Formal inconsistency and evolutionary databases
DOI:
https://doi.org/10.12775/LLP.2000.008Abstract
This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning.References
A. Avron, “On an implication connective of RM,” Notre Dame Journal of Formal Logic 27(2), 201–209, 1986.
D. Batens, “Paraconsistent extensional propositional logics,” Logique et Analyse 90–91, 195–234, 1980.
D. Batens, K. De Clercq and N. Kurtonina, “Embedding and interpolation for some paralogics. The propositional case,” Reports on Mathematical Logic 33, 29–44, 1999.
N. D. Belnap, “A useful four-valued logic,” pp. 7–37 in Modern Uses of Multiple-Valued Logic, J.M. Dunn and G. Epstein (Eds.). D. Reidel Publishing Co, Boston, 1977.
J.-Y. Béziau, “The paraconsistent logic Z. A possible solution to Jaśkowski’s problem,” submitted to publication.
J.-Y. Béziau, “S5 is a paraconsistent logic and so is classical first-order logic,” submitted to publication.
W.A. Carnielli, “Systematization of finite many-valued logics through the method of tableaux,” The Journal of Symbolic Logic 52(2), 473–493, 1987.
W.A. Carnielli, “Possible-translations semantics for paraconsistent logics,” pp. 149–63 in Frontiers in Paraconsistent Logic: Proc. of the I World Congress on Paraconsistency, Ghent, Belgium, D. Batens et alia (Eds.), King’s College Publications, 2000.
W.A. Carnielli, W.A. and I.M.L. D’Ottaviano, “Translations between logical systems: a manifesto,” Logique et Analyse 157, 67–81, 1997.
W.A. Carnielli and S. de Amo, “A logic-based system for controlling inconsistencies in evolutionary databases,” pp. 89–101 in Proc. of the VI Workshop on Logic, Language, Information and Computation (WoLLIC’99), Itatiaia, Brazil, 1999.
W.A. Carnielli and M. Lima-Marques, “Reasoning under inconsistent knowledge,” Journal of Applied Non-Classical Logics 2(1), 49–79, 1992.
W.A. Carnielli and J. Marcos, “Limits for paraconsistent calculi,” Notre Dame Journal of Formal Logic, 40(3), 375–390, 1999.
W.A. Carnielli and J. Marcos, “A taxonomy of C-systems,” in W.A. Carnielli, M.E. Coniglio, and I.M.L. D’Ottaviano, editors, Paraconsistency: The logical way to the inconsistent. Proceedings of the II World Congress on Paraconsistency (WCP’2000), pp. 1–94. Marcel Dekker, 2001.
W.A. Carnielli and J. Marcos, “Semantics for C-systems,” forthcoming.
E.F Codd, “A relational model of data for large shared data banks,” Communications of the ACM 13(6), 377–387, 1970.
S. de Amo, W.A. Carnielli, and J. Marcos, “A logical framework for integrating inconsistent information in multiple databases,” to appear in Proceedings of the Second International Symposium on Fundations of Information and Knowledge Systems (FoIKS 2002), Schloß Sulzan, Germany, February 2002.
N.C.A. da Costa, “On the theory of inconsistent formal systems,” Notre Dame Journal of Formal Logic 11, 497–510, 1974.
N.C.A. da Costa, J.-Y. Béziau and O. Bueno, “Aspects of paraconsistent logic,” Bulletin of the IGPL 3(4), 597–614, 1995.
J.J. da Silva, I.M.L. D’Ottaviano, and A.M. Sette, “Translations between logics,” pp. 435–48 in Models, Algebras and Proofs: Proceedings of the X Latin American Symposium on Mathematical Logic, X. Caicedo, C.H. Montenegro (Ed.), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1998.
I.M.L. D’Ottaviano, “The completeness and compactness of a three-valued first-order logic”, in Proceedings of the V Latin American Symposium on Mathematical Logic, Bogotá, 1981. Revista Colombiana de Matemáticas 19(1–2), 77–94, 1985.
I.M.L. D’Ottaviano and N.C.A. da Costa, “Sur un probleme de Jaśkowski,” Comptes Rendus de l’Academie de Sciences de Paris sér. A–B 270, 1349–1353, 1970.
M. Fitting, “Bilattices and the semantics of logic programming,” Journal of Logic Programming 11, 91–116, 1991.
S. Jaśkowski, “Rachunek zdan dla systemów dedukcyjnych sprzecznych,” Studia Soc. Scient. Torunensis sec. A1(5), 55–77, 1948 (English translation in Studia Logica 24, 143–157, 1969.)
A.S. Karpenko, “A maximal paraconsistent logic: the combination of two three-valued isomorphs of classical propositional logic,” pp. 181–187 in Frontiers in Paraconsistent Logic: Proc. of the I World Congress on Paraconsistency, Ghent, Belgium, D. Batens et alia (Eds.), King’s College Publications, 2000.
M. Kifer and E.L. Lozinskii, “A logic for reasoning with inconsistency,” Journal of Automated Reasoning 9(2), 179–215, 1992.
J. Marcos, Possible-Translations Semantics (in Portuguese), Thesis, State University of Campinas, IFCH, 1999. (ftp://www.cle.unicamp.br/pub/thesis/J.Marcos/)
J. Marcos, “8K solutions and semi-solutions to a problem of da Costa,” forthcoming.
G. Priest, R. Routley, and J. Norman (Eds.), Paraconsistent Logic: essays on the inconsistent, Philosophia Verlag, München, 1989.
N. Rescher and R. Manor, “On inference from inconsistent premises,” Theory and Decision 1, 179–219, 1970.
A.M. Sette, “On the propositional calculus P 1 ,” Mathematica Japonicae 18, 173–80, 1973.
B.A. Trakhtenbrot, “The impossibility of an algorithm for the decision problem for finite domains” (in Russian), Doklady Akademii Nauk SSSR (N.S.) 70, 569–572, 1950.
Downloads
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 586
Number of citations: 0
