Skip to main content
SpringerLink
Log in

Qualitative and Semi-Quantitative Inductive Reasoning with Conditionals

Technical Project Report

  • Research Project
  • Published:
KI - Künstliche Intelligenz Aims and scope Submit manuscript

Abstract

Conditionals like “birds fly—if bird then fly” are crucial for commonsense reasoning. In this technical project report we show that conditional logics provide a powerful formal framework that helps understanding if-then sentences in a way that is much closer to human reasoning than classical logic and allows for high-quality reasoning methods. We describe methods that inductively generate models from conditional knowledge bases. For this, we use both qualitative (like preferential models) and semi-quantitative (like Spohn’s ranking functions) semantics. We show similarities and differences between the resulting inference relations with respect to formal properties. We further report on two graphical methods on top of the ranking approaches which allow to decompose the models into smaller, more feasible components and allow for local inferences.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. i.e., we use numbers in a qualitative way.

  2. http://airconditionals.cs.tu-dortmund.de.

References

  1. Adams EW (1975) The Logic of Conditionals: An Application of Probability to Deductive Logic. Synthese Library. Springer Science+Business Media, Dordrecht

    Book  MATH  Google Scholar 

  2. Benferhat S, Cayrol C, Dubois D, Lang J, Prade H (1993) Inconsistency Management and Prioritized Syntax-Based Entailment. In: Proceedings of the International Joint Conferences on Artificial Intelligence. pp. 640–647

  3. R Da Silva Neves, Bonnefon JF, Raufaste E (2002) An Empirical Test of Patterns for Nonmonotonic Inference. Ann Math Artif Intell 34(1–3):107–130

    MathSciNet  Google Scholar 

  4. Dietz EA, Hölldobler S, Ragni M (2013) A computational logic approach to the abstract and the social case of the selection task. In: Proceedings of the 11th International Symposium on Logical Formalizations of Commonsense Reasoning

  5. Donald N, Cross CB (2002) Conditional logic. In: Gabbay DM, Guenther F (eds) Handbook of Philosophical Logic, 2nd edn., vol. 4. Kluwer Academic Publishers, Dordrecht, pp. 1–98

  6. Eichhorn C, Kern-Isberner G (2014) LEG Networks for Ranking Functions. In: Logics in Artificial Intelligence. In: Proceedings of the 14th European Conference on Logics in Artificial Intelligence (JELIA’14), Lecture Notes in Computer Science, vol. 8761. pp. 210–223

  7. Eichhorn C, Kern-Isberner G (2015) Using inductive reasoning for completing OCF-networks. In: FLAIRS Uncertain Reasoning special issue (in print), Journal of Applied Logic

  8. Furbach U, Schon C (2015) Deontic Logic for Human Reasoning. In: Eiter T, Strass H, Truszczynski M, Woltran S (eds) Advances in Knowledge Representation, Logic Programming, and Abstract Argumentation—Essays Dedicated to Gerhard Brewka on the Occasion of His 60th Birthday, no. 9060 in Lecture Notes in Artificial Intelligence. Springer International Publishing, Wiesbaden

  9. Goldszmidt M, Pearl J (1991) System-Z+: A formalism for reasoning with variable-strength defaults. In: Proceedings of the ninth National conference on Artificial intelligence (AAAI’91), vol. 1. pp 399–404

  10. Goldszmidt M, Pearl J (1996) Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artif Intell 84(1–2):57–112

    Article  MathSciNet  Google Scholar 

  11. Halpern JY (2005) Reasoning About Uncertainty. MIT Press, Cambridge

    Google Scholar 

  12. Kern-Isberner G (2001) Conditionals in Nonmonotonic Reasoning and Belief Revision—Considering Conditionals as Agents. No. 2087 in Lecture Notes in Computer Science. Springer, Berlin

  13. Kern-Isberner G (2004) A thorough axiomatization of a principle of conditional preservation in belief revision. Ann Math Artif Intell 40:127–164

    Article  MathSciNet  MATH  Google Scholar 

  14. Kern-Isberner G, Eichhorn C (2012) A structural base for conditional reasoning. In: Human Reasoning and Automated Deduction—KI 2012 Workshop Proceedings. pp. 25–32

  15. Kern-Isberner G, Eichhorn C (2013) Intensional combination of rankings for OCF-networks. In: Proceedings of the 26th International FLAIRS Conference. pp. 615–620

  16. Kern-Isberner G, Eichhorn C (2013) OCF-networks with missing values. In: Proceedings of the 4th Workshop on Dynamics of Knowledge and Belief. pp. 46–60

  17. Kern-Isberner G, Eichhorn C (2014) Structural inference from conditional knowledge bases. In: Unterhuber M, Schurz G (eds) Studia Logica Special Issue Logic and Probability: Reasoning in Uncertain Environments, vol 102(4). Springer Science+Business Media, Dordrecht

    Google Scholar 

  18. Kraus S, Lehmann DJ, Magidor M (1990) Nonmonotonic reasoning, preferential models and cumulative logics. Artif Intell 44(1–2):167–207

    Article  MathSciNet  MATH  Google Scholar 

  19. Kuhnmünch G, Ragni M (2014) Can Formal Non-monotonic Systems Properly Describe Human Reasoning? Proc Cognitive Sci Conf 2014:1806–1811

    Google Scholar 

  20. Lehmann DJ, Magidor M (1992) What does a conditional knowledge base entail? Artif Intell 55(1):1–60

    Article  MathSciNet  Google Scholar 

  21. Lewis D (1976) Probabilities of conditionals and conditional probabilities. Philos Rev 85:297–315

    Article  MATH  Google Scholar 

  22. Lukasiewicz T (2005) Weak nonmonotonic probabilistic logics. Artif Intell 168(1–2):119–161

    Article  MathSciNet  MATH  Google Scholar 

  23. Makinson D (1994) General patterns in nonmonotonic reasoning. In: Gabbay DM, Hogger CJ, Robinson JA (eds) Handbook of Logic in Artificial Intelligence and Logic Programming, vol 3. Oxford University Press, New York, pp 35–110

    Google Scholar 

  24. Meyer CH (1998) Korrektes Schließen bei unvollständiger Information: Anwendung des Prinzips der maximalen Entropie in einem probabilistischen Expertensystem. 41. Peter Lang Publishing Inc, Pieterlen (in German)

  25. Oaksford M, Chater N (1995) Information gain explains relevance which explains the selection task. Cognition 57:97–108

    Article  Google Scholar 

  26. Pearl J (1988) Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann Publishers Inc., San Francisco

    Google Scholar 

  27. Pearl J (1990) System Z: A natural ordering of defaults with tractable applications to nonmonotonic reasoning. In: Proceedings of the 3rd conference on Theoretical aspects of reasoning about knowledge (TARK’90). pp. 121–135

  28. Pfeifer N, Kleiter GD (2005) Coherence and nonmonotonicity in human reasoning. Synthese 146(1–2):93–109

    Article  MathSciNet  MATH  Google Scholar 

  29. Sperber D, Cara F, Girotto V (1995) Relevance theory explains the selection task. Cognition 57:31–95

    Article  Google Scholar 

  30. Spohn W (1988) Ordinal conditional functions: A dynamic theory of epistemic states. In: Harper W, Skyrms B (eds) Causation in Decision, Belief Change and Statistics, The University of Western Ontario Series in Philosophy of Science, vol 42. Springer Science+Business Media, Dordrecht, pp 105–134

    Chapter  Google Scholar 

  31. Spohn W (2012) The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford University Press, New York

    Book  Google Scholar 

  32. Thimm M (2014) Tweety—A Comprehensive Collection of Java Libraries for Logical Aspects of Artificial Intelligence and Knowledge Representation. In: Proceedings of the 14th International Conference on Principles of Knowledge Representation and Reasoning (KR’14)

  33. Thorn PD, Eichhorn C, Kern-Isberner G, Schurz G (2015) Qualitative Probabilistic Inference with Default Inheritance for Exceptional Subclasses. In: progic 2015: The Seventh Workshop on Combining Probability and Logic

  34. Wason PC (1968) Reasoning about a rule. Q J Exp Psychol 20(3):273–281

    Article  Google Scholar 

  35. Wason PC, Johnson-Laird P (1990) Psychology of Reasoning: structure and Content. Harvard University Press

Download references

Author information

Authors and Affiliations

  1. Technische Universität Dortmund, Lehrstuhl Informatik 1, Dortmund, Germany

    Christian Eichhorn & Gabriele Kern-Isberner

Authors
  1. Christian Eichhorn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriele Kern-Isberner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Eichhorn.

Additional information

We thank the anonymous referees for their valuable hints that helped us improving the paper. This work was supported by grant KI\(\;1413/5-1\) to Gabriele Kern-Isberner from the Deutsche Forschungsgemeinschaft (DFG) as part of the priority program “New Frameworks of Rationality” (SPP 1516). Christian Eichhorn is supported by this Grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eichhorn, C., Kern-Isberner, G. Qualitative and Semi-Quantitative Inductive Reasoning with Conditionals. Künstl Intell 29, 279–289 (2015). https://doi.org/10.1007/s13218-015-0376-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13218-015-0376-x

Keywords

Search

Navigation