8.8 Combining Probability and Counterfactuals


8.8 Combining Probability and Counterfactuals

Subtleties similar in spirit to those discussed in Sections 6.2 and 8.2 that arise when combining knowledge and probability or knowledge and belief arise when combining other modalities. The combination of modalities brings into sharp focus the question of what "possible worlds" really are, and what their role is in the reasoning process. I conclude this chapter with a brief discussion of one more example—combining counterfactuals and probability.

To reason about both counterfactuals and probability requires that an agent have two different sets of possible worlds at a world w, say, Wpw and Wcw, where Wpw is used when doing probabilistic reasoning and Wcw is used for doing counterfactual reasoning. How are these sets related?

It seems reasonable to require that Wpw be a subset of Wcw—the worlds considered possible for probabilistic reasoning should certainly all be considered possible for counterfactual reasoning—but the converse may not hold. It might also seem reasonable to require, if a partial preorder is used to model similarity to w, that worlds in Wpw be closer to w than worlds not in Wpw. That is, it may seem that worlds that are ascribed positive probability should be considered closer than worlds that are considered impossible. However, some thought shows that this may not be so. For example, suppose that there are three primitive propositions, p, q, and r, and the agent knows that p is true if and only if exactly one of q or r is true. Originally, the agent considers two worlds possible, w1 and w2, and assigns each of them probability 1/2; the formula p q is true in w1, while p q is true in w2. Now what is the closest world to w1 where q is false? Is it necessarily w2? That depends. Suppose that, intuitively, w1 is a world where p's truth value is determined by q's truth value (so that p is true if q is true and p is false if q is false) and, in addition, q happens to be true, making p true as well. The agent may well say that, even though he considers it quite possible that the actual world is w2, where q is true, if the actual world were w1, then the closest world to w1 where q is not true is the world w3 where p q is true. It may be reasonable to take the closest world to w1 to be one that preserves the property that p and q have the same truth value, even though this world is considered impossible.

I leave it to the reader to consider other possible connections between Wpw and Wcw. The real moral of this discussion is simply that these issues are subtle. However, I stress that, given a structure for probability and counterfactuals, there is no difficulty giving semantics to formulas involving probability and counterfactuals.




Reasoning About Uncertainty
Reasoning about Uncertainty
ISBN: 0262582597
EAN: 2147483647
Year: 2005
Pages: 140

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