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On sharp and single-conclusion justification models 关于尖锐的和单结论的论证模型
J. Log. Comput. Pub Date : 2020-12-19 DOI: 10.1093/logcom/exaa074
V. Krupski
{"title":"On sharp and single-conclusion justification models","authors":"V. Krupski","doi":"10.1093/logcom/exaa074","DOIUrl":"https://doi.org/10.1093/logcom/exaa074","url":null,"abstract":"\u0000 Justification awareness models (JAMs) were proposed by S. Artemov as a tool for modelling epistemic scenarios such as Russell’s prime minister example. It was demonstrated that the sharpness and the single-conclusion property of a model play an essential role in the epistemic usage of JAMs. The problem to axiomatize these properties using the propositional justification language was left opened. We propose the solution and define a decidable justification logic $textsf{J}_{textit{ref}}$ that is sound and complete with respect to the class of all sharp single-conclusion justification models. We also provide the complete axiomatizations for the classes of all single-conclusion justification models and all sharp justification models.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"19-20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127175627","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}
引用次数: 0
Explicit analyses of proof/refutation interaction for constructible falsity and Heyting-Brouwer logic 可构造虚性与heyting - browwer逻辑的证明/反驳交互作用的显式分析
J. Log. Comput. Pub Date : 2020-12-10 DOI: 10.1093/LOGCOM/EXAA047
T. M. Ferguson
{"title":"Explicit analyses of proof/refutation interaction for constructible falsity and Heyting-Brouwer logic","authors":"T. M. Ferguson","doi":"10.1093/LOGCOM/EXAA047","DOIUrl":"https://doi.org/10.1093/LOGCOM/EXAA047","url":null,"abstract":"\u0000 Nelson’s logic of constructible falsity  $textsf{N}$ and Rauszer’s Heyting–Brouwer logic  $textsf{HB}$ are well-known cases of extensions of intuitionistic logic $textsf{Int}$ enriched with novel connectives. Wansing has suggested that Gödel’s provability interpretation of $textsf{Int}$ can be extended to these systems by pairing the category of formal proofs with a distinct category of formal refutations. In this paper, we extend the framework of Artemov’s justification logic to provide explicit analyses of $textsf{N}$ and $textsf{HB}$ (and the dual-intuitionistic logic $textsf{DualInt}$) that respect a distinction between proofs and refutations. The application distinguishes the categories by reinterpreting the agents of multiple-agent justification logic as devices that operate exclusively on one or the other category. The analyses reveal that differences between $textsf{N}$ and $textsf{HB}$ can be reduced to competing interaction principles characterizing the coordination between proofs and refutations. We conclude by reappraising some of the unusual features of $textsf{HB}$ in light of the explicit analysis of $textsf{HB}$.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115786145","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}
引用次数: 2
Epistemic predicates in the arithmetical context 算术语境中的认知谓词
J. Log. Comput. Pub Date : 2020-12-10 DOI: 10.1093/logcom/exaa051
V. A. Peluce
{"title":"Epistemic predicates in the arithmetical context","authors":"V. A. Peluce","doi":"10.1093/logcom/exaa051","DOIUrl":"https://doi.org/10.1093/logcom/exaa051","url":null,"abstract":"\u0000 In this paper, we investigate epistemic predicates in extensions of arithmetic. We use as our case study Kurt Gödel’s 1951 thesis that either the power of the human mind surpasses that of any finite machine or there are absolutely unsolvable problems. Because Gödel also claimed that his disjunction was a mathematically established fact, we must ask the following: what sort of syntactical object should formalize human reason? In this paper, we lay the foundations for a predicate treatment of this epistemic feature. We begin with a very general examination of the Gödel sentence in the arithmetical context. We then discuss two systems of modal predicates over arithmetic. The first, called coreflective arithmetic or ${textsf{CoPA}}$, extends ${textsf{PA}}$ with a coreflective modal predicate but does not contain a consistency statement. The second, called doxastic arithmetic or ${textsf{DA}}$, has as its characteristic feature the consistency statement but does not contain coreflection or its instance, the ${textsf{4}}$ axiom. We examine the logical properties of, motivations for and criticisms of both systems. We close with a brief comparison of the systems in the context of Gödel’s disjunction.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127088441","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}
引用次数: 1
An abstract form of the first epsilon theorem 第一定理的抽象形式
J. Log. Comput. Pub Date : 2020-12-10 DOI: 10.1093/logcom/exaa044
M. Baaz, A. Leitsch, Anela Lolic
{"title":"An abstract form of the first epsilon theorem","authors":"M. Baaz, A. Leitsch, Anela Lolic","doi":"10.1093/logcom/exaa044","DOIUrl":"https://doi.org/10.1093/logcom/exaa044","url":null,"abstract":"\u0000 We present a new method of computing Herbrand disjunctions. The up-to-date most direct approach to calculate Herbrand disjunctions is based on Hilbert’s epsilon formalism (which is in fact also the oldest framework for proof theory). The algorithm to calculate Herbrand disjunctions is an integral part of the proof of the extended first epsilon theorem. This paper introduces a more abstract form of epsilon proofs, the function variable proofs. This leads to a computational improved version of the extended first epsilon theorem, which allows a nonelementary speed up of the computation of Herbrand disjunctions. As an application, sequent calculus proofs are translated into function variable proofs and a variant of the axiom of global choice is shown to be removable from proofs in Neumann–Bernays–Gödel set theory.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116276622","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}
引用次数: 0
Kripke-style Semantics and Completeness for Full Simply Typed Lambda Calculus 完全简单类型Lambda演算的kripke风格语义和完备性
J. Log. Comput. Pub Date : 2020-12-10 DOI: 10.1093/LOGCOM/EXAA055
Simona Kasterovic, S. Ghilezan
{"title":"Kripke-style Semantics and Completeness for Full Simply Typed Lambda Calculus","authors":"Simona Kasterovic, S. Ghilezan","doi":"10.1093/LOGCOM/EXAA055","DOIUrl":"https://doi.org/10.1093/LOGCOM/EXAA055","url":null,"abstract":"\u0000 Full simply typed lambda calculus is the simply typed lambda calculus extended with product types and sum types. We propose a Kripke-style semantics for full simply typed lambda calculus. We then prove soundness and completeness of type assignment in full simply typed lambda calculus with respect to the proposed semantics. The key point in the proof of completeness is the notion of a canonical model.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"197 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115655140","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}
引用次数: 4
Herbrand Proofs and Expansion Proofs as Decomposed Proofs 作为分解证明的赫氏证明和展开证明
J. Log. Comput. Pub Date : 2020-10-24 DOI: 10.1093/logcom/exaa052
B. Ralph
{"title":"Herbrand Proofs and Expansion Proofs as Decomposed Proofs","authors":"B. Ralph","doi":"10.1093/logcom/exaa052","DOIUrl":"https://doi.org/10.1093/logcom/exaa052","url":null,"abstract":"\u0000 The reduction of undecidable first-order logic to decidable propositional logic via Herbrand’s theorem has long been of interest to theoretical computer science, with the notion of a Herbrand proof motivating the definition of expansion proofs. In this paper we construct simple deep inference systems for first-order logic, both with and without cut, such that ‘decomposed’ proofs—proofs where the contractive and non-contractive behaviour of the proof is separated—in each system correspond to either expansion proofs or Herbrand proofs. Translations between proofs in this system, expansion proofs and Herbrand proofs are given, retaining much of the structure in each direction.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127084330","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}
引用次数: 0
Founded semantics and constraint semantics of logic rules 逻辑规则的建立语义和约束语义
J. Log. Comput. Pub Date : 2020-10-05 DOI: 10.1093/logcom/exaa056
Yanhong A. Liu, S. Stoller
{"title":"Founded semantics and constraint semantics of logic rules","authors":"Yanhong A. Liu, S. Stoller","doi":"10.1093/logcom/exaa056","DOIUrl":"https://doi.org/10.1093/logcom/exaa056","url":null,"abstract":"\u0000 Logic rules and inference are fundamental in computer science and have been studied extensively. However, prior semantics of logic languages can have subtle implications and can disagree significantly, on even very simple programs, including in attempting to solve the well-known Russell’s paradox. These semantics are often non-intuitive and hard-to-understand when unrestricted negation is used in recursion. This paper describes a simple new semantics for logic rules, founded semantics, and its straightforward extension to another simple new semantics, constraint semantics, that unify the core of different prior semantics. The new semantics support unrestricted negation, as well as unrestricted existential and universal quantifications. They are uniquely expressive and intuitive by allowing assumptions about the predicates, rules and reasoning to be specified explicitly, as simple and precise binary choices. They are completely declarative and relate cleanly to prior semantics. In addition, founded semantics can be computed in linear time in the size of the ground program.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121524208","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}
引用次数: 1
Fraïssé-Hintikka theorem in institutions Fraïssé-Hintikka制度定理
J. Log. Comput. Pub Date : 2020-09-03 DOI: 10.1093/logcom/exaa042
Daniel Găină, T. Kowalski
{"title":"Fraïssé-Hintikka theorem in institutions","authors":"Daniel Găină, T. Kowalski","doi":"10.1093/logcom/exaa042","DOIUrl":"https://doi.org/10.1093/logcom/exaa042","url":null,"abstract":"We generalize the characterization of elementary equivalence by Ehrenfeucht-Fräıssé games to arbitrary institutions whose sentences are finitary. These include many-sorted first-order logic, higher-order logic with types, as well as a number of other logics arising in connection to specification languages. The gain for the classical case is that the characterization is proved directly for all signatures, including infinite ones.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114478652","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}
引用次数: 0
A model-theoretic approach to descriptive general frames: the van Benthem characterization theorem 描述一般框架的模型理论方法:van Benthem表征定理
J. Log. Comput. Pub Date : 2020-08-26 DOI: 10.1093/logcom/exaa040
N. Bezhanishvili, Tim Henke
{"title":"A model-theoretic approach to descriptive general frames: the van Benthem characterization theorem","authors":"N. Bezhanishvili, Tim Henke","doi":"10.1093/logcom/exaa040","DOIUrl":"https://doi.org/10.1093/logcom/exaa040","url":null,"abstract":"\u0000 The celebrated van Benthem characterization theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper, we prove an analogue of the van Benthem characterization theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a ‘continuous’ binary relation. The proof of our theorem generalizes Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem and of the upward Löwenheim–Skolem theorem.1","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133540557","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}
引用次数: 0
Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages 有限Kripke框架一阶模态逻辑的算法性质
J. Log. Comput. Pub Date : 2020-08-26 DOI: 10.1093/logcom/exaa041
M. Rybakov, D. Shkatov
{"title":"Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages","authors":"M. Rybakov, D. Shkatov","doi":"10.1093/logcom/exaa041","DOIUrl":"https://doi.org/10.1093/logcom/exaa041","url":null,"abstract":"\u0000 We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $textbf{GL}$, $textbf{Grz}$ and $textbf{KTB}$—among them, $textbf{K}$, $textbf{T}$, $textbf{D}$, $textbf{KB}$, $textbf{K4}$ and $textbf{S4}$.","PeriodicalId":398763,"journal":{"name":"J. Log. Comput.","volume":"148 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132184668","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}
引用次数: 15
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