# PROBABILITY AND CONDITIONALS: BELIEF REVISION AND RATIONAL DECISION.

PROBABILITY AND CONDITIONALS: BELIEF REVISION AND RATIONAL DECISION. Edited by ELLERY EELLS and BRIAN SKYRMS. Cambridge University Press, 1994. Pp. vii, 207

This collection of essays is a Festschrift for Ernest W. Adams, and is based on a symposium that was held in his honor in 1993. As the title suggests, most of the essays focus on probability and the logic of conditionals, and the relationship between them; they draw their inspiration from Adams's seminal work on the subject. As a computer scientist, I was struck by just how much the topics discussed play a major role in much recent work in computer science, and how relevant much recent work in computer science is to some of the issues discussed here.

The volume starts and ends with contributions by Patrick Suppes, who was Adams's thesis advisor. (In fact, Adams was Suppes's first student.) Suppes' first contribution, "Some questions about Adams' conditionals," considers some issues related to Adams's probabilistic analysis of conditionals. The last contribution, "A brief survey of Adams' contributions to philosophy," does just what its title says.

The central theme of six of the essays is the analysis of the CCCP--the conditional construal of conditional probability. A reader coming to this topic for the first time is best advised to start by reading "The hypothesis of the conditional construal of conditional possibility," by Alan Hajek and Ned Hall, the sixth essay in the volume. It lays out the central problem very clearly. Writing A [is greater than] B for the conditional "if A then B", the CCCP roughly states that P(A [is greater than] B) = P(B|A); that is, the probability of the conditional is just the conditional probability. Hajek and Hall describe why the problem is of interest, the reasons to believe CCCP, and the reasons to be suspicious of it. To my mind, the reasons to believe the hypothesis all boil down to the "it sounds right" argument; as van Fraassen pointed out back in 1976, it seems that the probability of the assertion "if I throw an even number then it will be a six" seems to be precisely the conditional probability of six given that I throw an even number. All the other reasons to believe the CCCP seem to me to appeal to intuitions based on this line of reasoning.

As is well known, this reasoning is completely correct if (and only if) A is probabilistically independent of A [is greater than] B. (Of course, to make sense of this statement, we need an underlying formal model of conditionals and probability. For the statement to be true, we must also assume that conditionals satisfy some standard properties. I take for granted in the remainder of this discussion that we have such models and that they satisfy the standard assumptions.) It seems that for many examples that arise in practice, we can indeed assume that A is probabilistically independent of A [is greater than] B. To my mind, that is what accounts for the intuitive appeal of the CCCP. However, there are well-known examples of situations where A and A [is greater than] B are not independent. Perhaps the best known is Newcomb's problem (Nozick 1969).(1)

While these examples may seem to settle the issue, they have not stopped various attempts to prove positive and negative results. The positive results involve coming up with variants of the CCCP that are valid; the negative results involve proving triviality results, showing that proposed variants of the CCCP hold only in trivial models. (The precise notion of "trivial" varies from paper to paper.) The remainder of the Hajek and Hall paper, as well as the following two essays by Hajek and Hall individually ("Triviality on the cheap?" and "Back in the CCCP") are devoted to proving various such triviality results. Three essays that come earlier in the collection ("Adams conditionals," by Brian Skyrms, "Letter to Brian Skyrms," by Robert Stalnaker, and "Conditionals as random variables," by Robert Stalnaker and Richard Jeffries) are devoted to proving positive results. The key idea in all of these contributions is that of viewing conditional chances (in the case of Skyrms) and conditional sentences (in the case of Stalnaker and Jeffrey) as random variables on possible worlds.

The fifth essay, "From Adams' conditionals to default expressions, causal conditions, and counterfactuals," is by a computer scientist, Judea Pearl. It takes as its point of departure work on defeasible and non-monotonic reasoning in the Al community (McCarthy 1980, Reiter 1980). The goal of this work is to provide semantics for statements like "birds typically/ normally/by default fly." We want to be able to conclude from such statements that, for example, if all we know about Tweety is that Tweety is a bird, then Tweety flies, while still allowing it to be consistent that Oscar the bird does not fly.

Many different approaches have been suggested for capturing default reasoning. Recently, a number of approaches have been suggested--based on preference orders (Kraus, Lehmann, and Magidor 1990), Spohn's (1988) ordinal conditional functions, nonstandard analysis (Lehmann and Magidor 1992), possibility measures (Dubois and Prade 1991), and limiting probabilities (called e-semantics in the literature) (Pearl 1989)--that are all characterized by the same axioms, ones originally introduced by Adams (1966, 1975) for conditional logic.(2) In retrospect, this should not be surprising. A defeasible statement like "birds typically fly" can be viewed as the conditional statement "if it is a bird then it flies." Indeed, Pearl's [Epsilon]-semantics is precisely Adams's semantics for conditionals. Adams's axioms are well known to license only a few of the conclusions we would hope to derive. For example, they do not allow transitivity: if as are typically bs and bs are typically cs, we cannot conclude that as are typically cs. Pearl discusses approaches for going beyond Adams's axioms that would allow such conclusions, including techniques involving maximum entropy, for example. He also connects these ideas to counterfactuals and belief revision. Perhaps it is worth mentioning here that Pearl has more recently done work on the probabilistic evaluation of counterfactual queries (Balke and Pearl 1994; see also Pearl 2000), work that is quite related to the concerns of Skyrms's article in this volume.

The ninth essay, "The Howson-Urbach proofs of Bayesian principles," by Charles Chihara, criticizes the arguments presented by Howson and Urbach (1989) in defense of the Bayesian approach, in particular, the Principle of Conditionalization. Part of the problem, as Chihara notes, is the lack of a formal statement of the Principle. What becomes clear reading Chihara's essay is that another part of the problem is the lack of good models for representing how knowledge and beliefs change over time, as new information is acquired. I believe that some of the models used in computer science--while intended to model simpler settings than those typically considered in the philosophy literature--may help in this regard. At the risk of pointing too much to my own work, I direct readers to the models of runs and systems in (Fagin et al. 1995, chap. 11), which are applied in a probabilistic setting in (Halpern and Tuttle 1993).

The tenth essay, "Learning the Impossible," by Vann McGee, considers the problem of updating on evidence that has prior probability 0. He considers two approaches, one based on Popper functions, the other based on nonstandard probabilities, and proves them equivalent. Interestingly, as I mentioned earlier, nonstandard probabilities arise in a very similar way in the analysis of default reasoning (Lehmann and Magidor 1992).

I'd like to conclude this review on a more personal note. I first met Ernest Adams in 1994. We discovered that we were using very similar technical approaches to solve problems that were seemingly unrelated. I was working with some colleagues on finding approaches for going from statistical information to degrees of belief (this work is reported in Bacchus et al. 1996); as it happens, our results also have relevance to nonmonotonic reasoning and conditional logic, although that wasn't our primary motivation). Ernest was working with his son Jim, who has a Ph.D. in political science and at that time was a postdoctoral fellow, on extending some of the ideas in his approach to conditional logic to dealing with some voting anomalies in political theory. Among other things, the hope was to get a notion of conditional logic that allowed transitivity. In both cases the key idea was to assume a uniform prior on a set of states. (As shown in Bacchus et al. 1996, this idea is closely related to Pearl's use of maximum entropy to achieve the same effect.) In the course of discussing these issues, it became clear how many points of contact there are between the work in this area in AI and computer science and the work in philosophy. This volume, with its discussions of probability, conditionals, counterfactuals, belief revision, and default reasoning--all issues of concern in both communities-helps bring out the connections clearly.

(1) Recall that in Newcomb's problem, there are two opaque boxes. In Box 1, there is \$1,000. Box 2 either contains \$1,000,000 or is empty. You can choose just Box 2 or both Box 1 and Box 2. You get the contents of the box(es) that you choose. It seems obvious that you should choose both boxes, but there is a catch. A demon, who you believe can predict your behavior exactly (or, at least, extremely well), has chosen what to put in Box 2. The demon put \$1,000,000 in Box 2 exactly if he expected you to take only Box 2; otherwise he left Box 2 empty. The question here is whether "I choose both boxes" is independent of "there is \$1,000,000 in Box 1, given that I choose both boxes."

(2) For yet another approach that provides an explanation of why all these approaches are characterized by the same axioms, see (Friedman and Halpern 1998).

References

Adams, E. 1966. "Probability and the Logic of Conditionals." In Aspects of Inductive Logic, ed. J. Hintikka and P. Suppes. Amsterdam: North Holland.

--. 1975. The Logic of Conditionals. Dordrecht: D. Reidel.

Bacchus, F., A.J. Grove, J. Y., Halpern, and D. Koller. 1996. "From Statistical Knowledge Bases to Degrees of Belief." Artificial Intelligence 87:75-143.

Balke, A., and J. Pearl. 1994. "Probabilistic Evalution of Counterfactual Queries." Proceedings of the Twelfth National Conference on Artificial Intelligence (AAAI), 1994, 230-37.

Dubois, D., and H. Prade. 1991. "Possibilistic Logic, Preferential Models, Non-monotonicity and Related Issues." Proceedings of the Twelfth International Joint Conference on Artificial Intelligence (IJCAI), 1991, 419-24.

Fagin, R., J. Y. Halpern, Y. Moses, and M. Y. Vardi. 1995. Reasoning about Knowledge. Cambridge: MIT Press.

Friedman, N., and J. Y. Halpern. 1998. "Plausibility Measures and Default Reasoning." Forthcoming in Journal of the ACM.

Halpern, J. Y., and M. R. Tuttle. 1993. "Knowledge, Probability, and Adversaries." Journal of the ACM 40:917-62.

Howson, C., and P. Urbach. 1989. Scientific Reasoning: The Bayesian Approach. La Salle, Ill.: Open Court.

Kraus, S., D. Lehmann, and M. Magidor. 1990. "Nonmonotonic Reasoning, Preferential Models and Cumulative Logics." Artificial Intelligence 44:167-207.

Lehmann, D., and M. Magidor. 1992. "What Does a Conditional Knowledge Base Entail?" Artificial Intelligence 55:1-60.

McCarthy, J. 1980. "Circumscription--A Form of Non-monotonic Reasoning." Artificial Intelligence 13:27-39.

Nozick, R. 1969. "Newcomb's Problem and Two Principles of Choice." In Essays in Honor of Carl G. Hempel, ed. N. Rescher et al., 130-31. Dordrecht: D. Reidel.

Pearl, J. 1989. "Probabilistic Semantics for Nonmonotonic Reasoning: A Survey." In Proceedings of the First International Conference on Principles of Knowledge Representation and Reasoning (KR), 1989, ed. R.J. Brachman, H. J. Levesque, and R. Reiter, 505-16. Reprinted in Readings in Uncertain Reasoning, ed. G. Shafer and J. Pearl, 699-710 (San Francisco: Morgan Kaufmann, 1990).

--. 2000. Causality: Models, Reasoning; and Inference. New York: Cambridge University Press.

Reiter, R. 1980. "A Logic for Default Reasoning." Artificial Intelligence 13: 81-132.

Spohn, W. 1988. "Ordinal Conditional Functions: A Dynamic Theory of Epistemic States." In Causation in Decision, Belief Change, and Statistics, vol. 2, ed. W. Harper and B. Skyrms, 105-34. Dordrecht: D. Reidel.

JOSEPH Y. HALPERN

Cornell University, Computer Science