Notre Dame Journal of Formal Logic

Limits for Paraconsistent Calculi

Walter A. Carnielli and João Marcos

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}$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.

Article information

Source
Notre Dame J. Formal Logic Volume 40, Number 3 (1999), 375-390.

Dates
First available in Project Euclid: 28 May 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1022615617

Mathematical Reviews number (MathSciNet)
MR1845624

Digital Object Identifier
doi:10.1305/ndjfl/1022615617

Zentralblatt MATH identifier
1007.03028

Subjects
Primary: 03B53: Paraconsistent logics

Citation

Carnielli, Walter A.; Marcos, João. Limits for Paraconsistent Calculi. Notre Dame J. Formal Logic 40 (1999), no. 3, 375--390. doi:10.1305/ndjfl/1022615617. http://projecteuclid.org/euclid.ndjfl/1022615617.


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