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Vol. 5(1), 2005 (Section Logic)
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Logics of Formal Inconsistency
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Walter Carnielli
Personal web-page
Marcelo E. Coniglio
Personal web-page
João Marcos
Personal web-page
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Date Posted: February, 22nd 2005
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Revised Version - July 3rd, 2007   [PDF]   [PS]
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The final version of this paper has been published
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Carnielli, W., Coniglio, M.E., and Marcos, J., Logics of Formal Inconsistency. Handbook of Philosophical Logic, 2nd edition, volume 14, pages 1-93. Springer-Verlag, 2007.


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.Abstract
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The Logics of Formal Inconsistency (LFIs) are logical systems that treat consistency  and inconsistency as mathematical objects. These logics permit the internalization of  the notions of consistency and inconsistency at the object-language level, resulting  in very expressive logical systems whose fundamental feature is the ability of recovering all consistent reasoning, while still allowing for some inconsistency to be represented without leading to deductive trivialization.
The paper defines and exemplifies LFIs in all detail, expliciting the concepts and definitions dealing with the property of explosion and showing how this reflects on the principles of logic.
The C-systems are introduced as the subclass of LFIs where consistency can be expressed as a unary connective. Further, the dC-systems are introduced as the C-systems in which the consistency connective is explicitly definable  in terms of other usual connectives. Particular cases of dC-systems are da Costa's logics Cn, 1 £ nw, Jaskowski's logic D2, and most normal modal logics under convenient formulation.
Starting from a fundamental example of LFI, the logic mbC, we show how to introduce a large family of logics by controlling the propagation of consistency, clarifying a procedure  that allows one to define tailor-suited LFIs. The paper dedicates a good deal of attention to semantical tools (valuations and possible-translations semantics), showing several  results about uncharacterizability by finite matrices, and also discusses some possibilities of algebraizing LFIs. Some perspectives on the research about LFIs including applications to such diverse topics as epistemic paradoxes and database theory are also discussed.
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