Contribution
A survey, not a research paper. It sets out the AGM model (contraction, expansion, revision; partial meet and entrenchment-based contraction; Grove’s spheres), the debate over Recovery, belief bases and their operations, non-prioritized change, other operations (update, replacement, multiple, indeterministic, relevance-sensitive, resource-bounded, strength-changing), iterated change, alternative accounts (learning theory, dynamic doxastic logic, descriptor revision) and Bayesian change. Results are stated, not proved; the author’s own work (belief bases, descriptor revision, hyperreal probability) is prominent.
Key insight
A set of current conclusions, closed under consequence, does not fix how those conclusions should change. Which of them are held in their own right, and which only because others are, decides what survives a contraction and what a revision yields. The entry’s term for this is that two bases can be “statically equivalent” and not “dynamically equivalent”.
Assumptions
- Beliefs are sentences of a classical truth-functional language with a Tarskian consequence operator Cn (inclusion, monotony, iteration), supraclassical. “Sentences do not capture all aspects of belief, but they are the best general-purpose representation that is presently available.”
- Belief sets are closed, which is “clearly an unrealistic idealization, since it means that the agent is taken to be “logically omniscient””. Following Levi, a belief set is what one “is committed to believe”.
- Models are input-assimilating: “No explicit representation of time is included. Instead, the characteristic mathematical constituent is a function that, to each pair of a belief set and an input, assigns a new belief set.”
- Full belief only in sections 1–9: “In formal models of belief change, the belief set represents the set of full beliefs. Uncertain beliefs are not represented in these models.”
- Belief bases are in general “not (except as a limiting case) closed under logical consequence”, and the base results cited are for finite bases.
Key results
Belief sets vs belief bases (section 5)
- What bases keep. “In the approaches discussed above, all beliefs in the belief set are treated equally in the sense that they are all taken seriously as beliefs in their own right. However, due to its logical closure the belief set contains many elements that are not worth taking seriously.” 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”, i.e., they have no independent standing.”
- Derived beliefs fall with their support. “The underlying intuition is that the merely derived beliefs are not worth retaining for their own sake. If one of them loses the support that it had in basic beliefs, then it will be automatically discarded.”
- Bases carry more than their closure. “one and the same belief set can be represented by different belief bases. In this sense, belief bases have more expressive power than belief sets.” {p, q} and {p, p ↔ q} “have the same logical closure. They are therefore statically equivalent, in the sense of representing the same beliefs. On the other hand, the following example shows that they are not dynamically equivalent in the sense of behaving in the same way under operations of change. They can be taken to represent different ways of holding the same beliefs.” Abe holds p and q as basic, Bob p and p ↔ q. Both revise by ¬p: “Abe believes that q whereas Bob believes that ¬q.”
- How belief-set models answer this. “(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. Abe has a selection mechanism that gives priority to q over p ↔ q, whereas Bob’s selection mechanism has the opposite priorities.)” The difference can be carried by an ordering attached to the closed set instead of by a base.
- Inconsistent states. “There is only one inconsistent belief set (logically closed inconsistent set), namely the whole language. On the other hand there are, in any non-trivial logic, many different inconsistent belief bases. Therefore, belief bases make it possible to distinguish between different inconsistent belief states.” So consolidation (A! = A ÷ ⊥) has no counterpart for sets: “Once an inconsistent belief set has been obtained, all distinctions have been lost, and consolidation cannot restore them.” Likewise external revision, (A +′ p) ÷ ¬p, “does not exist for belief sets”, and semi-revision as (A +′ p)! “cannot be used for belief sets”.
- Not foundationalism vs coherentism. “In belief revision theory it has mostly been taken for granted that belief sets correspond to a coherentist epistemology, whereas belief bases represent foundationalism (Fuhrmann 1991, Nayak 1994b). However, the logical relationships among the elements of a logically closed set do not adequately represent epistemic coherence.” Coherentists “hardly mean this to apply to merely derived beliefs”. “Therefore, the distinction between operations on belief bases and operations on belief sets should not be equated with that between foundationalism and coherentism.”
- Base-generated contraction on sets (5.5) is characterised: an operation on a consistent belief set is generated by partial meet contraction on some finite base for it iff it satisfies Closure, Success, Inclusion, Vacuity, Extensionality, Finitude, Symmetry and Conservativity. Recovery is not among them.
Recovery and contraction (sections 2.1, 2.3, 5.2)
- Recovery: K ⊆ (K ÷ p) + p. “According to Recovery, so much is retained after p has been removed that everything will be recovered by reinclusion (through expansion) of p.” It is part of the AGM representation theorem for partial meet contraction on belief sets.
- “Recovery is the most debated postulate of belief change.” The counterexamples (Cleopatra’s children; George the criminal, mass murderer and shoplifter) are cases where what is reinstated should not bring back what came with it. Core-retainment, which looks weaker, implies Recovery for belief sets. Severe withdrawal avoids Recovery but is Expulsive. “The construction of a plausible operation of contraction for belief sets that does not satisfy Recovery is still an open issue.”
- “Recovery does not hold for partial meet contraction of belief bases.” Levi’s coin: base {p_1, …, p_n, c, h} with h implying c. Contracting by c removes h; reinserting c gives Cn({p_1, …, p_n, c}), “which does not contain h, as desired.” On the set, Recovery forces h back.
- Partial meet contraction on bases is characterised (Hansson 1999) by Success, Inclusion, Relevance and Uniformity. “The Relevance postulate has much the same function as Recovery has for belief sets, namely to prevent unnecessary losses of beliefs.” Kernel contraction (removing at least one element of every minimal p-implying subset) generalises it.
- Descriptor revision (9.3) also fails Recovery: “Notably, the recovery postulate that creates problems in the AGM framework does not hold in the descriptor revision framework.”
Iterated revision (section 8, with 7.8 and 10)
- “The preceding sections have only dealt with changes of one and the same belief set or belief base. This is clearly a severe limitation.”
- One AGM selection function can serve all belief sets once the definition of partial meet contraction is adjusted for the empty remainder set; the result is global contraction and revision that “does not impose any new properties on contraction or revision” (Hansson 2012).
- “However, most of the discussion on iterated change has been based on the assumption that such additional properties are plausible and indeed desirable.” The Darwiche–Pearl postulates DP1–DP4 express “that when we revise by a sentence p, then the ordering among p-worlds should be unchanged, and so should the ordering among ¬p-worlds.” “A large number of proposals for such shifts have been put forward, but opinions differ widely on the adequacy of these proposals.”
- Nayak (1994a) and Fermé and Rott (2004) take the belief state to be an entrenchment ordering: “from an entrenchment ordering ≤ and such a generalized input a new entrenchment ordering ≤′ is obtained that contains the information needed to construct the new belief set.”
- 7.8: “An operation of change can raise or lower the position of a sentence in an entrenchment ordering without affecting the belief set (but affecting how the belief state responds to new inputs).”
- Bayesian (10.1): conditioning on s gives p′(¬s) = 0, “and consequently p′ cannot be revised by ¬s”. Dyadic probability (10.2) can remove a full belief once, but “did not provide us with a new dyadic probability function to replace P. Therefore, iterated belief change with this method is not possible.”
- The entry does not say whether bases handle iteration better. Section 8 speaks of “any belief set (belief base)” alike and gives no base-specific iterated result.
Justifications, reasons, truth maintenance
- The only mention of truth maintenance is historical (1.1): “The truth maintenance systems developed by Jon Doyle (1979) were important in this development”, the development being models of database updating. Doyle 1979 is in the bibliography, unannotated. Doyle 2013 is cited only among models for resource-bounded agents (7.7).
- The entry never discusses recording why a belief is held, its source, its derivation or defeated alternatives. The base-level notion is binary: a sentence is basic (“held independently”) or derived. The “support that it had in basic beliefs” is logical implication from the base, computed, not stored.
- Sources appear only as inputs: merge “can represent a process of conflict resolution through selective combination of information from two agents or sources” (6.4). Time appears only in update (Katsuno and Mendelzon’s time-indexed sentences, 7.1) and in Governatori and Rotolo’s legal model (7.9).
- 7.7 ties bases to bounded resources: “The use of finite belief bases (section 5) can be seen as a way to avoid the operations on infinite belief sets that are standard in AGM.”
Small faults in the text
- 2.1 refers to “Section 6.5” for operations that fail Success; there is no 6.5 (shielded contraction is 6.3).
- 5.1 says “models such as AGM that use belief sets as representations of belief sets”, evidently meaning “of belief states”.
Bearing on the record
The words the record attributes to the entry are not in it. The LIT entry LIT-1072 says A10 cited it for the distinction “between belief states represented as closed sets” and “belief bases retaining additional structure”. Neither phrase occurs in the revision of 29 July 2026. They are A10’s paraphrase, and a fair one. The entry’s own words are “statically equivalent” versus “dynamically equivalent”, and beliefs “held independently” versus “merely derived”. The LIT summary (“keep the beliefs held in their own right apart from their consequences”) matches the entry.
On QUESTION-105: the entry gives the logical starting point the question expects, and narrows what that point is.
- It confirms that current conclusions underdetermine revision and names three things the closed set loses: which beliefs are basic (Abe and Bob), the difference between inconsistent states (consolidation), and the ability to contract without Recovery bringing back what rode on the retracted belief (Levi’s coin).
- It also shows the same revision behaviour can be kept without a base, by a selection mechanism or entrenchment ordering attached to the closed set (the Abe and Bob parenthetical; 2.2; 7.8; Nayak and Fermé–Rott in 8). So the logic offers at least two retained states of different form, a base and an ordering, for the same family of single revisions. Which is smaller, or which an optimum resembles, is not addressed. For iterated revision the literature it reports mostly keeps an ordering, and “opinions differ widely” on how the ordering should shift.
- A base is not provenance, dependencies or defeated alternatives in the question’s sense. It marks sentences as basic and recomputes everything else. It does not store what supports what, where a basic belief came from, or what was given up. The question’s candidate structures are nearer to Doyle’s truth maintenance systems, which the entry names once, historically, and does not discuss.
- The entry treats only full belief (and, in 10, credences). Its own Bayesian section shows a probability function can lose the ability to revise (p′(¬s) = 0, the dyadic case), which is the probabilistic face of the same problem.
On CLAIM-294: the entry gives a qualitative analogue, not support for the probabilistic construction. Abe and Bob agree on every belief and revise by ¬p to opposite conclusions about q. That is the same non-implication in full-belief logic, in a survey that attributes it to standard belief-base theory. It does not check or bear on CASE-093’s arithmetic, which is about marginals and conditionals. The entry’s parenthetical, that a belief-set model handles the case by a different selection mechanism on the same set, parallels the claim’s own caveat: a richer current state (there the joint, here the ordering) suffices.
On CLAIM-295: consistent with the first half, silent on the second.
- Read with the parenthetical and 5.5, base structure is information a suitably equipped current state can also carry. That fits the claim’s view that a record of history adds nothing for decisions a sufficient state covers; the entry does not frame it so.
- Where bases do better than sets, it is by widening the family of operations that remain well defined (inconsistent states, external revision, semi-revision, contraction without Recovery). That is close to the claim’s “wider family of decisions”, but it is an argument for a base, not for a record of justifications.
- 7.7 and the remark in 9.3 that remainders and possible worlds are “logically infinite (if the language is so), and therefore cognitively inaccessible” bear on the computational value the claim names: a finite base or a choice over outcomes is what a bounded agent can operate on. The entry asserts this; it gives no measure.
- Nothing here addresses justification graphs, sources, audit or option value, and nothing weighs storage against benefit.