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Bulletin of the Section of Logic最新文献

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On Implicative and Positive Implicative GE Algebras 关于隐含和正隐含GE代数
Bulletin of the Section of Logic Pub Date : 2023-09-25 DOI: 10.18778/0138-0680.2023.21
Andrzej Walendziak
{"title":"On Implicative and Positive Implicative GE Algebras","authors":"Andrzej Walendziak","doi":"10.18778/0138-0680.2023.21","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.21","url":null,"abstract":"GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebrasverifying the antisymmetry) are a generalization of Hilbert algebras. Here some properties and characterizations of these algebras are investigated. Connections between GE algebras and other classes of algebras of logic are studied. The implicative and positive implicative properties are discussed. It is shown that the class of positive implicative GE algebras (resp. the class of implicative aGE algebras) coincides with the class of generalized Tarski algebras (resp. the class of Tarski algebras). It is proved that for any aGE algebra the property of implicativity is equivalent to the commutative property. Moreover, several examples to illustrate the results are given. Finally, the interrelationships between some classes of implicative and positive implicative algebras are presented.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135864231","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
Cut Elimination for Extended Sequent Calculi 扩展序列微积分的切消法
Bulletin of the Section of Logic Pub Date : 2023-09-25 DOI: 10.18778/0138-0680.2023.22
Simone Martini, Andrea Masini, Margherita Zorzi
{"title":"Cut Elimination for Extended Sequent Calculi","authors":"Simone Martini, Andrea Masini, Margherita Zorzi","doi":"10.18778/0138-0680.2023.22","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.22","url":null,"abstract":"We present a syntactical cut-elimination proof for an extended sequent calculus covering the classical modal logics in the (mathsf{K}), (mathsf{D}), (mathsf{T}), (mathsf{K4}), (mathsf{D4}) and (mathsf{S4}) spectrum. We design the systems uniformly since they all share the same set of rules. Different logics are obtained by “tuning” a single parameter, namely a constraint on the applicability of the cut rule and on the (left and right, respectively) rules for (Box) and (Diamond). Starting points for this research are 2-sequents and indexed-based calculi (sequents and tableaux). By extending and modifying existing proposals, we show how to achieve a syntactical proof of the cut-elimination theorem that is as close as possible to the one for first-order classical logic.In doing this, we implicitly show how small is the proof-theoretical distance between classical logic and the systems under consideration.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135864087","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
Extended BCK-Ideal Based on Single-Valued Neutrosophic Hyper BCK-Ideals 基于单值Neutrosophic超BCK理想的扩展BCK理想
Bulletin of the Section of Logic Pub Date : 2023-08-10 DOI: 10.18778/0138-0680.2023.20
M. Hamidi
{"title":"Extended BCK-Ideal Based on Single-Valued Neutrosophic Hyper BCK-Ideals","authors":"M. Hamidi","doi":"10.18778/0138-0680.2023.20","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.20","url":null,"abstract":"This paper introduces the concept of single-valued neutrosophic hyper (BCK)-subalgebras as a generalization and alternative of hyper (BCK)-algebras and on any given nonempty set constructs at least one single-valued neutrosophic hyper (BCK)-subalgebra and one a single-valued neutrosophic hyper (BCK)-ideal. In this study level subsets play the main role in the connection between single-valued neutrosophic hyper (BCK)-subalgebras and hyper (BCK)-subalgebras and the connection between single-valued neutrosophic hyper (BCK)-ideals and hyper (BCK)-ideals. The congruence and (strongly) regular equivalence relations are the important tools for connecting hyperstructures and structures, so the major contribution of this study is to apply and introduce a (strongly) regular relation on hyper (BCK)-algebras and to investigate their categorical properties (quasi commutative diagram) via single-valued neutrosophic hyper (BCK)-ideals. Indeed, by using the single-valued neutrosophic hyper (BCK)-ideals, we define a congruence relation on (weak commutative) hyper (BCK)-algebras that under some conditions is strongly regular and the quotient of any (single-valued neutrosophic)hyper (BCK)-(sub)algebra via this relation is a (single-valued neutrosophic)(hyper (BCK)-subalgebra) (BCK)-(sub)algebra.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48765688","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
Fundamental Relation on HvBE-Algebras hvbe代数上的基本关系
Bulletin of the Section of Logic Pub Date : 2023-08-09 DOI: 10.18778/0138-0680.2023.10
Farzad Iranmanesh, M. Ghadiri, A. Borumand Saeid
{"title":"Fundamental Relation on HvBE-Algebras","authors":"Farzad Iranmanesh, M. Ghadiri, A. Borumand Saeid","doi":"10.18778/0138-0680.2023.10","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.10","url":null,"abstract":"In this paper, we are going to introduce a fundamental relation on (H_{v}BE)-algebra and investigate some of properties, also construct new ((H_{v})BE)-algebras via this relation. We show that quotient of any (H_{v}BE)-algebra via a regular regulation is an (H_{v}BE)-algebra and this quotient, via any strongly relation is a (BE)-algebra. Furthermore we consider that under what conditions some relations on (H_{v}BE)-algebra are transitive.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44508257","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
Core Type Theory 核心类型理论
Bulletin of the Section of Logic Pub Date : 2023-08-09 DOI: 10.18778/0138-0680.2023.19
Emma van Dijk, David Ripley, J. Gutierrez
{"title":"Core Type Theory","authors":"Emma van Dijk, David Ripley, J. Gutierrez","doi":"10.18778/0138-0680.2023.19","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.19","url":null,"abstract":"Neil Tennant’s core logic is a type of bilateralist natural deduction system based on proofs and refutations. We present a proof system for propositional core logic, explain its connections to bilateralism, and explore the possibility of using it as a type theory, in the same kind of way intuitionistic logic is often used as a type theory. Our proof system is not Tennant’s own, but it is very closely related, and determines the same consequence relation. The difference, however, matters for our purposes, and we discuss this. We then turn to the question of strong normalization, showing that although Tennant’s proof system for core logic is not strongly normalizing, our modified system is.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42032495","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
Fractional-Valued Modal Logic and Soft Bilateralism 分数值模态逻辑和软双边主义
Bulletin of the Section of Logic Pub Date : 2023-08-09 DOI: 10.18778/0138-0680.2023.17
M. Piazza, G. Pulcini, Matteo Tesi
{"title":"Fractional-Valued Modal Logic and Soft Bilateralism","authors":"M. Piazza, G. Pulcini, Matteo Tesi","doi":"10.18778/0138-0680.2023.17","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.17","url":null,"abstract":"In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic (mathbf{K}), whose values lie in the closed interval ([0,1]) of rational numbers. In this paper, after clarifying our conception of bilateralism -- dubbed ``soft bilateralism\" -- we generalize the fractional method to encompass extensions and weakenings of (mathbf{K}). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic (mathbf{D}) and the non-normal modal logics (mathbf{E}) and (mathbf{M}) and thoroughly investigate their structural properties.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41754131","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
Bilateral Rules as Complex Rules 双边规则作为复杂规则
Bulletin of the Section of Logic Pub Date : 2023-07-18 DOI: 10.18778/0138-0680.2023.13
Leonardo Ceragioli
{"title":"Bilateral Rules as Complex Rules","authors":"Leonardo Ceragioli","doi":"10.18778/0138-0680.2023.13","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.13","url":null,"abstract":"Proof-theoretic semantics is an inferentialist theory of meaning originally developed in a unilateral framework. Its extension to bilateral systems opens both opportunities and problems. The problems are caused especially by Coordination Principles (a kind of rule that is not present in unilateral systems) and mismatches between rules for assertion and rules for rejection. In this paper, a solution is proposed for two major issues: the availability of a reduction procedure for tonk and the existence of harmonious rules for the paradoxical zero-ary connective (bullet). The solution is based on a reinterpretation of bilateral rules as complex rules, that is, rules that introduce or eliminate connectives in a subordinate position. Looking at bilateral rules from this perspective, the problems faced by bilateralism can be seen as special cases of general problems of complex systems, which have been already analyzed in the literature. In the end, a comparison with other proposed solutions underlines the need for further investigation in order to complete the picture of bilateral proof-theoretic semantics.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44746795","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
Harmony and Normalisation in Bilateral Logic 双边逻辑的和谐与规范
Bulletin of the Section of Logic Pub Date : 2023-07-18 DOI: 10.18778/0138-0680.2023.14
Pedro Del Valle-Inclan
{"title":"Harmony and Normalisation in Bilateral Logic","authors":"Pedro Del Valle-Inclan","doi":"10.18778/0138-0680.2023.14","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.14","url":null,"abstract":"In a recent paper del Valle-Inclan and Schlöder argue that bilateral calculi call for their own notion of proof-theoretic harmony, distinct from the usual (or ‘unilateral’) ones. They then put forward a specifically bilateral criterion of harmony, and present a harmonious bilateral calculus for classical logic.\u0000In this paper, I show how del Valle-Inclan and Schlöder’s criterion of harmony suggests a notion of normal form for bilateral systems, and prove normalisation for two (harmonious) bilateral calculi for classical logic, HB1 and HB2. The resulting normal derivations have the usual desirable features, like the separation and subformula properties. HB1-normal form turns out to be strictly stronger that the notion of normal form proposed by Nils Kürbis, and HB2-normal form is neither stronger nor weaker than a similar proposal by Marcello D’Agostino, Dov Gabbay, and Sanjay Modgyl.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49431125","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
On Synonymy in Proof-theoretic Semantics: The Case of (mathtt{2Int}) 论证明论语义中的同义词：以（mathtt｛2Int｝）为例
Bulletin of the Section of Logic Pub Date : 2023-07-18 DOI: 10.18778/0138-0680.2023.18
Sara Ayhan, H. Wansing
{"title":"On Synonymy in Proof-theoretic Semantics: The Case of (mathtt{2Int})","authors":"Sara Ayhan, H. Wansing","doi":"10.18778/0138-0680.2023.18","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.18","url":null,"abstract":"We consider an approach to propositional synonymy in proof-theoretic semantics that is defined with respect to a bilateral G3-style sequent calculus (mathtt{SC2Int}) for the bi-intuitionistic logic (mathtt{2Int}). A distinctive feature of (mathtt{SC2Int}) is that it makes use of two kind of sequents, one representing proofs, the other representing refutations. The structural rules of (mathtt{SC2Int}), in particular its cut-rules, are shown to be admissible. Next, interaction rules are defined that allow transitions from proofs to refutations, and vice versa, mediated through two different negation connectives, the well-known implies-falsity negation and the less well-known co-implies-truth negation of (mathtt{2Int}). By assuming that the interaction rules have no impact on the identity of derivations, the concept of inherited identity between derivations in (mathtt{SC2Int}) is introduced and the notions of positive and negative synonymy of formulas are defined. Several examples are given of distinct formulas that are either positively or negatively synonymous. It is conjectured that the two conditions cannot be satisfied simultaneously.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47721948","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
Supposition: A Problem for Bilateralism 假设：双边主义的一个问题
Bulletin of the Section of Logic Pub Date : 2023-07-18 DOI: 10.18778/0138-0680.2023.07
Nils Kürbis
{"title":"Supposition: A Problem for Bilateralism","authors":"Nils Kürbis","doi":"10.18778/0138-0680.2023.07","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.07","url":null,"abstract":"In bilateral logic formulas are signed by + and -, indicating the speech acts assertion and denial. I argue that making an assumption is also speech act. Speech acts cannot be embedded within other speech acts. Hence we cannot make sense of the notion of making an assumption in bilateral logic. Some attempts at a solution to this problem are considered and rejected. ","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47532041","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
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