Abstract
This is the first of two chapters devoted to problems in the currently dominant model of belief change (the AGM model) that justify the development of alternative frameworks for belief change. In this chapter the focus is on the selection mechanisms that are used to determine which previous beliefs are retained and which are given up in operations of change. In the AGM model, such epistemic choices are assumed to be performed in two steps (the select-and-intersect method). First, a selection is made among a set of logically infinite objects of choice (remainders or possible worlds) that are not themselves representations of plausible belief states. This is followed by a second step in which the outcome is obtained by intersecting the objects that were chosen in the first step. The plausibility of this sequence of operations is critically examined, and the formal limits to its applicability are also pointed out. One of its problems is that epistemic choiceworthiness is not in general preserved under intersection. It is argued that epistemic choices should instead be represented as choices directly among the potential outcomes of the operation of change, i.e., choices among potential belief states.
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Notes
- 1.
Arrow introduced choice functions in economics. He said: “We do not want to prescribe that C(S) contains only a single element; for example, S may contain two elements between which the chooser is indifferent.” [7, p. 4]. At that time choice functions were already used in logic, but the standard definition in logic was different. A choice function for a set \(\mathfrak {X}\) of non-empty sets was defined as a function C such that \(C(X)\in X\) for all \(X\in \mathfrak {X}\). [147] − On the use of choice functions in logic, see also [138].
- 2.
A few studies have been devoted to indeterministic belief change operations. These are operations that deliver, for each input, a set that may contain more than one possible outcome [66, 169].
- 3.
This difference would seem to have implications for the view that the use of choice functions in both areas reveals an underlying unity between practical and theoretical reasoning. On that view, see [205, 215, 217].
- 4.
See Section 4.4 for a formal characterization of the preservation of success conditions under intersection of belief sets.
- 5.
On the implausibility of maxichoice contraction of belief sets, see also [1, 99, pp. 76–77], and [109, p. 33]. Maxichoice contraction is less implausible for belief bases (that are not logically closed) than for belief sets, see [175] and [99, p. 77].
- 6.
A language is syntactically finite if it has only a finite number of non-identical sentences. All syntactically finite languages are logically finite, but the converse does not hold. For instance, a language that contains the atom a and the conjunction sign is syntactically infinite since it contains the infinite set of sentences \( \{a,a \& a, a \& a \& a,\dots \}\). Contrary to logical finiteness, syntactic finiteness is a property of the language itself (rather than a property of the logic).
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Hansson, S.O. (2017). Inside the Black Box. In: Descriptor Revision. Trends in Logic, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-53061-1_2
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DOI: https://doi.org/10.1007/978-3-319-53061-1_2
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