pp. 1345–1365. - Rea Golan, There is no tenable notion of global metainferential validity. Analysis, vol. 81 (2021), no. 3, pp. 411–420. - Volume 29 Issue 3","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135688180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christopher Henney-Turner, P. Holy, Philipp Schlicht, Philip Welch
{"title":"ASYMMETRIC CUT AND CHOOSE GAMES","authors":"Christopher Henney-Turner, P. Holy, Philipp Schlicht, Philip Welch","doi":"10.1017/bsl.2023.31","DOIUrl":"https://doi.org/10.1017/bsl.2023.31","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"429 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91460681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus “grassroots mathematics”. We begin with a brief review of FO(#), first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, MFO(#). We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced. Next, we investigate a series of strengthenings of MFO(#), again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system ML(#) that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of ML(#), and provide a new kind of bisimulation matching the expressive power of the language. As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of FO(#) by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of “mass” or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables. Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite “semantic automata”. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.
{"title":"INTERLEAVING LOGIC AND COUNTING","authors":"J. van Benthem, Thomas F. Icard","doi":"10.1017/bsl.2023.30","DOIUrl":"https://doi.org/10.1017/bsl.2023.30","url":null,"abstract":"Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus “grassroots mathematics”. We begin with a brief review of FO(#), first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, MFO(#). We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced. Next, we investigate a series of strengthenings of MFO(#), again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system ML(#) that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of ML(#), and provide a new kind of bisimulation matching the expressive power of the language. As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of FO(#) by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of “mass” or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables. Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite “semantic automata”. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86423408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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