Pub Date : 2018-10-30DOI: 10.18778/0138-0680.47.3.02
Zofia Kostrzycka
We prove that there is infinitely many tabular modal logics extending KB.Alt(2) which have interpolation.
我们证明了扩展KB.Alt（2）的表格式模态逻辑有无穷多个具有插值性质。
{"title":"On interpolation in NEXT(KB.Alt(2))","authors":"Zofia Kostrzycka","doi":"10.18778/0138-0680.47.3.02","DOIUrl":"https://doi.org/10.18778/0138-0680.47.3.02","url":null,"abstract":"We prove that there is infinitely many tabular modal logics extending KB.Alt(2) which have interpolation.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45211319","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 : 2018-10-30DOI: 10.18778/0138-0680.47.3.05
Mateusz Klonowski
One of the logic defined by Richard Epstein in a context of an analysis of subject matter relationship is Symmetric Relatedness Logic S. In the monograph [2] we can find some open problems concerning relatedness logic, a Post-style completeness theorem for logic S is one of them. Our paper introduces a solution of this metalogical issue.
Richard Epstein在分析主题关系的背景下定义的逻辑之一是对称关系逻辑S。在专著[2]中，我们可以发现一些关于关系逻辑的开放问题，逻辑S的后式完备性定理就是其中之一。我们的论文介绍了这个金属学问题的解决方案。
{"title":"A Post-style proof of completeness theorem for symmetric relatedness Logic S","authors":"Mateusz Klonowski","doi":"10.18778/0138-0680.47.3.05","DOIUrl":"https://doi.org/10.18778/0138-0680.47.3.05","url":null,"abstract":"One of the logic defined by Richard Epstein in a context of an analysis of subject matter relationship is Symmetric Relatedness Logic S. In the monograph [2] we can find some open problems concerning relatedness logic, a Post-style completeness theorem for logic S is one of them. Our paper introduces a solution of this metalogical issue.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43572201","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 : 2018-10-30DOI: 10.18778/0138-0680.47.3.01
M. Manzano, M. C. Moreno
This article is a continuation of our promenade along the winding roads of identity, equality, nameability and completeness. We continue looking for a place where all these concepts converge. We assume that identity is a binary relation between objects while equality is a symbolic relation between terms. Identity plays a central role in logic and we have looked at it from two different points of view. In one case, identity is a notion which has to be defined and, in the other case, identity is a notion used to define other logical concepts. In our previous paper, [16], we investigated whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic with standard semantics a reliable definition of identity is possible. In the present study we have moved to modal logic and realized that here we can distinguish in the formal language between two different equality symbols, the first one shall be interpreted as extensional genuine identity and only applies for objects, the second one applies for non rigid terms and has the characteristic of synonymy. We have also analyzed the hybrid modal logic where we can introduce rigid terms by definition and can express that two worlds are identical by using the nominals and the @ operator. We finish our paper in the kingdom of identity where the only primitives are lambda and equality. Here we show how other logical concepts can be defined in terms of the identity relation. We have found at the end of our walk a possible point of convergence in the logic Equational Hybrid Propositional Type Theory (EHPTT), [14] and [15].
{"title":"Identity, equality, nameability and completeness. Part II","authors":"M. Manzano, M. C. Moreno","doi":"10.18778/0138-0680.47.3.01","DOIUrl":"https://doi.org/10.18778/0138-0680.47.3.01","url":null,"abstract":"This article is a continuation of our promenade along the winding roads of identity, equality, nameability and completeness. We continue looking for a place where all these concepts converge. We assume that identity is a binary relation between objects while equality is a symbolic relation between terms. Identity plays a central role in logic and we have looked at it from two different points of view. In one case, identity is a notion which has to be defined and, in the other case, identity is a notion used to define other logical concepts. In our previous paper, [16], we investigated whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic with standard semantics a reliable definition of identity is possible. In the present study we have moved to modal logic and realized that here we can distinguish in the formal language between two different equality symbols, the first one shall be interpreted as extensional genuine identity and only applies for objects, the second one applies for non rigid terms and has the characteristic of synonymy. We have also analyzed the hybrid modal logic where we can introduce rigid terms by definition and can express that two worlds are identical by using the nominals and the @ operator. We finish our paper in the kingdom of identity where the only primitives are lambda and equality. Here we show how other logical concepts can be defined in terms of the identity relation. We have found at the end of our walk a possible point of convergence in the logic Equational Hybrid Propositional Type Theory (EHPTT), [14] and [15].","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44457544","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 : 2018-10-30DOI: 10.18778/0138-0680.47.3.03
Berhanu Assaye, Mihret Alemneh, Gerima Tefera
The paper introduces the concept of B-Almost distributive fuzzy lattice (BADFL) in terms of its principal ideal fuzzy lattice. Necessary and sufficient conditions for an ADFL to become a B-ADFL are investigated. We also prove the equivalency of B-algebra and B-fuzzy algebra. In addition, we extend PSADL to PSADFL and prove that B-ADFL implies PSADFL.
{"title":"B-almost distributive fuzzy lattice","authors":"Berhanu Assaye, Mihret Alemneh, Gerima Tefera","doi":"10.18778/0138-0680.47.3.03","DOIUrl":"https://doi.org/10.18778/0138-0680.47.3.03","url":null,"abstract":"The paper introduces the concept of B-Almost distributive fuzzy lattice (BADFL) in terms of its principal ideal fuzzy lattice. Necessary and sufficient conditions for an ADFL to become a B-ADFL are investigated. We also prove the equivalency of B-algebra and B-fuzzy algebra. In addition, we extend PSADL to PSADFL and prove that B-ADFL implies PSADFL.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43701912","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 : 2018-06-30DOI: 10.18778/0138-0680.47.2.03
Guillermo Badia
We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.
我们刻画了具有变量共享性质的非平凡子结构逻辑及其强版本。为此，我们找到了这些逻辑性质的代数对应项。
{"title":"Variable Sharing in Substructural Logics: an Algebraic Characterization","authors":"Guillermo Badia","doi":"10.18778/0138-0680.47.2.03","DOIUrl":"https://doi.org/10.18778/0138-0680.47.2.03","url":null,"abstract":"We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46704356","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":"Grzegorczyk Algebras Revisited","authors":"Michał M. Stronkowski","doi":"10.18778/0138-0680.47.2.05","DOIUrl":"https://doi.org/10.18778/0138-0680.47.2.05","url":null,"abstract":"We provide simple algebraic proofs of two important facts, due to Zakharyaschev and Esakia, about Grzegorczyk algebras.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47739376","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 : 2018-06-30DOI: 10.18778/0138-0680.47.2.04
A. Walendziak
In this paper we study pseudo-BCH algebras which are semilattices or lattices with respect to the natural relations ≤; we call them pseudo-BCH join-semilattices, pseudo-BCH meet-semilattices and pseudo-BCH lattices, respectively. We prove that the class of all pseudo-BCH join-semilattices is a variety and show that it is weakly regular, arithmetical at 1, and congruence distributive. In addition, we obtain the systems of identities defininig pseudo-BCH meet-semilattices and pseudo-BCH lattices.
{"title":"Pseudo-BCH Semilattices","authors":"A. Walendziak","doi":"10.18778/0138-0680.47.2.04","DOIUrl":"https://doi.org/10.18778/0138-0680.47.2.04","url":null,"abstract":"In this paper we study pseudo-BCH algebras which are semilattices or lattices with respect to the natural relations ≤; we call them pseudo-BCH join-semilattices, pseudo-BCH meet-semilattices and pseudo-BCH lattices, respectively. We prove that the class of all pseudo-BCH join-semilattices is a variety and show that it is weakly regular, arithmetical at 1, and congruence distributive. In addition, we obtain the systems of identities defininig pseudo-BCH meet-semilattices and pseudo-BCH lattices.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43114477","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 : 2018-06-30DOI: 10.18778/0138-0680.47.2.02
A. Figallo, Nora Oliva, A. Ziliani
Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].
在A.V.Figallo，N.Oliva，A.Ziliani，Modal pseudo-complemented De Morgan代数，Acta Univ.Palacki中研究了模态伪补的De Morgan代数（或mpM代数）。Olomuc。，Fac。rer。nat.，Mathematica 53，1（2014），pp.65-79，并且它们构成了H.Sankappanavar在1987年研究的满足x∧（~x）*=（~x∧。本文继续对这些代数进行研究。更准确地说，给出了mpM同余的新刻画。特别地，其中一个是通过考虑在这些代数上定义为弱蕴涵的蕴涵运算来确定的。此外，还考虑了有限mpM代数，并给出了它们的因子分解定理。最后，得到了自由有限生成mpM代数的结构，并建立了用自由生成元数计算其基数的公式。关于有限生成的自由德摩根代数、自由布尔-德摩根代数和自由德摩根拟格的刻画，参见：[16，17，18]。
{"title":"Free Modal Pseudocomplemented De Morgan Algebras","authors":"A. Figallo, Nora Oliva, A. Ziliani","doi":"10.18778/0138-0680.47.2.02","DOIUrl":"https://doi.org/10.18778/0138-0680.47.2.02","url":null,"abstract":"Modal pseudocomplemented De Morgan algebras (or mpM-algebras) were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1 (2014), pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ(∼x)* = (∼(xΛ(∼x)*))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18].","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46590357","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 : 2018-06-30DOI: 10.18778/0138-0680.47.2.01
G. Voutsadakis
This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is only available for self-extensional π-institutions by a previous result of the author obtained as an extension of a classical result of Wójcicki. Finally, it is shown that it is possible to replace an arbitrary pseudoreferential matrix system semantics by a discrete pseudo-referential matrix system semantics.
{"title":"Categorical Abstract Algebraic Logic: Pseudo-Referential Matrix System Semantics","authors":"G. Voutsadakis","doi":"10.18778/0138-0680.47.2.01","DOIUrl":"https://doi.org/10.18778/0138-0680.47.2.01","url":null,"abstract":"This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is only available for self-extensional π-institutions by a previous result of the author obtained as an extension of a classical result of Wójcicki. Finally, it is shown that it is possible to replace an arbitrary pseudoreferential matrix system semantics by a discrete pseudo-referential matrix system semantics.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45088745","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 : 2015-01-01DOI: 10.18778/0138-0680.44.1.2.02
Marcin Łazarz, Krzysztof Siemienczuk
Using known facts we give a simple characterization of the distributivity of lattices of finite length.
利用已知的事实，给出了有限长度格的分布性的一个简单表征。
{"title":"A Note on some Characterization of Distributive Lattices of Finite Length","authors":"Marcin Łazarz, Krzysztof Siemienczuk","doi":"10.18778/0138-0680.44.1.2.02","DOIUrl":"https://doi.org/10.18778/0138-0680.44.1.2.02","url":null,"abstract":"Using known facts we give a simple characterization of the distributivity of lattices of finite length.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67609789","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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