cricket download as we will explain in a moment 11 In the special case Y X completeness within Y reduces to completeness simpliciter and the conjunction of consistency and Cr closure can be replaced by the simple clause that Cr B 0 So the relation then reduces to the one defined in above The nonlooseness violation of the present credence belief relation is due to the fact that the relation never specifies which of two propositions p is to be believed in the binary sense except when Cr p is 0 or 1 This is so even if Cr p is much greater than or vice versa In response one might refine the relation by demanding that Cr B be as high as possible Formally for all Cr and all B This relation strikes a balance in the aforementioned tradeoff between the completeness of the belief set B and the overall credence in it Again if Y X the definition of can be simplified Completenesswithin Y reduces to completeness simpliciter and it is sufficient to require deductive closure rather than Cr closure So the definition of becomes the following Distance Based Relations Our next example is a variant of a proposal discussed in our earlier work 12 It captures the idea that binary beliefs should approximate degrees of belief as closely as possible subject to the constraint of consistency and deductive closure We need one preliminary definition For any belief set B define a membership function B X 0 1 where for each p in X Now we stipulate that a belief set B is coherent with a credence function Cr if and only if B s membership function best approximates Cr or in different words it minimizes the distance from Cr Formally for any Cr and any B define distance Cr B p X B p Cr p Then our credence belief relation is defined as follows for all Cr and all B In our earlier work we called this proposal the Hamming rule in light of its structural similarity to an equally named proposal in judgment aggregation theory It satisfies all of our conditions except locality and propositionwise certainty preservation The proposal does not satisfy functionality but displays violations of functionality only in special cases namely when several distinct belief sets equally minimize the distance from a given credence function Note that in the present construction we could also use other definitions of distance such as quadratic rather than absolute ones We have chosen the current definition of distance Cr B just for illustrative purposes An alternative variant of the present proposal replaces the idea of optimizing the distance between binary beliefs and degrees of belief with the idea of satisficing Here a belief set B is deemed coherent with a credence function Cr just in case B is sufficiently rather than maximally close to Cr Formally fix a tolerance parameter δ 1 Then Stability Theoretic Relations Our final example of a holistic credence belief relation builds on Hannes Leitgeb s work 13 Leitgeb proposes a way of relating degrees of belief to binary beliefs First he defines a proposition p X to be stable relative to a credence function Cr for short Cr stable if for any proposition q X that is consistent with p and satisfies Cr q 0 14 It is easy to see that Cr stable propositions always exist For instance any proposition p X with Cr p 1 is Cr stable The term stable refers to the fact that the credence in a non empty Cr stable proposition always exceeds 1 2 and continues to exceed 1 2 after Bayesian conditionalization on any other proposition that is consistent with it Now the credence belief relation can be defined as follows for all Cr and all B As Leitgeb notes on this proposal belief does indeed correspond to high enough degree of belief but what counts as high enough may depend on the credence function in question The credence belief relation we have just defined satisfies all of our conditions except locality and nonlooseness In that sense it captures a holistic relationship between degrees of belief and binary beliefs Leitgeb recognizes this holism which he describes as a strong form of sensitivity of belief to context 15 That there is such holism is not accidental at all As our impossibility result shows if there is to be any relation between degrees of belief and binary beliefs which satisfies universality belief consistency and deductive closure propositionwise certainty preservation and nonlooseness then it must be holistic A Further Generalization Up to this point we have assumed that the set X of propositions on which beliefs are held has the structure of an algebra i e it is a non empty set of propositions that is closed under both union and complement and thereby also intersection We now want to extend our analysis to more general sets of propositions The assumption that X is closed under union and intersection is somewhat restrictive For instance we might assign credences to propositions about the weather and to propositions about inflation without assigning credences to any conjunctions or disjunctions of those propositions In such a case the set X does not satisfy the given closure conditions We will show that our impossibility result does not depend on the assumption that X is an algebra but applies to any set of propositions that satisfies some less demanding structural conditions For a start let us assume that X is a non empty and for present purposes again finite set of propositions that is closed under complement i e for every p X we also have but not necessarily under conjunction and disjunction Recall that a credence function Cr assigns to each proposition p in X a number Cr p in the interval from 0 to 1 where this assignment is probabilistically coherent In the present case probabilistic coherence means that Cr is extendable to a probability function with the standard properties on an algebra

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