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Duality Theory and Abstract Algebraic Logic

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Duality Theory and Abstract Algebraic Logic Duality Theory and Abstract Algebraic Logic María Esteban ADVERTIMENT. La consulta d’aquesta tesi queda condicionada a l’acceptació de les següents condicions d'ús: La difusió d’aquesta tesi per mitjà del servei TDX (www.tdx.cat) i a través del Dipòsit Digital de la UB (diposit.ub.edu) ha estat autoritzada pels titulars dels drets de propietat intel·lectual únicament per a usos privats emmarcats en activitats d’investigació i docència. No s’autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d’un lloc aliè al servei TDX ni al Dipòsit Digital de la UB. No s’autoritza la presentació del seu contingut en una finestra o marc aliè a TDX o al Dipòsit Digital de la UB (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. En la utilització o cita de parts de la tesi és obligat indicar el nom de la persona autora. ADVERTENCIA. La consulta de esta tesis queda condicionada a la aceptación de las siguientes condiciones de uso: La difusión de esta tesis por medio del servicio TDR (www.tdx.cat) y a través del Repositorio Digital de la UB (diposit.ub.edu) ha sido autorizada por los titulares de los derechos de propiedad intelectual únicamente para usos privados enmarcados en actividades de investigación y docencia. No se autoriza su reproducción con finalidades de lucro ni su difusión y puesta a disposición desde un sitio ajeno al servicio TDR o al Repositorio Digital de la UB. No se autoriza la presentación de su contenido en una ventana o marco ajeno a TDR o al Repositorio Digital de la UB (framing). Esta reserva de derechos afecta tanto al resumen de presentación de la tesis como a sus contenidos. En la utilización o cita de partes de la tesis es obligado indicar el nombre de la persona autora. WARNING. On having consulted this thesis you’re accepting the following use conditions: Spreading this thesis by the TDX (www.tdx.cat) service and by the UB Digital Repository (diposit.ub.edu) has been authorized by the titular of the intellectual property rights only for private uses placed in investigation and teaching activities. Reproduction with lucrative aims is not authorized nor its spreading and availability from a site foreign to the TDX service or to the UB Digital Repository. Introducing its content in a window or frame foreign to the TDX service or to the UB Digital Repository is not authorized (framing). Those rights affect to the presentation summary of the thesis as well as to its contents. In the using or citation of parts of the thesis it’s obliged to indicate the name of the author. DUALITY THEORY AND ABSTRACT ALGEBRAIC LOGIC Mar´ıa Esteban PhD program: Pure and Applied Logic Supervisors: Prof. dr. Ramon Jansana and Prof. dr. Sergio A. Celani Department of Logic, History and Philosophy of Science Faculty of Philosophy Mar´ıaEsteban: Duality theory and Abstract Algebraic Logic, September 2013 supervisors: Prof. dr. Ramon Jansana Prof. dr. Sergio A. Celani committee members: Prof. dr. Josep Maria Font i Llovet Prof. dr. Mai Gehrke Prof. dr. Alessandra Palmigiano This research was supported by the Spanish Ministry of Education, Culture and Sport (FPU Program) grant AP2008-03838 A mis padres y a mis hermanos A Juan Abstract In this thesis we present the results of our research on duality theory for non- classical logics under the point of view of Abstract Algebraic Logic. Firstly, we propose an abstract Spectral-like duality and an abstract Priestley-style duality for every filter distributive finitary congruential logic with theorems. This proposal aims to unify the various dualities for concrete logics that we find in the literature, by showing the abstract template in which all of them fit. Secondly, the dual correspondence of some logical properties is examined. This serves to reveal the connection between our abstract dualities and the concrete dualities related to concrete logics. We apply those results to get new dualities for suitable expansions of a well-known logic: the implicative fragment of intuitionistic logic. Finally, we develop a new technique that can be modularly applied to simplify some of the obtained dualities. Resumen En esta tesis presentamos los resultados de nuestra investigaci´onacerca de la teor´ıade la dualidad para l´ogicasno cl´asicasdesde el punto de vista de la L´ogica Algebr´aicaAbstracta. En primer lugar, proponemos una dualidad abstracta de tipo espectral y otra dualidad abstracta de tipo Priestley para cada l´ogicacongru- encial, filtro distributiva, finitaria y con teoremas. Esta propuesta pretende unificar las distintas dualidades para l´ogicasno cl´asicasque encontramos en la literatura, mostrando el esquema abstracto en el que todas ellas encajan. En segundo lugar, la correspondencia dual de algunas propiedades l´ogicases examinada. Esto sirve para revelar la conexi´onque existe entre nuestras dualidades abstractas y las du- alidades concretas relacionadas con l´ogicasconcretas. Aplicamos estos resultados para obtener nuevas dualidades para expansiones apropiadas de una l´ogicabien conocida: el fragmento implicativo de la l´ogicaintuicionista. Finalmente, desarrol- lamos una nueva t´ecnicaque puede ser aplicada de forma modular para simplificar algunas de las dualidades obtenidas. v Acknowledgments I would like to express my deepest gratitude to my supervisor, Ramon Jansana, for his excellent guidance, caring, patience, and providing me with an excellent atmosphere for doing research, not to mention his unsurpassed knowledge of duality theory and Abstract Algebraic Logic. The good advice, support and friendship of my second supervisor, Sergio A. Celani, has been invaluable on both an academic and a personal level, for which I am extremely grateful. I would also like to express my gratitude to the members of my thesis com- mittee: Josep Mar´ıaFont, Mai Gehrke and Alessandra Palmigiano. Thank you very much for having accepted the task of critically reading this thesis. I have also profited from discussion with the members of the Barcelona research group on Non-classical logics, and with audiences in Oxford, Sevilla, Nashville and Orange. Thanks to a Master fellowship from Obra Social Caja Espa~na,I could come to study the Master in Barcelona, and thanks to a FPU doctoral grant from the spanish Ministry of Education, Culture and Sport, I could stay in Barcelona four more years and finish my PhD. During these years I met a lot of people along the way, that made that the task that lay ahead was more enjoyable. This dissertation could not have been written without the support and friendship found at Barcelona. Pablo welcomed me so charming at the very beginning and Marta and Mar´ıahelped me a lot at the very end. Nuria and Manel provided me with a home, and it was so nice to have Bernat and Nuno as neighbors. I will miss so much lunch at el office with Luz, Pepe, Viviane, Blanca, Sergi, Luciano, Daniel, Gon¸calo,...I would like to thank you all for being there, and also the people of the Logic Team and of Logos. My research has gotten a great benefit from the travels I have undertaken. Apart from the conferences I attended, during my PhD I have made three stays abroad that have been very profitable from both an academic and a personal per- spective. The first one was at the ILLC in Amsterdam during fall 2010, under the supervision of the professor Alessandra Palmigiano. The second one was in Tandil during fall 2011, under the supervision of my co-supervisor Sergio A. Celani. The third one was in Paris during winter 2013, under the supervision of the professor Mai Gehrke. I feel very lucky to have had the opportunity of working in such diffe- rent environments (philosophy, mathematics and computer science). Any student who has the opportunity should take advantage of it. In Amsterdam (and also later on in Barcelona) I had the opportunity of dis- cussing with Alessandra, whose work has been very inspiring to this dissertation. I could also feel the great atmosphere of the ILLC, and I met many people that made the stay so great. I would like to thank all of them, and specially Margot. In Tandil (and also later on in Barcelona) I had the pleasure of working with Sergio, that also shared with me asados and walks. Ramon, Ismael, Felix and Carles helped to make the stay nicer. In Paris I had the chance of discussing and working with Mai, whose enthusiasm and devotion about mathematics makes any conversation with her amazing. I also enjoyed a lot the discussions with Sam, that definitely vii transmits the same passion about what he does. Besides, Margot and my flatmates at T´el´egraphemade so pleasant my stay there. I would like to acknowledge the financial, academic and technical support of the University of Barcelona and its staff, specially Pablo R´ıo,Oscar Cabaco and Paco Murcia, who were always willing to help me. Luckily I could count on the help of logalg, thank you very much to all who make it possible, notably to Felix Bou. I would also like to thank the online LATEX community that has made the actual process of writing and typesetting this dissertation so much easier, by developing the software and helping newbies like me to understand and benefit from them. Por ´ultimo,querr´ıadar las gracias a mi familia y amigos. Especialmente a mi t´ıaVirginia, porque me ayud´oenormemente al leer y revisar algunas partes del manuscrito. Tambi´ena mis padres, Antonio y Visi, porque siempre han cre´ıdoen m´ıy me han ayudado a alcanzar mis metas.
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