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Actions on belief. (English) Zbl 0803.03012

Drawing on earlier work by R. C. Moore, it is shown how to plausibly define various belief changing actions, such as finding that, finding whether, finding \(a\), and evaluating a term \(t\) in a certain multi-modal logic which combines epistemic and dynamic logic. These actions are completely characterized by their effects upon belief. Let BEL be the epistemic accessibility relation used to interpret a belief operator, and let ; be the composition of relations. An action \(A\) is information [ignorance] preserving iff \((\text{BEL}; A)\supseteq (A;\text{BEL})\) \([(\text{BEL}; A)\subseteq (A;\text{BEL})]\). The interpretation of finding that \(S\) in a model \(\mathcal M\) under an assignment \(g\) is defined as \(\bigcup \{X\mid (\text{BEL};[[S?]]; X)\supseteq(X; \text{BEL})\), where \([[S?]]\) is the interpretation of \(S\)? in \(\mathcal M\) under \(g\). The following can be shown: belief that if \(S\) is true, then after finding that \(S\) the formula \(U\) is true implies that after finding that \(S\) it is believed that \(U\) is true. The action finding whether is defined by \([[\mathbf{find whether} S]]= [[\text\textbf{find that} S]]\cup [[\text\textbf{find that}\neg S]]\), and finding \(a\) is defined by \([[\mathbf{find a} \Phi]]= \bigcup_{d\in \text{Dom}}[[\text\textbf{find that} \Phi(x)]][d/x]\). Evaluating a term \(t\) is defined by \([[\mathbf{evaluate} t]]= \bigcup_{d\in \text{Dom}} [[\text\textbf{find that} x= t]] [d/x]\). Another action that is defined in the paper is finding that \(S\) accepting a theory \(T\). The problems of axiomatizability and completeness are not addressed.

MSC:

03B60 Other nonclassical logic
68T27 Logic in artificial intelligence
03B45 Modal logic (including the logic of norms)
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