Bulletin of the Section of Logic最新文献

{"title":"Weakly Free Multialgebras","authors":"M. Coniglio, Guilherme Vicentin de Toledo","doi":"10.18778/0138-0680.2021.19","DOIUrl":"https://doi.org/10.18778/0138-0680.2021.19","url":null,"abstract":"In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in exploring the foundations of multialgebras applied to the study of logic systems.\u0000It is well known from universal algebra that, for every signature (Sigma), there exist algebras over (Sigma) which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of (Sigma)-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of (Sigma)-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor (mathcal{U}), from the category of (Sigma)-multialgebras to Set, does not have a left adjoint.\u0000In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that (mathcal{U}) does not have a left adjoint.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46141835","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":"An Arithmetically Complete Predicate Modal Logic","authors":"G. Tourlakis, Yunge Hao","doi":"10.18778/0138-0680.2021.18","DOIUrl":"https://doi.org/10.18778/0138-0680.2021.18","url":null,"abstract":"This paper investigates a first-order extension of GL called (textup{ML}^3). We outline briefly the history that led to (textup{ML}^3), its key properties and some of its toolbox: the emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness (with respect to finite reverse well-founded Kripke models) is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that (textup{ML}^3) is arithmetically complete. As expanded below, (textup{ML}^3) is a first-order modal logic that along with its built-in ability to simulate general classical first-order provability―\"(Box)\" simulating the the informal classical \"(vdash)\"―is also arithmetically complete in the Solovay sense.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49172280","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":"A Note on Gödel-Dummet Logic LC","authors":"G. Robles, J. Méndez","doi":"10.18778/0138-0680.2021.15","DOIUrl":"https://doi.org/10.18778/0138-0680.2021.15","url":null,"abstract":"Let (A_{0},A_{1},...,A_{n}) be (possibly) distintict wffs, (n) being an odd number equal to or greater than 1. Intuitionistic Propositional Logic IPC plus the axiom ((A_{0}rightarrow A_{1})vee ...vee (A_{n-1}rightarrow A_{n})vee (A_{n}rightarrow A_{0})) is equivalent to Gödel-Dummett logic LC. However, if (n) is an even number equal to or greater than 2, IPC plus the said axiom is a sublogic of LC.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47591689","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":"Reasoning about Social Phenomena","authors":"Tomasz Jarmużek, F. Ju, P. Kulicki, B. Liao","doi":"10.18778/0138-0680.2021.14","DOIUrl":"https://doi.org/10.18778/0138-0680.2021.14","url":null,"abstract":"<jats:p>n/a</jats:p>","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44148385","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":"Tableau Systems for Epistemic Positional Logics","authors":"Mateusz Klonowski, K. Krawczyk, Bożena Piȩta","doi":"10.18778/0138-0680.2021.06","DOIUrl":"https://doi.org/10.18778/0138-0680.2021.06","url":null,"abstract":"The goal of the article is twofold. The first one is to provide logics based on positional semantics which will be suitable for the analysis of epistemic modalities such as ‘agent ... knows/beliefs that ...’. The second one is to define tableau systemsfor such logics. Firstly, we present the minimal positional logic MR. Then, we change the notion of formulas and semantics in order to consider iterations of the operator of realization and “free” classical formulas. After that, we move on to weaker logics in order to avoid the well known problem of logical omniscience. At the same time, we keep the positional counterparts of modal axioms (T), (4) and (5). For all of the considered logics we present sound and complete tableau systems.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46493022","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":"The category of EQ-algebras","authors":"R. Borzooei, N. Akhlaghinia, M. Kologani, X. Xin","doi":"10.18778/0138-0680.2021.01","DOIUrl":"https://doi.org/10.18778/0138-0680.2021.01","url":null,"abstract":"\u0000 \u0000 \u0000EQ-algebras were introduced by Nova ́k in [15] as an algebraic structure of truth values for fuzzy type theory (FFT). In this paper, we studied the category of EQ-algebras and showed that it is complete, but it is not cocomplete, in general. We proved that multiplicatively relative EQ-algebras have coequlizers and we calculate coprodut and pushout in a special case. Also, we construct a free EQ-algebra on a singleton. \u0000 \u0000 \u0000","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49388266","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":"On complete representations and minimal completions in algebraic logic, both positive and negative results","authors":"T. Sayed Ahmed","doi":"10.18778/0138-0680.2021.17","DOIUrl":"https://doi.org/10.18778/0138-0680.2021.17","url":null,"abstract":"Fix a finite ordinal (ngeq 3) and let (alpha) be an arbitrary ordinal. Let (mathsf{CA}_n) denote the class of cylindric algebras of dimension (n) and (sf RA) denote the class of relation algebras. Let (mathbf{PA}_{alpha}(mathsf{PEA}_{alpha})) stand for the class of polyadic (equality) algebras of dimension (alpha). We reprove that the class (mathsf{CRCA}_n) of completely representable (mathsf{CA}_n)s, and the class (sf CRRA) of completely representable (mathsf{RA})s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety (sf V) between polyadic algebras of dimension (n) and diagonal free (mathsf{CA}_n)s. We show that that the class of completely and strongly representable algebras in (sf V) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class (sf CRRA) is not closed under (equiv_{infty,omega}). In contrast, we show that given (alphageq omega), and an atomic (mathfrak{A}in mathsf{PEA}_{alpha}), then for any (n<omega), (mathfrak{Nr}_nmathfrak{A}) is a completely representable (mathsf{PEA}_n). We show that for any (alphageq omega), the class of completely representable algebras in certain reducts of (mathsf{PA}_{alpha})s, that happen to be varieties, is elementary. We show that for (alphageq omega), the the class of polyadic-cylindric algebras dimension (alpha), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension (n) and (mathsf{PA}_{alpha})s for (alpha) an infinite ordinal, proving negative results for the first and positive ones for the second.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67609588","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":"From Intuitionism to Brouwer's Modal Logic","authors":"Zofia Kostrzycka","doi":"10.18778/0138-0680.2020.22","DOIUrl":"https://doi.org/10.18778/0138-0680.2020.22","url":null,"abstract":"We try to translate the intuitionistic propositional logic INT into Brouwer's modal logic KTB. Our translation is motivated by intuitions behind Brouwer's axiom p →☐◊p The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sentences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which is the Lindenbaum algebra of monadic formulas in INT. Unfortunately, INT is not embedded by this mapping into KTB.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43317326","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":"New New Modification of the Subformula Property for a Modal Logic","authors":"M. Takano","doi":"10.18778/0138-0680.2020.15","DOIUrl":"https://doi.org/10.18778/0138-0680.2020.15","url":null,"abstract":"A modified subformula property for the modal logic KD with the additional axiom \u0000$BoxDiamond(Avee B)supsetBoxDiamond AveeBoxDiamond B$ is shown. \u0000A new modification of the notion of subformula is proposed for this purpose. \u0000This modification forms a natural extension of our former one \u0000on which modified subformula property for the modal logics K5, K5D and S4.2 has been shown \u0000(Bull Sect Logic 30:115--122, 2001 and 48:19--28, 2019). \u0000The finite model property as well as decidability for the logic follows from this.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46738084","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":"Equality Logic","authors":"S. Ghorbani","doi":"10.18778/0138-0680.2020.14","DOIUrl":"https://doi.org/10.18778/0138-0680.2020.14","url":null,"abstract":"Abstract: In this paper, we introduce and study a corresponding logic toequality-algebras and obtain some basic properties of this logic. We provethe soundness and completeness of this logic based on equality-algebrasand local deduction theorem. Then we introduce the concept of (prelinear)equality-algebras and investigate some related properties. Also, westudy -deductive systems of equality-algebras. In particular, we provethat every prelinear equality-algebra is a subdirect product of linearly orderedequality-algebras. Finally, we construct prelinear equality logicand prove the soundness and strong completeness of this logic respect toprelinear equality-algebras.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48692988","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}