We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas. Our framework uses the coinductive models and, hence, the results apply to a wide class of modal logics including, for example, hybrid logics. Our main result shows that the symmetries of a modal formula preserve entailment.
{"title":"Symmetries in Modal Logics","authors":"C. Areces, Guillaume Hoffmann, Ezequiel Orbe","doi":"10.4204/EPTCS.113.6","DOIUrl":"https://doi.org/10.4204/EPTCS.113.6","url":null,"abstract":"We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas. Our framework uses the coinductive models and, hence, the results apply to a wide class of modal logics including, for example, hybrid logics. Our main result shows that the symmetries of a modal formula preserve entailment.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2013-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72379201","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}
what exactly are these logical This essay is the text of a retiring presidential address to the Association for Symbolic Logic. For a number of historical and sociological reasons, the largely mathematical membership of the Association is aware of the three great schools in the philosophy of mathematics at the turn of the 20th century—Platonism, Formalism, and Intuitionism—but unfamiliar with any school of thought in the philosophy of logic. This address was an effort to introduce the range of philosophical views about logic by rough analogy with the big three about mathematics. Along the way, it sketches the positions of Kant, the 19th-century German scientific materialists, Frege, Mill, early Wittgenstein, Carnap, Ayer, Quine, and Putnam, with gestures toward Descartes, Bolzano, and Russell, and ends with a second-philosophical alternative.
{"title":"The philosophy of logic","authors":"P. Maddy","doi":"10.2178/BSL.1804010","DOIUrl":"https://doi.org/10.2178/BSL.1804010","url":null,"abstract":"what exactly are these logical This essay is the text of a retiring presidential address to the Association for Symbolic Logic. For a number of historical and sociological reasons, the largely mathematical membership of the Association is aware of the three great schools in the philosophy of mathematics at the turn of the 20th century—Platonism, Formalism, and Intuitionism—but unfamiliar with any school of thought in the philosophy of logic. This address was an effort to introduce the range of philosophical views about logic by rough analogy with the big three about mathematics. Along the way, it sketches the positions of Kant, the 19th-century German scientific materialists, Frege, Mill, early Wittgenstein, Carnap, Ayer, Quine, and Putnam, with gestures toward Descartes, Bolzano, and Russell, and ends with a second-philosophical alternative.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84513836","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}
We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.
我们概述了在建立描述性集论二分定理中使用色数的思想。
{"title":"The graph-theoretic approach to descriptive set theory","authors":"B. D. Miller","doi":"10.2178/bsl.1804030","DOIUrl":"https://doi.org/10.2178/bsl.1804030","url":null,"abstract":"We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75828463","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}
Vopěnka [2] proved long ago that every set of ordinals is set-generic over HOD, Godel’s inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over (HOD, S), and indeed over the even smaller inner model S = (L[S], S), where S is the Stability predicate. We refer to the inner model S as the Stable Core of V . The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals which are not set-generic but preserve the Stable Core (this is not possible for HOD by Vopěnka’s theorem). For an infinite cardinal α, H(α) consists of those sets whose transitive closure has size less than α. Let C denote the closed unbounded class of all infinite cardinals β such that H(α) has cardinality less than β whenever α is an infinite cardinal less than β. Definition 1 For a finite n > 0, we say that α is n-Stable in β iff α < β, α and β are limit points of C and (H(α), C ∩ α) is Σn elementary in (H(β), C ∩ β). The Stability predicate S places the Stability notion into a single predicate. S consists of all triples (α, β, n) such that α is n-Stable in β. The ∆2 definable predicate S describes the “core” of V , in the following sense. ∗The author wishes to thank the Austrian Science Fund (FWF) for its generous support through Project P 22430-N13.
{"title":"The stable core","authors":"S. Friedman","doi":"10.2178/bsl/1333560807","DOIUrl":"https://doi.org/10.2178/bsl/1333560807","url":null,"abstract":"Vopěnka [2] proved long ago that every set of ordinals is set-generic over HOD, Godel’s inner model of hereditarily ordinal-definable sets. Here we show that the entire universe V is class-generic over (HOD, S), and indeed over the even smaller inner model S = (L[S], S), where S is the Stability predicate. We refer to the inner model S as the Stable Core of V . The predicate S has a simple definition which is more absolute than any definition of HOD; in particular, it is possible to add reals which are not set-generic but preserve the Stable Core (this is not possible for HOD by Vopěnka’s theorem). For an infinite cardinal α, H(α) consists of those sets whose transitive closure has size less than α. Let C denote the closed unbounded class of all infinite cardinals β such that H(α) has cardinality less than β whenever α is an infinite cardinal less than β. Definition 1 For a finite n > 0, we say that α is n-Stable in β iff α < β, α and β are limit points of C and (H(α), C ∩ α) is Σn elementary in (H(β), C ∩ β). The Stability predicate S places the Stability notion into a single predicate. S consists of all triples (α, β, n) such that α is n-Stable in β. The ∆2 definable predicate S describes the “core” of V , in the following sense. ∗The author wishes to thank the Austrian Science Fund (FWF) for its generous support through Project P 22430-N13.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2012-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83894791","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 paper (Tarski:Lesfondementsdelageometriedescorps,AnnalesdelaSoci´´ Polonaise de Math´ ematiques, pp. 29-34, 1929) is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Lephilosophy and systems do not play the significant role that one may be tempted to assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica. This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in (Gruszczyand Pietruszczak: Full development of Tarski's Geometry of Solids, The Bulletin of Symbolic Logic, vol. 4 (2008), no. 4, pp. 481-540) according to which Tarski has made a certain mistake.
这篇论文(Tarski:Lesfondementsdelageometriedescorps,AnnalesdelaSoci ' ' Polonaise de Math ' ematiques, pp. 29-34, 1929)在许多方面都是非凡的。我们将讨论三个历史哲学问题，这些问题迫使读者不得不面对。首先，我们认为，在这篇论文中，塔斯基没有实现他自己的方法论理想，而是展示了一种更加实用的方法。其次，我们表明，哲学和系统并没有发挥重要的作用，人们可能会倾向于赋予他们第一眼。特别是背景逻辑的作用必须至少部分地分配给罗素的数学原理系统。第三，这种分析将我们引向话语领域在理论中体现的技术方式的三重区别。有了所有这些，我们讨论为什么我们必须拒绝(Gruszczyand Pietruszczak: Tarski's Geometry of Solids的全面发展，《符号逻辑通报》，vol. 4 (2008)， no. 5)中的论点。根据塔斯基的说法，他犯了一个错误。
{"title":"On Tarski's foundations of the geometry of solids","authors":"A. Betti, I. Loeb","doi":"10.2178/bsl/1333560806","DOIUrl":"https://doi.org/10.2178/bsl/1333560806","url":null,"abstract":"The paper (Tarski:Lesfondementsdelageometriedescorps,AnnalesdelaSoci´´ Polonaise de Math´ ematiques, pp. 29-34, 1929) is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Lephilosophy and systems do not play the significant role that one may be tempted to assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica. This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in (Gruszczyand Pietruszczak: Full development of Tarski's Geometry of Solids, The Bulletin of Symbolic Logic, vol. 4 (2008), no. 4, pp. 481-540) according to which Tarski has made a certain mistake.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2012-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84949563","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 paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Godel's relative consistency result.
{"title":"Early history of the Generalized Continuum Hypothesis: 1878 - 1938","authors":"Gregory H. Moore","doi":"10.2178/BSL/1318855631","DOIUrl":"https://doi.org/10.2178/BSL/1318855631","url":null,"abstract":"This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Godel's relative consistency result.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2011-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88751068","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}
We survey the theory of Muycnik (weak) and Medvedev (strong) degrees of subsets of ! ! with particular attention to the degrees of � 0 subsets of ! 2. Later sections present proofs, some more complete than others, of the major results of the subject.
{"title":"A survey of Mučnik and Medvedev degrees","authors":"P. Hinman","doi":"10.2178/bsl/1333560805","DOIUrl":"https://doi.org/10.2178/bsl/1333560805","url":null,"abstract":"We survey the theory of Muycnik (weak) and Medvedev (strong) degrees of subsets of ! ! with particular attention to the degrees of � 0 subsets of ! 2. Later sections present proofs, some more complete than others, of the major results of the subject.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2010-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85839367","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}
We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnkova–Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V → V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V → V known to be equivalent to the Axiom of Infinity.
{"title":"The Axiom of Infinity and transformations j: V -> V","authors":"P. Corazza","doi":"10.2178/bsl/1264433797","DOIUrl":"https://doi.org/10.2178/bsl/1264433797","url":null,"abstract":"We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnkova–Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V → V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V → V known to be equivalent to the Axiom of Infinity.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2178/bsl/1264433797","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72488781","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}
In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics co-logic and /?-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Q-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. This paper is concerned with strong logics of first and second order. At the most abstract level, a strong logic of first-order has the following general form: Let L be a first-order language and let (x) be a formula that defines a class of L-structures. Then, for a recursively enumerable set T of sentences of L, and for a sentence ip of L set
{"title":"Strong logics of first and second order","authors":"P. Koellner","doi":"10.2178/bsl/1264433796","DOIUrl":"https://doi.org/10.2178/bsl/1264433796","url":null,"abstract":"In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics co-logic and /?-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Q-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. This paper is concerned with strong logics of first and second order. At the most abstract level, a strong logic of first-order has the following general form: Let L be a first-order language and let (x) be a formula that defines a class of L-structures. Then, for a recursively enumerable set T of sentences of L, and for a sentence ip of L set","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2010-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72953192","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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