Minimising disjunctive information
Keywordsparaconsistency, disjunctive information, inconsistencies
AbstractIn [5, 6], Belnap proposed a number of amendments to Rescher’s strategy for reasoning with maximal consistent subsets. More recently in , Horty explicitly endorsed Belnap’s amendment to address a related problem in handling inconsistent instructions and commands. In this paper, we’ll examine Belnap’s amendment and point out that Belnap’s suggestion in the use of conjunctive containment is open to the very objection he raised. We’ll propose a way out. The strategy turns on the use of First Degree Entailment in combination with Quine’s notion of prime implicate.
A.E. Anderson and N.D. Belnap, Entailment: the Logic of Relevance and Necessity, Vol 1. Princeton University Press, 1975.
A.E. Anderson, N.D. Belnap, and J.M. Dunn, Entailment: the Logic of Relevance and Necessity, Vol 2. Princeton University Press, 1992.
O. Arieli and M. Denecker, “Modelling paraconsistent reasoning by classical logic”, pages 1–14 in: T. Eiter and K-D. Schewe, editors, Proceedings of the Second International Symposium on Foundations of Information and Knowledge Systems (FoIKS), Lecture Notes in Computer Science 2284. Springer Verlag, 2002.
O. Arieli and M. Denecker, “Reducing preferential paraconsistent reasoning to classical entailment”, Journal of Logic and Computation 13, 4 (2003): 557–580.
N.D. Belnap, “Rescher’s hypothetical reasoning”, pages 19–28 in: E. Sosa, editor, The Philosophy of Nicholas Rescher: Discussion and Replies. D. Reidel Pub., 1979.
N.D. Belnap, “Conjunctive containment”, pages 145–156 in: J. Norman and R. Sylvan, editors, Directions in Relevant Logic. Kluwer Academic Pub., 1989.
P. Besnard and A. Hunter, “Quasi-classical logic: Non-trivializable classical reasoning from inconsistent information”, pages 44–51 in: Symbolic and Quantitative Approaches to Reasoning and Uncertainty 95, Lecture Notes in Artificial Intelligence 946. Springer Verlag, 1995.
P. Besnard and T. H. Schaub, “Signed systems for paraconsistent reasoning”, Journal of Automated Reasoning 20 (1998): 191–213.
J. de Kleer, “Focusing on probable diagnoses”, pages 842–848 in: Proceedings of the National Conference on Artificial Intelligence (AAAI 1991). Morgan Kaufmann, 1991.
J. de Kleer, A.K. Mackworth, and R. Reiter, “Characterizing diagnoses”, pages 324–330 in: Proceedings of the National Conference on Artificial Intelligence (AAAI 1990). Morgan Kaufmann, 1990.
J. de Kleer, A.K. Mackworth, and R. Reiter, “Characterizing diagnoses and systems”, Artificial Intelligence 56, 2–3 (1992): 197–222.
J.M. Dunn, “Relevance logic and entailment”, pages 117–224 in: D.M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, Volume 3: Alternatives To Classical Logic. D. Reidel Pub., 1986.
K. Gemes, “A new theory of content I: Basic content”, Journal of Philosophical Logic, 23, 6 (1994): 595–620.
K. Gemes, “A new theory of content II: Model theory and some alternatives”, Journal of Philosophical Logic 26, 4 (1997): 449–476.
G. Gottlob, “Complexity results for nonmonotonic logics”, Journal of Logic and Computation 2 (1992): 397–425.
C. Hewitt, “Large-scale organisational computing requires unstratified reflection and strong paraconsistency”, pages 110–124 in: Coordination, Organizations, Institutions, and Norms in Agent Systems III: COIN 2007 International Workshops, Lecture Notes in Artificial Intelligence 4870, 2008.
C. Hewitt, “Common sense for concurrency and inconsistency tolerance using Direct Logic TM and the Actor Model”. Published online http://arxiv.org/abs/0812.4852
J.F. Horty, “Nonmonotonic foundations for deontic logic”, pages 17–44 in: D. Nute, editor, Defeasible Deontic Logic, Studies in Epistemology, Logic, Methodology, and Philosophy of Science vol 263. Kluwer Academic Pub., 1997.
A. Hunter, “Reasoning with contradictory information using quasi-classical logic”, Journal of Logic and Computation 10, 5 (2000): 677–703.
A. Hunter, “A semantic tableau version of first-order quasi-classical logic”, pages 544–555 in: S. Benferhat and P. Besnard, editors, Symbolic and Quantitative Approaches to Reasoning with Uncertainty, 6th European Conference, ECSQARU 2001, Toulouse, France, September 19–21, 2001, Proceedings, Lecture Notes in Artificial Intelligence 2143. Springer Verlag, 2001.
A. Hunter, “Measuring inconsistency in knowledge via quasi-classical models”, pages 68–73 in: Proceedings of the 18th National Conference of Artificial Intelligence (AAAI 2002), 2002.
H. Kautz and B. Selman, “A general framework for knowledge compilation”, pages 287–300 in: H. Richter and M. Richter, editors, Proceedings of International Workshop on Processing Delarative Knowledge, Lecture Notes in Artificial Intelligence 567. Springer Verlag, 1991.
H. Kautz and B. Selman, “Knowledge compilation and theory approximation”, Journal of the ACM 43, 2 (1996): 193–224.
P. Marquis, “Consequence finding algorithms”, pages 41–145 in: S. Moral and J. Kohlas, editors, Algorithms for Defeasible and Uncertain Reasoning, volume 5 of Handbook on Defeasible Reasoning and Uncertainty Management Systems, chapter 2. Kluwer Academic Pub., 2000.
B. Nebel, “Belief revision and default reasoning: Syntax-based approaches”, pages 417–428 in: J.A. Allen, R. Fikes, and E. Sandewall, editors, Principle of Knowledge Representation and Reasoning: Proceedings of the Second International Conference (KR91). Morgan Kaufmann, 1991.
B. Nebel, “Syntax-based approaches to belief revision”, pages 52–88 in: P. Gärdenfors, editor, Belief Revision. Cambridge University Press, 1992.
W.V.O. Quine, “The problem of simplifying truth functions”, American Mathematical Monthly, 59 (1952): 521–531.
W.V.O. Quine, “A way to simplify truth functions”, American Mathematical Monthly 62 (1955): 627–631.
W.V.O. Quine, “On cores and prime implicants of truth functions”, American Mathematical Monthly 66 (1959): 755–760.
A. Ramesh, G. Becker, and N.V. Murray, “CNF and DNF considered harmful for computing prime implicants/implicates”, Journal of AutomatedReasoning 18 (1997): 337–356.
A. Ramesh, B. Beckert, R. Hähnle, and N. V. Murray, “Fast subsumption checks using anti-links”, Journal of Automated Reasoning 18 (1997): 47–83.
A.G. Ramesh, Some Applications of Non Clausal Deduction, PhD thesis, Department of Computer Science, State University of New York at Albany, 1995.
R. Reiter, “A logic for default reasoning”, Artificial Intelligence 13 (1980): 81–132.
N. Rescher, Hypothetical Reasoning, North-Holland, 1964.
N. Rescher, The Coherence Theory of Truth, Oxford University Press, 1973.
N. Rescher and R. Manor, “On inference from inconsistent premisses”, Theory and Decision 1 (1970): 179–217.
T.H. Schaub, “The family of default logics”, pages 77–134 in: P. Besnard and A. Hunter, editors, Handbook of Defeasible Reasoning and Uncertain Information Volume 2, Reasoning wiht Actual and Potential Contradictions. Kluwer Academic Pub., 1998.
J. Stillman, “It’s not my default: The complexity of membership problems in restricted propositional default logics”, pages 571–578 in: Proceedings of the 8th National Conference on Artificial Intelligence, AAAI. MIT Press, 1990.
N. Tennant, “Frege’s content-principle and relevant deducibility”, Journal of Philosophical Logic 32 (2003): 245–258.
Ch. Umans, “The minimum equivalent DNF problem and shortest implicants”, page 556 in: FOCS ’98: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, Washington, DC, USA, 1998. IEEE Computer Society.
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