Investigations into the role of translations in abstract algebraic logic
[eng] This memoir is divided into two parts, devoted to two topics in (ab-stract) algebraic logic. In the first part we develop a hierarchy in which propositional logics “L” are classified according to the definability conditions enjoyed by the truth sets of the matrix semantics Mod* L. More precise...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2016 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/102248 |
| Acceso en línea: | https://hdl.handle.net/2445/102248 http://hdl.handle.net/10803/394028 |
| Access Level: | acceso abierto |
| Palabra clave: | Lògica algebraica Abstracció Deducció Algebraic logic Abstraction Deduction (Logic) |
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Investigations into the role of translations in abstract algebraic logicMoraschini, TommasoLògica algebraicaAbstraccióDeduccióAlgebraic logicAbstractionDeduction (Logic)[eng] This memoir is divided into two parts, devoted to two topics in (ab-stract) algebraic logic. In the first part we develop a hierarchy in which propositional logics “L” are classified according to the definability conditions enjoyed by the truth sets of the matrix semantics Mod* L. More precisely, we focus on conditions belonging to the conceptual framework of the Leibniz hierarchy, meaning that they can be characterized by means of the order-theoretic behaviour of the Leibniz operator. We study the class of logics such that truth is definable in Mod* L by means of universally quantified equations leaving one variable free. Then we study logics for which truth is implicitly definable in Mod* L and show that the injectivity of the Leibniz operator does not transfer in general from theories to filters over arbitrary algebras. Finally we consider an intermediate condition on the truth sets in Mod* L that corresponds to the order-reflection of the Leibniz operator. We conclude this part of the memoir by taking a computational glimpse to the Leibniz and Frege hierarchies. In the second part of this memoir we present an algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences. This correspondence provides a general explanation of the correspondence that appears in some well-known trans-lations between logics, e.g., Godel's translation of intuitionistic logic into the gobal modal logic 84 corresponds to the functor that takes an interior algebra to the Heyting algebra of its open elements and Kolmogorov's translation of classical logic into intuitionistic logic corresponds to the functor that takes a Heyting algebra to the Boolean algebra of its regular elements.Universitat de BarcelonaFont Llovet, Josep MariaJansana, RamonUniversitat de Barcelona. Departament de Lògica, Història i Filosofia de la Ciència2016info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/102248http://hdl.handle.net/10803/394028Tesis Doctorals - Departament - Lògica, Història i Filosofia de la Ciènciareponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaIngléscc-by-nc-nd, (c) Moraschini,, 2016http://creativecommons.org/licenses/by-nc-nd/3.0/info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1022482026-05-27T06:46:51Z |
| dc.title.none.fl_str_mv |
Investigations into the role of translations in abstract algebraic logic |
| title |
Investigations into the role of translations in abstract algebraic logic |
| spellingShingle |
Investigations into the role of translations in abstract algebraic logic Moraschini, Tommaso Lògica algebraica Abstracció Deducció Algebraic logic Abstraction Deduction (Logic) |
| title_short |
Investigations into the role of translations in abstract algebraic logic |
| title_full |
Investigations into the role of translations in abstract algebraic logic |
| title_fullStr |
Investigations into the role of translations in abstract algebraic logic |
| title_full_unstemmed |
Investigations into the role of translations in abstract algebraic logic |
| title_sort |
Investigations into the role of translations in abstract algebraic logic |
| dc.creator.none.fl_str_mv |
Moraschini, Tommaso |
| author |
Moraschini, Tommaso |
| author_facet |
Moraschini, Tommaso |
| author_role |
author |
| dc.contributor.none.fl_str_mv |
Font Llovet, Josep Maria Jansana, Ramon Universitat de Barcelona. Departament de Lògica, Història i Filosofia de la Ciència |
| dc.subject.none.fl_str_mv |
Lògica algebraica Abstracció Deducció Algebraic logic Abstraction Deduction (Logic) |
| topic |
Lògica algebraica Abstracció Deducció Algebraic logic Abstraction Deduction (Logic) |
| description |
[eng] This memoir is divided into two parts, devoted to two topics in (ab-stract) algebraic logic. In the first part we develop a hierarchy in which propositional logics “L” are classified according to the definability conditions enjoyed by the truth sets of the matrix semantics Mod* L. More precisely, we focus on conditions belonging to the conceptual framework of the Leibniz hierarchy, meaning that they can be characterized by means of the order-theoretic behaviour of the Leibniz operator. We study the class of logics such that truth is definable in Mod* L by means of universally quantified equations leaving one variable free. Then we study logics for which truth is implicitly definable in Mod* L and show that the injectivity of the Leibniz operator does not transfer in general from theories to filters over arbitrary algebras. Finally we consider an intermediate condition on the truth sets in Mod* L that corresponds to the order-reflection of the Leibniz operator. We conclude this part of the memoir by taking a computational glimpse to the Leibniz and Frege hierarchies. In the second part of this memoir we present an algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences. This correspondence provides a general explanation of the correspondence that appears in some well-known trans-lations between logics, e.g., Godel's translation of intuitionistic logic into the gobal modal logic 84 corresponds to the functor that takes an interior algebra to the Heyting algebra of its open elements and Kolmogorov's translation of classical logic into intuitionistic logic corresponds to the functor that takes a Heyting algebra to the Boolean algebra of its regular elements. |
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2016 |
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2016 |
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info:eu-repo/semantics/doctoralThesis info:eu-repo/semantics/publishedVersion |
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doctoralThesis |
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publishedVersion |
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https://hdl.handle.net/2445/102248 http://hdl.handle.net/10803/394028 |
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https://hdl.handle.net/2445/102248 http://hdl.handle.net/10803/394028 |
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Inglés |
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Inglés |
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cc-by-nc-nd, (c) Moraschini,, 2016 http://creativecommons.org/licenses/by-nc-nd/3.0/ info:eu-repo/semantics/openAccess |
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cc-by-nc-nd, (c) Moraschini,, 2016 http://creativecommons.org/licenses/by-nc-nd/3.0/ |
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openAccess |
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application/pdf |
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Universitat de Barcelona |
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Universitat de Barcelona |
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Tesis Doctorals - Departament - Lògica, Història i Filosofia de la Ciència reponame:Dipòsit Digital de la UB instname:Universidad de Barcelona |
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Universidad de Barcelona |
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Dipòsit Digital de la UB |
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Dipòsit Digital de la UB |
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