Relations of epistemic proximity for belief change
The author proposed in [Stud. Log. 102, No. 5, 955--980 (2014; Zbl 1339.03017)] a new belief change operation known as descriptor revision which has the most well-known change operations (namely, revision, contraction, multiple contraction, etc.) as special cases. This approach relies on a metalinguistic operator \(\mathfrak{B}\) which is interpreted as the belief predicate in the sense that, for any sentence \(p\) in the object language, \(\mathfrak{B} p\) means that \(p\) is believed. A descriptor is then defined as a set of truth-functional combinations of expressions of the form \(\mathfrak{B} p\) and the descriptor revision is presented as an operation which is such that the result of the descriptor revision of a belief set \(K\) by a descriptor \(\Psi\), denoted \(K \circ \Psi\), is a belief set that satisfies the (success) condition expressed by the descriptor \(\Psi\). In the mentioned paper, a class of descriptor revision operations -- namely the relational descriptor revisions -- was introduced. The definition of these functions makes use of a total ordering on the set composed by the belief sets which are outcomes of revision by some descriptor.NEWLINENEWLINEIn the paper under review, a certain class of binary relations on descriptors, called relations of epistemic proximity is formally introduced and a way of defining descriptor revisions by means of these relations is provided. One of the main results of the paper is the equivalence between the descriptor revisions based on relations of epistemic proximity introduced in this paper and the relational descriptor revisions mentioned above. Moreover, this fact is proven by presenting explicit constructions of a relation of epistemic proximity \(\succeq\) from a total ordering \(\leqq\) on the set of descriptor revision outcomes and vice versa, which are such that the descriptor revisions based on \(\succeq\) coincides with the relational descriptor revisions defined by means of \(\leqq\).NEWLINENEWLINEThe author observes that this result can be considered the analogous, in the context of descriptor revision, to the very well-known fact that the class of transitively relational partial meet contractions of \textit{C. E. Alchourrón} et al. [J. Symb. Log. 50, 510--530 (1985; Zbl 0578.03011)], which are defined by means of a relation on sets of sentences, coincides with the class of contraction functions proposed in [\textit{P. Gärdenfors}, Knowledge in flux. Modeling the dynamics of epistemic states. Cambridge, MA: MIT Press (1988; Zbl 1229.03008)] and [\textit{P. Gärdenfors} and \textit{D. Makinson}, in: Theoretical aspects of reasoning about knowledge, Proc. 2nd Conf., Pacific Grove/CA (USA). Los Altos, CA: Morgan Kaufmann. 83--95 (1988; Zbl 0711.03009)], which are defined by means of a relation on sentences that is called epistemic entrenchment relation.NEWLINENEWLINEFurthermore, in this paper two special classes of restrictions of the relations of epistemic proximity are studied in detail, namely the binary relations which result of restricting the relations of epistemic proximity to descriptors of the form \(\{\mathfrak{B} p\}\) and those which result of restricting the relations of epistemic proximity to descriptors of the form \(\{\neg \mathfrak{B} p\}\). In what follows, we briefly summarize the main results presented in the paper in this regard.NEWLINENEWLINEConsider a belief set \(K\) and let \(\succeq\) be a relation of epistemic proximity and \(\leqq\) be a total ordering on the set of outcomes of descriptor revisions of \(K\) which give rise to the same descriptor revision on \(K\), say \(\circ\). Now, consider the sentential operations \(\ast\) and \(-_{.}\) on \(K\) defined, for all sentences \(p\), by \(K \ast p = K \circ \{\mathfrak{B} p\}\) and \(K -_{.} p = K \circ \{\neg \mathfrak{B} p\}\), respectively. Furthermore, let \({\leqq}^{\ast}\), \({\leqq}^{-}\), \({\succeq}^{\ast}\) and \(\leq_{.}\) be the restrictions of \(\leqq\) and \(\succeq\) defined as follows: {\parindent=0.5cm\begin{itemize}\item[--] \({\leqq}^{\ast}\) is called additive restriction of \(\leqq\) and is the restriction of \(\leqq\) to those elements of its domain that can be obtained as the outcome of revising \(K\) by some descriptor of the form \(\{\mathfrak{B} p\}\); \item[--] \({\succeq}^{\ast}\) is called believability relation based on \(\succeq\) and is the relation on sentences such that \(p {\succeq}^{\ast} q\) iff \(\{\mathfrak{B} p\} \succeq \{\mathfrak{B} q\}\); \item[--] \({\leqq}^{-}\) is called subtractive restriction of \(\leqq\) and is the restriction of \(\leqq\) to those elements of its domain that can be obtained as the outcome of revising \(K\) by some descriptor of the form \(\{\neg \mathfrak{B} p\}\); \item[--] \(\leq_{.}\) is called entrenchment relation derived from \(\succeq\) and is the relation on sentences such that \(p \leq_{.} q\) iff \(\{\neg \mathfrak{B} p\} \succeq \{\neg \mathfrak{B} q\}\). NEWLINENEWLINE\end{itemize}} It was shown in [the author, loc. cit.] that \(\ast\) is a semirevision and \(-_{.}\) is a revocation. In the paper under review, the following results regarding the interrelations among these operations and the above described restrictions of \(\leqq\) and \(\succeq\) were obtained: {\parindent=0.6cm\begin{itemize}\item[(i)] \(\ast\) and \({\succeq}^{\ast}\) can be derived from \({\leqq}^{\ast}\), but neither of the converses of those derivations is possible. Furthermore, neither \(\ast\) nor \({\succeq}^{\ast}\) can be derived from the other. \item[(ii)] \({\leqq}^{-}\) and \(-_{.}\) are interdefinable and \(\leq_{.}\) can be derived from each of these. However, neither \({\leqq}^{-}\) nor \(-_{.}\) can be derived from \(\leq_{.}\). NEWLINENEWLINE\end{itemize}} Additionally, it is proven in this paper that the entrenchment relation \(\leq_{.}\) satisfies all the standard properties of epistemic entrenchment. For this reason, epistemic proximity can be seen as a generalization of epistemic entrenchment. However, although all revocation operators that are obtainable (as indicated above) from an operator of relational descriptor revision have an entrenchment relation associated with them, this is a different category of operations than those that were proposed by Gärdenfors and Makinson [loc. cit.] (which are defined by means of an epistemic entrenchment relation). The author clarifies this by highlighting that the contraction operators proposed in this paper cannot in general be reconstructed from its associated entrenchment relation. More precisely, it is shown that an entrenchment relation may be obtainable from different belief set orderings and, furthermore, an exact characterization of these belief set orderings is presented.
scientific article; zbMATH DE number 6362036