×
Carnielli, Walter A.; Marcos, João

Limits for paraconsistent calculi. (English) Zbl 1007.03028

Notre Dame J. Formal Logic 40, No. 3, 375-390 (1999).
Summary: 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 \({\mathcal C}_\omega\), 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 \({\mathcal C}_\omega\), 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 \({\mathcal C}_{\text{Lim}}\), 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 \({\mathcal C}_{\min}\).

Cited in 24 Documents

MSC:

03B53 Paraconsistent logics

Keywords:

deductive limits; possible-translation semantics; decision procedure; da Costa’s hierarchy
Cite Review PDF
Full Text: DOI

References:

[1] Alves, E. H., Logic and Inconsistency: A Study of the Calculi \(\mathcalC_n, 1\leq n\leq \omega\) (in Portuguese), thesis, State University of São Paulo, FFLCH, 1976.
[2] Alves, E. H., “Paraconsistent logic and model theory,” Studia Logica , vol. 43 (1984), pp. 17–32. · Zbl 0582.03014 · doi:10.1007/BF00935737
[3] Arruda, A. I., “Remarques sur les systèmes \(\mathcalC_n\),” Comptes Rendus de l’Academie de Sciences de Paris , Séries A-B (1975) t. 280, pp. 1253–6. · Zbl 0304.02011
[4] Baaz, M., “Kripke-type semantics for da Costa’s paraconsistent logic \(\mathcalC_\omega\),” The Notre Dame Journal of Formal Logic , vol. 27 (1986), pp. 523–27. · Zbl 0622.03013 · doi:10.1305/ndjfl/1093636764
[5] Blackburn, P., and M. de Rijke, “Zooming in, zooming out,” The Journal of Logic, Language, and Information , vol. 6 (1997), pp. 5–31. · Zbl 0872.03008 · doi:10.1023/A:1008204403391
[6] Carnielli, W. A., “Many-valued logics and plausible reasoning,” pp. 328–35 in Proceedings of the Twentieth International Congress on Many-Valued Logics , IEEE Computer Society, University of Charlotte, North Carolina, 1990.
[7] Carnielli, W. A., “Possible-translations semantics for paraconsistent logics,” pp. 149–63 in Frontiers in Paraconsistent Logic: Proceedings of the First World Congress on Paraconsistency, Ghent, 1998 , edited by D. Batens et al., King’s College Publications, 2000. · Zbl 1002.03525
[8] Carnielli, W. A., and M. E. Coniglio, “A categorial approach to the combination of logics,” Manuscrito , vol. 22 (1999), pp. 69–94.
[9] Carnielli, W. A., and I. M. L. D’Ottaviano, “Translations between logical systems: a manifesto,” Logique & Analyse , vol. 157 (1997), pp. 67–81. · Zbl 0982.03006
[10] Carnielli, W. A., and M. Lima-Marques, “Society semantics and multiple-valued logics,” pp. 33–52 in Advances in Contemporary Logic and Computer Science: Proceedings of the Eleventh Brazilian Conference on Mathematical Logic , edited by W. A. Carnielli and I. M. L. D’Ottaviano, Contemporary Mathematics, vol. 235, American Mathematical Society, 1999. · Zbl 0942.03034
[11] Carnielli, W. A., and J. Marcos, “A taxonomy of C -systems,” forthcoming in Paraconsistency: The Logical Way to the Inconsistent , Proceedings of the Second World Congress on Paraconsistency (WCP 2000), edited by W. A. Carnielli, M. E. Coniglio, and I. M. L. D’Ottaviano, Marcel Dekker, New York, 2001. · Zbl 1036.03022
[12] Carnielli, W. A., and J. Marcos, “Possible-translations semantics and dual logics,” forthcoming in Soft Computing . · Zbl 1002.03525
[13] da Costa, N. C. A., Inconsistent Formal Systems (in Portuguese), thesis, Universidade Federal do Paraná, 1963. Curitiba: Editora UFPR, 1993. · Zbl 0798.03025
[14] da Costa, N. C. A., “On the theory of inconsistent formal systems,” Notre Dame Journal of Formal Logic , vol. 15 (1974), pp. 497–510. · doi:10.1305/ndjfl/1093891487
[15] da Costa, N. C. A., and E. H. Alves, “A semantical analysis of the calculi \(\mathcalC_n\),” Notre Dame Journal of Formal Logic , vol. 18 (1977), pp. 621–30. · Zbl 0349.02022 · doi:10.1305/ndjfl/1093888132
[16] Jaśkowski, S., “Propositional calculus for contradictory deductive systems,” Studia Logica , vol. 24 (1969), pp. 143–57. · Zbl 0244.02004 · doi:10.1007/BF02134311
[17] Loparić, A., “A semantical study of some propositional calculi,” The Journal of Non-Classical Logic , vol. 3 (1986), pp. 73–95. · Zbl 0626.03017
[18] Loparić, A., and E. H. Alves, “The semantics of the systems \(\mathcalC_n\) of da Costa,” pp. 161–72 in Proceedings of the Third Brazilian Conference on Mathematical Logic , edited by A. I. Arruda, N. C. A. da Costa, and A. M. Sette, Sociedade Brasileira de Lógica, São Paulo, 1980. · Zbl 0448.03015
[19] Marcos, J., Possible-Translations Semantics (in Portuguese), thesis, State University of Campinas, IFCH, 1999, (ftp://www.cle.unicamp.br/pub/thesis/J.Marcos/).
[20] Mortensen, C., “Paraconsistency and \(\mathcalC_n\),” pp. 289–305 in Paraconsistent Logic: Essays on the Inconsistent , edited by G. Priest, R. Routley, and J. Norman, Philosophia Verlag, München, 1989. · Zbl 0692.03015
[21] Queiroz, G. S., On the Duality Between Intuitionism and Paraconsistency (in Portuguese), Ph.D. thesis, State University of Campinas, IFCH, 1997.
[22] Sette, A. M., “On the propositional calculus \(\mathcalP^1\),” Mathematica Japonicae , vol. 18 (1973), pp. 173–80. · Zbl 0289.02013
[23] Sette, A. M., and W. A. Carnielli, “Maximal weakly-intuitionistic logics,” Studia Logica , vol. 55 (1995), pp. 181–203. · Zbl 0841.03009 · doi:10.1007/BF01053037
[24] Sylvan, R., “Variations on da Costa \(\mathcalC\) systems and dual-intuitionistic logics I: Analyses of \(\mathcalC_\omega\) and \(\mathcalC\mathcalC_\omega\),” Studia Logica , vol. 49 (1990), pp. 47–65. · Zbl 0733.03016 · doi:10.1007/BF00401553
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.