Formal inconsistency and evolutionary databases

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.

[1]  N. Rescher,et al.  On inference from inconsistent premisses , 1970 .

[2]  A. Sette,et al.  ON THE PROPOSITIONAL CALCULUS P1. , 1973 .

[3]  Newton C. A. da Costa,et al.  On the theory of inconsistent formal systems , 1974, Notre Dame J. Formal Log..

[4]  Nuel D. Belnap,et al.  A Useful Four-Valued Logic , 1977 .

[5]  D. Batens.  Paraconsistent extensional propositional logics , 1980 .

[6]  I. L. D'Ottaviano.  The completeness and compactness of a three-valued first-order logic , 1985 .

[7]  Arnon Avron,et al.  On an implication connective of RM , 1986, Notre Dame J. Formal Log..

[8]  Walter Alexandre Carnielli,et al.  Systematization of finite many-valued logics through the method of tableaux , 1987, Journal of Symbolic Logic.

[9]  Richard L. Epstein.  Translations Between Logics , 1990 .

[10]  Lorenzo Peña,et al.  Paraconsistent logic: essays on the inconsistent , 1990 .

[11]  Melvin Fitting,et al.  Bilattices and the Semantics of Logic Programming , 1991, J. Log. Program..

[12]  Mamede Lima-Marques,et al.  Reasoning under Inconsistent Knowledge , 1992, J. Appl. Non Class. Logics.

[13]  Newton C. A. da Costa,et al.  Aspects of Paraconsistent Logic , 1995, Log. J. IGPL.

[14]  X. Caicedo.  X Latin American Symposium on Mathematical Logic , 1996, Bulletin of Symbolic Logic.

[15]  Walter Carnielli,et al.  Translations between logical systems: a manifesto , 1997 .

[16]  Walter Alexandre Carnielli,et al.  Limits for Paraconsistent Calculi , 1999, Notre Dame J. Formal Log..

[17]  Diderik Batens,et al.  Embedding and Interpolation for some Paralogics. The Propositional Case , 1999, Reports Math. Log..

[18]  F. E.,et al.  A Relational Model of Data Large Shared Data Banks , 2000 .

[19]  S5 is a Paraconsistent Logic and so is Classical First-Order Logic , 2000 .

[20]  João Marcos.  8K solutions and semi-solutions to a problem of da Costa , 2000 .

[21]  W. Carnielli,et al.  A Taxonomy of C-systems , 2001 .

[22]  Walter Alexandre Carnielli,et al.  A Logical Framework for Integrating Inconsistent Information in Multiple Databases , 2002, FoIKS.

[23]  Jean-Yves Béziau,et al.  S5 is a paraconsistent logic and so is first-order classical logic , 2002 .

[24]  Itala M. Loffredo D'Ottaviano,et al.  Paraconsistency: The Logical Way to the Inconsistent , 2002 .

[25]  Michael Kifer,et al.  A logic for reasoning with inconsistency , 1992, Journal of Automated Reasoning.