It has been pointed out to me by a reviewer that the definition of Peano assemblage given in example 1 on p. 24 of my paper “Incompleteness in a General Setting” [1] is flawed. Unfortunately I overlooked the fact that Gödel numbering cannot be automatically extended to equivalence classes of formulas under provability so as to preserve the recursive nature of the coding. A corrected version of the paper is available on my website at http://publish.uwo.ca/~jbell/.
一位审稿人向我指出，我的论文“一般设置中的不完备性”(Incompleteness in a General Setting)[1]第24页示例1中给出的Peano集合的定义是有缺陷的。不幸的是，我忽略了一个事实，即Gödel编号不能在可证明性下自动扩展到公式的等价类，以保持编码的递归性质。论文的更正版可以在我的网站http://publish.uwo.ca/~jbell/上找到。
{"title":"Corrigendum to “Incompleteness in a General Setting”","authors":"J. Bell","doi":"10.2178/BSL/1208358848","DOIUrl":"https://doi.org/10.2178/BSL/1208358848","url":null,"abstract":"It has been pointed out to me by a reviewer that the definition of Peano assemblage given in example 1 on p. 24 of my paper “Incompleteness in a General Setting” [1] is flawed. Unfortunately I overlooked the fact that Gödel numbering cannot be automatically extended to equivalence classes of formulas under provability so as to preserve the recursive nature of the coding. A corrected version of the paper is available on my website at http://publish.uwo.ca/~jbell/.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"122 - 122"},"PeriodicalIF":0.6,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68345725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic.
{"title":"On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics","authors":"Ian Pratt-Hartmann","doi":"10.2178/bsl/1208358842","DOIUrl":"https://doi.org/10.2178/bsl/1208358842","url":null,"abstract":"Abstract The numerically definite syllogistic is the fragment of English obtained by extending the language of the classical syllogism with numerical quantifiers. The numerically definite relational syllogistic is the fragment of English obtained by extending the numerically definite syllogistic with predicates involving transitive verbs. This paper investigates the computational complexity of the satisfiability problem for these fragments. We show that the satisfiability problem (= finite satisfiability problem) for the numerically definite syllogistic is strongly NP-complete, and that the satisfiability problem (= finite satisfiability problem) for the numerically definite relational syllogistic is NEXPTIME-complete, but perhaps not strongly so. We discuss the related problem of probabilistic (propositional) satisfiability, and thereby demonstrate the incompleteness of some proof-systems that have been proposed for the numerically definite syllogistic.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"1 - 28"},"PeriodicalIF":0.6,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/bsl/1208358842","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68345248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2008-03-01DOI: 10.1017/S1079898600001888
A. Urquhart
but they are always thought provoking. In the reviewer’s opinion, the most glaring omission in the choice of topics in the text is the absence of any material on computability theory and incompleteness theorems. Of course there is only so much one can accomplish during a mere semester, and there is always the option of reaching for supplementary materials and texts, but I hope that the author would consider enriching the future editions of this text with a treatment of these two important topics. I wish to conclude this review with a brief discussion of Farsi logic textbooks in order to provide a background for thework reviewedhere. The firstmodern logic textbook inFarsiwas published in Iran more than half a century ago in 1955. It was authored by G. H. Mosaheb, a gifted and admirably tireless mathematician, educator, and man of letters. Thanks to the industry of Alonzo Church, the book was reviewed two years later inThe Journal of Symbolic Logic by none other than L. A. Zadeh, the founder of fuzzy logic; see JSL, vol. 22, no. 4 (Dec. 1957), pp. 354–355. The next series of notable mathematical logic texts to arrive on the Iranian scene were the Farsi translations of Enderton’s A Mathematical Introduction to Logic, and Hamilton’s Logic for Mathematicians, which appeared three decades later in the mid-1980’s and helped to introduce the subject to a new generation of students. In the early 1990’s Zia Movahed (an expert in philosophical logic, who is also well-known in Iran as a literary scholar and poet) authored a respectable introductory textbook on modern logic, and more recently (2002) he published a substantial textbook on modal logic, both of which will hopefully be reviewed by BSL in the near future. Ali Enayat Department of Mathematics and Statistics, 4400 Mass. Ave. NW, American University, Washington, DC 20016-8050, USA. enayat@american.edu.
{"title":"Truth and games: Essays in honour of Gabriel Sandu. edited by Tuomo Aho and Ahti-Veikko Pietarinen, Acta Philosophica Fennica, vol. 78. Societas Philosophica Fennica, Helsinki, 2006, vi + 322 pp.","authors":"A. Urquhart","doi":"10.1017/S1079898600001888","DOIUrl":"https://doi.org/10.1017/S1079898600001888","url":null,"abstract":"but they are always thought provoking. In the reviewer’s opinion, the most glaring omission in the choice of topics in the text is the absence of any material on computability theory and incompleteness theorems. Of course there is only so much one can accomplish during a mere semester, and there is always the option of reaching for supplementary materials and texts, but I hope that the author would consider enriching the future editions of this text with a treatment of these two important topics. I wish to conclude this review with a brief discussion of Farsi logic textbooks in order to provide a background for thework reviewedhere. The firstmodern logic textbook inFarsiwas published in Iran more than half a century ago in 1955. It was authored by G. H. Mosaheb, a gifted and admirably tireless mathematician, educator, and man of letters. Thanks to the industry of Alonzo Church, the book was reviewed two years later inThe Journal of Symbolic Logic by none other than L. A. Zadeh, the founder of fuzzy logic; see JSL, vol. 22, no. 4 (Dec. 1957), pp. 354–355. The next series of notable mathematical logic texts to arrive on the Iranian scene were the Farsi translations of Enderton’s A Mathematical Introduction to Logic, and Hamilton’s Logic for Mathematicians, which appeared three decades later in the mid-1980’s and helped to introduce the subject to a new generation of students. In the early 1990’s Zia Movahed (an expert in philosophical logic, who is also well-known in Iran as a literary scholar and poet) authored a respectable introductory textbook on modern logic, and more recently (2002) he published a substantial textbook on modal logic, both of which will hopefully be reviewed by BSL in the near future. Ali Enayat Department of Mathematics and Statistics, 4400 Mass. Ave. NW, American University, Washington, DC 20016-8050, USA. enayat@american.edu.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"119 - 121"},"PeriodicalIF":0.6,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S1079898600001888","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57303564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper deals with the problem of giving a principled characterization of the class of logical constants. According to the so-called Tarski–Sher thesis, an operation is logical iff it is invariant under permutation. In the model-theoretic tradition, this criterion has been widely accepted as giving a necessary condition for an operation to be logical. But it has been also widely criticized on the account that it counts too many operations as logical, failing thus to provide a sufficient condition. Our aim is to solve this problem of overgeneration by modifying the invariance criterion. We introduce a general notion of invariance under a similarity relation and present the connection between similarity relations and classes of invariant operations. The next task is to isolate a similarity relation well-suited for a definition of logicality. We argue that the standard arguments in favor of invariance under permutation, which rely on the generality and the formality of logic, should be modified. The revised arguments are shown to support an alternative to Tarski's criterion, according to which an operation is logical iff it is invariant under potential isomorphism.
{"title":"Logicality and Invariance","authors":"D. Bonnay","doi":"10.2178/bsl/1208358843","DOIUrl":"https://doi.org/10.2178/bsl/1208358843","url":null,"abstract":"Abstract This paper deals with the problem of giving a principled characterization of the class of logical constants. According to the so-called Tarski–Sher thesis, an operation is logical iff it is invariant under permutation. In the model-theoretic tradition, this criterion has been widely accepted as giving a necessary condition for an operation to be logical. But it has been also widely criticized on the account that it counts too many operations as logical, failing thus to provide a sufficient condition. Our aim is to solve this problem of overgeneration by modifying the invariance criterion. We introduce a general notion of invariance under a similarity relation and present the connection between similarity relations and classes of invariant operations. The next task is to isolate a similarity relation well-suited for a definition of logicality. We argue that the standard arguments in favor of invariance under permutation, which rely on the generality and the formality of logic, should be modified. The revised arguments are shown to support an alternative to Tarski's criterion, according to which an operation is logical iff it is invariant under potential isomorphism.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"29 - 68"},"PeriodicalIF":0.6,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/bsl/1208358843","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68345295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is complete. This result and proof have a number of nice corollaries: the Scott rank of ε is + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly orbit (from the proof).
{"title":"The Complexity of Orbits of Computably Enumerable Sets","authors":"Peter A. Cholak, R. Downey, L. Harrington","doi":"10.2178/bsl/1208358844","DOIUrl":"https://doi.org/10.2178/bsl/1208358844","url":null,"abstract":"Abstract The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is complete. This result and proof have a number of nice corollaries: the Scott rank of ε is + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly orbit (from the proof).","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"69 - 87"},"PeriodicalIF":0.6,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68345346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties. The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA. The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory. Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.
{"title":"Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics","authors":"M. Viale","doi":"10.2178/bsl/1208358846","DOIUrl":"https://doi.org/10.2178/bsl/1208358846","url":null,"abstract":"The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties. The first example I will treat is the proof that the proper forcing axiom PFA implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA. The second example aims to show that the transfer principle (ℵω+1, ℵω) ↠ (ℵ2, ℵ1) fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory. Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"99 - 113"},"PeriodicalIF":0.6,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/bsl/1208358846","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"14th Workshop on Logic, Language, Information and Computation (WoLLIC 2007)","authors":"D. Leivant, Ruy J. G. B. de Queroz","doi":"10.2178/BSL/1208358850","DOIUrl":"https://doi.org/10.2178/BSL/1208358850","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"160 - 161"},"PeriodicalIF":0.6,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/BSL/1208358850","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68346454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.
{"title":"The Complexity of Propositional Proofs","authors":"Nathan Segerlind","doi":"10.2178/bsl/1203350879","DOIUrl":"https://doi.org/10.2178/bsl/1203350879","url":null,"abstract":"Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"35 1","pages":"417-481"},"PeriodicalIF":0.6,"publicationDate":"2007-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85010143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.
这是一篇关于波兰空间中闭集的世系的描述集合论的综述论文。本文主要讨论闭集或紧集的理想和σ-理想。
{"title":"Descriptive Set Theory of Families of Small Sets","authors":"É. Matheron, M. Zelený","doi":"10.2178/bsl/1203350880","DOIUrl":"https://doi.org/10.2178/bsl/1203350880","url":null,"abstract":"This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"71 1","pages":"482-537"},"PeriodicalIF":0.6,"publicationDate":"2007-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77150004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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