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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.

DOI: 10.1007/978-3-319-40220-8_8

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