Abstract
It is well-known that knowledgebases may contain inconsistencies. We provide a measure to quantify the inconsistency of a knowledgebase, thereby allowing for the comparison of the inconsistency of various knowledgebases, represented as first-order logic formulas. We use quasi-classical (QC) logic for this purpose. QC logic is a formalism for reasoning and analysing inconsistent information. It has been used as the basis of a framework for measuring inconsistency in propositional theories. Here we extend this framework, by using a first-order logic version of QC logic for measuring inconsistency in first-order theories. We motivate the QC logic approach by considering some formulae as database or knowledgebase integrity constraints. We then define a measure of extrinsic inconsistency that can be used to compare the inconsistency of different knowledgebases. This measure takes into account both the language used and the underlying domain. We show why this definition also captures the intrinsic inconsistency of a knowledgebase. We also provide a formalization of paraconsistent equality, called quasi-equality, and we use this in an extended example of an application for measuring inconsistency between heterogeneous sources of information and integrity constraints prior to merging.
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References
Arieli, O., & Avron, A. (1998). The value of the four values. Artificial Intelligence, 102, 97–141.
Baral, C., Kraus, S., Minker, J., & Subrahmanian, V. (1992). Combining knowledgebases of first-order theories. Computational Intelligence, 8, 45–71.
Belnap, N. (1977). A useful four-valued logic. In G. Epstein (Ed.), Modern uses of multiple-valued logic (pp. 8–37). Reidel.
Bertossi, L., & Chomicki, J. (2003). Query answering in inconsistent databases. In G. Saake, J. Chomicki, & R. van der Meyden (Eds.), Logics for emerging applications of databases. Springer.
Besnard, P., & Hunter, A. (1995). Quasi-classical logic: Non-trivializable classical reasoning from inconsistent information. In Symbolic and quantitative approaches to uncertainty, vol. 946 of LNCS (pp. 44–51).
Carnielli, W., & Marcos, J. (2002). A taxonomy of C systems. In Paraconsistency: The Logical Way to the Inconsistent (pp. 1–94). Marcel Dekker.
da Costa, N. C. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15, 497–510.
Dubois, D., Lang, J., & Prade, H. (1994). Possibilistic logic. In Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3 (pp. 439–513). Oxford University Press.
Grant, J. (1978). Classifications for inconsistent theories. Notre Dame Journal of Formal Logic, 19, 435–444.
Grant, J., & Subrahmanian, V. S. (2000). Applications of paraconsistency in data and knowledge bases. Synthese, 125, 121–132.
Halevy, A. (2001). Answering queries using views: A survey. VLDB Journal, 10(4), 270–294.
Hunter, A. (1998). Paraconsistent logics. In Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 2 (pp. 11–36). Kluwer.
Hunter, A. (2000a). Reasoning with conflicting information using quasi-classical logic. Journal of Logic and Computation, 10, 677–703.
Hunter, A. (2000b). Reasoning with inconsistency in structured text. Knowledge Engineering Review, 15, 317–337.
Hunter, A. (2001). A semantic tableau version of first-order quasi-classical logic. In Symbolic and Quantitative Approaches to Uncertainty, vol. 2143 of LNCS (pp. 544–556).
Hunter, A. (2002). Measuring inconsistency in knowledge via quasi-classical models. In Proceedings of the National Conference on Artificial Intelligence (AAAI'02) (pp. 68–73). MIT Press.
Hunter, A. (2003). Evaluating significance of inconsistencies. In Proceedings of the 18th International Joint Conference on Artificial Intellignce (IJCAI'03) (pp. 468–473).
Hunter, A., & Nuseibeh, B. (1998). Managing inconsistent specifications: Reasoning, analysis and action. ACM Transactions on Software Engineering and Methodology, 7, 335–367.
Konieczny, S., & Pino Perez, R. (1998). On the logic of merging. In Proceedings of the Sixth International Conference on Principles of Knowledge Representation and Reasoning (KR98) (pp. 488–498). Morgan Kaufmann.
Konieczny, S., Lang, J., & Marquis, P. (2003). Quantifying information and contradiction in propositional logic through epistemic actions. In Proceedings of the 18th International Joint Conference on Artificial Intellignce (IJCAI'03) (pp. 106–111).
Levesque, H. (1984). A logic of implicit and explicit belief. In Proceedings of the National Conference on Artificial Intelligence (AAAI'84) (pp. 198–202).
Lozinskii, E. (1994). Information and evidence in logic systems. Journal of Experimental and Theoretical Artificial Intelligence, 6, 163–193.
Marquis, P., & Porquet, N. (2001). Computational aspects of quasi-classical entailment. Journal of Applied Non-classical Logics, 11, 295–312.
Miarka, R., Derrick, J., & Boiten, E. (2002). Handling inconsistencies in z using quasi-classical logic. In D. Bert, J. Bowen, M. Henson, & K. Robinson (Eds.), ZB2002: Formal Specification and Development in Z and B, vol. 2272 of Lecture Notes in Computer Science (pp. 204–225). Springer.
Priest, G. (1989). Reasoning about truth. Artificial Intelligence, 39, 231–244.
Priest, G. (2002). Paraconsistent logic. In Handbook of philosophical logic, vol. 6. Kluwer.
Reiter, R. (1978). Equality and domain closure. Journal of the ACM, 27, 235–249.
Sheth, A., & Larson, J. (1990). Federated database systems for managing distributed, heterogeneous, and autonomous databases. ACM Computing Surveys, 22, 183–236.
Smullyan, R. (1968). First-order logic. Springer.
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Grant, J., Hunter, A. Measuring inconsistency in knowledgebases. J Intell Inf Syst 27, 159–184 (2006). https://doi.org/10.1007/s10844-006-2974-4
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DOI: https://doi.org/10.1007/s10844-006-2974-4