8.1 Belief


8.1 Belief

For the purposes of this section, I use "belief" in the sense that it is used in a sentence like "I believe that Alice is at home today." If p is believed, then the speaker thinks that p is almost surely true. Although I have spoken loosely in previous chapters about representing an agent's beliefs using, say, a probability measure, in this section, if belief is modeled probabilistically, an event is said to be believed iff it has probability 1. Typically, it is assumed that (1) beliefs are closed under conjunction (at least, finite conjunction) so that if Bob believes p1, , pn, then Bob also believes their conjunction, p1 pn and (2) beliefs are closed under implication, so that if Bob believes p, and p implies q, then Bob believes q.

Many ways of representing beliefs have been proposed. Perhaps the two most common are using probability, where an event is believed if it has probability 1 (so that Alice believes formula p in a probability structure M if μA(pM) = 1), and using epistemic frames, where Alice believes U at world w if A (w) U. Note that, in the latter case, the definition of belief is identical to that of knowledge. This is by design. The difference between knowledge and belief is captured in terms of the assumptions made on the relation. For knowledge, as I said in Section 6.1, the relation is taken to be reflexive. For belief it is not; however, it is usually taken to be serial. (The relation is also typically assumed to be Euclidean and transitive when modeling belief, but that is not relevant to the discussion in this section.)

A yet more general model of belief uses filters.Given a set W of possible worlds, a filter F is a nonempty set of subsets of W that (1) is closed under supersets (so that if U F and U U, then U F), (2) is closed under finite intersection (so that if U, U F, then U U F), and (3) does not contain the empty set. Given a filter F, an agent is said to believe U iff U F. Note that the set of sets that are given probability 1 by a probability measure form a filter; the events believed by agent i at world w in an epistemic frame (i.e., those events U such that i (w) U) are also easily seen to be a filter (Exercise 8.1). Similarly, given an epistemic frame, the events that are believed at world w (i.e., those sets U such that Ki (w) U) clearly form a filter. Conversely, if each agent's beliefs at each world are characterized by a filter, then it is easy to construct an epistemic frame representing the agent's beliefs: take i (w) to be the intersection of all the sets in agent i's filter at w. (This will not work in general in an infinite space; see Exercise 8.2.)

The use of filters can be viewed as a descriptive approach to modeling belief; the filter describes what an agent's beliefs are by listing the events believed. The requirement that filters be closed under supersets and under intersection corresponds precisely to the requirement that beliefs be closed under implication and conjunction. (Recall from Exercise 7.4 that M p q iff pM qM.) However, filters do not give any insight into where beliefs are coming from. It turns out that plausibility measures are a useful framework for getting a general understanding of belief.

Given a plausibility space (W, , PI), say that an agent believes U if Pl(U) > Pl(U); that is, the agent believes U if U is more plausible than not. It easily follows from Pl3 that this definition satisfies closure under implication: if U V and Pl(U) > Pl(U), then Pl(V) > Pl(V) (Exercise 8.3). However, in general, this definition does not satisfy closure under conjunction. In the case of probability, for example, this definition just says that U is believed if the probability of U is greater than 1/2. What condition on a plausibility measure Pl is needed to guarantee that this definition of belief is closed under conjunction? Simple reverse engineering shows that the following restriction does the trick:

I actually want a stronger version of this property, to deal with conditional beliefs. An agent believes U conditional on V, if given V, U is more plausible than U, that is, if Pl(U | V) > Pl(U | V). In the presence of the coherency condition CPl5 from Section 3.9 (which I implicitly assume for this section), if V , then Pl(U | V) > Pl(U | V) iff Pl(U V) > Pl(U V) (Exercise 8.4). In this case, conditional beliefs are closed under conjunction if the following condition holds:

Pl4 is somewhat complicated. In the presence of Pl3, there is a much simpler property that is equivalent to Pl4. It is a variant of a property that we have seen before in the context of partial preorders: the qualitative property (see Section 2.7).

In words, Pl4 says that if U0 U1 is more plausible than U2 and if U0 U2 is more plausible than U1, then U0 by itself is already more plausible than U1 U2.

Proposition 8.1.1

start example

A plausibility measure satisfies Pl4 if and only if it satisfies Pl4.

end example

Proof See Exercise 8.5.

Thus, for plausibility measures, Pl4 is necessary and sufficient to guarantee that conditional beliefs are closed under conjunction. (See Exercise 8.27 for other senses in which Pl4 is necessary and sufficient.) Proposition 8.1.1 helps explain why all the notions of belief discussed earlier are closed under conjunction. More precisely, for each notion of belief discussed earlier, it is trivial to construct a plausibility measure Pl satisfying Pl4 that captures it: Pl gives plausibility 1 to the events that are believed and plausibility 0 to the rest. Perhaps more interesting, Proposition 8.1.1 shows that it is possible to define other interesting notions of belief. In particular, it is possible to use a preference order on worlds, taking U to be believed if U s U. As Exercise 2.49 shows, s satisfies the qualitative property, and hence Pl4. Moreover, since possibility measures and ranking functions also satisfy the qualitative property (see Exercises 2.52 and 2.53), they can also be used to define belief. For example, given a possibility measure on W, defining belief in U as Poss(U) > Poss(U) gives a notion of belief that is closed under conjunction.

Pl4 is necessary and sufficient for beliefs to be closed under finite intersection, but it does not guarantee closure under infinite intersection. This is a feature: beliefs are not always closed under infinite intersection. The classic example is the lottery paradox.

Example 8.1.2

start example

Consider a situation with infinitely many individuals, each of whom holds a ticket to a lottery. It seems reasonable to believe that, for each i, individual i will not win, and yet to believe that someone will win. If Ui is the event that individual i does not win, this amounts to believing U1, U2, U3, and also believing iUi (and not believing iUi). It is easy to capture this with a plausibility measure. Let W = {w1, w2, }, where wi is the world where individual i wins (so that Ui = W {wi}). Let Pllot be a plausibility measure that assigns plausibility 0 to the empty set, plausibility 1/2 to all finite sets, and plausibility 1 to all infinite sets. Pllot satisfies Pl4 (Exercise 8.6); nevertheless, each event Ui is believed according to Pllot, asis iUi.

end example

The key property that guarantees that (conditional) beliefs are closed under infinite intersection is the following generalization of Pl4:

PL4*. For any index set I such that 0 I and |I| 2, if {Ui : i I} are pairwise disjoint sets, U = iI Ui, and Pl(U Ui) > Pl(Ui) for all i I {0}, then Pl(U0) > Pl(U U0).

Pl4 is the special case of Pl4* where I = {0, 1, 2}. Because Pl4* does not hold for Pllot, it can be used to represent the lottery paradox. On the other hand, Pl4* does hold for the plausibility measure corresponding to beliefs in epistemic frames; thus, belief in epistemic frames is closed under infinite conjunction. A countable version of Pl4* holds for σ-additive probability measures, which is why probability-1 beliefs are closed under countable conjunctions (but not necessarily under arbitrary infinite conjunctions). I defer further discussion of Pl4* to Section 10.4.




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

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net