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Summer 1999 Limits for Paraconsistent Calculi
Walter A. Carnielli, João Marcos
Notre Dame J. Formal Logic 40(3): 375-390 (Summer 1999). DOI: 10.1305/ndjfl/1022615617

Abstract

This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $ \mathcal {C}$n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$$\scriptstyle \omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is to define the deductive limit to this hierarchy, that is, its greatest lower deductive bound. The calculus $ \mathcal {C}$min, stronger than $ \mathcal {C}$$\scriptstyle \omega$, is first presented as a step toward this limit. As an alternative to the bivaluation semantics of $ \mathcal {C}$min presented thereupon, possible-translations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possible-translations semantics are then used to provide both a semantics and a decision procedure for $\mathcal{C}_{\rm Lim}$, the real deductive limit of da Costa's hierarchy. Possible-translations semantics also make it possible to characterize a precise sense of duality: as an example, $ \mathcal {D}$min is proposed as the dual to $ \mathcal {C}$min.

Citation

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Walter A. Carnielli. João Marcos. "Limits for Paraconsistent Calculi." Notre Dame J. Formal Logic 40 (3) 375 - 390, Summer 1999. https://doi.org/10.1305/ndjfl/1022615617

Information

Published: Summer 1999
First available in Project Euclid: 28 May 2002

zbMATH: 1007.03028
MathSciNet: MR1845624
Digital Object Identifier: 10.1305/ndjfl/1022615617

Subjects:
Primary: 03B53

Rights: Copyright © 1999 University of Notre Dame

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Vol.40 • No. 3 • Summer 1999
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