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QUESTION-289: Is an inquiry's current set of conclusions, or its logically closed belief set, enough to retain to fix its future revisions, and if not, what must a single AGM change of a closed theory have besides the theory?

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Properties5
StatusAnswered
Tagsepistemology, information-theory
Linenucleating-argument
RefinesQUESTION-105 — What is the least an evolving inquiry must retain, beyond its current conclusions and a closed belief set, which do not fix its revisions, to support a stated family of future revisions, and does the optimal retained state look like provenance, dependencies and defeated alternatives, or like something else?
Cited as “answers” byCLAIM-294 — Two inquiry histories can agree on every current conclusion and still call for different revisions of it, so a current marginal belief is not a sufficient state for updating on evidence about what it depended on, THEORY-347 — Belief bases with the same logical closure can revise to opposite conclusions on the same input, so a closed belief set does not fix its own revision; partial meet contraction of a base need not satisfy Recovery, and bases keep distinct inconsistent states apart, while a belief-set model reproduces the base difference only by attaching a different selection mechanism to the same set, THEORY-353 — For a logically closed theory, the contractions satisfying the six basic AGM postulates are exactly the partial meet contractions, and those satisfying the two supplementary postulates as well are exactly the ones whose selection of remainders is generated by a transitive relation; the theory alone does not fix the contraction, and the results concern one change of one fixed theory

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The question

QUESTION-105 asks for the least an inquiry must retain to support a family of future revisions. Its body names the lower edge: whether current conclusions alone are enough. This question asks that, and what the belief-revision literature adds for one change of full belief.

The answer

Three Active entries settle it, in the negative and then for one change:

  • CLAIM-294 answers for graded belief. Two joint distributions share P(H) = 0.9, and learning the same condition sends one to 1.0 and the other to 0.8. Keeping only the current marginal merges histories that need different updates.
  • THEORY-347 answers for full belief. The bases {p, q} and {p, p ↔ q} have one closure, yet revised by ¬p one yields q and the other ¬q. A closed belief set does not fix its own revision. A base, marking which beliefs are held in their own right, is one state that does.
  • THEORY-353 says what a closed theory needs instead. The contractions meeting the six basic AGM postulates are exactly the partial meet contractions, and with the two supplementary postulates exactly those whose selection of remainders is generated by a transitive relation. So a selection function or ordering must be added to the theory, and in the finite case distinct ones give distinct contractions. Revision follows by the Levi identity.

All three concern one change, not iterated revision. What the least retained state is for a family of revisions, and whether it looks like provenance, dependencies and defeated alternatives, remain QUESTION-105’s.

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