Minds and Machines

, Volume 27, Issue 1, pp 37–77 | Cite as

Reasoning in Non-probabilistic Uncertainty: Logic Programming and Neural-Symbolic Computing as Examples

  • Tarek R. Besold
  • Artur d’Avila Garcez
  • Keith Stenning
  • Leendert van der Torre
  • Michiel van Lambalgen


This article aims to achieve two goals: to show that probability is not the only way of dealing with uncertainty (and even more, that there are kinds of uncertainty which are for principled reasons not addressable with probabilistic means); and to provide evidence that logic-based methods can well support reasoning with uncertainty. For the latter claim, two paradigmatic examples are presented: logic programming with Kleene semantics for modelling reasoning from information in a discourse, to an interpretation of the state of affairs of the intended model, and a neural-symbolic implementation of input/output logic for dealing with uncertainty in dynamic normative contexts.


Uncertainty in reasoning Interpretation Logic programming Dynamic norms Neural-symbolic integration 



We want to thank the following people for their indispensable contributions to different parts of the work reported in this article: Guido Boella, Silvano Colombo Tosatto, Valerio Genovese, Laura Martignon, Alan Perotti, and Alexandra Varga.


  1. Alchourrón, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic, 50(02), 510–530.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Antoniou, G., Billington, D., & Maher, M. (1998). Sceptical logic programming based default reasoning: defeasible logic rehabilitated. In R. Miller, M. Shanahan (Eds.), COMMONSENSE 98, The 4th symposium on logical formalizations of commonsense reasoning, London.Google Scholar
  3. Apt, K. R., & Pedreschi, D. (1993). Reasoning about termination of pure prolog programs. Information and Computation, 106, 109–157.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Baggio, G., Stenning, K., & van Lambalgen, M. (2016). The cognitive interface. In M. Aloni & P. Dekker (Eds.), Cambridge handbook of formal semantics. Cambridge: Cambridge University Press.Google Scholar
  5. Boella, G, & van der Torre, L (2005). Permission and authorization in normative multiagent systems. In Procs. of int. conf. on artificial intelligence and law ICAIL (pp. 236–237).Google Scholar
  6. Boella, G., & van der Torre, L. (2006). A game theoretic approach to contracts in multiagent systems. IEEE Transactions on Systems, Man, and Cybernetics, Part C, 36(1), 68–79.CrossRefGoogle Scholar
  7. Boella, G., Pigozzi, G., & van der Torre, L. (2009). Normative framework for normative system change. In 8th Int. joint conf. on autonomous agents and multiagent systems AAMAS 2009, IFAAMAS (pp. 169–176).Google Scholar
  8. Bradley, R., & Drechsler, M. (2014). Types of uncertainty. Erkenntnis, 79, 1225–1248.MathSciNetCrossRefGoogle Scholar
  9. Doets, K. (1994). From logic to logic programming. Cambridge, MA: MIT Press.zbMATHGoogle Scholar
  10. Gabbay, D., Horty, J., Parent, X., van der Meyden, R., & van der Torre, L. (Eds.). (2013). Handbook of deontic logic and normative systems. London: College Publications.zbMATHGoogle Scholar
  11. Garcez, A., Broda, K., & Gabbay, D. M. (2001). Symbolic knowledge extraction from trained neural networks: A sound approach. Artificial Intelligence, 125, 155–207.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Garcez, A., Broda, K., & Gabbay, D. (2002). Neural-symbolic learning systems: Foundations and applications. Perspectives in neural computing. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  13. Garcez, A., Gabbay, D., & Lamb, L. (2005). Value-based argumentation frameworks as neural-symbolic learning systems. Journal of Logic and Computation, 15(6), 1041–1058.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Garcez, A., Lamb, L. C., & Gabbay, D. M. (2009). Neural-symbolic cognitive reasoning. Berlin: Springer.zbMATHGoogle Scholar
  15. Garcez, A., Besold, T.R., de Raedt, L., Földiak, P., Hitzler, P., Icard, et al. (2015). Neural-symbolic learning and reasoning: Contributions and challenges. In: AAAI Spring 2015 symposium on knowledge representation and reasoning: Integrating symbolic and neural approaches, AAAI technical reports (vol SS-15-03). AAAI Press.Google Scholar
  16. Gelfond, M., & Lifschitz, V. (1988). The stable model semantics for logic programming. In Proceedings of the 5th logic programming symposium, MIT Press (pp. 1070–1080).Google Scholar
  17. Gelfond, M., & Lifschitz, V. (1991). Classical negation in logic programs and disjunctive databases. New Generation Computing, 9, 365–385.CrossRefzbMATHGoogle Scholar
  18. Gigerenzer, G., Todd, P. M., & The ABC Research Group. (1999). Simple heuristics that make us smart. Oxford: Oxford University Press.Google Scholar
  19. Gigerenzer, G., Hertwig, R., & Pachur, T. (2011). Heuristics: The foundations of adaptive behavior. Oxford: Oxford University Press.CrossRefGoogle Scholar
  20. Graves, A., Mohamed, A., & Hinton, G.E. (2013). Speech recognition with deep recurrent neural networks. CoRR arXiv:abs/1303.5778.
  21. Halpern, J. (2005). Reasoning about uncertainty. Cambridge, MA: MIT Press.zbMATHGoogle Scholar
  22. Hansen, J. (2006). Deontic logics for prioritized imperatives. Artificial Intelligence and Law, 14(1–2), 1–34.Google Scholar
  23. Haykin, S. (1999). Neural networks: A comprehensive foundation. Upper Saddle River: Prentice Hall.zbMATHGoogle Scholar
  24. Horty, J. F. (1993). Deontic logic as founded on nonmonotonic logic. Annals of Mathematics and Artificial Intelligence, 9(1–2), 69–91.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Jörgensen, J. (1937). Imperatives and logic. Erkenntnis, 7, 288–296.Google Scholar
  26. Juslin, P., Nilsson, Håkan, & Winman, A. (2009). Probability theory, not the very guide of life. Psychological Review, 116(4), 856–874.CrossRefGoogle Scholar
  27. Kahneman, D., & Tversky, A. (1982). The concept of probability in psychological experiments. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), The concept of probability in psychological experiments (pp. 509–520). Cambridge: Cambridge University Press.Google Scholar
  28. Kern-Isberner, G., & Lukasiewicz, T. (2017). Many facets of reasoning under uncertainty, inconsistency, vagueness, and preferences: A brief survey. Künstliche Intelligenz. doi: 10.1007/s13218-016-0480-6.Google Scholar
  29. Knight, F. (1921). Risk, uncertainty and profit. New York: Hart, Schaffner and Marx.Google Scholar
  30. Kowalski, R. A. (1988). The early years of logic programming. Communications of the ACM, 31, 38–42.CrossRefGoogle Scholar
  31. Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1), 167–207.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Lindahl, L., & Odelstad, J. (2003). Normative systems and their revision: An algebraic approach. Artificial Intelligence and Law, 11(2–3), 81–104.CrossRefGoogle Scholar
  33. Lloyd, J. W. (1987). Foundations of logic programming. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  34. Makinson, D., & van der Torre, L. (2000). Input/output logics. Journal of Philosophical Logic, 29(4), 383–408.MathSciNetCrossRefzbMATHGoogle Scholar
  35. Makinson, D., & van der Torre, L. (2001). Constraints for input-output logics. Journal of Philosophical Logic, 30(2), 155–185.MathSciNetCrossRefzbMATHGoogle Scholar
  36. Makinson, D., & van der Torre, L. (2003a). Permissions from an input-output perspective. Journal of Philosophical Logic, 32(4), 391–416.MathSciNetCrossRefzbMATHGoogle Scholar
  37. Makinson, D., & van der, Torre L. (2003b). What is input/output logic? In B. Löwe, W. Malzkorn & T. Räsch (Eds.), Foundations of the formal sciences II: Applications of mathematical logic in philosophy and linguistics, trends in logic (Vol. 17). Kluwer.Google Scholar
  38. McCarthy, J. (1980). Circumscription: A form of non-monotonic reasoning. Artificial Intelligence, 13(1), 27–39.MathSciNetCrossRefzbMATHGoogle Scholar
  39. Minsky, M. (1974). A framework for representing knowledge. Tech. Rep. 306, AI Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA.Google Scholar
  40. Mousavi, S., & Gigerenzer, G. (2014). Risk, uncertainty, and heuristics. Journal of Business Research, 67, 1671–1678.CrossRefGoogle Scholar
  41. Mozina, M., Zabkar, J., & Bratko, I. (2007). Argument based machine learning. Artificial Intelligence, 171(10–15), 922–937.MathSciNetCrossRefzbMATHGoogle Scholar
  42. Nilsson, N. J. (1986). Probabilistic logic. Artificial intelligence, 28(1), 71–87.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Nute, D. (1994). Defeasible logic. In D. Gabbay & J. Robinson (Eds.), Handbook of logic in artificial intelligence and logic programming (Vol. 3, pp. 353–396). Oxford: Oxford University Press.Google Scholar
  44. Nute, D. (Ed.). (1997). Defeasible deontic logic, synthese library (Vol. 263). Alphen aan den Rijn: Kluwer.zbMATHGoogle Scholar
  45. Oaksford, M., & Chater, N. (1998). Rationality in an uncertain world: Essays in the cognitive science of human understanding. Hove: Psychology Press.CrossRefGoogle Scholar
  46. Pearl, J. (2000). Causality: Models, reasoning, and inferece. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  47. Pijnacker, J., Geurts, B., van Lambalgen, M., Buitelaar, J., & Hagoort, P. (2010). Exceptions and anomalies: An ERP study on context sensitivity in autism. Neuropsychologia, 48, 2940–2951.CrossRefGoogle Scholar
  48. Pinosio, R. (in prep.) A common core shared by logic programming and probabilistic causal models.Google Scholar
  49. Prakken, H., & Sergot, M. (1997). Dyadic deontic logic and contrary-to-duty obligations. In D. Nute (Ed.), Defeasible deontic logic (pp. 223–262). Berlin: Springer.Google Scholar
  50. Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations by error propagation. In D. Rumelhart, J. McClelland & PDP Research Group (Eds.), Parallel Distributed Processing (Vol 1. pp. 318–362). Cambridge: MIT Press.Google Scholar
  51. Sen, S., & Airiau, S. (2007). Emergence of norms through social learning. In Procs. of the 20th International Joint Conference on Artificial Intelligence—IJCAI (pp. 1507–1512).Google Scholar
  52. Shanahan, M. (2002). Reinventing Shakey. In J. Minker (Ed.), Logic-based artificial intelligence. Dordrecht: Kluwer.Google Scholar
  53. Shoham, Y. (1987). A semantical approach to non-monotonic logics. In Proceedings of the tenth international joint conference on artificial intelligence (IJCAI) (pp. 388–392).Google Scholar
  54. Shoham, Y., & Tennenholtz, M. (1997). On the emergence of social conventions: Modeling, analysis, and simulations. Artificial Intelligence, 94(1–2), 139–166.CrossRefzbMATHGoogle Scholar
  55. Sloman, S., & Lagnado, D. (2015). Causality in thought. The Annual Review of Psychology, 66, 1–25.CrossRefGoogle Scholar
  56. Stenning, K., & van Lambalgen, M. (2008). Human reasoning and Cognitive Science. Cambridge, MA: MIT Press.Google Scholar
  57. Stenning, K., & van Lambalgen, M. (2010). The logical response to a noisy world. In M. Oaksford (Ed.), Cognition and conditionals: Probability and logic in human thought (pp. 85–102). Oxford: Oxford University Press.CrossRefGoogle Scholar
  58. Stenning, K., & Varga, A. (2016). Many logics for the many things that people do in reasoning. In L. Ball & V. Thompson (Eds.), International Handbook of Thinking and Reasoning. Abingdon-on-Thames: Psychology Press.Google Scholar
  59. Stenning, K., Martignon, L., & Varga, A. (2017). Adaptive reasoning: integrating fast and frugal heuristics with a logic of interpretation. Decision.Google Scholar
  60. Tosatto, S. C., Boella, G., van der Torre, L., & Villata, S. (2012). Abstract normative systems: Semantics and proof theory. In G. Brewka, T. Eiter, & S. A. McIlraith (Eds.), Principles of knowledge representation and reasoning: Proceedings of the thirteenth international conference. AAAI Press.Google Scholar
  61. Towell, G. G., & Shavlik, J. W. (1994). Knowledge-based artificial neural networks. Artificial Intelligence, 70(1), 119–165.CrossRefzbMATHGoogle Scholar
  62. Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131.CrossRefGoogle Scholar
  63. Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90(4), 293.CrossRefGoogle Scholar
  64. van der Torre, L. (1997). Reasoning about obligations. PhD thesis, Erasmus University Rotterdam.Google Scholar
  65. van der Torre, L., & Tan, Y. (1999). Deontic update semantics. In P. McNamara & H. Prakken (Eds.), Norms, logics and information systems. new studies on deontic logic and computer science. Amsterdam: IOS Press.Google Scholar
  66. van der Torre, L. (2010). Deontic redundancy: A fundamental challenge for deontic logic. In Deontic Logic in Computer Science, 10th International Conference ( \(\Delta\) EON 2010).Google Scholar
  67. van Lambalgen, M., & Hamm, F. (2004). The proper treatment of events. Oxford: Blackwell.Google Scholar
  68. Varga, A. (2013). A formal model of infants’ acquisition of practical knowledge from observation. PhD thesis, Central European University, Budapest.Google Scholar
  69. von Wright, G. H. (1951). Deontic logic. Mind, 60, 1–15.CrossRefGoogle Scholar
  70. Weston, J., Chopra, S., & Bordes, A. (2014). Memory networks. CoRR arXiv:abs/1410.3916.

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Tarek R. Besold
    • 1
  • Artur d’Avila Garcez
    • 2
  • Keith Stenning
    • 3
  • Leendert van der Torre
    • 4
  • Michiel van Lambalgen
    • 5
  1. 1.Digital Media Lab, Center for Computing and Communication Technologies (TZI)University of BremenBremenGermany
  2. 2.Department of Computer ScienceCity University LondonLondonUK
  3. 3.School of InformaticsUniversity of EdinburghEdinburghScotland, UK
  4. 4.Computer Science and CommunicationUniversity of LuxembourgEsch-sur-AlzetteLuxembourg
  5. 5.Faculty of Humanities, Logic and LanguageUniversity of AmsterdamAmsterdamThe Netherlands

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