Choice revision (Q2008635)

Let \(K\) be a set of propositional formulas, conceived of as a belief set. A choice revision operator \(\ast_c\) maps \(K\) and a finite set \(\{\varphi_1,\dots,\varphi_n\}\) of propositional formulas to a new set of formulas \(K\ast_c\{\varphi_1,\dots,\varphi_n\}\) that contains at least of one \(\varphi_1\), \dots, \(\varphi_n\) if \(n\geq 1\) -- so accepting at least one of those formulas as a new belief --, and is equal to \(K\) if \(n=0\). The paper first considers the case where \(\ast_c\) is determined by a \textit{relational model}, e.g., a pair \(( \mathbb{X},\leqq)\) where \(\mathbb{X}\) is a set of belief sets that contains \(K\), \(\leqq\) is an ordering of \(\mathbb{X}\) with \(K\) as a maximal element, and for all \(n>0\) and formulas \(\varphi_1\), \dots, \(\varphi_n\), if some member of \(\mathbb{X}\) contains at least of one \(\varphi_1\), \dots, \(\varphi_n\) then a unique \(\leqq\)-minimal member \(X\) of \(\mathbb{X}\) contains one of \(\varphi_1\), \dots, \(\varphi_n\); in that case, \(K\ast_c\{\varphi_1,\dots,\varphi_n\}\) is defined as \(X\), otherwise, \(K\ast_c\{\varphi_1,\dots,\varphi_n\}\) is defined as \(K\). The first main result of the paper is that, assuming there are only finitely many propositional atoms, a choice revision operator \(\ast_c\) is determined by a relational model iff it satisfies the following properties for all formulas \(\varphi\): \begin{itemize} \item[\(\ast\)-closure:] \(\mathrm{Cn}(K\ast\varphi) = K\ast\varphi\); \item[\(\ast\)-relative success:] if \(K\ast\varphi\neq K\) then \(\varphi\in K\ast\varphi\); \item[\(\ast\)-confirmation:] if \(\varphi\in K\ast\varphi\) then \(K\ast\varphi=K\); \item[\(\ast\)-regularity:] for all formulas \(\psi\) with \(\psi\in K\ast\varphi\), \(\psi\in K\ast\psi\); \item[\(\ast\)-reciprocity:] for all formulas \(\psi\) with \(\psi\in K\ast\varphi\) and \(\varphi\in K\ast\psi\), \(K\ast\varphi=K\ast\psi\). \end{itemize} Another result provides a similar equivalence with respect to the choice revision operators that satisfy \(\ast\)-closure, \(\ast\)-confirmation and \(\ast\)-reciprocity together with \begin{itemize} \item[\(\ast\)-success:] if \(A\neq\emptyset\) then \(A\cap(K\ast_c A)\neq\emptyset\); \item[\(\ast\)-vacuity:] \(K\ast_c\emptyset=K\); \item[\(\ast\)-consistency:] if \(A\) is consistent then \(K\ast_c A\) is consistent; \end{itemize} and the class of relational models that itself satisfies appropriate conditions. The second main result of the paper characterises again the choice revision operators that satisfy the first five conditions, but without assuming that the language contains finitely many propositional atoms only. It involves multi-believability relations, e.g., binary relations on the set of all finite sets of formulas. Given two finite sets of formulas \(X\) and \(Y\), write \(X\simeq_c Y\) for \(X\preceq_c Y\) and \(Y\preceq_c X\), and \(X\varowedge Y\) for \(\{\varphi\wedge\psi\mid\varphi\in X,\,\psi\in Y\}\). Then a multi-believability relation \(\preceq_c\) is said to determine a choice revision operator \(\ast_c\) iff for all finite sets \(A\) of formulas, \(K\ast_c A\) is defined as \(\bigl\{\varphi\mid A\simeq_c A\varowedge\{\varphi\}\bigr\}\) in case both \(A\preceq\emptyset\) and \(\emptyset\not\preceq A\), and it is defined as \(K\) otherwise. The result then reads: a choice revision operator \(\ast_c\) satisfies the first five conditions iff it is determined by a multi-believability relation \(\preceq_c\) that itself satisfies the following conditions, all involving finite sets of formulas: \begin{itemize} \item[\(\preceq_c\)-transitivity] If \(A\preceq_c B\) and \(B\preceq_c C\) then \(A\preceq_c C\); \item[\(\preceq_c\)-weak coupling] If \(A\simeq_c A\varowedge B\) and \(A\simeq_c A\varowedge C\) then \(A\simeq_c A\varowedge(B\varowedge C)\); \item[\(\preceq_c\)-counter dominance] If for every \(\varphi\in B\) there exists \(\psi\in A\) with \(\varphi\models\psi\) then \(A\preceq_c B\); \item[\(\preceq_c\)-minimality] \(A\preceq_c B\) for all \(B\) iff \(A\cap K\neq\emptyset\); \item[\(\preceq_c\)-union] \(A\preceq_c B\) or \(B\preceq_c A\). \end{itemize} In parallel with the first characterisation, another result provides a similar equivalence with respect to the set of choice revision operators that satisfy \(\ast\)-closure, \(\ast\)-confirmation, \(\ast\)-reciprocity, \(\ast\)-success, \(\ast\)-vacuity and \(\ast\)-consistency, and the class of multi-believability relations that itself satisfies appropriate extra conditions.