G. Bezhanishvili, C. Franks, Selwyn Ng, Dima Sinapova, M. Thomas, Paddy Blanchette, Peter A. Cholak, J. Knight
{"title":"2021 NORTH AMERICAN ANNUAL MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC","authors":"G. Bezhanishvili, C. Franks, Selwyn Ng, Dima Sinapova, M. Thomas, Paddy Blanchette, Peter A. Cholak, J. Knight","doi":"10.1017/bsl.2021.50","DOIUrl":"https://doi.org/10.1017/bsl.2021.50","url":null,"abstract":"of the invited 32nd Annual Gödel Lecture MATTHEW FOREMAN, Gödel diffeomorphisms. Department of Mathematics, University of California, Irvine, CA, USA. E-mail: mforeman@math.uci.edu Motivated by problems in physics, solutions to differential equations were studied in the late 19th and early 20th centuries by people like Birkhoff, Poincaré and von Neumann. Poincaré’s work was described by Smale in the 1960s as the qualitative study and von Neumann’s own description was the study of the statistical aspects of differential equations. The explicit goal was to classify this behavior. A contemporaneous problem was whether time forwards could be distinguished from time backwards. The modern formulation of these problems is to classify diffeomorphisms of smooth manifolds up to topological conjugacy and measure isomorphism and to ask, for a given diffeomorphism, whether T ∼= T –1. Very significant progress was made on both classes of problems, in the first case by people like Birkhoff, Morse and Smale and in the second case by Birkhoff, Poincare, von Neumann, Halmos, Kolmogorov, Sinai, Ornstein and Furstenberg. This talk applies techniques developed by Kechris, Louveau and Hjorth to these problems to show that the relevant equivalence relations are complete analytic. Moreover the collection of T that are measure theoretically isomorphic to their inverses is also complete analytic. Finally, the whole story can be miniaturized to show that the collection of diffeomorphisms of the two-torus that are measure theoretically isomorphic to their inverses is Π1-hard. 30","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"2017 1","pages":"298 - 330"},"PeriodicalIF":0.0,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86735493","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}
{"title":"Automating the Repair of Faulty Logical Theories","authors":"Xue Li","doi":"10.1017/bsl.2021.43","DOIUrl":"https://doi.org/10.1017/bsl.2021.43","url":null,"abstract":"Abstract This thesis aims to develop a domain-independent system for repairing faulty Datalog-like theories by combining three existing techniques: abduction, belief revision, and conceptual change. Accordingly, the proposed system is named the ABC repair system (ABC). Given an observed assertion and a current theory, abduction adds axioms, which explain that observation by making the corresponding assertion derivable from the expanded theory. Belief revision incorporates a new piece of information which conflicts with the input theory by deleting old axioms. Conceptual change uses the reformation algorithm for blocking unwanted proofs or unblocking wanted proofs. The former two techniques change an axiom as a whole, while reformation changes the language in which the theory is written. These three techniques are complementary. But they have not previously been combined into one system. We are working on aligning these three techniques in ABC, which is capable of repairing logical theories with better result than each individual technique alone. In addition, ABC extends abduction and belief revision to operate on preconditions: the former deletes preconditions from rules, and the latter adds preconditions to rules. Datalog is used as the underlying logic of theories in this thesis, but the proposed system has the potential to be adapted to theories in other logics. Abstract prepared by Xue Li by taking directly from the thesis. E-mail: xuerr.lee@gmail.com","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"259 1","pages":"530 - 530"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85747689","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}
{"title":"PROJECTIVE DUALITY AND THE RISE OF MODERN LOGIC","authors":"Günther Eder","doi":"10.1017/bsl.2021.40","DOIUrl":"https://doi.org/10.1017/bsl.2021.40","url":null,"abstract":"Abstract The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Specifically, we want to see in which ways geometers’ preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been influenced by reflections on duality.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"26 1","pages":"351 - 384"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82260772","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}
{"title":"KURT GÖDEL ON LOGICAL, THEOLOGICAL, AND PHYSICAL ANTINOMIES","authors":"Tim Lethen","doi":"10.1017/bsl.2021.41","DOIUrl":"https://doi.org/10.1017/bsl.2021.41","url":null,"abstract":"Abstract This paper presents hitherto unpublished writings of Kurt Gödel concerning logical, epistemological, theological, and physical antinomies, which he generally considered as “the most interesting facts in modern logic,” and which he used as a basis for his famous metamathematical results. After investigating different perspectives on the notion of the logical structure of the antinomies and presenting two “antinomies of the intensional,” a new kind of paradox closely related to Gödel’s ontological proof for the existence of God is introduced and completed by a compilation of further theological antinomies. Finally, after a presentation of unpublished general philosophical remarks concerning the antinomies, Gödel’s type-theoretic variant of Leibniz’ Monadology, discovered in his notes on the foundations of quantum mechanics, is examined. Most of the material presented here has been transcribed from the Gabelsberger shorthand system for the first time.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"46 1","pages":"267 - 297"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80476281","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}
{"title":"LARGE CARDINALS AS PRINCIPLES OF STRUCTURAL REFLECTION","authors":"J. Bagaria","doi":"10.1017/bsl.2023.2","DOIUrl":"https://doi.org/10.1017/bsl.2023.2","url":null,"abstract":"Abstract After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principle of Structural Reflection are equivalent to well-known large cardinal axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"23 1","pages":"19 - 70"},"PeriodicalIF":0.0,"publicationDate":"2021-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82526497","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}
{"title":"SEALING OF THE UNIVERSALLY BAIRE SETS","authors":"G. Sargsyan, Nam Trang","doi":"10.1017/bsl.2021.29","DOIUrl":"https://doi.org/10.1017/bsl.2021.29","url":null,"abstract":"Abstract A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. \u0000${sf Sealing}$\u0000 is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The \u0000${sf Largest Suslin Axiom}$\u0000 ( \u0000${sf LSA}$\u0000 ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let \u0000${sf LSA}$\u0000 - \u0000${sf over}$\u0000 - \u0000${sf uB}$\u0000 be the statement that in all (set) generic extensions there is a model of \u0000$sf {LSA}$\u0000 whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, \u0000$sf {Sealing}$\u0000 is equiconsistent with \u0000$sf {LSA}$\u0000 - \u0000$sf {over}$\u0000 - \u0000$sf {uB}$\u0000 . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that \u0000$sf {Sealing}$\u0000 is weaker than the theory “ \u0000$sf {ZFC}$\u0000 + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of \u0000$sf {Sealing}$\u0000 by Woodin. A variation of \u0000$sf {Sealing}$\u0000 , called \u0000$sf {Tower Sealing}$\u0000 , is also shown to be equiconsistent with \u0000$sf {Sealing}$\u0000 over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then \u0000$sf {Sealing}$\u0000 holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that \u0000$sf {Sealing}$\u0000 holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that \u0000$sf {LSA}$\u0000 - \u0000$sf {over}$\u0000 - \u0000$sf {uB}$\u0000 is not equivalent to \u0000$sf {Sealing}$\u0000 .","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"19 1","pages":"254 - 266"},"PeriodicalIF":0.0,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75692506","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}
{"title":"LOGICALITY AND MODEL CLASSES","authors":"J. Kennedy, Jouko Vaananen","doi":"10.1017/bsl.2021.42","DOIUrl":"https://doi.org/10.1017/bsl.2021.42","url":null,"abstract":"Abstract We ask, when is a property of a model a logical property? According to the so-called Tarski–Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim–Skolem theorem, Completeness theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic. We also offer a refinement of the result of McGee that logical properties of models can be expressed in \u0000$L_{infty infty }$\u0000 if the expression is allowed to depend on the cardinality of the model, based on replacing \u0000$L_{infty infty }$\u0000 by a “tamer” logic.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"8 1","pages":"385 - 414"},"PeriodicalIF":0.0,"publicationDate":"2021-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88338776","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}
{"title":"Semantics modulo satisfiability with applications: function representation, probabilities and game theory","authors":"Sandro Preto","doi":"10.1017/bsl.2022.2","DOIUrl":"https://doi.org/10.1017/bsl.2022.2","url":null,"abstract":"Abstract In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed. We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic. Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected. Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence. Abstract prepared by Sandro Márcio da Silva Preto. E-mail: spreto@ime.usp.br URL: https://doi.org/10.11606/T.45.2021.tde-17062021-163257","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"1 1","pages":"264 - 265"},"PeriodicalIF":0.0,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78944949","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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