Abstract
The knowledge representation and reasoning of both humans and artificial systems often involves conditionals. A conditional connects a consequence which holds given a precondition. It can be easily recognized in natural languages with certain key words, like “if” in English. A vast amount of literature in both fields, both artificial intelligence and psychology, deals with the questions of how such conditionals can be best represented and how these conditionals can model human reasoning. On the other hand, findings in the psychology of reasoning, such as those in the Suppression Task, have led to a paradigm shift from the monotonicity assumptions in human inferences towards nonmonotonic reasoning. Nonmonotonic reasoning is sensitive for information change, that is, inferences are drawn cautiously such that retraction of previous information is not required with the addition of new information. While many formalisms of nonmonotonic reasoning have been proposed in the field of Artificial Intelligence, their capability to model properties of human reasoning has not yet been extensively investigated. In this paper, we analyzed systematically from both a formal and an empirical perspective the power of formal nonmonotonic systems to model (i) possible explicit defeaters, as in the Suppression Task, and (ii) more implicit conditional rules that trigger nonmonotonic reasoning by the keywords in such rules. The results indicated that the classical evaluation for the correctness of inferences has to be extended in the three major aspects (i) regarding the inference system, (ii) the knowledge base, and (iii) possible assumed exceptions for the rule.
Similar content being viewed by others
Notes
The expression weak completion is used to denote the difference to completion processes that consider a mapping of undefined atoms to \(\bot \) (cp. Dietz et al. 2015).
For instance, it is without problem to introduce variables F and M to a logical system and, without stating any conditions that these may be mutually exclusive, have states of worlds or individuals for which, one, both, or neither of these variables are true, despite the fact that the user of the formal system intended to encode individuals to female or male with these variables.
That is, “in general one cannot be...”. One might have a key, have been (by accident or deliberately) locked in, have broken in, etc. But in the plausible situations, this rule and thus this connection between the variables holds.
References
Adams, E. (1965). The logic of conditionals. Inquiry, 8(1–4), 166–197.
Antoniou, G. (1997). Nonmonotonic reasoning. Cambridge, MA: MIT Press.
Baratgin, J., Over, D., & Politzer, G. (2014). New psychological paradigm for conditionals and general de finetti tables. Mind & Language, 29(1), 73–84.
Batchelder, W. H., & Riefer, D. M. (1999). Theoretical and empirical review of multinomial process tree modeling. Psychonomic Bulletin & Review, 6(1), 57–86.
Beierle, C., Eichhorn, C., & Kern-Isberner, G. (2016). Sceptical Inference based on C-representation and its characterization as a constraint satisfaction problem. In Proceedings of the 9th International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2016), Lecture Notes of Computer Science (Vol. 9616, pp. 65–82). Berlin, DE: Springer.
Beierle, C., & Kern-Isberner, G. (2014). Methoden wissensbasierter Systeme, 5. überarbeitete und erweiterte Auflage. Wiesbaden, DE: Springer Vieweg (in German).
Bochman, A. (2001). A logical theory of nonmonotonic inference and belief change. Berlin, DE: Springer.
Bonnefon, J. F., Da Silva Neves, R., Dubois, D., & Prade, H. (2006). Background default knowledge and causality ascriptions. In: G. Brewka, S. Coradeschi, A. Perini, & P. Traverso (Eds.), Proceedings of the 17th European Conference on Artificial Intelligence (ECAI 2006), Frontiers in Artificial Intelligence and Applications (Vol. 141, pp. 11–15). Amsterdam, NL: IOS Press.
Byrne, R. M. (1989). Suppressing valid inferences with conditionals. Cognition, 31, 61–83.
Byrne, R. M. (1991). Can valid inferences be suppressed? Cognition, 39(1), 71–78.
Byrne, R. M., Espino, O., & Santamaria, C. (1999). Counterexamples and the suppression of inferences. Journal of Memory and Language, 40(3), 347–373.
Da Silva Neves, R., Bonnefon, J. F., & Raufaste, E. (2002). An empirical test of patterns for nonmonotonic inference. Annals of Mathematics and Artificial Intelligence, 34(1–3), 107–130.
Dietz, E., Hölldobler, S., & Ragni, M. (2012). A computational approach to the suppression task. In N. Miyake, D. Peebles, & R. Cooper (Eds.), Proceedings of the 34th Annual Conference of the Cognitive Science Society (pp. 1500–1505). Austin, TX: Cognitive Science Society.
Dietz, E. A., & Hölldobler, S. (2015). A new computational logic approach to reason with conditionals. In: International conference on logic programming and nonmonotonic reasoning (pp. 265–278). New York: Springer.
Dubois, D., & Prade, H. (1996). Conditional objects as nonmonotonicConsequence relations. In Principles of knowledge representationand reasoning: Proceedings of the 4th international conference (KR’94) (pp. 170–177). San Francisco, CA: Morgan Kaufmann Publishers.
Dubois, D., & Prade, H. (2015). Possibility theory and its applications: Where do we stand?. Berlin: Springer.
Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2), 321–357.
de Finetti, B. (1974). Theory of probability (Vol. 1,2). New York, NY: Wiley.
Fitting, M. (1985). A Kripke–Kleene semantics for logic programs. Journal of Logic Programming, 2(4), 295–312.
Gabbay, D. (1985). Theoretical foundations for non-monotonic reasoning in expert systems. In K. R. Apt (Ed.), Logics and models of concurrent systems (pp. 439–457). New York, NY: Springer.
García, A. J., & Simari, G. R. (2004). Defeasible logic programming: An argumentative approach. Theory and Practice of Logic Programming, 4, 95–138.
Gazzo Castañeda, E. L., & Knauff, M. (2016). Defeasible reasoning with legal conditionals. Memory & Cognition, 44(3), 499–517. doi:10.3758/s13421-015-0574-7.
Goldszmidt, M., & Pearl, J. (1996). Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artificial Intelligence, 84(1–2), 57–112.
Halpern, J. Y. (2005). Reasoning about uncertainty. Cambridge, MA: MIT Press.
Hölldobler, S., & Kencana Ramli, C. D. (2009a). Logic programs under three-valued Łukasiewicz semantics. In P. M. Hill, & D. S. Warren (Eds.), Logic programming, 25th international conference, ICLP 2009, LNCS (Vol. 5649, pp. 464–478). Heidelberg: Springer.
Hölldobler, S., & Kencana Ramli, C. D. (2009b). Logics and networks for human reasoning. In C. Alippi, M. M. Polycarpou, C. G.Panayiotou, & G. Ellinas (Eds.), International conference on artificial neural networks, ICANN 2009, part II, LNCS (Vol. 5769, pp. 85–94). Heidelberg: Springer.
Johnson-Laird, P., & Byrne, R. (2002). Conditionals: A theory of meaning, pragmatics, and inference. Psychological Review, 109(4), 646–677.
Kern-Isberner, G. (2001). Conditionals in nonmonotonic reasoning and belief revision. No. 2087 in LNCS. Berlin, DE: Springer.
Kern-Isberner, G. (2004). A thorough axiomatization of a principle of conditional preservation in belief revision. Annals of Mathematics and Artificial Intelligence, 40, 127–164.
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).
Klauer, K. C., Singmann, H., & Kellen, D. (2015). Parametric order constraints in multinomial processing tree models: An extension of Knapp and Batchelder (2004). Journal of Mathematical Psychology, 64, 1–7.
Klauer, K. C., Stahl, C., & Erdfelder, E. (2007). The abstract selection task: New data and an almost comprehensive model. Journal of Experimental Psychology: Learning, Memory, and Cognition, 33(4), 680–703.
Kleene, S. C. (1952). Introduction to metamathematics. Amsterdam: Bibl. Matematica.
Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1–2), 167–207.
Kripke, S. A. (1972). Naming and necessity. Cambridge, MA: Harvard University Press.
Kuhnmünch, G., & Ragni, M. (2014). Can formal non-monotonic systems properly describe human reasoning? In P. Bello, M. Guarini, M. McShane & B. Scassellati (Eds.), Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp. 1222–1227). Austin, TX: Cognitive Science Society.
Lehmann, D. J., & Magidor, M. (1992). What does a conditional knowledge base entail? Artificial Intelligence, 55(1), 1–60.
Lewis, D. K. (1986). On the plurality of worlds. Hoboken, NJ: Blackwell Publishers.
Łukasiewicz, J. (1920). O logice trójwartościowej. Ruch Filozoficzny5, 169–171 (1920). English translation: On three-valued logic. In J. Łukasiewicz & L. Borkowski (Eds.) (1990). Selected works (pp. 87–88). Amsterdam: North Holland.
Łukasiewicz, T. (2005). Weak nonmonotonic probabilistic logics. Artificial Intelligence, 168(1–2), 119–161.
Makinson, D. (1994). General patterns in nonmonotonic reasoning. In D. M. Gabbay, C. J. Hogger, & J. A. Robinson (Eds.), Handbook of logic in artificial intelligence and logic programming (Vol. 3, pp. 35–110). New York, NY: Oxford University Press.
Oaksford, M., & Chater, N. (2007). Bayesian rationality: The probabilistic approach to human reasoning. Oxford: Oxford University Press.
Oaksford, M., & Chater, N. (2016). Probabilities, causation, and logic programming in conditional reasoning: Reply to Stenning and van Lambalgen. Thinking & Reasoning, 22(3), 336–354.
Pearl, J. (1990). System Z: A natural ordering of defaults with tractable applications to nonmonotonic reasoning. In R. Parikh (Ed.), Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning about Knowledge (TARK1990) (pp. 121–135). San Francisco, CA: Morgan Kaufmann Publishers Inc.
Pfeifer, N., & Kleiter, G. D. (2005). Coherence and nonmonotonicity in human reasoning. Synthese, 146(1–2), 93–109.
Politzer, G. (2005). Uncertainty and the suppression of inferences. Thinking & Reasoning, 11(1), 5–33.
Ragni, M., Eichhorn, C., & Kern-Isberner, G. (2016). Simulating human inferences in the light of new information: A formal analysis. In S. Kambhampati (Ed.). Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI’16) (pp. 2604–2610).
Ragni, M., & Knauff, M. (2013). A theory and a computational model of spatial reasoning with preferred mental models. Psychological Review, 120(3), 561–588.
Ragni, M., Singmann, H., & Steinlein, E. M. (2014). Theory comparison for generalized quantifiers. In P. Bello, M. Guarini, M. McShane & B. Scassellati (Eds.), Proceedings of the 36th Annual Conference of the Cognitive Science Society (pp. 1330–1335). Austin, TX: Cognitive Science Society.
Ramsey, F. P. (1929). General propositions and causality. In Philosophical papers (pp. 145–163). Cambridge, UK: Cambridge University Press.
Reiter, R. (1980). A logic for default reasoning. Artificial Intelligence, 13(1–2), 81–132.
Singmann, H., & Kellen, D. (2013). MPTinR: Analysis of multinomial processing tree models in R. Behavior Research Methods, 45(2), 560–575.
Skovgaard-Olsen, N., Singmann, H., & Klauer, K. C. (2016). The relevance effect and conditionals. Cognition, 150, 26–36.
Spohn, W. (1988). Ordinal conditional functions: A dynamic theory of epistemic states. In Causation in decision, belief change and statistics: Proceedings of the Irvine Conference on Probability and Causation, the Western Ontario Series in Philosophy of Science (Vol. 42, pp. 105–134). Dordrecht, NL: Springer.
Spohn, W. (2012). The laws of belief: Ranking theory and its philosophical applications. Oxford, UK: Oxford University Press.
Stenning, K., & Lambalgen, M. (2008). Human reasoning and cognitive science. Cambridge, MA: MIT Press.
Thorn, P. D., Eichhorn, C., Kern-Isberner, G., & Schurz, G. (2015). Qualitative probabilistic inference with default inheritance for exceptional subclasses. In C. Beierle, G. Kern-Isberner, M. Ragni, & F. Stolzenburg (Eds.), Proceedings of the 5th Workshop on Dynamics of Knowledge and Belief (DKB-2015) and the 4th Workshop KI & Kognition (KIK-2015) co-located with 38th German Conference on Artificial Intelligence (KI-2015), CEUR Workshop Proceedings (Vol. 1444).
Wason, P. C. (1968). Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20(3), 273–281.
Acknowledgements
This work is supported by DFG-Grants KI1413/5-1 to G. Kern-Isberner and by RA1934 2/1, RA 1934 4/1 and Heisenberg DFG fellowship RA1934 3/1 to M. Ragni. T. Bock and C. Eichhorn are supported by Grant KI1413/5-1 and A. P. P. Tse by RA1934 2/1. The authors would like to thank Richard Niland and Daniel Lux for discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ragni, M., Eichhorn, C., Bock, T. et al. Formal Nonmonotonic Theories and Properties of Human Defeasible Reasoning. Minds & Machines 27, 79–117 (2017). https://doi.org/10.1007/s11023-016-9414-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11023-016-9414-1