We introduce crossed modules in cycloids, as a generalization of cycloids, which are algebraic logical structures arising in the context of the quantum Yang–Baxter equation. As a spacial case, we in particular focus on the crossed modules of $L-$algebras. These types of crossed modules are exceptional, since the category of $L-$algebras is not protomodular, nor Barr-exact, but it nevertheless has natural semidirect products that have not been described in category theoretic terms. We identify crossed ideals of crossed module in $L-$algebras, and obtain some characteristics of these objects that are normally not encountered on crossed modules of groups or algebras. As a consequence, we characterize crossed self-similarity completely in terms of properties of $L-$algebras and the boundary map forming the crossed module.
{"title":"A characterization of crossed self-similarity on crossed modules in L-algebras","authors":"Selim Çetin, Utku Gürdal","doi":"10.1093/jigpal/jzae003","DOIUrl":"https://doi.org/10.1093/jigpal/jzae003","url":null,"abstract":"We introduce crossed modules in cycloids, as a generalization of cycloids, which are algebraic logical structures arising in the context of the quantum Yang–Baxter equation. As a spacial case, we in particular focus on the crossed modules of $L-$algebras. These types of crossed modules are exceptional, since the category of $L-$algebras is not protomodular, nor Barr-exact, but it nevertheless has natural semidirect products that have not been described in category theoretic terms. We identify crossed ideals of crossed module in $L-$algebras, and obtain some characteristics of these objects that are normally not encountered on crossed modules of groups or algebras. As a consequence, we characterize crossed self-similarity completely in terms of properties of $L-$algebras and the boundary map forming the crossed module.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140011420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Usually in logic, proof systems are defined having in mind proving properties like validity and semantic consequence. It seems worthwhile to address the problem of having proof systems where satisfiability is a primitive notion in the sense that a formal derivation means that a finite set of formulas is satisfiable. Moreover, it would be useful to cover within the same framework as many logics as possible. We consider Kripke semantics where the properties of the constructors are provided by valuation constraints as the common ground of those logics. This includes for instance intuitionistic logic, paraconsistent Nelson’s logic ${textsf{N4}}$, paraconsistent logic ${textsf{imbC}}$ and modal logics among others. After specifying a logic by those valuation constraints, we show how to induce automatically and from scratch an existential proof system for that logic. The rules of the proof system are shown to be invertible. General results of soundness and completeness are proved and then applied to the logics at hand.
{"title":"Labelled proof systems for existential reasoning","authors":"Jaime Ramos, João Rasga, Cristina Sernadas","doi":"10.1093/jigpal/jzad030","DOIUrl":"https://doi.org/10.1093/jigpal/jzad030","url":null,"abstract":"Usually in logic, proof systems are defined having in mind proving properties like validity and semantic consequence. It seems worthwhile to address the problem of having proof systems where satisfiability is a primitive notion in the sense that a formal derivation means that a finite set of formulas is satisfiable. Moreover, it would be useful to cover within the same framework as many logics as possible. We consider Kripke semantics where the properties of the constructors are provided by valuation constraints as the common ground of those logics. This includes for instance intuitionistic logic, paraconsistent Nelson’s logic ${textsf{N4}}$, paraconsistent logic ${textsf{imbC}}$ and modal logics among others. After specifying a logic by those valuation constraints, we show how to induce automatically and from scratch an existential proof system for that logic. The rules of the proof system are shown to be invertible. General results of soundness and completeness are proved and then applied to the logics at hand.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a uniform proof-theoretic proof of the Gödel–McKinsey–Tarski embedding for a class of first-order intuitionistic theories. This is achieved by adapting to the case of modal logic the methods of proof analysis in order to convert axioms into rules of inference of a suitable sequent calculus. The soundness and the faithfulness of the embedding are proved by induction on the height of the derivations in the augmented calculi. Finally, we define an extension of the modal system for which the result holds with respect to geometric intuitionistic.
{"title":"Constructive theories through a modal lens","authors":"Matteo Tesi","doi":"10.1093/jigpal/jzad029","DOIUrl":"https://doi.org/10.1093/jigpal/jzad029","url":null,"abstract":"We present a uniform proof-theoretic proof of the Gödel–McKinsey–Tarski embedding for a class of first-order intuitionistic theories. This is achieved by adapting to the case of modal logic the methods of proof analysis in order to convert axioms into rules of inference of a suitable sequent calculus. The soundness and the faithfulness of the embedding are proved by induction on the height of the derivations in the augmented calculi. Finally, we define an extension of the modal system for which the result holds with respect to geometric intuitionistic.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"93 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139064409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that both the $n$-density and the bounded $n$-width of Kripke frames can be modally defined not only with natural and well-known Sahlqvist formulae containing a linear number of different propositional variables but also with formulae of polynomial length with a logarithmic number of different propositional variables and then we prove that this exponential decrease in the number of variables leads us outside the class of Sahlqvist formulae.
{"title":"On the number of different variables required to define the n-density or the bounded n-width of Kripke frames with some consequences for Sahlqvist formulae","authors":"Petar Iliev","doi":"10.1093/jigpal/jzad026","DOIUrl":"https://doi.org/10.1093/jigpal/jzad026","url":null,"abstract":"We show that both the $n$-density and the bounded $n$-width of Kripke frames can be modally defined not only with natural and well-known Sahlqvist formulae containing a linear number of different propositional variables but also with formulae of polynomial length with a logarithmic number of different propositional variables and then we prove that this exponential decrease in the number of variables leads us outside the class of Sahlqvist formulae.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"7 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The product of two $textbf {Alt}$ logics possesses the polynomial product finite model property and its membership problem is $textbf {coNP}$-complete. Using a reduction from an undecidable domino-tiling problem, we prove that its admissibility problem is undecidable.
{"title":"Undecidability of admissibility in the product of two Alt logics","authors":"Philippe Balbiani, Çiğdem Gencer","doi":"10.1093/jigpal/jzad021","DOIUrl":"https://doi.org/10.1093/jigpal/jzad021","url":null,"abstract":"Abstract The product of two $textbf {Alt}$ logics possesses the polynomial product finite model property and its membership problem is $textbf {coNP}$-complete. Using a reduction from an undecidable domino-tiling problem, we prove that its admissibility problem is undecidable.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"27 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135166313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J Berger, Douglas Bridges, Hannes Diener, Helmet Schwichtenberg
Abstract The notions of permutable and weak-permutable convergence of a series $sum _{n=1}^{infty }a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD- $mathbb {N}$ implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but BD- $mathbb{N}$ does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD- $mathbb {N}$ . We show that this is the case when the property is weak-permutable convergence.
{"title":"Constructive aspects of Riemann’s permutation theorem for series","authors":"J Berger, Douglas Bridges, Hannes Diener, Helmet Schwichtenberg","doi":"10.1093/jigpal/jzad024","DOIUrl":"https://doi.org/10.1093/jigpal/jzad024","url":null,"abstract":"Abstract The notions of permutable and weak-permutable convergence of a series $sum _{n=1}^{infty }a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD- $mathbb {N}$ implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but BD- $mathbb{N}$ does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD- $mathbb {N}$ . We show that this is the case when the property is weak-permutable convergence.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135322692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element $0$, then the relative pseudocomplement with respect to $0$ is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with $0$ satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation $x^{0}$ and implication $xrightarrow y$ as the set of all maximal elements $z$ satisfying $xwedge z=0$ and as the set of all maximal elements $z$ satisfying $xwedge zleq y$, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry $x$ or to two entries $x$ and $y$ belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.
{"title":"Algebraic structures formalizing the logic with unsharp implication and negation","authors":"Ivan Chajda, Helmut Länger","doi":"10.1093/jigpal/jzad023","DOIUrl":"https://doi.org/10.1093/jigpal/jzad023","url":null,"abstract":"Abstract It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element $0$, then the relative pseudocomplement with respect to $0$ is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with $0$ satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation $x^{0}$ and implication $xrightarrow y$ as the set of all maximal elements $z$ satisfying $xwedge z=0$ and as the set of all maximal elements $z$ satisfying $xwedge zleq y$, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry $x$ or to two entries $x$ and $y$ belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"238 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135888004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is intended first for the formal argumentation community (see https://comma.csc.liv.ac.uk/). This community develops logics and systems modelling argumentation and dialogues. The community is in search of major applications areas for their models. One such application area e.g. is Law. The message of this paper is that there is another major application area for formal argumentation. There is an international community of sex offender therapist that is well established and well funded, and their therapy methods use (methods that can be modelled by) formal argumentation and logic. This community presents a natural application area for formal argumentation. We thus describe in this paper how the sex offender therapists work, to give the formal argumentation researcher a view of this application area. What is especially important about this application area is that in order to model it and learn from it, the formal argumentation community have to evolve their formal methods and adapt to this new application. Part of this enhancement is to modify and import certain methods from other areas of Logic e.g. from Non-Monotonic logic. The members of the formal argumentation community are not familiar, on average, with other areas of logic, and so we also describe in this paper, what we need from neighbouring logics. This makes this paper of interest also to sex offender therapist as well. They may be already familiar with their own practices, but the additional logics described will be of interest to them.
{"title":"The use of logic and argumentation in therapy of sex offenders","authors":"Dov Gabbay, Gadi Rozenberg, Lydia Rivlin","doi":"10.1093/jigpal/jzad022","DOIUrl":"https://doi.org/10.1093/jigpal/jzad022","url":null,"abstract":"Abstract This paper is intended first for the formal argumentation community (see https://comma.csc.liv.ac.uk/). This community develops logics and systems modelling argumentation and dialogues. The community is in search of major applications areas for their models. One such application area e.g. is Law. The message of this paper is that there is another major application area for formal argumentation. There is an international community of sex offender therapist that is well established and well funded, and their therapy methods use (methods that can be modelled by) formal argumentation and logic. This community presents a natural application area for formal argumentation. We thus describe in this paper how the sex offender therapists work, to give the formal argumentation researcher a view of this application area. What is especially important about this application area is that in order to model it and learn from it, the formal argumentation community have to evolve their formal methods and adapt to this new application. Part of this enhancement is to modify and import certain methods from other areas of Logic e.g. from Non-Monotonic logic. The members of the formal argumentation community are not familiar, on average, with other areas of logic, and so we also describe in this paper, what we need from neighbouring logics. This makes this paper of interest also to sex offender therapist as well. They may be already familiar with their own practices, but the additional logics described will be of interest to them.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136034586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this work, we illustrate applications of a semantic framework for non-congruential modal logic based on hyperintensional models. We start by discussing some philosophical ideas behind the approach; in particular, the difference between the set of possible worlds in which a formula is true (its intension) and the semantic content of a formula (its hyperintension), which is captured in a rigorous way in hyperintensional models. Next, we rigorously specify the approach and provide a fundamental completeness theorem. Moreover, we analyse examples of non-congruential systems that can be semantically characterized within this framework in an elegant and modular way. Finally, we compare the proposed framework with some alternatives available in the literature. In the light of the results obtained, we argue that hyperintensional models constitute a basic, general and unifying semantic framework for (non-congruential) modal logic.
{"title":"Hyperintensional models for non-congruential modal logics","authors":"Matteo Pascucci, Igor Sedlár","doi":"10.1093/jigpal/jzad018","DOIUrl":"https://doi.org/10.1093/jigpal/jzad018","url":null,"abstract":"Abstract In this work, we illustrate applications of a semantic framework for non-congruential modal logic based on hyperintensional models. We start by discussing some philosophical ideas behind the approach; in particular, the difference between the set of possible worlds in which a formula is true (its intension) and the semantic content of a formula (its hyperintension), which is captured in a rigorous way in hyperintensional models. Next, we rigorously specify the approach and provide a fundamental completeness theorem. Moreover, we analyse examples of non-congruential systems that can be semantically characterized within this framework in an elegant and modular way. Finally, we compare the proposed framework with some alternatives available in the literature. In the light of the results obtained, we argue that hyperintensional models constitute a basic, general and unifying semantic framework for (non-congruential) modal logic.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136237794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Journal Article Correction to: Decidability of interpretability logics ILM0 and ILW* Get access Luka Mikec, Luka Mikec University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Tin Perkov, Tin Perkov University of Zagreb E-mail: tin.perkov@ufzg.hr Search for other works by this author on: Oxford Academic Google Scholar Mladen Vukoviĉ Mladen Vukoviĉ University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Logic Journal of the IGPL, jzad020, https://doi.org/10.1093/jigpal/jzad020 Published: 14 September 2023 Article history Received: 07 September 2023 Published: 14 September 2023
期刊文章更正:可解释性逻辑ILM0和ILW的可决定性*访问Luka Mikec, Luka Mikec萨格勒布大学搜索本作者的其他作品:牛津学术谷歌学者Tin Perkov, Tin Perkov萨格勒布大学E-mail: tin.perkov@ufzg.hr搜索本作者的其他作品:牛津学术谷歌学者Mladen vukovii Mladen vukovii萨格勒布大学搜索本作者的其他作品:牛津学术谷歌学者逻辑IGPL期刊，jzad020, https://doi.org/10.1093/jigpal/jzad020发布日期:2023年9月14日文章历史收稿日期:2023年9月07日发布日期:2023年9月14日
{"title":"Correction to: Decidability of interpretability logics <b>IL</b> <tt>M</tt>0 and <b>IL</b> <tt>W</tt>*","authors":"Luka Mikec, Tin Perkov, Mladen Vukoviĉ","doi":"10.1093/jigpal/jzad020","DOIUrl":"https://doi.org/10.1093/jigpal/jzad020","url":null,"abstract":"Journal Article Correction to: Decidability of interpretability logics ILM0 and ILW* Get access Luka Mikec, Luka Mikec University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Tin Perkov, Tin Perkov University of Zagreb E-mail: tin.perkov@ufzg.hr Search for other works by this author on: Oxford Academic Google Scholar Mladen Vukoviĉ Mladen Vukoviĉ University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Logic Journal of the IGPL, jzad020, https://doi.org/10.1093/jigpal/jzad020 Published: 14 September 2023 Article history Received: 07 September 2023 Published: 14 September 2023","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134913976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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