{"title":"BSL volume 28 issue 4 Cover and Back matter","authors":"","doi":"10.1017/bsl.2022.41","DOIUrl":"https://doi.org/10.1017/bsl.2022.41","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82235689","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}
Abstract Locality is a property of logics, whose origins lie in the works of Hanf and Gaifman, having their utility in the context of finite model theory. Such a property is quite useful in proofs of inexpressibility, but it is also useful in establishing normal forms for logical formulas. There are generally two forms of locality: (i’) if two structures $mathfrak {A}$ and $mathfrak {B}$ realize the same multiset of types of neighborhoods of radius d, then they agree on a given sentence $Phi $ . Here d depends only on $Phi $ ; (ii’) if the d-neighborhoods of two tuples $vec {a}_1$ and $vec {a}_2$ in a structure $mathfrak {A}$ are isomorphic, then $mathfrak {A} models Phi (vec {a}_1) Leftrightarrow Phi (vec {a}_2)$ . Again, d depends on $Phi $ , and not on $mathfrak {A}$ . Form (i’) originated from Hanf’s works. Form (ii’) came from Gaifman’s theorem. There is no doubt about the usefulness of the notion of locality, which as seen applies to a huge number of situations. However, there is a deficiency in such a notion: all versions of the notion of locality refer to isomorphism of neighborhoods, which is a fairly strong property. For example, where structures simply do not have sufficient isomorphic neighborhoods, versions of the notion of locality obviously cannot be applied. So the question that immediately arises is: would it be possible to weaken such a condition and maintain Hanf/Gaifman-localities? Arenas, Barceló, and Libkin establish a new condition for the notions of locality, weakening the requirement that neighborhoods should be isomorphic, establishing only the condition that they must be indistinguishable in a given logic. That is, instead of requiring $N_d(vec {a}) cong N_d(vec {b})$ , you should only require $N_d(vec {a}) equiv _k N_d(vec {b})$ , for some $k geq 0$ . Using the fact that logical equivalence is often captured by Ehrenfeucht–Fraïssé games, the authors formulate a game-based framework in which logical equivalence-based locality can be defined. Thus, the notion defined by the authors is that of game-based locality. Although quite promising as well as easy to apply, the game-based framework (used to define locality under logical equivalence) has the following problem: if a logic $mathcal {L}$ is local (Hanf-, or Gaifman-, or weakly) under isomorphisms, and $mathcal {L}'$ is a sub-logic of $mathcal {L}$ , then $mathcal {L}'$ is local as well. The same, however, is not true for game-based locality: properties of games guaranteeing locality need not be preserved if one passes to weaker games. The question that immediately arises is: is it possible to define the notion of locality under logical equivalence without resorting to game-based frameworks? In this thesis, I present a homotopic variation for locality under logical equivalence, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k. Abstract pr
{"title":"Quillen Model Categories-Based Notions of Locality of Logics over Finite Structures","authors":"Hendrick Maia","doi":"10.1017/bsl.2021.37","DOIUrl":"https://doi.org/10.1017/bsl.2021.37","url":null,"abstract":"Abstract Locality is a property of logics, whose origins lie in the works of Hanf and Gaifman, having their utility in the context of finite model theory. Such a property is quite useful in proofs of inexpressibility, but it is also useful in establishing normal forms for logical formulas. There are generally two forms of locality: (i’) if two structures \u0000$mathfrak {A}$\u0000 and \u0000$mathfrak {B}$\u0000 realize the same multiset of types of neighborhoods of radius d, then they agree on a given sentence \u0000$Phi $\u0000 . Here d depends only on \u0000$Phi $\u0000 ; (ii’) if the d-neighborhoods of two tuples \u0000$vec {a}_1$\u0000 and \u0000$vec {a}_2$\u0000 in a structure \u0000$mathfrak {A}$\u0000 are isomorphic, then \u0000$mathfrak {A} models Phi (vec {a}_1) Leftrightarrow Phi (vec {a}_2)$\u0000 . Again, d depends on \u0000$Phi $\u0000 , and not on \u0000$mathfrak {A}$\u0000 . Form (i’) originated from Hanf’s works. Form (ii’) came from Gaifman’s theorem. There is no doubt about the usefulness of the notion of locality, which as seen applies to a huge number of situations. However, there is a deficiency in such a notion: all versions of the notion of locality refer to isomorphism of neighborhoods, which is a fairly strong property. For example, where structures simply do not have sufficient isomorphic neighborhoods, versions of the notion of locality obviously cannot be applied. So the question that immediately arises is: would it be possible to weaken such a condition and maintain Hanf/Gaifman-localities? Arenas, Barceló, and Libkin establish a new condition for the notions of locality, weakening the requirement that neighborhoods should be isomorphic, establishing only the condition that they must be indistinguishable in a given logic. That is, instead of requiring \u0000$N_d(vec {a}) cong N_d(vec {b})$\u0000 , you should only require \u0000$N_d(vec {a}) equiv _k N_d(vec {b})$\u0000 , for some \u0000$k geq 0$\u0000 . Using the fact that logical equivalence is often captured by Ehrenfeucht–Fraïssé games, the authors formulate a game-based framework in which logical equivalence-based locality can be defined. Thus, the notion defined by the authors is that of game-based locality. Although quite promising as well as easy to apply, the game-based framework (used to define locality under logical equivalence) has the following problem: if a logic \u0000$mathcal {L}$\u0000 is local (Hanf-, or Gaifman-, or weakly) under isomorphisms, and \u0000$mathcal {L}'$\u0000 is a sub-logic of \u0000$mathcal {L}$\u0000 , then \u0000$mathcal {L}'$\u0000 is local as well. The same, however, is not true for game-based locality: properties of games guaranteeing locality need not be preserved if one passes to weaker games. The question that immediately arises is: is it possible to define the notion of locality under logical equivalence without resorting to game-based frameworks? In this thesis, I present a homotopic variation for locality under logical equivalence, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k. Abstract pr","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87148493","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}
Abstract Logic is traditionally considered to be a purely syntactic discipline, at least in principle. However, prof. David Isles has shown that this ideal is not yet met in traditional logic. Semantic residue is present in the assumption that the domain of a variable should be fixed in advance of a derivation, and also in the notion that a numerical notation must refer to a number rather than be considered a mathematical object in and of itself. Based on his work, the central question of this thesis is what kind of logic, if any, results from removing this semantic residue from traditional logic. We differ from traditional logic in two significant ways. The first is that the assumption that a numerical notation must refer to a number is denied. Numerical notations are considered as mathematical objects in their own right, related to each other by means of rewrite rules. The traditional notion of reference is then replaced by the notion of reduction (by means of the rewrite rules) to a normal form. Two numerical notations that reduce to the same normal form would traditionally be considered identical, as they would refer to the same number, and hence they would be interchangeable salva veritate. In the new system, called Buridan-Volpin (BV), the numerical notations themselves are the elements of the domains of variables, and two numerical notations that reduce to the same normal form need not be interchangeable salva veritate, except when they are syntactically identical (i.e., have the same Gödel number). The second is that we do away with the assumption that the domains of variables need to be fixed in advance of a derivation. Instead we focus on what is needed to guarantee preservation of truth in every step of a derivation. These conditions on the domains of the variables, accumulated in the course of a derivation, are combined in a reference grammar. Whereas traditionally a derivation is considered valid when the conclusion follows from the premisses by way of the derivation rules (and possibly axioms), in the BV system a derivation must meet the extra condition that no inconsistency occurs within the reference grammar. For if the reference grammar were to give rise to inconsistency (i.e., it would be impossible to assign domains to all the variables without breaking at least one of the conditions placed on them in the reference grammar), there is no longer a guarantee that truth has been preserved in every step of the derivation, and hence the truth of the conclusion is not guaranteed by the derivation. In Chapter 2 the BV system is introduced in some formal detail. Chapter 3 gives some examples of derivations, notably totality of addition, multiplication and exponentiation, as well as a lemma needed for the proof of Euclid’s Theorem. These examples, taken from prof. Isles’ First-Order Reasoning and Primitive Recursive Natural Number Notations, show that there is a real proof-theoretical difference between traditional logic and the BV syste
{"title":"The Buridan-Volpin Derivation System; Properties and Justification","authors":"Sven Storms","doi":"10.1017/bsl.2022.35","DOIUrl":"https://doi.org/10.1017/bsl.2022.35","url":null,"abstract":"Abstract Logic is traditionally considered to be a purely syntactic discipline, at least in principle. However, prof. David Isles has shown that this ideal is not yet met in traditional logic. Semantic residue is present in the assumption that the domain of a variable should be fixed in advance of a derivation, and also in the notion that a numerical notation must refer to a number rather than be considered a mathematical object in and of itself. Based on his work, the central question of this thesis is what kind of logic, if any, results from removing this semantic residue from traditional logic. We differ from traditional logic in two significant ways. The first is that the assumption that a numerical notation must refer to a number is denied. Numerical notations are considered as mathematical objects in their own right, related to each other by means of rewrite rules. The traditional notion of reference is then replaced by the notion of reduction (by means of the rewrite rules) to a normal form. Two numerical notations that reduce to the same normal form would traditionally be considered identical, as they would refer to the same number, and hence they would be interchangeable salva veritate. In the new system, called Buridan-Volpin (BV), the numerical notations themselves are the elements of the domains of variables, and two numerical notations that reduce to the same normal form need not be interchangeable salva veritate, except when they are syntactically identical (i.e., have the same Gödel number). The second is that we do away with the assumption that the domains of variables need to be fixed in advance of a derivation. Instead we focus on what is needed to guarantee preservation of truth in every step of a derivation. These conditions on the domains of the variables, accumulated in the course of a derivation, are combined in a reference grammar. Whereas traditionally a derivation is considered valid when the conclusion follows from the premisses by way of the derivation rules (and possibly axioms), in the BV system a derivation must meet the extra condition that no inconsistency occurs within the reference grammar. For if the reference grammar were to give rise to inconsistency (i.e., it would be impossible to assign domains to all the variables without breaking at least one of the conditions placed on them in the reference grammar), there is no longer a guarantee that truth has been preserved in every step of the derivation, and hence the truth of the conclusion is not guaranteed by the derivation. In Chapter 2 the BV system is introduced in some formal detail. Chapter 3 gives some examples of derivations, notably totality of addition, multiplication and exponentiation, as well as a lemma needed for the proof of Euclid’s Theorem. These examples, taken from prof. Isles’ First-Order Reasoning and Primitive Recursive Natural Number Notations, show that there is a real proof-theoretical difference between traditional logic and the BV syste","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86963350","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}
Valentine Kabanets, J. Moore, Rehana Patel, S. Shieh, J. Knight, Philipp Hieronymi, Joel Nagloo, Christopher Porter, J. Zapletal
The Connes Embedding Problem is one of the most famous open problems in the theory of von Neumann algebras and can be stated in purely model-theoretic terms: do all II 1 factors have the same universal theory? Here, a II 1 factor is an infinite-dimensional von Neumann algebra that
{"title":"2022 NORTH AMERICAN ANNUAL MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC Cornell University Ithaca, NY, USA April 7–10, 2022","authors":"Valentine Kabanets, J. Moore, Rehana Patel, S. Shieh, J. Knight, Philipp Hieronymi, Joel Nagloo, Christopher Porter, J. Zapletal","doi":"10.1017/bsl.2022.24","DOIUrl":"https://doi.org/10.1017/bsl.2022.24","url":null,"abstract":"The Connes Embedding Problem is one of the most famous open problems in the theory of von Neumann algebras and can be stated in purely model-theoretic terms: do all II 1 factors have the same universal theory? Here, a II 1 factor is an infinite-dimensional von Neumann algebra that","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85349999","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":"2022 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM 2022 Reykjavík University Reykjavík, Iceland June 27 – July 1, 2022","authors":"","doi":"10.1017/bsl.2022.38","DOIUrl":"https://doi.org/10.1017/bsl.2022.38","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86948408","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}
G. Bezhanishvili, A. Enayat, K. Bimbó, Øystein Linnebo, Paola D’Aquino Anca Muscholl, P. Dybjer, A. Pauly, Albert Atserias, D. Macpherson, M. Atten, Antonio Montalbán, B. V. D. Berg, Christian Retoré, Clinton Conley, Marion Scheepers, B. Hart, Nam Trang, Christian Rosendal
{"title":"BSL volume 28 issue 4 Cover and Front matter","authors":"G. Bezhanishvili, A. Enayat, K. Bimbó, Øystein Linnebo, Paola D’Aquino Anca Muscholl, P. Dybjer, A. Pauly, Albert Atserias, D. Macpherson, M. Atten, Antonio Montalbán, B. V. D. Berg, Christian Retoré, Clinton Conley, Marion Scheepers, B. Hart, Nam Trang, Christian Rosendal","doi":"10.1017/bsl.2022.42","DOIUrl":"https://doi.org/10.1017/bsl.2022.42","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73878178","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}
Abstract Intuitionistic logic formalises the foundational ideas of L.E.J. Brouwer’s mathematical programme of intuitionism. It is one of the earliest non-classical logics, and the difference between classical and intuitionistic logic may be interpreted to lie in the law of the excluded middle, which asserts that either a proposition is true or its negation is true. This principle is deemed unacceptable from the constructive point of view, in whose understanding the law means that there is an effective procedure to determine the truth of all propositions. This understanding of the distinction between the two logics supports the view that negation plays a vital role in the formulation of intuitionistic logic. Nonetheless, the formalisation of negation in intuitionistic logic has not been universally accepted, and many alternative accounts of negation have been proposed. Some seek to weaken or strengthen the negation, and others actively supporting negative inferences that are impossible with it. This thesis follows this tradition and investigates various aspects of negation in intuitionistic logic. Firstly, we look at a problem proposed by H. Ishihara, which asks how effectively one can conserve the deducibility of classical theorems into intuitionistic logic, by assuming atomic classes of non-constructive principles. The classes given in this section improve a previous class given by K. Ishii in two respects: (a) instead of a single class for the law of the excluded middle, two classes are given in terms of weaker principles, allowing a finer analysis and (b) the conservation now extends to a subsystem of intuitionistic logic called Glivenko’s logic. This section also discusses the extension of Ishihara’s problem to minimal logic. Secondly, we study the relationship between two frameworks for weak constructive negation, the approach of D. Vakarelov on one hand and the framework of subminimal negation by A. Colacito, D. de Jongh, and A. L. Vargas on the other hand. We capture a version of Vakarelov’s logic with the semantics of the latter framework, and clarify the relationship between the two semantics. This also provides proof-theoretic insights, which results in the formulation of a cut-free sequent calculus for the aforementioned system. Thirdly, we investigate the ways to unify the formalisations of some logics with contra-intuitionistic inferences. The enquiry concerns paraconsistent logics by R. Sylvan and A. B. Gordienko, as well as the logic of co-negation by G. Priest and of empirical negation by M. De and H. Omori. We take Sylvan’s system as basic, and formulate the frame conditions of the defining axioms of the other systems. The conditions are then used to obtain cut-free labelled sequent calculi for the systems. Finally, we consider L. Humberstone’s actuality operator for intuitionistic logic, which can be seen as the dualisation of a contra-intuitionistic negation. A compete axiomatisation of intuitionistic logic with actuality opera
直观主义逻辑形式化了布朗尔的直观主义数学纲领的基本思想。它是最早的非经典逻辑之一，古典逻辑与直觉逻辑之间的区别可以解释为排中律，排中律主张命题为真或其否定为真。从建设性的观点来看，这一原则被认为是不可接受的，在建设性的观点中，对法律的理解意味着存在一个有效的程序来确定所有命题的真实性。这种对两种逻辑之间区别的理解支持了否定在直觉主义逻辑的形成中起着至关重要作用的观点。尽管如此，直觉主义逻辑中否定的形式化并没有被普遍接受，并且已经提出了许多关于否定的替代说法。一些人试图削弱或加强否定，而另一些人则积极支持不可能的否定推论。本文沿袭这一传统，探讨了直觉主义逻辑中否定的各个方面。首先，我们看一下H. Ishihara提出的一个问题，这个问题是通过假设非构造原理的原子类，如何有效地将经典定理的可演绎性保存到直觉逻辑中。本节给出的类在两个方面改进了K. Ishii先前给出的类:(a)不是排除中间定律的单一类，而是根据较弱的原则给出的两个类，允许更精细的分析;(b)守恒现在扩展到直觉逻辑的子系统，称为Glivenko逻辑。本节还讨论了将石原问题扩展到最小逻辑的问题。其次，我们研究了两种弱建设性否定框架，即D. Vakarelov的方法和A. Colacito、D. de Jongh和A. L. Vargas的次极小否定框架之间的关系。我们用后一种框架的语义捕捉了Vakarelov逻辑的一个版本，并阐明了这两个语义之间的关系。这也提供了证明理论的见解，这导致了上述系统的无切割序列演算的公式。第三，我们研究了用反直觉推理统一某些逻辑形式化的方法。该研究涉及R. Sylvan和A. B. Gordienko的副一致逻辑，以及G. Priest的共同否定逻辑和M. De和H. Omori的经验否定逻辑。以Sylvan系统为基本，给出了其他系统定义公理的框架条件。然后利用这些条件得到系统的无切割标记序演算。最后，我们考虑了L. Humberstone对于直觉逻辑的现实性算子，它可以看作是一个反直觉否定的二元化。给出了直觉逻辑与现实算子的竞争公理化，并对相关算子进行了比较。摘要由Satoru Niki准备。电子邮件:Satoru.Niki@rub.de
{"title":"Investigations into intuitionistic and other negations","authors":"Satoru Niki","doi":"10.1017/bsl.2022.29","DOIUrl":"https://doi.org/10.1017/bsl.2022.29","url":null,"abstract":"Abstract Intuitionistic logic formalises the foundational ideas of L.E.J. Brouwer’s mathematical programme of intuitionism. It is one of the earliest non-classical logics, and the difference between classical and intuitionistic logic may be interpreted to lie in the law of the excluded middle, which asserts that either a proposition is true or its negation is true. This principle is deemed unacceptable from the constructive point of view, in whose understanding the law means that there is an effective procedure to determine the truth of all propositions. This understanding of the distinction between the two logics supports the view that negation plays a vital role in the formulation of intuitionistic logic. Nonetheless, the formalisation of negation in intuitionistic logic has not been universally accepted, and many alternative accounts of negation have been proposed. Some seek to weaken or strengthen the negation, and others actively supporting negative inferences that are impossible with it. This thesis follows this tradition and investigates various aspects of negation in intuitionistic logic. Firstly, we look at a problem proposed by H. Ishihara, which asks how effectively one can conserve the deducibility of classical theorems into intuitionistic logic, by assuming atomic classes of non-constructive principles. The classes given in this section improve a previous class given by K. Ishii in two respects: (a) instead of a single class for the law of the excluded middle, two classes are given in terms of weaker principles, allowing a finer analysis and (b) the conservation now extends to a subsystem of intuitionistic logic called Glivenko’s logic. This section also discusses the extension of Ishihara’s problem to minimal logic. Secondly, we study the relationship between two frameworks for weak constructive negation, the approach of D. Vakarelov on one hand and the framework of subminimal negation by A. Colacito, D. de Jongh, and A. L. Vargas on the other hand. We capture a version of Vakarelov’s logic with the semantics of the latter framework, and clarify the relationship between the two semantics. This also provides proof-theoretic insights, which results in the formulation of a cut-free sequent calculus for the aforementioned system. Thirdly, we investigate the ways to unify the formalisations of some logics with contra-intuitionistic inferences. The enquiry concerns paraconsistent logics by R. Sylvan and A. B. Gordienko, as well as the logic of co-negation by G. Priest and of empirical negation by M. De and H. Omori. We take Sylvan’s system as basic, and formulate the frame conditions of the defining axioms of the other systems. The conditions are then used to obtain cut-free labelled sequent calculi for the systems. Finally, we consider L. Humberstone’s actuality operator for intuitionistic logic, which can be seen as the dualisation of a contra-intuitionistic negation. A compete axiomatisation of intuitionistic logic with actuality opera","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89847350","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}
Abstract We present a proof system for a multimode and multimodal logic, which is based on our previous work on modal Martin-Löf type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e., a small 2-category. The logic is extended to a lambda calculus, establishing a Curry–Howard correspondence.
{"title":"UNDER LOCK AND KEY: A PROOF SYSTEM FOR A MULTIMODAL LOGIC","authors":"G. A. Kavvos, Daniel Gratzer","doi":"10.1017/bsl.2023.14","DOIUrl":"https://doi.org/10.1017/bsl.2023.14","url":null,"abstract":"Abstract We present a proof system for a multimode and multimodal logic, which is based on our previous work on modal Martin-Löf type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e., a small 2-category. The logic is extended to a lambda calculus, establishing a Curry–Howard correspondence.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82504919","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}
Abstract Two famous negative results about da Costa’s paraconsistent logic ${mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${mathscr {C}}_1$ . On the one hand, we strengthen the negative results about ${mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${mathscr {C}}_1$ covered in the literature. We prove that for extensions ${mathcal {S}}$ such as ${mathcal {C}ilo}$ [26], every algebra in ${mathsf {Alg}}^*({mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${mathcal {S}}$ such as , , or [16, 53], every subdirectly irreducible algebra in ${mathsf {Alg}}^*({mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${mathsf {Alg}}^*({mathcal {S}})$ and the intrinsic variety $mathbb {V}({mathcal {S}})$ , with , , and .
{"title":"AN ABSTRACT ALGEBRAIC LOGIC STUDY OF DA COSTA’S LOGIC \u0000${mathscr {C}}_1$\u0000 AND SOME OF ITS PARACONSISTENT EXTENSIONS","authors":"Hugo Albuquerque, Carlos Caleiro","doi":"10.1017/bsl.2022.36","DOIUrl":"https://doi.org/10.1017/bsl.2022.36","url":null,"abstract":"Abstract Two famous negative results about da Costa’s paraconsistent logic \u0000${mathscr {C}}_1$\u0000 (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed \u0000${mathscr {C}}_1$\u0000 seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic \u0000${mathscr {C}}_1$\u0000 . On the one hand, we strengthen the negative results about \u0000${mathscr {C}}_1$\u0000 by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, \u0000${mathscr {C}}_1$\u0000 is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of \u0000${mathscr {C}}_1$\u0000 covered in the literature. We prove that for extensions \u0000${mathcal {S}}$\u0000 such as \u0000${mathcal {C}ilo}$\u0000 [26], every algebra in \u0000${mathsf {Alg}}^*({mathcal {S}})$\u0000 contains a Boolean subalgebra, and for extensions \u0000${mathcal {S}}$\u0000 such as , , or [16, 53], every subdirectly irreducible algebra in \u0000${mathsf {Alg}}^*({mathcal {S}})$\u0000 has cardinality at most 3. We also characterize the quasivariety \u0000${mathsf {Alg}}^*({mathcal {S}})$\u0000 and the intrinsic variety \u0000$mathbb {V}({mathcal {S}})$\u0000 , with , , and .","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76442278","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}
We prove various results connected together by the common thread of computability theory. First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension. We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable subsets. We demonstrate how this notion of $Pi ^0_1$ -immunity is connected to other immunity notions, and construct $Pi ^0_1$ -immune reals throughout the high/low and Ershov hierarchies. We also study those degrees that cannot compute or cannot co-enumerate a $Pi ^0_1$ -immune set. Finally, we discuss a recently discovered truth-table reduction for transforming a Kolmogorov–Loveland random input into a Martin-Löf random output by exploiting the fact that at least one half of such a KL-random is itself ML-random. We show that there is no better algorithm relying on this fact, in the sense that there is no positive, linear, or bounded truth-table reduction which does this. We also generalize these results to the problem of outputting randomness from infinitely many inputs, only some of which are random. Abstract prepared by David J. Webb. E-mail: dwebb@math.hawaii.edu URL: https://arxiv.org/pdf/2209.05659.pdf
我们证明了由可计算性理论的共同线索联系在一起的各种结果。首先，我们研究了一种新的算法维数的概念——不可逃避维数，它位于有效豪斯多夫维数和包装维数之间。我们还研究了它的推广，得到了图灵度嵌入到维数概念中的方法。然后，我们研究了在前面的研究过程中产生的一个新的可计算理论免疫的概念，即没有共枚举子集的自然数集。我们证明了$Pi ^0_1$ -免疫的概念如何与其他免疫概念联系起来，并在整个高/低和Ershov层次中构造$Pi ^0_1$ -免疫实数。我们还研究了那些不能计算或不能共枚举$Pi ^0_1$免疫集的度。最后，我们讨论了最近发现的将Kolmogorov-Loveland随机输入转换为Martin-Löf随机输出的真值表约简，该约简利用了这样一个KL-random的至少一半本身是ML-random的事实。我们证明，没有更好的算法依赖于这个事实，在某种意义上，没有正的，线性的，或有界的真值表约简做到了这一点。我们也将这些结果推广到从无限多个输入输出随机性的问题，其中只有一些是随机的。David J. Webb撰写的摘要。电子邮件:dwebb@math.hawaii.edu URL: https://arxiv.org/pdf/2209.05659.pdf
{"title":"On New Notions of Algorithmic Dimension, Immunity, and Medvedev Degree","authors":"David J. Webb","doi":"10.1017/bsl.2022.33","DOIUrl":"https://doi.org/10.1017/bsl.2022.33","url":null,"abstract":"We prove various results connected together by the common thread of computability theory. First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension. We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable subsets. We demonstrate how this notion of \u0000$Pi ^0_1$\u0000 -immunity is connected to other immunity notions, and construct \u0000$Pi ^0_1$\u0000 -immune reals throughout the high/low and Ershov hierarchies. We also study those degrees that cannot compute or cannot co-enumerate a \u0000$Pi ^0_1$\u0000 -immune set. Finally, we discuss a recently discovered truth-table reduction for transforming a Kolmogorov–Loveland random input into a Martin-Löf random output by exploiting the fact that at least one half of such a KL-random is itself ML-random. We show that there is no better algorithm relying on this fact, in the sense that there is no positive, linear, or bounded truth-table reduction which does this. We also generalize these results to the problem of outputting randomness from infinitely many inputs, only some of which are random. Abstract prepared by David J. Webb. E-mail: dwebb@math.hawaii.edu URL: https://arxiv.org/pdf/2209.05659.pdf","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76838231","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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