In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $textbf{0}^{prime }$ (theories with Turing degree $textbf{0}^{prime }$), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties $P$ and $Q$ in the above list, we examine whether $P$ implies $Q$.
{"title":"On the relationships between some meta-mathematical properties of arithmetical theories","authors":"Yong Cheng","doi":"10.1093/jigpal/jzad015","DOIUrl":"https://doi.org/10.1093/jigpal/jzad015","url":null,"abstract":"\u0000 In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $textbf{0}^{prime }$ (theories with Turing degree $textbf{0}^{prime }$), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties $P$ and $Q$ in the above list, we examine whether $P$ implies $Q$.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47083373","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 well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with $wedge $, $vee $ and $sim $ as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer’s basic logic B and its useful disjunctive extension B$^{textrm {d}}$. It is to be noted that Boolean negation (so, classical propositional logic) is definable in the strongest element in the said class.
{"title":"The lattice of all 4-valued implicative expansions of Belnap–Dunn logic containing Routley and Meyer’s basic logic B<i>d</i>","authors":"Gemma Robles, José M Méndez","doi":"10.1093/jigpal/jzad005","DOIUrl":"https://doi.org/10.1093/jigpal/jzad005","url":null,"abstract":"Abstract The well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with $wedge $, $vee $ and $sim $ as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer’s basic logic B and its useful disjunctive extension B$^{textrm {d}}$. It is to be noted that Boolean negation (so, classical propositional logic) is definable in the strongest element in the said class.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"231 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002173","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 this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics $textbf {GL}$, $textbf {Grz}$, $textbf {wGrz}$ and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between $textbf {QwGrz}$ and $textbf {QGL.3}$ or between $textbf {QwGrz}$ and $textbf {QGrz.3}$ is $Pi ^1_1$-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics $textbf {GL}$, $textbf {Grz}$, $textbf {wGrz}$ are not Kripke complete. Both $Pi ^1_1$-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula.
{"title":"Predicate counterparts of modal logics of provability: High undecidability and Kripke incompleteness","authors":"M. Rybakov","doi":"10.1093/jigpal/jzad002","DOIUrl":"https://doi.org/10.1093/jigpal/jzad002","url":null,"abstract":"\u0000 In this paper, the predicate counterparts, defined both axiomatically and semantically by means of Kripke frames, of the modal propositional logics $textbf {GL}$, $textbf {Grz}$, $textbf {wGrz}$ and their extensions are considered. It is proved that the set of semantical consequences on Kripke frames of every logic between $textbf {QwGrz}$ and $textbf {QGL.3}$ or between $textbf {QwGrz}$ and $textbf {QGrz.3}$ is $Pi ^1_1$-hard even in languages with three (sometimes, two) individual variables, two (sometimes, one) unary predicate letters, and a single proposition letter. As a corollary, it is proved that infinite families of modal predicate axiomatic systems, based on the classical first-order logic and the modal propositional logics $textbf {GL}$, $textbf {Grz}$, $textbf {wGrz}$ are not Kripke complete. Both $Pi ^1_1$-hardness and Kripke incompleteness results of the paper do not depend on whether the logics contain the Barcan formula.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47614232","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 Standard unary modal logic and binary modal logic, i.e. modal logic with one binary operator, are shown to be definitional extensions of one another when an additional axiom $U$ is added to the basic axiomatization of the binary side. This is a strengthening of our previous results. It follows that all unary modal logics extending Classical Modal Logic, in other words all unary modal logics with a neighborhood semantics, can equivalently be seen as binary modal logics. This in particular applies to standard modal logics, which can be given simple natural axiomatizations in binary form. We illustrate this in the logic K. We call such logics binary expansions of the unary modal logics. There are many more such binary expansions than the ones given by the axiom $U$. We initiate an investigation of the properties of these expansions and in particular of the maximal binary expansions of a logic. Our results directly imply that all sub- and superintuitionistic logics with a standard modal companion also have binary modal companions. The latter also applies to the weak subintuitionistic logic WF of our previous papers. This logic doesn’t seem to have a unary modal companion.
{"title":"Binary modal logic and unary modal logic","authors":"Dick de Jongh, Fatemeh Shirmohammadzadeh Maleki","doi":"10.1093/jigpal/jzac083","DOIUrl":"https://doi.org/10.1093/jigpal/jzac083","url":null,"abstract":"Abstract Standard unary modal logic and binary modal logic, i.e. modal logic with one binary operator, are shown to be definitional extensions of one another when an additional axiom $U$ is added to the basic axiomatization of the binary side. This is a strengthening of our previous results. It follows that all unary modal logics extending Classical Modal Logic, in other words all unary modal logics with a neighborhood semantics, can equivalently be seen as binary modal logics. This in particular applies to standard modal logics, which can be given simple natural axiomatizations in binary form. We illustrate this in the logic K. We call such logics binary expansions of the unary modal logics. There are many more such binary expansions than the ones given by the axiom $U$. We initiate an investigation of the properties of these expansions and in particular of the maximal binary expansions of a logic. Our results directly imply that all sub- and superintuitionistic logics with a standard modal companion also have binary modal companions. The latter also applies to the weak subintuitionistic logic WF of our previous papers. This logic doesn’t seem to have a unary modal companion.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136198035","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 the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders $mathcal {L}( @, {downarrow })$. We define the class of restricted Sahlqvist formulas for $mathcal {L}( @, {downarrow })$ following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for every restricted Sahlqvist formula $varphi $ and its hybrid pure correspondence $pi $, $textbf {K}_{mathcal {H}( @, {downarrow })}+varphi $ proves $pi $; therefore, $textbf {K}_{mathcal {H}( @, {downarrow })}+varphi $ is complete with respect to the class of frames defined by $pi $, using a modified version $textsf {ALBA}^{{downarrow }}_{textsf {Modified}}$ of the algorithm $textsf {ALBA}^{{downarrow }}$ defined in Zhao (2021, Logic J. IGPL).
{"title":"Sahlqvist Completeness Theory for Hybrid Logic with Downarrow Binder","authors":"Zhiguang Zhao","doi":"10.1093/jigpal/jzac079","DOIUrl":"https://doi.org/10.1093/jigpal/jzac079","url":null,"abstract":"Abstract In the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders $mathcal {L}( @, {downarrow })$. We define the class of restricted Sahlqvist formulas for $mathcal {L}( @, {downarrow })$ following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for every restricted Sahlqvist formula $varphi $ and its hybrid pure correspondence $pi $, $textbf {K}_{mathcal {H}( @, {downarrow })}+varphi $ proves $pi $; therefore, $textbf {K}_{mathcal {H}( @, {downarrow })}+varphi $ is complete with respect to the class of frames defined by $pi $, using a modified version $textsf {ALBA}^{{downarrow }}_{textsf {Modified}}$ of the algorithm $textsf {ALBA}^{{downarrow }}$ defined in Zhao (2021, Logic J. IGPL).","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135694078","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 We study an extension of first-degree entailment (FDE) by Dunn and Belnap with a non-contingency operator $blacktriangle phi $ which is construed as ‘$phi $ has the same value in all accessible states’ or ‘all sources give the same information on the truth value of $phi $’. We equip this logic dubbed $textbf {K}^blacktriangle _{textbf {FDE}}$ with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the $blacktriangle $ operator modelling search for inconsistencies in the provided information. We construct an analytic cut system for the logic and show its soundness and completeness. We prove that $blacktriangle $ is not definable via the necessity modality $Box $ of $textbf {K}_{textbf{FDE}}$. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, $textbf {S4}$ and $textbf {S5}$ (among others) frames are definable.
{"title":"Non-contingency in a Paraconsistent Setting","authors":"Daniil Kozhemiachenko, Liubov Vashentseva","doi":"10.1093/jigpal/jzac081","DOIUrl":"https://doi.org/10.1093/jigpal/jzac081","url":null,"abstract":"Abstract We study an extension of first-degree entailment (FDE) by Dunn and Belnap with a non-contingency operator $blacktriangle phi $ which is construed as ‘$phi $ has the same value in all accessible states’ or ‘all sources give the same information on the truth value of $phi $’. We equip this logic dubbed $textbf {K}^blacktriangle _{textbf {FDE}}$ with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the $blacktriangle $ operator modelling search for inconsistencies in the provided information. We construct an analytic cut system for the logic and show its soundness and completeness. We prove that $blacktriangle $ is not definable via the necessity modality $Box $ of $textbf {K}_{textbf{FDE}}$. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, $textbf {S4}$ and $textbf {S5}$ (among others) frames are definable.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135694089","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}
Hernán Díaz, J. J. Palacios, Irene Díaz, C. R. Vela, I. G. Rodríguez
{"title":"Robust schedules for tardiness optimization in job shop with interval uncertainty","authors":"Hernán Díaz, J. J. Palacios, Irene Díaz, C. R. Vela, I. G. Rodríguez","doi":"10.1093/jigpal/jzac016","DOIUrl":"https://doi.org/10.1093/jigpal/jzac016","url":null,"abstract":"","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"31 1","pages":"240-254"},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60952912","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}
Majid Alizadeh, Mohammad Ardeshir, P. Balbiani, M. Mojtahedi
{"title":"Unification types in Euclidean modal logics","authors":"Majid Alizadeh, Mohammad Ardeshir, P. Balbiani, M. Mojtahedi","doi":"10.1093/jigpal/jzab036","DOIUrl":"https://doi.org/10.1093/jigpal/jzab036","url":null,"abstract":"","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"31 1","pages":"422-440"},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60952143","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}
Francisco Zayas-Gato, Álvaro Michelena, Héctor Quintián-Pardo, Esteban Jove, J. Casteleiro-Roca, Paulo Leitão, J. Calvo-Rolle
{"title":"A novel method for anomaly detection using beta Hebbian learning and principal component analysis","authors":"Francisco Zayas-Gato, Álvaro Michelena, Héctor Quintián-Pardo, Esteban Jove, J. Casteleiro-Roca, Paulo Leitão, J. Calvo-Rolle","doi":"10.1093/jigpal/jzac026","DOIUrl":"https://doi.org/10.1093/jigpal/jzac026","url":null,"abstract":"","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"31 1","pages":"390-399"},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60952759","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}
S. Simic, S. Sakac, Z. Bankovic, J. Villar, J. Calvo-Rolle, S. Simic, D. Simić
{"title":"A three-stage hybrid clustering system for diagnosing children with primary headache disorder","authors":"S. Simic, S. Sakac, Z. Bankovic, J. Villar, J. Calvo-Rolle, S. Simic, D. Simić","doi":"10.1093/jigpal/jzac020","DOIUrl":"https://doi.org/10.1093/jigpal/jzac020","url":null,"abstract":"","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":"31 1","pages":"300-313"},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60953047","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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