Pub Date : 2024-05-20DOI: 10.18778/0138-0680.2024.05
Patrycja Kupś, Szymon Chlebowski
We study natural deduction systems for a fragment of intuitionistic logic with propositional identity from the point of view of proof-theoretic semantics. We argue that the identity connective is a natural operator to be treated under the elimination rules as basic approach.
{"title":"Meaning is Use: the Case of Propositional Identity","authors":"Patrycja Kupś, Szymon Chlebowski","doi":"10.18778/0138-0680.2024.05","DOIUrl":"https://doi.org/10.18778/0138-0680.2024.05","url":null,"abstract":"We study natural deduction systems for a fragment of intuitionistic logic with propositional identity from the point of view of proof-theoretic semantics. We argue that the identity connective is a natural operator to be treated under the elimination rules as basic approach.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141123010","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 : 2024-05-20DOI: 10.18778/0138-0680.2024.08
Sławomir Przybyło, K. Słomczyńska
We construct free algebras in the variety generated by the equivalential algebra with conjunction on dense elements and compute the formula for the free spectrum of this variety. Moreover, we describe the decomposition of free algebras into directly indecomposable factors.
{"title":"Free Spectra of Equivalential Algebras with Conjunction on Dense Elements","authors":"Sławomir Przybyło, K. Słomczyńska","doi":"10.18778/0138-0680.2024.08","DOIUrl":"https://doi.org/10.18778/0138-0680.2024.08","url":null,"abstract":"We construct free algebras in the variety generated by the equivalential algebra with conjunction on dense elements and compute the formula for the free spectrum of this variety. Moreover, we describe the decomposition of free algebras into directly indecomposable factors.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141121533","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 : 2024-05-20DOI: 10.18778/0138-0680.2024.07
Andrzej Walendziak
In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown.
{"title":"On pre-Hilbert and positive implicative pre-Hilbert algebras","authors":"Andrzej Walendziak","doi":"10.18778/0138-0680.2024.07","DOIUrl":"https://doi.org/10.18778/0138-0680.2024.07","url":null,"abstract":"In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141119130","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 : 2023-12-18DOI: 10.18778/0138-0680.2023.31
R. Borzooei, M. Aaly Kologani, M. Mohseni Takallo, Y. B. Jun
In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of ((in, in)) and ((in, in! vee , {q}))-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of ((alpha,beta))-fuzzy subalgebras and ((alpha,beta))-fuzzy ideals on equality algebras. Also, by using the notion of ((in, in))-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of (X), and we proved that the set of all classes of (X) by these order is a poset.
{"title":"Fuzzy Sub-Equality Algebras Based on Fuzzy Points","authors":"R. Borzooei, M. Aaly Kologani, M. Mohseni Takallo, Y. B. Jun","doi":"10.18778/0138-0680.2023.31","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.31","url":null,"abstract":"In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of ((in, in)) and ((in, in! vee , {q}))-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of ((alpha,beta))-fuzzy subalgebras and ((alpha,beta))-fuzzy ideals on equality algebras. Also, by using the notion of ((in, in))-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of (X), and we proved that the set of all classes of (X) by these order is a poset.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139174604","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 : 2023-12-15DOI: 10.18778/0138-0680.2023.30
Hamzeh Mohammadi
A many-valued modal logic, called linear abelian modal logic (rm {mathbf{LK(A)}}) is introduced as an extension of the abelian modal logic (rm mathbf{K(A)}). Abelian modal logic (rm mathbf{K(A)}) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic (rm mathbf{LK(A)}) is axiomatized by extending (rm mathbf{K(A)}) with the modal axiom schemas (Box(varphiveepsi)rightarrow(BoxvarphiveeBoxpsi)) and ((BoxvarphiwedgeBoxpsi)rightarrowBox(varphiwedgepsi)). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, the correspondence between hypersequent calculi and axiomatization is investigated.
{"title":"Linear Abelian Modal Logic","authors":"Hamzeh Mohammadi","doi":"10.18778/0138-0680.2023.30","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.30","url":null,"abstract":"A many-valued modal logic, called linear abelian modal logic (rm {mathbf{LK(A)}}) is introduced as an extension of the abelian modal logic (rm mathbf{K(A)}). Abelian modal logic (rm mathbf{K(A)}) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic (rm mathbf{LK(A)}) is axiomatized by extending (rm mathbf{K(A)}) with the modal axiom schemas (Box(varphiveepsi)rightarrow(BoxvarphiveeBoxpsi)) and ((BoxvarphiwedgeBoxpsi)rightarrowBox(varphiwedgepsi)). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, the correspondence between hypersequent calculi and axiomatization is investigated.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138996136","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 : 2023-12-04DOI: 10.18778/0138-0680.2023.27
S. Saidi Goraghani, Rajab Ali Borzooei
In this paper, considering L-algebras, which include a significant number of other algebraic structures, we present a definition of modules on L-algebras (L-modules). Then we provide some examples and obtain some results on L-modules. Also, we present definitions of prime ideals of L-algebras and L-submodules (prime L-submodules) of L-modules, and investigate the relationship between them. Finally, by proving a number of theorems, we provide some conditions for having prime L-submodules.
本文考虑l -代数中包含大量其他代数结构，给出了l -代数上模的定义(l -模)。然后给出了一些例子，得到了关于l模的一些结果。给出了l -代数的素理想和l -模的l -子模(素- l -子模)的定义，并研究了它们之间的关系。最后，通过对若干定理的证明，给出了l子模存在素数的若干条件。
{"title":"L-Modules","authors":"S. Saidi Goraghani, Rajab Ali Borzooei","doi":"10.18778/0138-0680.2023.27","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.27","url":null,"abstract":"In this paper, considering L-algebras, which include a significant number of other algebraic structures, we present a definition of modules on L-algebras (L-modules). Then we provide some examples and obtain some results on L-modules. Also, we present definitions of prime ideals of L-algebras and L-submodules (prime L-submodules) of L-modules, and investigate the relationship between them. Finally, by proving a number of theorems, we provide some conditions for having prime L-submodules.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138604431","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 : 2023-11-22DOI: 10.18778/0138-0680.2023.26
S. Celani
In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic (mathbf{IL}). We will introduce the logics (mathbf{BIL}) and (mathbf{BIL}^{+}) in the propositional language with a modal operator (square) and a binary operator (Rightarrow) such that (mathbf{BIL}subseteqmathbf{BIL}^{+}subseteqmathbf{IL}). The logic (mathbf{BIL}) is generated by the relational structures (left), called basic frames, where (left) is a Kripke frame and (left) is a neighborhood frame. We will prove that the logic (mathbf{BIL}^{+}) is generated by the basic frames where the binary relation (R) is definable by the neighborhood relation (N) and, therefore, the neighborhood semantics is suitable to study the logic (mathbf{BIL}^{+}) and its extensions. We shall also study some axiomatic extensions of (mathsf{mathbf{BIL}}) and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames.
{"title":"Some Logics in the Vicinity of Interpretability Logics","authors":"S. Celani","doi":"10.18778/0138-0680.2023.26","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.26","url":null,"abstract":"In this paper we shall define semantically some families of propositional modal logics related to the interpretability logic (mathbf{IL}). We will introduce the logics (mathbf{BIL}) and (mathbf{BIL}^{+}) in the propositional language with a modal operator (square) and a binary operator (Rightarrow) such that (mathbf{BIL}subseteqmathbf{BIL}^{+}subseteqmathbf{IL}). The logic (mathbf{BIL}) is generated by the relational structures (left), called basic frames, where (left) is a Kripke frame and (left) is a neighborhood frame. We will prove that the logic (mathbf{BIL}^{+}) is generated by the basic frames where the binary relation (R) is definable by the neighborhood relation (N) and, therefore, the neighborhood semantics is suitable to study the logic (mathbf{BIL}^{+}) and its extensions. We shall also study some axiomatic extensions of (mathsf{mathbf{BIL}}) and we will prove that these extensions are sound and complete with respect to a certain classes of basic frames.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139249879","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 : 2023-11-20DOI: 10.18778/0138-0680.2023.28
M. Aaly Kologani, G. Rezaei
The main goal of this paper is to introduce the notion of stabilizers in (L)-algebras and develop stabilizer theory in (L)-algebras. In this paper, we introduced the notions of left and right stabilizers and investigated some related properties of them. Then, we discussed the relations among stabilizers, ideal and co-annihilators. Also, we obtained that the set of all ideals of a (CKL)-algebra forms a relative pseudo-complemented lattice. In addition, we proved that right stabilizers in (CKL)-algebra are ideals. Then by using the right stabilizers we produced a basis for a topology on (L)-algebra. We showed that the generated topology by this basis is Baire, connected, locally connected and separable and we investigated the other properties of this topology.
{"title":"Stabilizers on (L)-algebras","authors":"M. Aaly Kologani, G. Rezaei","doi":"10.18778/0138-0680.2023.28","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.28","url":null,"abstract":"The main goal of this paper is to introduce the notion of stabilizers in (L)-algebras and develop stabilizer theory in (L)-algebras. In this paper, we introduced the notions of left and right stabilizers and investigated some related properties of them. Then, we discussed the relations among stabilizers, ideal and co-annihilators. Also, we obtained that the set of all ideals of a (CKL)-algebra forms a relative pseudo-complemented lattice. In addition, we proved that right stabilizers in (CKL)-algebra are ideals. Then by using the right stabilizers we produced a basis for a topology on (L)-algebra. We showed that the generated topology by this basis is Baire, connected, locally connected and separable and we investigated the other properties of this topology.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139255596","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 : 2023-09-30DOI: 10.18778/0138-0680.2023.24
Sara Ayhan
{"title":"Introduction: Bilateralism and Proof-Theoretic Semantics (Part II)","authors":"Sara Ayhan","doi":"10.18778/0138-0680.2023.24","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.24","url":null,"abstract":"","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135127135","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 : 2023-09-28DOI: 10.18778/0138-0680.2023.25
Tin Adlešić, Vedran Čačić
In this paper we rigorously prove the existence of type-level ordered pairs in Quine's New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU+Inf+AC). The proof uses Tarski's theorem about choice, which is a theorem of NFU+Inf+AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatization is NFU+Inf+AC+Tarski, a conservative extension of NFU+Inf+AC.
{"title":"Tarski's theorem about choice and the alternative axiomatic extension of NFU","authors":"Tin Adlešić, Vedran Čačić","doi":"10.18778/0138-0680.2023.25","DOIUrl":"https://doi.org/10.18778/0138-0680.2023.25","url":null,"abstract":"In this paper we rigorously prove the existence of type-level ordered pairs in Quine's New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU+Inf+AC). The proof uses Tarski's theorem about choice, which is a theorem of NFU+Inf+AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatization is NFU+Inf+AC+Tarski, a conservative extension of NFU+Inf+AC.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387289","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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