Source
Hansson, Logic of Belief Revision, LIT-1072, Stanford Encyclopedia of Philosophy, revision of 29 July 2026, read in full in NOTE-866 (sections 2, 5, 7.7–7.8 and 8). It is a survey: results are stated, not proved. Alchourrón, Gärdenfors and Makinson (1985), LIT-1050, read in NOTE-874, for Recovery over closed theories (Lemma 2.1).
What was actually shown
Static against dynamic equivalence. A base’s “elements represent beliefs that are held independently of any other belief or set of beliefs. Those elements of the belief set that are not in the belief base are “merely derived””. The bases {p, q} and {p, p ↔ q} “have the same logical closure. They are therefore statically equivalent, in the sense of representing the same beliefs.” Abe holds the first, Bob the second; both revise by ¬p, and “Abe believes that q whereas Bob believes that ¬q.” The example is checkable by hand: the only maximal subset of each base not implying p drops p and keeps the other element.
The belief-set reply. “(In belief set models, cases like these are taken care of by assuming that although Abe’s and Bob’s belief states are represented by the same belief set, this belief set is associated with different selection mechanisms in the two cases.)” The same difference can be carried by an ordering attached to the closed set (THEORY-353).
Recovery. Over a closed theory, even full meet contraction satisfies Recovery (LIT-1050, Lemma 2.1). “Recovery does not hold for partial meet contraction of belief bases.” In Levi’s coin example, contracting the base {p_1, …, p_n, c, h}, with h implying c, by c removes h, and adding c back does not restore h; on the closed set, Recovery would.
Inconsistent states. “There is only one inconsistent belief set (logically closed inconsistent set), namely the whole language.” Many inconsistent bases exist, so operations such as consolidation, external revision and semi-revision are defined on bases and have no counterpart on closed sets: “Once an inconsistent belief set has been obtained, all distinctions have been lost, and consolidation cannot restore them.”
What this does not say
- That a base records why a belief is held. A sentence is basic or derived, and support is logical implication from the base, computed, not stored. The survey does not discuss sources, derivations or defeated alternatives; truth maintenance systems appear once, as history.
- That bases are needed for the difference. For single revisions an ordering on the closed set does the same work. Which of the two retained states is smaller, or which an optimum resembles, is not addressed.
- That bases do better under iteration. The survey’s iterated-revision section treats sets and bases alike and gives no base-specific result. Most proposals there carry an ordering of worlds or an entrenchment relation.
- That rejecting Recovery is settled. For closed sets “The construction of a plausible operation of contraction for belief sets that does not satisfy Recovery is still an open issue.” The base characterizations the survey reports (Success, Inclusion, Relevance, Uniformity) are cited to Hansson 1999, which the record does not hold.
- That the split is foundationalism against coherentism. The survey says the distinction “should not be equated with that between foundationalism and coherentism.”
Connections
- For QUESTION-105 it narrows the one-step answer: two retained states of different form, a base or an ordering, each suffice for single revisions, and neither is provenance, dependencies or defeated alternatives.
- CLAIM-294 is the probabilistic form of the Abe and Bob case: same current conclusions, different revisions. CLAIM-295’s question, whether a record of justifications adds anything, is not answered here; bases widen the operations that stay well defined, which is not the same thing.