A solution to commutativity problems found in iterated belief revision

This paper mainly addresses the commutativity issue discovered in iterated belief revisions under the framework of AGM. Namely, given an iterated formula K ∗ x ∗ y, under what condition K ∗ x ∗ y = K ∗ (x ⁁ y) = K ∗ y ∗ x can hold needs to be reconsidered. As we shall demonstrate, AGM 7th and 8th postulates together falsely imply the commutativity existing in Rott's selection puzzle. On the other hand, Nayak et al.'s counteracting problem which demands commutativity is rejected by AGM postulates. The worse thing is the two realistic puzzles cannot be correctly simulated by AGM postulates under classic logics. Towards these issues, we suggest the inaccurate definition of consequence operation C
n
needs to be modified in order to accommodate part of Rott's puzzle. Furthermore, we illustrated neither of the two puzzles can be simulated under the AGM framework in the content level. Therefore, we present our approach as follows: first, we built a belief revision framework which satisfies the first six AGM postulates in the content level. Second, by setting appropriate OCF, our framework is able to accommodate the two puzzles in epistemic state level. Third, we prove the commutativity can be held in the epistemic state level if and only if (K ∗ x) ∩ (K ∗ y) ≠ φ and (K ∗ x) ∗ (K ∗ y) ├/┴ and thus eliminate the intrusion by the selection puzzle. Last, we demonstrate the hold of commutativity is only a special case of the selection criterion we suggest, that is, the most explainable interpretation(s) that satisfy the most recent evidence should be chosen to be the final revision result. In this sense, we can tell in the counteracting problems, namely, when only (K ∗ x) ∩ (K ∗ y) ≠ φ is satisfied, why the revision result had better be represented by K ∗ (x ∗ y) is because the interpretation(s) suggested in our selection criterion is only guaranteed to be in K ∗ (x ∗ y) rather than its iterated form. Hence, we unify both of puzzles with the modified AGM framework.