Pub Date : 2023-07-18DOI: 10.18778/0138-0680.2023.6
Greg Restall
Natural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proof system in which the connective rules are unmodified from the usual Prawitz introduction and elimination rules—the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules fixed, and varying purely structural rules.The key result of this paper is that the two principles that introduce kinds of irrelevance to natural deduction proofs: (a) the rule of explosion (from a contradiction, anything follows); and (b) the structural rule of vacuous discharge; are shown to be two sides of a single coin, in the same way that they correspond to the structural rule of weakening in the sequent calculus. The paper also includes a discussion of assumption classes, and how they can play a role in treating additive connectives in substructural natural deduction.
{"title":"Structural Rules in Natural Deduction with Alternatives","authors":"Greg Restall","doi":"10.18778/0138-0680.2023.6","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.6","url":null,"abstract":"Natural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proof system in which the connective rules are unmodified from the usual Prawitz introduction and elimination rules—the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules fixed, and varying purely structural rules.The key result of this paper is that the two principles that introduce kinds of irrelevance to natural deduction proofs: (a) the rule of explosion (from a contradiction, anything follows); and (b) the structural rule of vacuous discharge; are shown to be two sides of a single coin, in the same way that they correspond to the structural rule of weakening in the sequent calculus. The paper also includes a discussion of assumption classes, and how they can play a role in treating additive connectives in substructural natural deduction.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45193349","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}
Pub Date : 2023-06-30DOI: 10.18778/0138-0680.2023.12
Sara Ayhan
{"title":"Introduction: Bilateralism and Proof-Theoretic Semantics (Part I)","authors":"Sara Ayhan","doi":"10.18778/0138-0680.2023.12","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.12","url":null,"abstract":"","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367886","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}
Pub Date : 2023-04-25DOI: 10.18778/0138-0680.2023.11
Daniel Martinez, R. D. de Queiroz
One takes advantage of some basic properties of every homotopic (lambda)-model (e.g. extensional Kan complex) to explore the higher (betaeta)-conversions, which would correspond to proofs of equality between terms of a theory of equality of any extensional Kan complex. Besides, Identity types based on computational paths are adapted to a type-free theory with higher (lambda)-terms, whose equality rules would be contained in the theory of any $lambda$-homotopic model.
{"title":"The Theory of an Arbitrary Higher (lambda)-Model","authors":"Daniel Martinez, R. D. de Queiroz","doi":"10.18778/0138-0680.2023.11","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.11","url":null,"abstract":"One takes advantage of some basic properties of every homotopic (lambda)-model (e.g. extensional Kan complex) to explore the higher (betaeta)-conversions, which would correspond to proofs of equality between terms of a theory of equality of any extensional Kan complex. Besides, Identity types based on computational paths are adapted to a type-free theory with higher (lambda)-terms, whose equality rules would be contained in the theory of any $lambda$-homotopic model.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44217962","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}
Pub Date : 2023-04-21DOI: 10.18778/0138-0680.2023.05
Tore Fjetland Øgaard
An algebraic type of structure is shown forth which is such that if it is a characteristic matrix for a logic, then that logic satisfies Meyer's weak variable sharing property. As a corollary, it is shown that RM and all its odd-valued extensions (mathbf{RM}_{2nmathord{-}1}) satisfy the weak variable sharing property. It is also shown that a proof to the effect that the "fuzzy" version of the relevant logic R satisfies the property is incorrect.
{"title":"The weak variable sharing property","authors":"Tore Fjetland Øgaard","doi":"10.18778/0138-0680.2023.05","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.05","url":null,"abstract":"\u0000\u0000\u0000An algebraic type of structure is shown forth which is such that if it is a characteristic matrix for a logic, then that logic satisfies Meyer's weak variable sharing property. As a corollary, it is shown that RM and all its odd-valued extensions (mathbf{RM}_{2nmathord{-}1}) satisfy the weak variable sharing property. It is also shown that a proof to the effect that the \"fuzzy\" version of the relevant logic R satisfies the property is incorrect.\u0000\u0000\u0000","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45141430","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}
Pub Date : 2023-04-21DOI: 10.18778/0138-0680.2023.09
Z. Gyenis, Zalán Molnár, Övge Öztürk
In this paper we introduce the modelwise interpolation property of a logic that states that whenever (modelsphitopsi) holds for two formulas (phi) and (psi), then for every model (mathfrak{M}) there is an interpolant formula (chi$ formulated in the intersectionof the vocabularies of (phi$ and (psi), such that (mathfrak{M}models phitochi) and (mathfrak{M}modelschitopsi), that is, the interpolant formula in Craig interpolation may vary from model to model. We discuss examples and show that while the (n)-variable fragment of first order logic and difference logic have no Craig interpolation, they both have the modelwise interpolation property. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. In particular, the class of finite dimensional cylindric set algebras enjoys this weak form of superamalgamation.
{"title":"The Modelwise Interpolation Property of Semantic Logics","authors":"Z. Gyenis, Zalán Molnár, Övge Öztürk","doi":"10.18778/0138-0680.2023.09","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.09","url":null,"abstract":"In this paper we introduce the modelwise interpolation property of a logic that states that whenever (modelsphitopsi) holds for two formulas (phi) and (psi), then for every model (mathfrak{M}) there is an interpolant formula (chi$ formulated in the intersectionof the vocabularies of (phi$ and (psi), such that (mathfrak{M}models phitochi) and (mathfrak{M}modelschitopsi), that is, the interpolant formula in Craig interpolation may vary from model to model. We discuss examples and show that while the (n)-variable fragment of first order logic and difference logic have no Craig interpolation, they both have the modelwise interpolation property. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. In particular, the class of finite dimensional cylindric set algebras enjoys this weak form of superamalgamation.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41295599","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}
Pub Date : 2023-04-21DOI: 10.18778/0138-0680.2023.08
Beza Lamesgin Derseh, Berhanu Assaye Alaba, Y. G. Wondifraw
In this paper, we introduce the notion of intuitionistic fuzzy PMS-subalgebras under homomorphism and Cartesian product and investigate several properties. We study the homomorphic image and inverse image of the intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, which are also intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, and find some other interesting results. Furthermore, we also prove that the Cartesian product of intuitionistic fuzzy PMS- subalgebras is again an intuitionistic fuzzy PMS-subalgebra and characterize it in terms of its level sets. Finally, we consider the strongest intuitionistic fuzzy PMS-relations on an intuitionistic fuzzy set in a PMS-algebra and demonstrate that an intuitionistic fuzzy PMS-relation on an intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra if and only if the corresponding intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra of a PMS-algebra.
{"title":"On Homomorphism and Cartesian Product of Intuitionistic Fuzzy PMS-Subalgebras of a PMS-Algebra","authors":"Beza Lamesgin Derseh, Berhanu Assaye Alaba, Y. G. Wondifraw","doi":"10.18778/0138-0680.2023.08","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.08","url":null,"abstract":"In this paper, we introduce the notion of intuitionistic fuzzy PMS-subalgebras under homomorphism and Cartesian product and investigate several properties. We study the homomorphic image and inverse image of the intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, which are also intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, and find some other interesting results. Furthermore, we also prove that the Cartesian product of intuitionistic fuzzy PMS- subalgebras is again an intuitionistic fuzzy PMS-subalgebra and characterize it in terms of its level sets. Finally, we consider the strongest intuitionistic fuzzy PMS-relations on an intuitionistic fuzzy set in a PMS-algebra and demonstrate that an intuitionistic fuzzy PMS-relation on an intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra if and only if the corresponding intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra of a PMS-algebra.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49659834","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}
Pub Date : 2023-01-25DOI: 10.18778/0138-0680.2023.01
G. Rezaei, R. Borzooei, M. Aaly Kologani, Y. Jun
Rough is an excellent mathematical tool for the analysis of a vague description of actions in decision problems. Now, in this paper by considering the notion of an equality algebra, the notion of the lower and the upper approximations are introduced and some properties of them are given. Moreover, it is proved that the lower and the upper approximations are an interior operator and a closure operator, respectively. Also, using D-lower and D-upper approximation, conditions for a nonempty subset to be definable are provided and investigated that under which condition D-lower and D-upper approximation can be filter.
{"title":"Roughness of filters in equality algebras","authors":"G. Rezaei, R. Borzooei, M. Aaly Kologani, Y. Jun","doi":"10.18778/0138-0680.2023.01","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.01","url":null,"abstract":"Rough is an excellent mathematical tool for the analysis of a vague description of actions in decision problems. Now, in this paper by considering the notion of an equality algebra, the notion of the lower and the upper approximations are introduced and some properties of them are given. Moreover, it is proved that the lower and the upper approximations are an interior operator and a closure operator, respectively. Also, using D-lower and D-upper approximation, conditions for a nonempty subset to be definable are provided and investigated that under which condition D-lower and D-upper approximation can be filter.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45411244","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}
Pub Date : 2022-10-25DOI: 10.18778/0138-0680.2022.21
Oleg M. Grigoriev, D. Zaitsev
We consider an example of four valued semantics partially inspired by quantum computations and negation-like operations occurred therein. In particular we consider a representation of so called square root of negation within this four valued semantics as an operation which acts like a cycling negation. We define two variants of logical matrices performing different orders over the set of truth values. Purely formal logical result of our study consists in axiomatizing the logics of defined matrices as the systems of binary consequence relation and proving correctness and completeness theorems for these deductive systems.
{"title":"Basic Four-Valued Systems of Cyclic Negations","authors":"Oleg M. Grigoriev, D. Zaitsev","doi":"10.18778/0138-0680.2022.21","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.21","url":null,"abstract":"We consider an example of four valued semantics partially inspired by quantum computations and negation-like operations occurred therein. In particular we consider a representation of so called square root of negation within this four valued semantics as an operation which acts like a cycling negation. We define two variants of logical matrices performing different orders over the set of truth values. Purely formal logical result of our study consists in axiomatizing the logics of defined matrices as the systems of binary consequence relation and proving correctness and completeness theorems for these deductive systems.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47490943","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}
Pub Date : 2022-10-25DOI: 10.18778/0138-0680.2022.22
Slawomir Przybylo, K. Słomczyńska
We study the variety generated by the three-element equivalential algebra with conjunction on the dense elements. We prove the representation theorem which let us construct the free algebras in this variety.
{"title":"Equivalential algebras with conjunction on dense elements","authors":"Slawomir Przybylo, K. Słomczyńska","doi":"10.18778/0138-0680.2022.22","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.22","url":null,"abstract":"We study the variety generated by the three-element equivalential algebra with conjunction on the dense elements. We prove the representation theorem which let us construct the free algebras in this variety.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43777670","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}
Pub Date : 2022-10-14DOI: 10.18778/0138-0680.2022.20
G. Robles, J. Méndez
G3 is Gödelian 3-valued logic, G3(_text{Ł}^leq) is its paraconsistent counterpart and G3(_text{Ł}^1) is a strong extension of G3(_text{Ł}^leq). The aim of this paper is to endow each one of the logics just mentioned with a 2 set-up binary Routley semantics.
{"title":"A 2 Set-Up Binary Routley Semantics for Gödelian 3-Valued Logic G3 and Its Paraconsistent Counterpart G3(_text{Ł}^leq)","authors":"G. Robles, J. Méndez","doi":"10.18778/0138-0680.2022.20","DOIUrl":"https://doi.org/10.18778/0138-0680.2022.20","url":null,"abstract":"G3 is Gödelian 3-valued logic, G3(_text{Ł}^leq) is its paraconsistent counterpart and G3(_text{Ł}^1) is a strong extension of G3(_text{Ł}^leq). The aim of this paper is to endow each one of the logics just mentioned with a 2 set-up binary Routley semantics.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46727794","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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