Skip to main content
SpringerLink
Log in
Book cover

International Conference on Intelligent Decision Technologies

IDT 2017: Intelligent Decision Technologies pp 411–437Cite as

Reasoning and Decision Making in an Inconsistent World

Labeled Logical Varieties as a Tool for Inconsistency Robustness

  • Conference paper
  • First Online:

  • 1480 Accesses

Part of the book series: Smart Innovation, Systems and Technologies ((SIST,volume 39))

Abstract

The goal of this paper is the development of foundations for robust reasoning and decision-making in pervasively inconsistent theories and deductive databases. The pervasiveness of these inconsistencies is partly inherent to the human epistemic condition (i.e. the need to make decisions on the basis of perspective bound knowledge) and partly inherent to practical limitations (e.g. incomplete knowledge). In the first case, we do not want to eliminate the inconsistencies. In the second case, we cannot eliminate them. So, in both cases, we will have to incorporate them in our descriptions of reasoning and decision making. For this reason, inconsistency handling is one of the central problems in many areas of AI. There are different approaches to dealing with contradictions and other types of inconsistency. In this paper, we develop an approach based on logical varieties and prevarieties, which are complex structures constructed from logical calculi. Being locally isomorphic to a logical calculus, globally logical varieties form a logical structure, which allows representation of inconsistent knowledge in a consistent way and provides much more flexibility and efficacy for AI than standard logical methods. Logical varieties and prevarieties are efficiently used in database theory and practice, expert systems and knowledge representation and processing. To increase efficiency, flexibility and capabilities of logical varieties and prevarieties and to model perspective bound decision making, we introduce labeling of their elements and study labeled logical varieties and prevarieties. We illustrate the viability of this by an example of a labeled logical variety, the extended Logic of Reasonable Inferences, which is and has been applied in the legal domain.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Ackermann, W.: Zur Widerspruchsfreiheit der Zahlentheorie. Math. Ann. 117, pp. 162–194 (1940)

    Article  MathSciNet  Google Scholar 

  2. Amgoud, L., Cayrol, C.: On the acceptability of arguments in preference-based argumentation. In: Proceedings of the 14th Conference on Uncertainty in Artificial Intelligence (UAI’98), pp. 1–7 (1998)

    Google Scholar 

  3. Amir, E.: Dividing and Conquering Logic. Ph.D. thesis, Stanford University, Computer Science Department, 2002

    Google Scholar 

  4. Amir, E., McIlraith, S.: Partition-based logical reasoning for first-order and propositional theories. Artif. Intell. 162(1/2), 49–88 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bandemer, H., Gottwald, S.: Fuzzy Sets, Fuzzy Logic, Fuzzy Methods With Applications. Wiley, New York (1996)

    Google Scholar 

  6. Barker, C.: Continuations in natural language. In: Thielecke, H. (ed.), Proceedings of the 4th Continuations Workshop, pp. 55–64. Technical report CSR-04-1, School of Computer Science, University of Birmingham (2004)

    Google Scholar 

  7. Barwise, J., Seligman, J.: Information Flow: The Logic of Distributed Systems. Cambridge Tracts in Theoretical Computer Science 44, Cambridge University Press (1997)

    Google Scholar 

  8. Benferhat, S., Dubois, D., Prade. H.: Representing default rules in possibilistic logic. In: Proceedings of the 3rd International Conference of Principles of Knowledge Representation and Reasoning (KR’92), pp. 673–684 (1992)

    Google Scholar 

  9. Benferhat, S., Cayrol, C., Dubois, D., Lang, J., Prade, H.: Inconsistency management and prioritized syntax-based entailment. In: Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI’93), pp. 640–645 (1993)

    Google Scholar 

  10. Benferhat, S., Dubois, D., Prade, H.: How to infer from inconsistent beliefs without revising? In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI’95), pp. 1449–1455 (1995)

    Google Scholar 

  11. Benferhat, S., Garcia, L.: Handling locally stratified inconsistent knowledge bases. Stud. Logica. 70, 77–104 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bertossi, L.E., Hunter, A., Schaub, T. (eds.): Inconsistency Tolerance. LNCS, vol. 3300, Springer, Heidelberg (2005)

    Google Scholar 

  13. Besnard, P., Hunter, A.: Quasi-classical logic: non-trivializable classical reasoning from inconsistent information. In: Proceedings of ECSQARU’95, LNAI, vol. 946, pp. 44–51 (1995)

    Google Scholar 

  14. Brewka, G.: Preferred subtheories: an extended logical framework for default reason. In: Proceedings of the 11th Int Joint Conference on Artificial Intelligence (IJ CAI’89), pp. 1043–1048 (1989)

    Google Scholar 

  15. Brown, B., Priest, G.: Chunk and permeate: a paraconsistent inference strategy, part I: the infinitesimal calculus. J. Philos. Logic 33(4), 379–388 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Burgin, M.: Logical methods in artificial intelligent systems (in Russian). In: Vestnik of the Computer Society, No. 2, pp. 66–78 (1991)

    Google Scholar 

  17. Burgin, M.: Logical varieties and covarieties. (in Russian) In: Methodological and Theoretical Problems of Mathematics and Information and Computer Sciences, pp. 18–34. Kiev (1997)

    Google Scholar 

  18. Burgin, M.: Logical Tools for Program Integration and Interoperability. In: Procedings of the IASTED International Conference on Software Engineering and Applications, pp. 743–748, MIT, Cambridge (2004)

    Google Scholar 

  19. Burgin, M.: Super-recursive algorithms. Monographs in computer science. Springer, New York (2005). ISBN 0-387-95569-0

    Google Scholar 

  20. Burgin, M.: Languages, Algorithms, Procedures, Calculi, and Metalogic. Preprint in Mathematics LO/0701121, 31 pp. (electronic edition: http://arXiv.org) (2007)

  21. Burgin, M.: structural organization of temporal databases. In: Proceedings of the 17th International Conference on Software Engineering and Data Engineering (SEDE-2008), ISCA, pp. 68–73, Los Angeles, California (2008)

    Google Scholar 

  22. Burgin, M.: Theory of Named Sets. Nova Science Publishers, New York (2011)

    Google Scholar 

  23. Burgin, M., Debnath, N.: Reusability as design of second-level algorithms. In: Proceedings of the ISCA 25th International Conference “Computers and their Applications” (CATA-2010), ISCA, pp. 147–152. Honolulu, Hawaii (2010)

    Google Scholar 

  24. Burgin, M., Gupta, B.: Second-level algorithms, superrecursivity, and recovery problem in distributed systems. Theor. Comput. Syst. 50(4), 694–705 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Burgin, M., de Vey Mestdagh, C.N.J.: The Representation of Inconsistent Knowledge in advanced knowledge based systems. In: Koenig, A., Dengel, A., Hinkelmann, K., Kise, K., Howlett, R.J., Jain, L.C. (eds.) Knowlege-Based and Intelligent Information and Engineering Systems, vol. 2, pp. 524–537. Springer, ISBN 978-3-642-23862-8 (2011)

    Google Scholar 

  26. Burgin, M., de Vey Mestdagh, C.N.J.: Consistent structuring of inconsistent knowledge. In: Journal of Intelligent Information Systems, pp. 1–24. Springer US, Sept 2013

    Google Scholar 

  27. Church, A.: Introduction to Mathematical Logic. Princeton University Press, Princeton (1956)

    Google Scholar 

  28. Da Costa, N.C.A.: Calcul propositionnel pour les systemes formels inconsistants. Compte Rendu Academie des Sciences (Paris) 257, 3790–3792 (1963)

    MATH  Google Scholar 

  29. Dalal, M.: Investigations into a theory of knowledge base revision: preliminary report. In: Proceedings of the Seventh National Conference on Artificial Intelligence (AAAI’88), pp. 475–479 (1988)

    Google Scholar 

  30. DeWitt, B.S.: The Many-Universes interpretation of quantum mechanics. In: Foundations of Quantum Mechanics, pp. 167–218. Academic Press, New York (1971)

    Google Scholar 

  31. Dung, P.M.: On the acceptability of arguments and its fundamental role in non-monotonic reasoning, logic programming and n-person games. Artif. Intell. 77, 321–357 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  32. Everett, H.: ‘Relative State’ formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957)

    Article  MathSciNet  Google Scholar 

  33. Frege, F.L.G.: Über Sinn und Bedeutung. In: Zeitschrift für Philosophie und philosophische Kritik, pp. 25–50 (1892)

    Google Scholar 

  34. Friedman, N., Halpern, J.Y.: A knowledge-based framework for belief change, part II: revision and update. In: Proceedings of the Fourth International Conference on the Principles of Knowledge Representation and Reasoning (KR’94), pp. 190–200 (1994)

    Google Scholar 

  35. Gabbay, D.M.: Labelled deductive systems and the informal fallacies. In: Van Eemeren F.H. et al. (eds.). Proceedings of the 3rd International Conference on Argumentation, vol. 2: Analysis and Evaluation, Sponsored by ISSA, International Society for the Study of Argumentation, pp. 308–319 (1994)

    Google Scholar 

  36. Gabbay, D.M.: Labelled deductive systems. Oxford Logic Guides, vol. 33, Clarendon Press/Oxford Science Publications, Oxford (1996)

    Google Scholar 

  37. Gabbay, D.M., D’Agostino, M.: A generalization of analytic deduction via labelled deductive systems part 1: basic substructural logics. J. Autom. Reasoning 13, 243–281 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  38. Gabbay, D.M., Malod, G.: Naming worlds in modal and temporal logic. J. Log. Lang. Inf. 11, 29–65 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Gärdenfors, P., Rott, H.: Belief revision. In: Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, pp. 35–132. Oxford University Press (1995)

    Google Scholar 

  40. Gentzen, G.: Die Widerspruchfreiheit der reinen Zahlentheorie. Math. Ann. 112, 493–565 (1936)

    Article  MathSciNet  Google Scholar 

  41. Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatsh. Math. Phys. 38(1), 173–198 (1931–1932)

    Google Scholar 

  42. Gogoi, P., Das, R., Borah, B., Bhattacharyya, D.K.: Efficient rule set generation using rough set theory for classification of high dimensional data. Int. J. Smart Sens. Ad Hoc Netw. (IJSSAN) 1(2), 13–20 (2011)

    Google Scholar 

  43. Greco, S., Matarazzo, B., Slowinski, R., Stefanowski, J.: Variable consistency model of dominance-based rough sets approach. In: Rough Sets and Current Trends in Computing. Lecture Notes in Computer Science, vol. 2005, pp. 170–181 (2001)

    Google Scholar 

  44. Grzymala-Busse, J.W.: Knowledge acquisition under uncertainty-A rough set approach. J. Intell. Rob. Syst. 1, 3–16 (1988)

    Google Scholar 

  45. Grzymala-Busse, J.W.: Rough set theory with applications to data mining. Real World Appl. Comput. Intell. Stud. Fuzziness Soft Comput. 179, 221–244 (2005)

    Google Scholar 

  46. Hájek, P.: Metamathematics of Fuzzy Logic. kluwer Academic publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  47. Hunter, A., Liu, W.: Knowledge base stratification and merging based on degree of support. In: Quantitative and Qualitative Approaches to Reasoning and Uncertainty (ECSQARU’09), LNCS, vol. 5590, pp. 383–395. Springer (1994)

    Google Scholar 

  48. Jaśkowski, S.: Rachunek zdań dla systemów dedukcyjnych sprzecznych. Studia Societatis Scientiarun Torunesis (Sectio A) 1(5), 55–77 (1948)

    Google Scholar 

  49. Kong, H., Xue, G., He X., Yao, S.: A proposal to handle inconsistent ontology with fuzzy OWL. In: 2009 WRI World Congress on Computer Science and Information Engineering, pp. 599–603 (2009)

    Google Scholar 

  50. Lehmann, D.: Another perspective on default reasoning. Ann. Math. Artif. Intell. 15, 61–82 (1995)

    Google Scholar 

  51. Lehmann, D.: Belief revision, revised. In: Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (IJCAI’95), pp. 1534–1540 (1995)

    Google Scholar 

  52. MacCartney, B., McIlraith, S.A., Amir, A., Uribe, T.: Practical partition-based theorem proving for large knowledge bases. In: Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence (IJCAI-03), pp. 89–96 (2003)

    Google Scholar 

  53. Makinson, D.: Bridges from Classical to Nonmonotonic Logic. College Publications (2005)

    Google Scholar 

  54. Manna, Z., Waldinger, R.: The Deductive Foundations of Computer Programming. Addison-Wesley, Boston/New York/Toronto (1993)

    MATH  Google Scholar 

  55. Marek, W., Truszczynski, M.: Nonmonotonic Logics: Context-Dependent Reasoning. Springer, New York (1993)

    Book  MATH  Google Scholar 

  56. McDermott, D., Doyle, J.: Non-monotonic logic I. Artif. Intell. 25, 41–72 (1980)

    Article  MathSciNet  Google Scholar 

  57. McIlraith, S., Amir, E.: Theorem proving with structured theories. In: Proceedings of the 17th International Joint Conference on Artificial Intelligence, (IJCAI’01), pp. 624–631 (2001)

    Google Scholar 

  58. McNeill, D., Freiberger, P.: Fuzzy Logic. Simon and Schuster (1993)

    Google Scholar 

  59. Mollestad, T., Skowron, A.: A rough set framework for data mining of propositional default rules. In: Proceedings of the 9th International Symposium on Foundations of Intelligent Systems, pp. 448–457 (1996)

    Google Scholar 

  60. Nebel, B.: Belief revision and default reasoning: syntax-based approaches. In: Proceedings of the Second International Conference on the Principles of Knowledge Representation and Reasoning (KR’91), pp. 417–428 (1991)

    Google Scholar 

  61. Nebel, B.: Base revision operations and schemes: semantics, representation and complexity. In: Proceedings of the Eleventh European Conference on Artificial Intelligence (ECAI’94), pp. 341–345 (1994)

    Google Scholar 

  62. Nguen, N.T.: Inconsistency of knowledge and collective intelligence. Cybern. Syst. 39(6), 542–562 (2008)

    Article  Google Scholar 

  63. Papini, O.: A complete revision function in propositional calculus. In: Proceedings of the 10th European Conference on Artificial Intelligence (ECAI’92) (1992)

    Google Scholar 

  64. Partridge, D., Wilks, Y.: The Foundations of Artificial Intelligence. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  65. Pawlak, Z.: Rough set and intelligent data analysis. Int. J. Inform. Sci. 147, 1–12 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  66. Pollock, J.L., Gillies, A.: Belief revision and epistemology. Synthese 122, 69–92 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  67. Priest, G., Routley, R., Norman, J. (eds.): Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag, München (1989)

    MATH  Google Scholar 

  68. Prior, A.N.: Past, Present and Future. Clarendon Press, Oxford (1967)

    Book  MATH  Google Scholar 

  69. Rescher, N., Manor, R.: On inference from inconsistent premisses. Theor. Decis. 1(2), 179–217 (1970)

    Article  MATH  Google Scholar 

  70. Rescher, N.: Plausible reasoning: an introduction to the theory and practice of plausibilistic inference (1976)

    Google Scholar 

  71. Resconi, G., Hinde, C.J.: Active sets, fuzzy sets and inconsistency. In: IEEE International Conference on Fuzzy Systems (FUZZ), pp. 1–8, Barcelona, Spain (2010)

    Google Scholar 

  72. Ross, T.J.: Fuzzy Logic with Engineering Applications. McGraw-Hill P. C. (1994)

    Google Scholar 

  73. Routley, R., Plumwood, V., Meyer, R.K., Brady, R.T.: Relevant Logics and Their Rivals. Atascadero, Ridgeview, CA (1982)

    MATH  Google Scholar 

  74. Schütte, K.: Beweistheorie. Springer, Berlin (1960)

    MATH  Google Scholar 

  75. Shoenfield, J.R.: Mathematical Logic. Addison-Wesley, Reading (2001)

    MATH  Google Scholar 

  76. Smolin, L.: The Bekenstein bound, topological quantum field theory and pluralistic quantum field theory. Penn State preprint CGPG-95/8-7; Los Alamos Archives preprint in physics, gr-qc/9508064, electronic edition: http://arXiv.org (1995)

  77. Smullian, R.: What is the Name of this Book?. Prentice Hall, Englewood Cliffs (1978)

    Google Scholar 

  78. Toulmin, S.: The Uses of Argument. Cambridge University Press (1956)

    Google Scholar 

  79. de Vey Mestdagh, C.N.J., Verwaard, W., Hoepman, J.H.: The logic of reasonable inferences. In: Breuker, J.A., de Mulder, R.V., Hage, J.C. (eds.) Legal Knowledge Based Systems, Model-based legal reasoning, Proceedings of the 4th annual JURIX Conference on Legal Knowledge Based Systems, pp. 60–76. Vermande, Lelystad (1991)

    Google Scholar 

  80. de Vey Mestdagh, C.N.J.: Legal expert systems. Experts or Expedients? The Representation of Legal Knowledge in an Expert System for Environmental Permit Law. In: Ciampi, C., Marinai, E. (eds.) The Law in the Information Society, Conference Proceedings on CD-Rom, Firenze, 8 pp (1998)

    Google Scholar 

  81. de Vey Mestdagh, C.N.J., Hoepman, J.H.: Inconsistent knowledge as a natural phenomenon: the ranking of reasonable inferences as a computational approach to naturally inconsistent (legal) theories. In: Dodig-Crnkovic, G. & Burgin, M. (eds.) Information and Computation, pp. 439–476. World Scientific, New Jersey (2011)

    Google Scholar 

  82. Viganò, L., Volpe, M.: Labeled natural deduction systems for a family of tense logics. In: Demri, S., Christian S., Jensen, C.S. (eds.) 15th International Symposium on Temporal Representation and Reasoning (TIME 2008), pp. 118–126, University of Quebec, Montreal, Canada, 16–18 June 2008. IEEE Computer Society (2008)

    Google Scholar 

  83. Wang, G., Liu, F.: The inconsistency in rough set based rule generation. In: Rough Sets and Current Trends in Computing, Lecture Notes in Computer Science, vol. 2005, pp. 370–377 (2001)

    Google Scholar 

  84. Wassermann, R.: An algorithm for belief revision. In: Proceedings of the 7th International Conference of Principles of Knowledge Representation and Reasoning (KR’2000) (2000)

    Google Scholar 

  85. Weinzierl, A.: Comparing inconsistency resolutions in multi-context systems. In: Slavkovik, M. (ed.) Student Session of the European Summer School for Logic, Language, and Information, pp. 17–24 (2010)

    Google Scholar 

  86. Williams, M.A.: Transmutations of knowledge systems. In: Proceedings of the 4th International Conference of Principles of Knowledge Representation and Reasoning (KR’94), pp. 619–629 (1994)

    Google Scholar 

  87. Williams, M.A.: A practical approach to belief revision: reason-based change. In: Proceedings of the 5th International Confeence of Principles of Knowledge Representation and Reasoning (KR’96), pp. 412–421 (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Law & ICT, University of Groningen, Groningen, The Netherlands

    Cornelis N. J. de Vey Mestdagh

  2. Department of Computer Science, UCLA, Los Angeles, USA

    Mark Burgin

Authors
  1. Cornelis N. J. de Vey Mestdagh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark Burgin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cornelis N. J. de Vey Mestdagh .

Editor information

Editors and Affiliations

  1. FCT, Universidade Nova de Lisboa, Caparica, Portugal

    Rui Neves-Silva

  2. Faculty of Education, Science, Technology and Mathematics, University of Canberra, Canberra, Australia

    Lakhmi C. Jain

  3. KES International, Shoreham-by-sea, United Kingdom

    Robert J. Howlett

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

de Vey Mestdagh, C.N.J., Burgin, M. (2015). Reasoning and Decision Making in an Inconsistent World. In: Neves-Silva, R., Jain, L., Howlett, R. (eds) Intelligent Decision Technologies. IDT 2017. Smart Innovation, Systems and Technologies, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-19857-6_36

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-19857-6_36

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19856-9

  • Online ISBN: 978-3-319-19857-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation