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Reapproaching Ramsey: conditionals and iterated belief change in the spirit of AGM. (English) Zbl 1233.03015
Summary: According to the Ramsey test, conditionals reflect changes of beliefs: $$\alpha > \beta$$ is accepted in a belief state iff $$\beta$$ is accepted in the minimal revision of it that is necessary to accommodate $$\alpha$$. Since P. Gärdenfors’s seminal paper [“Belief revision and the Ramsey test for conditionals”, Philos. Rev. 95, 81–93 (1986)], a series of impossibility theorems (“triviality theorems”) has seemed to show that the Ramsey test is not a viable analysis of conditionals if it is combined with AGM-type belief revision models. I argue that it is possible to endorse that Ramsey test for conditionals while staying true to the spirit of AGM. A main focus lies on AGM’s condition of preservation according to which the original belief set should be fully retained after a revision by information that is consistent with it. I use concrete representations of belief states and (iterated) revisions of belief states as semantic models for (nested) conditionals. Among the four most natural qualitative models for iterated belief change, two are identified that indeed allow us to combine the Ramsey test with preservation in the language containing only flat conditionals of the form $$\alpha > \beta$$. It is shown, however, that preservation for this simple language enforces a violation of preservation for nested conditionals of the form $$\alpha > (\beta > \gamma )$$. In such languages, no two belief sets are ordered by strict subset inclusion. I argue that it has been wrong right from the start to expect that preservation holds in languages containing nested conditionals.

##### MSC:
 03A05 Philosophical and critical aspects of logic and foundations 03B42 Logics of knowledge and belief (including belief change) 03B48 Probability and inductive logic 68T27 Logic in artificial intelligence
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##### References:
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