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Interactions of Metaphysical and Epistemic Concepts Interactions of metaphysical and epistemic concepts Th`esepr´esent´ee`ala Facult´edes lettres et sciences humaines Universit´ede Neuchˆatel- CH Pour l’obtention du grade de docteur `eslettres Par Alexandre Fernandes Batista Costa Leite Accept´eesur proposition du jury: Prof. Jean-Yves B´eziau, Universit´ede Neuchˆatel, directeur de th`ese Prof. Pascal Engel, Universit´ede Gen`eve, rapporteur Prof. Paul Gochet, Universit´ede Li`ege,rapporteur Prof. Arnold Koslow, The City University of New York, rapporteur Soutenue le 03 Juillet 2007 Universit´ede Neuchˆatel 2007 lnlTT Faculté des lettres et sciences humaines Le doyen r EspaceLouis-Agassiz 1 I CH-2000Neuchâtel IMPRIMATUR La Facultédes lettreset scienceshumaines de I'Universitéde Neuchâtel,sur les rapportsde Mon- sieurJean-Yves Béziau, directeur de thèse,professeur assistant de psychologieà I'Université de Neu- châtel; M. PascalEngel, professeur à I'Universitéde Genève; M. Paul Gochet,professeur à I'Universitéde Liège;M. ArnoldKoslow, professeur à CityUniversity of NewYork autorise I'impres- sionde la thèseprésentée par Monsieur Alexandre Costa Leite, en laissantà I'auteurla responsabilité desopinions énoncées. .\A \.\ i \,^_ Neuchâtel,le 3 juillet2007 Le doyen J.-J.Aubert r Téléphone'.+41 32718 17 0O r Fax: +4132718 17 01 . E-mail: [email protected] www.unine.ch/lettres Abstract This work sets out the results of research on topics at the intersection of logic and philosophy. It shows how methods for combining logics can be applied to the study of epistemology and metaphysics. In a broader per- spective, it investigates interactions of modal concepts in order to create a bridge between metaphysics and epistemology. The first study of such in- teractions offers an analysis of a paradox in the limits of knowledge from the viewpoint of fusions and products of modal logics. We next examine the formal combination of an epistemic logic of knowledge with a contingency logic in order to determine whether contingent propositions can be known. The third example investigates a combination of logics for belief and con- tingency logics to determine in what sense the world is an object of belief. The work also studies contradictions in the context of modal operators and their relations with paraconsistent logics. In the final section, some gen- eral approaches to modalities are studied and some directions are presented. The concept of paraconsistentization of theories and logics, as well the idea of philosophical categorification are presented from an abstract viewpoint. The work presents some tools to deal with modalities and proposes some concepts for exploring problems in philosophy and logic. Mot cl´esen francais: combinaisons, connaissance, croyance, contin- gence, paraconsistentization, categorification philosophique Mot cl´esen anglais: combinations, knowledge, belief, contingency, paraconsistentization, philosophical categorification PhD Thesis: Interactions of metaphysical and epistemic concepts Research Areas: Philosophy and Logic Author: Alexandre Fernandes Batista Costa Leite Supervisor: Jean-Yves B´eziau Facult´edes lettres et sciences humaines Universit´ede Neuchˆatel July 2007 Work supported by a grant of the Swiss National Science Foundation To Confoederatio Helvetica CH Contents Introduction 1 1 Plug-ins: possibility and knowledge 28 1.1 Fusions of modal logics and Fitch’s paradox . 35 1.1.1 ChurchorFitch?..................... 36 1.1.2 The verification principle . 36 1.1.3 The collapse principle . 38 1.1.4 The nature of Fitch’s paradox . 38 1.1.5 Somanypeople...................... 45 1.2 Other formulations . 47 1.2.1 Philosophical foundations of the paradox . 48 1.3 Products of modal logics and the limits of knowledge . 50 1.3.1 The logic for knowability . 52 1.3.2 Combined modality? . 57 1.4 Imagination and possible knowledge . 58 2 Interplays: knowledge and contingency 63 2.1 Logical skepticism . 65 2.2 Combining knowledge and contingency . 70 2.2.1 Two notions of contingency . 71 2.2.2 Contingency logics . 74 2.2.3 Mixing knowledge and contingency . 78 2.3 Logical skepticism revisited . 90 2.3.1 Knowledge and reality . 90 2.3.2 Two basic criteria . 91 2.4 Fusions, skepticism and the border of concepts . 98 3 Connections: belief and contingency 100 3.1 Shaking knowledge and belief . 102 ii 3.2 Different conceptions of belief . 105 3.3 Interactions of knowledge and belief . 110 3.3.1 The knowledge-belief reduction . 115 3.4 Combining belief and contingency . 118 3.5 Fusions, skepticism and the border of concepts II . 123 4 Combinations: contradictions, contingency, knowledge and belief 125 4.1 Can contradictions be object of knowledge and belief? . 130 4.2 Paraconsistentization of logics . 134 4.2.1 The case of modal logics . 137 4.2.2 The problem of a paraconsistent negation . 141 4.2.3 Desparaconsistentization of logics? . 143 4.3 Paraconsistentization of classical propositional logic . 143 4.3.1 Fusions and paraconsistentization . 149 4.4 Paraconsistentizing fusions . 151 5 Network: a universal approach to concepts 154 5.1 Philosophical categorification . 157 5.1.1 A categorial approach . 157 5.1.2 Logic applied to philosophy . 158 5.1.3 Category theory and philosophy . 162 5.1.4 Categories and logics . 163 5.1.5 Categories and philosophy . 165 5.1.6 A system of philosophy based on category theory? . 167 5.2 The universal approach to modalities . 167 5.2.1 Structuralist theory of modalities . 168 5.2.2 The n-dimensional modal logics . 168 5.2.3 The n-opposition theory . 169 5.3 The duality of reality . 173 Conclusion 176 History and Interview 182 Bibliography 192 iii Acknowledgements This work reflects actions of a great variety of people and my interactions with everybody. Writing a PhD thesis is difficult; we need original ideas and some basic results. I would like to thank professor Jean-Yves B´eziauwho invited me to Switzerland, and who has helped me in the development of my ideas for a long time. His philosophy and way of thinking have been important to my own developments, and I am especially happy because Jean-Yves B´eziau allowed me to write this document developing my own ideas and contribu- tions. I have no doubt that he is one of the most important researchers in logic and philosophy working nowadays; he is deeply contributing to both areas. To Jean-Yves B´eziau,my eternal respect and gratitude. The grant of the Swiss National Science Foundation has been a necessary condition for the development of this work. Without this grant, the ideas developed here would not exist. Thanks to the members of the Jury: Prof. Arnold Koslow, Prof. Paul Gochet and Prof. Pascal Engel. They helped me a lot with their suggestions and corrections. I also would like to thanks all my Logic teachers: Wagner de Campos Sanz, Adriano Naves de Brito, Marcelo Coniglio, Walter Carnielli and Itala D’Ottavianno. With them, I learned the exact value of logic in philosophy. This is a list of people that I will be forever indebt for helping me in many different aspects: Saul Kripke (for coming to UNILOG’05), Carlos Caleiro (for accepting me in Lisbon), Maarten Marx (for accepting me in Amsterdam), Charles Alunni and Andrei Rodin (for accepting me in Paris), Eva Gl¨uck, C´ecileGuex-Joris (my beloved girlfriend), and people from the Institute of Logic at the University of Neuchˆatel: Prof. Denis Mi´eville, Nadine Gessler, Alessandro Facchini, C´edricD´egrange, Pierre Joray and Christiane Tripet. Thanks also to Prof. Anne-Nelly Perret-Clermont, from the Institute of Psychology at UniNE for her useful help in a very difficult moment. Another list of people who I am indebted can be found in the iv section History and Interview. Thanks also to many friends that I knew while writing this thesis, friends who will never be forgotten: Dietrich Choffat (my Swiss-Brazilian friend in Neuchˆatel),Ale Merch´an(el poeta maldito), Fabien Schang (who wrote a book where I am a character) and Alessio Moretti. Thanks to my logically-mathematically-philosophically-oriented colleagues Michael Wright (Mike), Mark Jago and Casey McGinnis for helping in the improvement of the English language of this book. It is not easy to write a book in a language different from one’s native language. Thanks to my family and God, for always protecting me against the flowing of reality. v Introduction “All men by nature desire to know. An indication of this is the delight we take in our senses; for even apart from their usefulness they are loved for themselves...” (ARISTOTLE in [2]) This book deals with concepts. It is a study on the interactions of metaphysical and epistemic modalities (i.e a particular family of concepts), and the relations between the notions of knowledge and belief, and the further connection of these concepts with the notions of possibility and contingency. Thus it deals with one aspect of the relations between metaphysics and epistemology. Below the reader find some explanations of its main objectives and purposes. At the end of this opening section, a general theory of the combination of logics is proposed. This serves to introduce the main topic of the work. Interactions This thesis started as an attempt to defend skepticism. That the world is not an object of knowledge but just the object of belief is a familiar philosophical thesis. This work aims to provide logical foundations for an account of the interactions of different modalities. The study of such interactions starts in old problems such as induction, the contingency of the world, and the epistemic status of agents in relation to such contingency. Inconsistencies in the borders of knowledge and belief are investigated, as is a universal logical framework for dealing with modalities. The concept of interaction, which appears in the title of the work, refers here to a logical formalism in which non-interdefinable modalities appear in the same formula.
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