Source
Alchourrón, Gärdenfors and Makinson (1985), LIT-1050, Journal of Symbolic Logic 50(2), read in full in NOTE-874 (Lemma 2.1, Observations 2.3–2.5, 3.1–3.4, 4.2–4.10, Corollary 4.5). Hansson, Logic of Belief Revision, LIT-1072, read in full in NOTE-866, for the standing of these results and of Recovery in the field since.
What was actually shown
The construction. For a theory A (a set closed under Cn) and a proposition x, A ⊥ x is the set of maximal subsets of A that do not imply x. A selection function γ picks a nonempty subset of them, and partial meet contraction keeps their intersection: A ∸ x = ⋂γ(A ⊥ x). Maxichoice (one remainder) is too large; its revision “will be complete whenever x is a proposition inconsistent with A”. Full meet (all remainders) is too small; its revision is just Cn(x).
The representation theorems. Over any theory A, ∸ is a partial meet contraction iff it satisfies closure, inclusion, vacuity, success, preservation and recovery (∸1)–(∸6) (Observation 2.5). A partial meet contraction satisfies the supplementary (∸7) and (∸8) iff γ is generated by a transitive relation over remainders (Corollary 4.5): “This result is the promised representation theorem for the collection of “basic” plus “supplementary” postulates”. Revision is defined by the Levi identity, A + x = Cn((A ∸ ¬x) ∪ {x}); that the revision postulates follow from the contraction postulates is cited to Gärdenfors, not proved here.
What the theory leaves open. Different selection functions give different contractions of the same A. In the finite case none is redundant: distinct selections always give distinct contractions (Observation 4.6). So what fixes a contraction is A together with γ, or under Corollary 4.5 a transitive relation over A’s remainders.
Recovery. Even full meet contraction satisfies recovery over a theory (Lemma 2.1): every ¬x ∨ a survives the contraction, so adding x back restores A.
What this does not say
- That the selection is explained. The paper defers what makes remainders “most important” to other work. The theorems say some γ exists for any operation obeying the postulates, not which γ is right.
- That it covers iterated change. γ is a selection function for one A. Nothing fixes the selection for the theory that results, and Hansson reports for iterated revision that “opinions differ widely on the adequacy of these proposals.”
- That the theorems hold for belief bases. The definitions are given for any set of propositions, but every theorem is for closed theories. Partial meet contraction of a base does not satisfy Recovery (THEORY-347).
- That the added state records reasons. γ is a preference over which remainders to keep. It records nothing about what a belief rested on, and the input to change is a sentence accepted outright, not evidence.
- That Recovery is right. It is assumed as a postulate and not defended. Hansson calls it “the most debated postulate of belief change”.
- That relational selection is characterized. Relational (not transitively relational) partial meet contraction has no representation by a condition on the contraction; the paper leaves it open.
Connections
- For QUESTION-105 it answers a one-step version: to support every single AGM contraction or revision of a fixed theory, a theory plus a transitive ordering of its remainders suffices, with nothing redundant in the finite case. It is silent on a family of revisions over time, and the state it keeps is none of provenance, dependencies or defeated alternatives.
- CLAIM-294 makes the probabilistic counterpart: current conclusions do not fix revision. Here the extra state is an ordering, which matches that claim’s caveat that a richer current state suffices.
- THEORY-347 is the belief-base alternative, which carries the same difference in which beliefs are basic rather than in an ordering.