From Paraconsistent Logic to Dialetheic Logic
The only condition for a logic to be paraconsistent is to invalidate the so-called explosion. However, the understanding of the only connective involved in the explosion, namely negation, is not shared among paraconsistentists. By returning to the modern
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Abstract The only condition for a logic to be paraconsistent is to invalidate the so-called explosion. However, the understanding of the only connective involved in the explosion, namely negation, is not shared among paraconsistentists. By returning to the modern origin of paraconsistent logic, this paper proposes an account of negation, and explores some of its implications. These will be followed by a consideration on underlying logics for dialetheic theories, especially those following the suggestion of Laura Goodship. More specifically, I will introduce a special kind of paraconsistent logic, called dialetheic logic, and present a new system of paraconsistent logic, which is dialetheic, by expanding the Logic of Paradox of Graham Priest. The new logic is obtained by combining connectives from different traditions of paraconsistency, and has some distinctive features such as its propositional fragment being Post complete. The logic is presented in a Hilbert-style calculus, and the soundness and completeness results are established.
Date: March 16, 2016. The author was a Postdoctoral Fellow for Research Abroad of the Japan Society for the Promotion of Science (JSPS) at the time of submission, and now a Postdoctoral Research Fellow of JSPS. I would like to thank Holger Andreas and Peter Verdée for their encouragement and patience. I would also like to thank Diderik Batens, Filippo Casati, Petr Cintula, Michael De, Graham Priest, Dilip Raghavan, Greg Restall, Daniel Skurt, Heinrich Wansing and Zach Weber for their valuable suggestions, comments and discussions. Earlier versions of the paper were presented to conference Paraconsistent Reasoning in Science and Mathematics in Munich, Workshop on Non-Classical [Meta]Mathematics in Otago and seminars in Ghent, Munich, Singapore and Melbourne, and many thanks go to organizers and audiences. Finally, I would like to thank the referees for their helpful comments and suggestions which substantially improved the paper. H. Omori The Graduate Center, City University of New York, New York, NY, USA H. Omori (B) Department of Philosophy, Kyoto University, Kyoto, Japan e-mail: [email protected] URL: https://sites.google.com/site/hitoshiomori/home © Springer International Publishing AG 2016 H. Andreas and P. Verdée (eds.), Logical Studies of Paraconsistent Reasoning in Science and Mathematics, Trends in Logic 45, DOI 10.1007/978-3-319-40220-8_8
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1 Introduction Dialetheism is the metaphysical view, not restricted to any of the specific topics, that some contradictions are true. At first sight, dialetheism looks not tenable at all, due to the popular view that no contradictions are true, which is based on the Law of Non-Contradiction since Aristotle. However, some philosophers, such as Jc Beall, Graham Priest and Richard Routley (later Sylvan), have challenged the Law of Non-Contradiction, and defended dialetheism.1 In this paper, I wish to pave the path towards satisfactory examination of dialetheism in the context of foundations of mathematics which is on
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