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2016 | OriginalPaper | Buchkapitel

From Paraconsistent Logic to Dialetheic Logic

verfasst von : Hitoshi Omori

Erschienen in: Logical Studies of Paraconsistent Reasoning in Science and Mathematics

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel Contradictoriness, Paraconsistent Negation and Non-intended Models of Classical Logic

Nächstes Kapitel Paradoxes of Expression

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Cf. [5, 30‐33].

Here, (Abs) and (Ext) are \(\exists y \forall x (x\in y \leftrightarrow B)\) and \(\forall x (x\in z\leftrightarrow x\in y)\rightarrow z=y\) respectively where B is any formula which does not contain y free, and \(\rightarrow \) and \(\leftrightarrow \) are suitable conditional and biconditional.

For an up-to-date survey on negation, see [16]. Note also that the following discussion focuses on the sentential negation since this is the key notion in the criteria for paraconsistent logics.

Cf. [17, p.38]. The notation of negation is adjusted.

Cf. [11, p.497]. The notation of negation is adjusted.

Note that we need the relativized truth and falsity conditions for modal logics, Nelson logics and relevant logics.

One may of course have some strong arguments against such a view on logic, and if that is the case, then the above expressivity requirement will not be substantial.

A simple way to see this is that ‘classical’ values are closed under the operations in LFI1, and thus the constant function mapping every argument to the intermediate value is not definable.

Note that connexive logics are not necessarily paraconsistent in general. But the idea imported in expanding LP relies on a kind of connexive logics that are also paraconsistent, and this is why I counted connexive logic as a tradition in paraconsistency.

See also [22] for a survey by Storrs McCall, one of the modern founders of connexive logics.

Note here that if expansions of BD is concerned, then not only that we obtain truth tables from truth and falsity conditions of relational semantics, but we can also go the other way around mechanically, namely to obtain truth and falsity conditions of relational semantics out of given any truth tables. For the details, see [27].

For an examination of the propagation of consistency in LFIs, see [29].

This was discovered after the first submission. I would like to thank Heinrich Wansing who informed me of Olkhovikov’s system, and Grigory Olkhovikov who sent me his paper and translated some of the results during a discussion.

For some discussions on classical negation in expansions of Belnap-Dunn logic, see [12].

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Titel: From Paraconsistent Logic to Dialetheic Logic
verfasst von: Hitoshi Omori
Verlag: Springer International Publishing
Buch: Logical Studies of Paraconsistent Reasoning in Science and Mathematics
Print ISBN: 978-3-319-40218-5

Electronic ISBN: 978-3-319-40220-8

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-40220-8_8

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