8.2 Knowledge and Belief


8.2 Knowledge and Belief

The previous section focused on a semantic characterization of belief. In this section, I consider an axiomatic characterization, and also examine the relationship between knowledge and belief.

Philosophers have long discussed the relationship between knowledge and belief. To distinguish them, I use the modal operator K for knowledge and B for belief (or Ki and Bi if there are many agents). Does knowledge entail belief; that is, does Kφ Bφ hold? (This has been called the entailment property.) Do agents know their beliefs; that is, do Bφ KBφ and Bφ KBφ hold? Are agents introspective with regard to their beliefs; that is, do Bφ BBφ and Bφ B Bφ hold? While it is beyond the scope of this book to go into the philosophical problems, it is interesting to see how notions like CONS, SDP, and UNIF, as defined in Section 6.2, can help illuminate them.

In this section, for definiteness, I model belief using plausibility measures satisfying Pl4. However, all the points I make could equally well be made in any of the other models of belief discussed in Section 8.1. Since I also want to talk about knowledge, I use epistemic belief structures, that is, epistemic plausibility structures where all the plausibility measures that arise satisfy Pl4. (See Exercise 8.7 for more on the relationship between using plausibility measures to model belief and using accessibility relations. The exercise shows that, if the set of worlds is finite, then the two approaches are equivalent. However, if the set of worlds is infinite, plausibility measures have more expressive power.)

If M = (W, 1, , n, 1, , n, π) is a measurable epistemic belief structure, then

where i (w) = (Ww, i, PIw, i). (In this chapter I follow the convention introduced in Chapter 2 of omitting the set of measurable sets from the description of the space when all sets are measurable.) The clause for Plw, i(Ww, i) = just takes care of the vacuous case where = according to Plw,i; in that case, everything is believed. This is the analogue of the case where i (w = ), when everything is vacuously known.

Analogues of CONS, UNIF, SDP, and CP can be defined in structures for knowledge and plausibility: simply replace i with i throughout. Interestingly, these properties are closely related to some of the issues regarding the relationship between knowledge and belief, as the following proposition shows:

Proposition 8.2.1

start example

Let M be an epistemic plausibility structure. Then

  1. if M satisfies CONS, then M Kiφ Biφ for all φ;

  2. if M satisfies SDP, then M Biφ KiBiφ and M Biφ Ki Biφ for all formulas φ;

  3. if M satisfies UNIF, then M Biφ BiBiφ and M Biφ Bi Biφ for all formulas φ.

end example

Proof See Exercise 8.9.

Thus, CONS gives the entailment property; with SDP, agents know their beliefs; and with UNIF, agents are introspective regarding their beliefs.




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

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