We study positively closed and strongly positively closed $mathcal{C}to Sh(B)$ models, where $mathcal{C}$ is a $(kappa ,kappa )$-coherent category and $B$ is a $(kappa ,kappa )$-coherent Boolean-algebra for some weakly compact $kappa $. We prove that if $mathcal{C}$ is coherent and $X$ is a Stone-space then positively closed, not strongly positively closed $mathcal{C}to Sh(X)$ models may exist.
{"title":"Positively closed parametrized models","authors":"Kristóf Kanalas","doi":"arxiv-2409.11231","DOIUrl":"https://doi.org/arxiv-2409.11231","url":null,"abstract":"We study positively closed and strongly positively closed $mathcal{C}to\u0000Sh(B)$ models, where $mathcal{C}$ is a $(kappa ,kappa )$-coherent category\u0000and $B$ is a $(kappa ,kappa )$-coherent Boolean-algebra for some weakly\u0000compact $kappa $. We prove that if $mathcal{C}$ is coherent and $X$ is a\u0000Stone-space then positively closed, not strongly positively closed\u0000$mathcal{C}to Sh(X)$ models may exist.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266727","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}
We look at the proofs of a fragment of Linear Logic as a whole: in fact, Linear Logic's coherent semantics interprets the proofs of a given formula $A$ as faces of an abstract simplicial complex, thus allowing us to see the set of the (interpretations of the) proofs of $A$ as a geometrical space, not just a set. This point of view has never been really investigated. For a ``webbed'' denotational semantics -- say the relational one --, it suffices to down-close the set of (the interpretations of the) proofs of $A$ in order to give rise to an abstract simplicial complex whose faces do correspond to proofs of $A$. Since this space comes triangulated by construction, a natural geometrical property to consider is its homology. However, we immediately stumble on a problem: if we want the homology to be invariant w.r.t. to some notion of type-isomorphism, we are naturally led to consider the homology functor acting, at the level of morphisms, on ``simplicial relations'' rather than simplicial maps as one does in topology. The task of defining the homology functor on this modified category can be achieved by considering a very simple monad, which is almost the same as the power-set monad; but, doing so, we end up considering not anymore the homology of the original space, but rather of its transformation under the action of the monad. Does this transformation keep the homology invariant ? Is this transformation meaningful from a geometrical or logical/computational point of view ?
{"title":"Denotational semantics driven simplicial homology?","authors":"Davide Barbarossa","doi":"arxiv-2409.11566","DOIUrl":"https://doi.org/arxiv-2409.11566","url":null,"abstract":"We look at the proofs of a fragment of Linear Logic as a whole: in fact,\u0000Linear Logic's coherent semantics interprets the proofs of a given formula $A$\u0000as faces of an abstract simplicial complex, thus allowing us to see the set of\u0000the (interpretations of the) proofs of $A$ as a geometrical space, not just a\u0000set. This point of view has never been really investigated. For a ``webbed''\u0000denotational semantics -- say the relational one --, it suffices to down-close\u0000the set of (the interpretations of the) proofs of $A$ in order to give rise to\u0000an abstract simplicial complex whose faces do correspond to proofs of $A$.\u0000Since this space comes triangulated by construction, a natural geometrical\u0000property to consider is its homology. However, we immediately stumble on a\u0000problem: if we want the homology to be invariant w.r.t. to some notion of\u0000type-isomorphism, we are naturally led to consider the homology functor acting,\u0000at the level of morphisms, on ``simplicial relations'' rather than simplicial\u0000maps as one does in topology. The task of defining the homology functor on this\u0000modified category can be achieved by considering a very simple monad, which is\u0000almost the same as the power-set monad; but, doing so, we end up considering\u0000not anymore the homology of the original space, but rather of its\u0000transformation under the action of the monad. Does this transformation keep the\u0000homology invariant ? Is this transformation meaningful from a geometrical or\u0000logical/computational point of view ?","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"96 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266725","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}
This article concerns with the Axiom of Choice (AC) and the well-ordering theorem (WO) in second-order predicate logic with Henkin interpretation (HPL). We consider a principle of choice introduced by Wilhelm Ackermann (1935) and discussed also by David Hilbert and Ackermann (1938), by G"unter Asser (1981), and by Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro (2020). Our discussion is restricted to so-called Henkin-Asser structures of second order. Here, we give the technical details of our proof of the independence of WO from the so-called Ackermann axioms in HPL presented at the Colloquium Logicum in 2022. Most of the definitions used here can be found in Sections 1, 2, and 3 in Part I.
本文关注亨金诠释（HPL）二阶谓词逻辑中的选择公理（AC）和井序定理（WO）。我们考虑了威廉-阿克曼（Wilhelm Ackermann，1935）提出的选择原则，大卫-希尔伯特和阿克曼（David Hilbert and Ackermann，1938）、本杰明-西斯金德（Benjamin Siskind）、保罗-曼科苏（Paolo Mancosu）和斯图尔特-夏皮罗（Stewart Shapiro，2020）也讨论了这一原则。我们的讨论仅限于所谓的二阶亨金-阿瑟结构。在此，我们给出了我们在 2022 年逻辑学术讨论会上提出的关于 WO 独立于 HPL 中所谓阿克曼公理的证明的技术细节。这里使用的大部分定义可以在第一部分的第 1、2 和 3 节中找到。
{"title":"AC and the Independence of WO in Second-Order Henkin Logic, Part II","authors":"Christine Gaßner","doi":"arxiv-2409.11126","DOIUrl":"https://doi.org/arxiv-2409.11126","url":null,"abstract":"This article concerns with the Axiom of Choice (AC) and the well-ordering\u0000theorem (WO) in second-order predicate logic with Henkin interpretation (HPL).\u0000We consider a principle of choice introduced by Wilhelm Ackermann (1935) and\u0000discussed also by David Hilbert and Ackermann (1938), by G\"unter Asser (1981),\u0000and by Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro (2020). Our\u0000discussion is restricted to so-called Henkin-Asser structures of second order.\u0000Here, we give the technical details of our proof of the independence of WO from\u0000the so-called Ackermann axioms in HPL presented at the Colloquium Logicum in\u00002022. Most of the definitions used here can be found in Sections 1, 2, and 3 in\u0000Part I.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266726","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}
Motivated by structural properties of differential field extensions, we introduce the notion of a theory $T$ being derivation-like with respect to another model-complete theory $T_0$. We prove that when $T$ admits a model-companion $T_+$, then several model-theoretic properties transfer from $T_0$ to $T_+$. These properties include completeness, quantifier-elimination, stability, simplicity, and NSOP$_1$. We also observe that, aside from the theory of differential fields, examples of derivation-like theories are plentiful.
{"title":"Neostability transfers in derivation-like theories","authors":"Omar Leon Sanchez, Shezad Mohamed","doi":"arxiv-2409.11248","DOIUrl":"https://doi.org/arxiv-2409.11248","url":null,"abstract":"Motivated by structural properties of differential field extensions, we\u0000introduce the notion of a theory $T$ being derivation-like with respect to\u0000another model-complete theory $T_0$. We prove that when $T$ admits a\u0000model-companion $T_+$, then several model-theoretic properties transfer from\u0000$T_0$ to $T_+$. These properties include completeness, quantifier-elimination,\u0000stability, simplicity, and NSOP$_1$. We also observe that, aside from the\u0000theory of differential fields, examples of derivation-like theories are\u0000plentiful.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"104 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269593","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}
Amirhossein Akbar Tabatabai, Majid Alizadeh, Masoud Memarzadeh
A $nabla$-algebra is a natural generalization of a Heyting algebra, unifying several algebraic structures, including bounded lattices, Heyting algebras, temporal Heyting algebras, and the algebraic representation of dynamic topological systems. In the prequel to this paper [3], we explored the algebraic properties of various varieties of $nabla$-algebras, their subdirectly-irreducible and simple elements, their closure under Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce $nabla$-spaces as a common generalization of Priestley and Esakia spaces, through which we develop a duality theory for certain categories of $nabla$-algebras. Then, we reframe these dualities in terms of spectral spaces and provide an algebraic characterization of natural families of dynamic topological systems over Priestley, Esakia, and spectral spaces. Additionally, we present a ring-theoretic representation for some families of $nabla$-algebras. Finally, we introduce several logical systems to capture different varieties of $nabla$-algebras, offering their algebraic, Kripke, topological, and ring-theoretic semantics, and establish a deductive interpolation theorem for some of these systems.
{"title":"On a Generalization of Heyting Algebras II","authors":"Amirhossein Akbar Tabatabai, Majid Alizadeh, Masoud Memarzadeh","doi":"arxiv-2409.10642","DOIUrl":"https://doi.org/arxiv-2409.10642","url":null,"abstract":"A $nabla$-algebra is a natural generalization of a Heyting algebra, unifying\u0000several algebraic structures, including bounded lattices, Heyting algebras,\u0000temporal Heyting algebras, and the algebraic representation of dynamic\u0000topological systems. In the prequel to this paper [3], we explored the\u0000algebraic properties of various varieties of $nabla$-algebras, their\u0000subdirectly-irreducible and simple elements, their closure under\u0000Dedekind-MacNeille completion, and their Kripke-style representation. In this sequel, we first introduce $nabla$-spaces as a common generalization\u0000of Priestley and Esakia spaces, through which we develop a duality theory for\u0000certain categories of $nabla$-algebras. Then, we reframe these dualities in\u0000terms of spectral spaces and provide an algebraic characterization of natural\u0000families of dynamic topological systems over Priestley, Esakia, and spectral\u0000spaces. Additionally, we present a ring-theoretic representation for some\u0000families of $nabla$-algebras. Finally, we introduce several logical systems to\u0000capture different varieties of $nabla$-algebras, offering their algebraic,\u0000Kripke, topological, and ring-theoretic semantics, and establish a deductive\u0000interpolation theorem for some of these systems.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269594","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}
We prove relative quantifier elimination for Pal's multiplicative valued difference fields with an added lifting map of the residue field. Furthermore, we generalize a $mathrm{NIP}$ transfer result for valued fields by Jahnke and Simon to $mathrm{NTP}_2$ to show that said valued difference fields are $mathrm{NTP}_2$ if and only if value group and residue field are.
{"title":"Tameness Properties in Multiplicative Valued Difference Fields with Lift and Section","authors":"Christoph Kesting","doi":"arxiv-2409.10406","DOIUrl":"https://doi.org/arxiv-2409.10406","url":null,"abstract":"We prove relative quantifier elimination for Pal's multiplicative valued\u0000difference fields with an added lifting map of the residue field. Furthermore,\u0000we generalize a $mathrm{NIP}$ transfer result for valued fields by Jahnke and\u0000Simon to $mathrm{NTP}_2$ to show that said valued difference fields are\u0000$mathrm{NTP}_2$ if and only if value group and residue field are.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266728","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}
In their article about distality in valued fields, Aschenbrenner, Chernikov, Gehret and Ziegler proved resplendent Ax-Kochen-Ershov principles for quantifier elimination in pure short exact sequences of Abelian structures. We study how their work relates to forking, and we prove Ax-Kochen-Ershov principles for forking and dividing in this setting.
{"title":"Transfer principles for forking and dividing in expansions of pure short exact sequences of Abelian groups","authors":"Akash Hossain","doi":"arxiv-2409.10148","DOIUrl":"https://doi.org/arxiv-2409.10148","url":null,"abstract":"In their article about distality in valued fields, Aschenbrenner, Chernikov,\u0000Gehret and Ziegler proved resplendent Ax-Kochen-Ershov principles for\u0000quantifier elimination in pure short exact sequences of Abelian structures. We\u0000study how their work relates to forking, and we prove Ax-Kochen-Ershov\u0000principles for forking and dividing in this setting.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266731","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}
Logics closed under classes of substitutions broader than class of uniform substitutions are known as hyperformal logics. This paper extends known results about hyperformal logics in two ways. First: we examine a very powerful form of hyperformalism that tracks, for bunched natural deduction systems, essentially all the intensional content that can possibly be tracked. We demonstrate that, after a few tweaks, the well-known relevant logic $mathbf{B}$ exhibits this form of hyperformalism. Second: we demonstrate that not only can hyperformalism be extended along these lines, it can also be extended to accommodate not just what is proved in a given logic but the proofs themselves. Altogether, the paper demonstrates that the space of possibilities for the study of hyperformalism is much larger than might have been expected.
{"title":"Hyperformalism for Bunched Natural Deduction Systems","authors":"Shay Allen Logan, Blane Worley","doi":"arxiv-2409.10418","DOIUrl":"https://doi.org/arxiv-2409.10418","url":null,"abstract":"Logics closed under classes of substitutions broader than class of uniform\u0000substitutions are known as hyperformal logics. This paper extends known results\u0000about hyperformal logics in two ways. First: we examine a very powerful form of\u0000hyperformalism that tracks, for bunched natural deduction systems, essentially\u0000all the intensional content that can possibly be tracked. We demonstrate that,\u0000after a few tweaks, the well-known relevant logic $mathbf{B}$ exhibits this\u0000form of hyperformalism. Second: we demonstrate that not only can hyperformalism\u0000be extended along these lines, it can also be extended to accommodate not just\u0000what is proved in a given logic but the proofs themselves. Altogether, the\u0000paper demonstrates that the space of possibilities for the study of\u0000hyperformalism is much larger than might have been expected.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266729","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}
We show how dinaturality plays a central role in the interpretation of directed type theory where types are interpreted as (1-)categories and directed equality is represented by $hom$-functors. We present a general elimination principle based on dinaturality for directed equality which very closely resembles the $J$-rule used in Martin-L"of type theory, and we highlight which syntactical restrictions are needed to interpret this rule in the context of directed equality. We then use these rules to characterize directed equality as a left relative adjoint to a functor between (para)categories of dinatural transformations which contracts together two variables appearing naturally with a single dinatural one, with the relative functor imposing the syntactic restrictions needed. We then argue that the quantifiers of such a directed type theory should be interpreted as ends and coends, which dinaturality allows us to present in adjoint-like correspondences to a weakening functor. Using these rules we give a formal interpretation to Yoneda reductions and (co)end calculus, and we use logical derivations to prove the Fubini rule for quantifier exchange, the adjointness property of Kan extensions via (co)ends, exponential objects of presheaves, and the (co)Yoneda lemma. We show transitivity (composition), congruence (functoriality), and transport (coYoneda) for directed equality by closely following the same approach of Martin-L"of type theory, with the notable exception of symmetry. We formalize our main theorems in Agda.
{"title":"Directed equality with dinaturality","authors":"Andrea Laretto, Fosco Loregian, Niccolò Veltri","doi":"arxiv-2409.10237","DOIUrl":"https://doi.org/arxiv-2409.10237","url":null,"abstract":"We show how dinaturality plays a central role in the interpretation of\u0000directed type theory where types are interpreted as (1-)categories and directed\u0000equality is represented by $hom$-functors. We present a general elimination\u0000principle based on dinaturality for directed equality which very closely\u0000resembles the $J$-rule used in Martin-L\"of type theory, and we highlight which\u0000syntactical restrictions are needed to interpret this rule in the context of\u0000directed equality. We then use these rules to characterize directed equality as\u0000a left relative adjoint to a functor between (para)categories of dinatural\u0000transformations which contracts together two variables appearing naturally with\u0000a single dinatural one, with the relative functor imposing the syntactic\u0000restrictions needed. We then argue that the quantifiers of such a directed type\u0000theory should be interpreted as ends and coends, which dinaturality allows us\u0000to present in adjoint-like correspondences to a weakening functor. Using these\u0000rules we give a formal interpretation to Yoneda reductions and (co)end\u0000calculus, and we use logical derivations to prove the Fubini rule for\u0000quantifier exchange, the adjointness property of Kan extensions via (co)ends,\u0000exponential objects of presheaves, and the (co)Yoneda lemma. We show\u0000transitivity (composition), congruence (functoriality), and transport\u0000(coYoneda) for directed equality by closely following the same approach of\u0000Martin-L\"of type theory, with the notable exception of symmetry. We formalize\u0000our main theorems in Agda.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266733","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}
This article concerns with the Axiom of Choice (AC) and the well-ordering theorem (WO) in second-order predicate logic with Henkin interpretation (HPL). We consider a principle of choice introduced by Wilhelm Ackermann (1935) and discussed also by David Hilbert and Ackermann (1938), by G"unter Asser (1981), and by Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro (2020). The discussion is restricted to so-called Henkin-Asser structures of second order. The language used is a many-sorted first-order language with identity. In particular, we give some of the technical details for a proof of the independence of WO from the so-called Ackermann axioms in HPL presented at the Colloquium Logicum in 2022.
本文关注具有亨金解释（HPL）的二阶谓词逻辑中的选择公理（AC）和井序定理（WO）。我们考虑了威廉-阿克曼（Wilhelm Ackermann，1935）提出的选择原则，大卫-希尔伯特和阿克曼（David Hilbert and Ackermann，1938）、本杰明-西斯金德（Benjamin Siskind）、保罗-曼科苏（Paolo Mancosu）和斯图尔特-夏皮罗（Stewart Shapiro，2020）也讨论了这一原则。讨论仅限于所谓的亨金-阿塞尔二阶结构。所使用的语言是多排序一阶语言，具有同一性。特别是，我们给出了在 2022 年逻辑学术讨论会（Colloquium Logicum）上提出的关于 WO 与 HPL 中所谓阿克曼公理的独立性证明的一些技术细节。
{"title":"AC and the Independence of WO in Second-Order Henkin Logic, Part I","authors":"Christine Gaßner","doi":"arxiv-2409.10276","DOIUrl":"https://doi.org/arxiv-2409.10276","url":null,"abstract":"This article concerns with the Axiom of Choice (AC) and the well-ordering\u0000theorem (WO) in second-order predicate logic with Henkin interpretation (HPL).\u0000We consider a principle of choice introduced by Wilhelm Ackermann (1935) and\u0000discussed also by David Hilbert and Ackermann (1938), by G\"unter Asser (1981),\u0000and by Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro (2020). The\u0000discussion is restricted to so-called Henkin-Asser structures of second order.\u0000The language used is a many-sorted first-order language with identity. In\u0000particular, we give some of the technical details for a proof of the\u0000independence of WO from the so-called Ackermann axioms in HPL presented at the\u0000Colloquium Logicum in 2022.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266730","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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