Skip to main content Accessibility help




Causal models defined in terms of structural equations have proved to be quite a powerful way of representing knowledge regarding causality. However, a number of authors have given examples that seem to show that the Halpern–Pearl (HP) definition of causality (Halpern & Pearl, 2005) gives intuitively unreasonable answers. Here it is shown that, for each of these examples, we can give two stories consistent with the description in the example, such that intuitions regarding causality are quite different for each story. By adding additional variables, we can disambiguate the stories. Moreover, in the resulting causal models, the HP definition of causality gives the intuitively correct answer. It is also shown that, by adding extra variables, a modification to the original HP definition made to deal with an example of Hopkins & Pearl (2003) may not be necessary. Given how much can be done by adding extra variables, there might be a concern that the notion of causality is somewhat unstable. Can adding extra variables in a “conservative” way (i.e., maintaining all the relations between the variables in the original model) cause the answer to the question “Is X = x a cause of Y = y?” to alternate between “yes” and “no”? It is shown that we can have such alternation infinitely often, but if we take normality into consideration, we cannot. Indeed, under appropriate normality assumptions. Adding an extra variable can change the answer from “yes’ to “no”, but after that, it cannot change back to “yes”.


Corresponding author



Hide All
Aleksandrowicz, G., Chockler, H., Halpern, J. Y., & Ivrii, A. (2014). The computational complexity of structure-based causality. In Brodley, C. E. and Stone, P., editors. Proceedings of Twenty-Eighth National Conference on Artificial Intelligence (AAAI ’14). Palo Alto, California: AAAI Press, pp. 974980.
Beer, I., Ben-David, S., Chockler, H., Orni, A., & Trefler, R. J. (2012). Explaining counterexamples using causality. Formal Methods in System Design, 40(1), 2040.
Blanchard, T., & Schaffer, J. (2013). Cause without default. In Beebee, H., Hitchcock, C., and Price, H., editors. Making a Difference. Oxford: Oxford University Press.
Cushman, F., Knobe, J., & Sinnott-Armstrong, W. (2008). Moral appraisals affect doing/allowing judgments. Cognition, 108(1), 281289.
Eberhardt, F. (2014). Direct causes and the trouble with soft intervention. Erkenntnis, 79(4), 755777.
Eiter, T., & Lukasiewicz, T. (2002). Complexity results for structure-based causality. Artificial Intelligence, 142(1), 5389.
Gerstenberg, T., & Lagnado, D. (2010). Spreading the blame: The allocation of responsibility amongst multiple agents. Cognition, 115, 166171.
Glymour, C., Danks, D., Glymour, B., Eberhardt, F., Ramsey, J., Scheines, R., Spirtes, P., Teng, C. M., & Zhang, J. (2010). Actual causation: A stone soup essay. Synthese, 175, 169192.
Hall, N. (2007). Structural equations and causation. Philosophical Studies, 132, 109136.
Halpern, J. Y. (2008). Defaults and normality in causal structures. In Brewka, G. and Lang, J., editors. Principles of Knowledge Representation and Reasoning: Proceedings of the Eleventh International Conference (KR ’08). Palo Alto, California: AAAI Press, pp. 198208.
Halpern, J. Y., & Hitchcock, C. (2010). Actual causation and the art of modeling. In Dechter, R., Geffner, H., and Halpern, J., editors. Causality, Probability, and Heuristics: A Tribute to Judea Pearl, pp. 383406. London: College Publications.
Halpern, J. Y., & Hitchcock, C. (2015). Graded causation and defaults. British Journal for the Philosophy of Science, 66(2), 413457.
Halpern, J. Y., & Pearl, J. (2001). Causes and explanations: A structural-model approach. Part I: Causes. In Breese, J. S. and Koller, D., editors. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI 2001). San Francisco, California: Morgan Kaufmann, pp. 194202.
Halpern, J. Y., & Pearl, J. (2005). Causes and explanations: A structural-model approach. Part I: Causes. British Journal for Philosophy of Science, 56(4), 843887.
Hiddleston, E. (2005). Causal powers. British Journal for Philosophy of Science, 56, 2759.
Hitchcock, C. (2001). The intransitivity of causation revealed in equations and graphs. Journal of Philosophy, XCVIII(6), 273299.
Hitchcock, C. (2007). Prevention, preemption, and the principle of sufficient reason. Philosophical Review, 116, 495532.
Hitchcock, C., & Knobe, J. (2009). Cause and norm. Journal of Philosophy, 106, 587612.
Hopkins, M. (2001). A proof of the conjunctive cause conjecture. Unpublished manuscript.
Hopkins, M., & Pearl, J. (2003). Clarifying the usage of structural models for commonsense causal reasoning. In Doherty, P., McCarthy, J., and Williams, M-A., editors. Proceedings of the AAAI Spring Symposium on Logical Formalization of Commonsense Reasoning. Palo Alto, California: AAAI Press.
Kahneman, D., & Miller, D. T. (1986). Norm theory: Comparing reality to its alternatives. Psychological Review, 94(2), 136153.
Knobe, J., & Fraser, B. (2008). Causal judgment and moral judgment: Two experiments. In Sinnott-Armstrong, W., editor. Moral Psychology: The Cognitive Science of Morality, Vol. 2, pp. 441447. Cambridge, MA: MIT Press.
Lagnado, D. A., Gerstenberg, T., & Zultan, R. (2013). Causal responsibility and counterfactuals. Cognitive Science, 37, 10361073.
Livengood, J. (2013). Actual causation in simple voting scenarios. Nous, 47(2), 316345.
Spohn, W. (2008). Personal email.
Strevens, M. (2008). Comments on woodward, making things happen. Philosophy and Phenomenology, 77(1), 171192.
Weslake, B. (2015). A partial theory of actual causation. British Journal for the Philosophy of Science. To appear.
Woodward, J. (2003). Making Things Happen: A Theory of Causal Explanation. Oxford, U.K.: Oxford University Press.




Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.