There are relatively few proposals for inconsistency measures for propositional belief bases. However inconsistency measures are potentially as important as information measures for artificial intelligence, and more generally for computer science. In particular, they can be useful to define various operators for belief revision, belief merging, and negotiation. The measures that have been proposed so far can be split into two classes. The first class of measures takes into account the number of formulae required to produce an inconsistency: the more formulae required to produce an inconsistency, the less inconsistent the base. The second class takes into account the proportion of the language that is affected by the inconsistency: the more propositional variables affected, the more inconsistent the base. Both approaches are sensible, but there is no proposal for combining them. We address this need in this paper: our proposal takes into account both the number of variables affected by the inconsistency and the distribution of the inconsistency among the formulae of the base. Our idea is to use existing inconsistency measures in order to define a game in coalitional form, and then to use the Shapley value to obtain an inconsistency measure that indicates the responsibility/contribution of each formula to the overall inconsistency in the base. This allows us to provide a more reliable image of the belief base and of the inconsistency in it. ⇤ This paper is a revised and extended version of the paper ”Shapley Inconcistency Values ” presented at KR’06. 1 1

Abstract. The dynamics of beliefs is one of the major components of any autonomous system, that should be able to incorporate new pieces of information. In this paper we give a quick overview of the main operators for belief change, in particular revision, update, and merging, when the beliefs are represented in propositional logic. And we discuss some works on belief change in more expressive frameworks. 1

While a large social-choice-theoretic literature discusses the aggregation of individual judgments into collective ones, there is relatively little formal work on the transformation of individual judgments in group deliberation. I develop a model of judgment transformation and prove a baseline impossibility result: Any judgment transformation function satisfying some initially plausible condition is the identity function, under which no opinion change occurs. I identify escape routes from this impossibility result and argue that successful group deliberation must be ‘holistic’: individuals cannot generally revise their judgments on a proposition based on judgments on that proposition alone but must take other propositions into account too. I discuss the significance of these findings for democratic theory.

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There are relatively few proposals for inconsistency measures for propositional belief bases. However inconsistency measures are potentially as important as information measures for artificial intelligence, and more generally for computer science. In particular, they can be useful to define various operators for belief revision, belief merging, and negotiation. The measures that have been proposed so far can be split into two classes. The first class of measures takes into account the number of formulae required to produce an inconsistency: the more formulae required to produce an inconsistency, the less inconsistent the base. The second class takes into account the proportion of the language that is affected by the inconsistency: the more propositional variables affected, the more inconsistent the base. Both approaches are sensible, but there is no proposal for combining them. We address this need in this paper: our proposal takes into account both the number of variables affected by the inconsistency and the distribution of the inconsistency among the formulae of the base. Our idea is to use existing inconsistency measures (ones that takes into account the proportion of the language affected by the inconsistency, and so allow us to look inside the formulae) in order to define a game in coalitional form, and then to use the Shapley value to obtain an inconsistency measure that indicates the responsibility/contribution of each formula to the overall inconsistency in the base. This allows us to provide a more reliable image of the belief base and of the inconsistency in it.

Abstract not found

There are relatively few proposals for inconsistency measures for propositional belief bases. However inconsistency measures are potentially as important as information measures for artificial intelligence, and more generally for computer science. In particular, they can be useful to define various operators for belief revision, belief merging, and negotiation. The measures that have been proposed so far can be split into two classes. The first class of measures takes into account the number of formulae required to produce an inconsistency: the more formulae required to produce an inconsistency, the less inconsistent the base. The second class takes into account the proportion of the language that is affected by the inconsistency: the more propositional variables affected, the more inconsistent the base. Both approaches are sensible, but there is no proposal for combining them. We address this need in this paper: our proposal takes into account both the number of variables affected by the inconsistency and the distribution of the inconsistency among the formulae of the base. Our idea is to use existing inconsistency measures (ones that takes into account the proportion of the language affected by the inconsistency, and so allow us to look inside the formulae) in order to define a game in coalitional form, and then to use the Shapley value to obtain an inconsistency measure that indicates the responsibility/contribution of each formula to the overall inconsistency in the base. This allows us to provide a more reliable image of the belief base and of the inconsistency in it.

The theory of belief revision and merging has recently been applied to judgement aggregation. In this paper I argue that judgements are best aggregated by merging the evidence on which they are based, rather than by directly merging the judgements themselves. This leads to a threestep strategy for judgement aggregation. First, merge the evidence bases of the various agents using some method of belief merging. Second, determine which degrees of belief one should adopt on the basis of this merged evidence base, by applying objective Bayesian theory. Third, determine which judgements are appropriate given these degrees of belief by applying a decision-theoretic account of rational judgement formation.

While a large social-choice-theoretic literature discusses the aggre-gation of individual judgments into collective ones, there is relatively little formal work on the transformation of individual judgments in group deliberation. I develop a model of judgment transformation and prove a baseline impossibility result: Any judgment transfor-mation function satisfying some initially plausible conditions is the identity function, under which no opinion change occurs. I identify escape routes from this impossibility result and argue that success-ful group deliberation must be holistic: individuals cannot generally revise their judgments on a proposition based on judgments on that proposition alone but must take other propositions into account too. I discuss the signi…cance of these …ndings for democratic theory. 1

The theory of belief revision and merging has recently been applied to judgement aggregation. In this article I argue that judgements are best aggregated by merging the evidence on which they are based, rather than by directly merging the judgements themselves. This leads to a three-step strategy for judgement aggregation. First, merge the evidence bases of the various agents using some method of belief merging. Second, determine which degrees of belief one should adopt on the basis of this merged evidence base, by applying objective Bayesian theory. Third, determine which judgements are appropriate given these degrees of belief by applying a decision-theoretic account of rational judgement formation.