{"title":"Editorial: Special Issue CISIS 2021","authors":"","doi":"10.1093/jigpal/jzae015","DOIUrl":"https://doi.org/10.1093/jigpal/jzae015","url":null,"abstract":"","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140247857","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}
Two new propositional non-classical logics, referred to as symmetric intuitionistic logic (SIL) and conflated intuitionistic logic (CIL), are introduced as indexed and non-indexed Gentzen-style sequent calculi. SIL is regarded as a natural hybrid logic combining intuitionistic and dual-intuitionistic logics, whereas CIL is regarded as a variant of intuitionistic paraconsistent logic with conflation and without paraconsistent negation. The cut-elimination theorems for SIL and CIL are proved. CIL is shown to be conservative over SIL.
本文介绍了两种新的命题非经典逻辑，分别称为对称直觉逻辑（SIL）和混同直觉逻辑（CIL），作为有索引和无索引的根岑式序列计算。SIL 被视为结合了直观逻辑和双直观逻辑的自然混合逻辑，而 CIL 则被视为直观准一致逻辑的一种变体，具有混淆性，但没有准一致的否定。证明了 SIL 和 CIL 的割除定理。证明了 CIL 比 SIL 保守。
{"title":"Symmetric and conflated intuitionistic logics","authors":"Norihiro Kamide","doi":"10.1093/jigpal/jzae001","DOIUrl":"https://doi.org/10.1093/jigpal/jzae001","url":null,"abstract":"Two new propositional non-classical logics, referred to as symmetric intuitionistic logic (SIL) and conflated intuitionistic logic (CIL), are introduced as indexed and non-indexed Gentzen-style sequent calculi. SIL is regarded as a natural hybrid logic combining intuitionistic and dual-intuitionistic logics, whereas CIL is regarded as a variant of intuitionistic paraconsistent logic with conflation and without paraconsistent negation. The cut-elimination theorems for SIL and CIL are proved. CIL is shown to be conservative over SIL.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140037889","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}
In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems $K$, $KT$, $K4$ and $S4$, with $square $ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between $square $ and a natural presentation of $lozenge $. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases.
{"title":"Base-extension semantics for modal logic","authors":"Timo Eckhardt, David J Pym","doi":"10.1093/jigpal/jzae004","DOIUrl":"https://doi.org/10.1093/jigpal/jzae004","url":null,"abstract":"In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems $K$, $KT$, $K4$ and $S4$, with $square $ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between $square $ and a natural presentation of $lozenge $. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140017349","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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