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Towards Belief Contraction without Compactness

Last modified: 2018-09-24

#### Abstract

In the AGM paradigm of belief change the background logic is taken to be a supra-classical logic satisfying compactness among other properties. Compactness requires that any conclusion drawn from a set of propositions X is implied by a finite subset of X.There are a number of interesting logics such as Computational Tree Logic (CTL, a temporal logic) which do not possess the compactness property, but are important from the belief change point of view. In this paper we explore AGM style belief contraction in non-compact logics as a starting point, with the expectation that the resulting account will facilitate development of corresponding accounts of belief revision. We show that, when the background logic does not satisfy compactness, as long as the language in question is closed under classical negation and disjunction, AGM style belief contraction functions (with appropriate adjustments) can be constructed. We provide such a constructive account of belief contraction that is characterised exactly by the eight AGM postulates of belief contraction. The primary difference between the classical AGM construction of belief contraction functions and the one presented here is that while the former employs remainders of the belief being removed, we use its complements.

#### Keywords

AGM Theory; Belief Contraction; Compactness

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