Traditional epistemology treats belief revision as a psychological process—something minds do, perhaps well or poorly, but not something governed by rigorous logical constraints. The Alchourrón-Gärdenfors-Makinson framework, developed in the 1980s, challenges this view fundamentally. AGM theory demonstrates that rational belief change admits precise formal characterization, that the space of permissible revisions is tightly constrained by a small number of rationality postulates.

The framework emerged from an unexpected source: legal theory. Carlos Alchourrón and David Makinson were investigating how legal codes change when statutes are repealed or amended. Peter Gärdenfors recognized that their formal apparatus applied to belief systems generally. The resulting theory provides representation theorems—mathematical proofs that any revision operation satisfying certain rationality conditions can be characterized by specific formal structures, and conversely.

For formal epistemologists, AGM theory occupies a peculiar position. It operates at the level of full belief—propositions accepted without qualification—rather than graded credences. This makes it both more tractable than Bayesian approaches and, arguably, less realistic. Understanding when and why these frameworks diverge illuminates fundamental questions about the nature of rational belief management. The tension between logical and probabilistic approaches to belief revision remains one of the most productive areas of contemporary formal epistemology.

AGM theory models an agent's belief state as a belief set K—a deductively closed set of propositions. Deductive closure means that if K contains propositions that logically entail some further proposition, K contains that proposition too. This idealization assumes logical omniscience: agents believe all logical consequences of their beliefs. The assumption is unrealistic but mathematically essential, separating the logic of belief revision from computational limitations.

The two primitive operations are expansion and contraction. Expansion K + φ adds a proposition φ to K along with all its logical consequences. This operation is straightforward: the expanded set is simply the deductive closure of K ∪ {φ}. No information is lost; the belief set only grows. Expansion models situations where new information is simply added to what we already believe, without creating inconsistency.

Contraction K ÷ φ removes a proposition φ from K while preserving consistency. This operation is far more complex. Simply deleting φ is insufficient—we must also remove enough other propositions to prevent φ from being rederived. The challenge is determining which propositions to remove. If I contract my belief that Socrates is mortal, must I abandon my belief that all humans are mortal, my belief that Socrates is human, or both?

The AGM postulates for contraction encode rationality constraints. Closure requires that K ÷ φ remain deductively closed. Inclusion demands that K ÷ φ ⊆ K—contraction never adds beliefs. Vacuity states that if φ ∉ K, then K ÷ φ = K—we cannot contract what we do not believe. Success requires that φ ∉ K ÷ φ when φ is not a logical truth—contraction actually removes the target belief. Recovery stipulates that K ⊆ (K ÷ φ) + φ—re-expanding by φ recovers the original belief set.

The recovery postulate has generated substantial controversy. Critics argue it is too strong: if I contract my belief that my keys are on the table, then add that belief back, I should not automatically recover my belief that I remember putting them there. The postulate conflates the informational content of propositions with their logical relationships. Various weakenings and alternatives have been proposed, generating a rich literature on the appropriate constraints for rational contraction.

Takeaway

Rational belief contraction is constrained by formal postulates that determine which propositions must be abandoned when retracting a belief—the choice is not arbitrary but governed by structural requirements on deductively closed belief sets.

Belief revision K * φ incorporates new information that may contradict existing beliefs. Unlike expansion, revision must handle inconsistency: when φ contradicts K, we cannot simply add φ without producing an inconsistent belief set. AGM theory's elegant solution is the Levi identity: revision reduces to contraction followed by expansion. Specifically, K * φ = (K ÷ ¬φ) + φ. First contract the negation of the new information, then expand by the new information itself.

The Levi identity embodies a principle of minimal mutilation: to accommodate new information inconsistent with current beliefs, remove just enough to make the new information consistent, then add it. The heavy lifting occurs in contraction—determining which beliefs to surrender when ¬φ must go. The representation theorems for contraction then induce corresponding results for revision.

The AGM revision postulates follow from the contraction postulates via the Levi identity, but they can be stated independently. Success requires φ ∈ K * φ—revision incorporates the new information. Consistency demands that K * φ be consistent when φ is consistent—revision does not introduce contradiction from consistent input. Inclusion states K * φ ⊆ K + φ—revision adds no beliefs beyond what expansion would add. Vacuity specifies that if ¬φ ∉ K, then K * φ = K + φ—revision coincides with expansion when no contradiction exists.

Two additional postulates govern iterated revision. Superexpansion: K * (φ ∧ ψ) ⊆ (K * φ) + ψ. Subexpansion: if ¬ψ ∉ K * φ, then (K * φ) + ψ ⊆ K * (φ ∧ ψ). These ensure that revising by a conjunction behaves coherently relative to revising by conjuncts separately. The postulates are jointly equivalent to a strong structural condition: the existence of a total preorder over possible worlds determining revision behavior.

The representation theorem establishes that any revision operation satisfying the AGM postulates can be characterized by a system of spheres—a nested family of sets of possible worlds centered on those compatible with current beliefs. Revision by φ selects the φ-worlds in the innermost sphere containing any φ-worlds. This possible worlds semantics provides both intuitive geometric interpretation and computational tractability, connecting the abstract postulates to concrete model-theoretic structures.

Takeaway

The Levi identity reveals that belief revision is not a primitive operation but decomposes into contraction followed by expansion—understanding how to rationally give up beliefs is the key to understanding how to rationally change them.

Bayesian epistemology and AGM theory both address rational belief change, but their formalisms differ fundamentally. Bayesianism models belief states as probability functions over propositions, with belief change governed by conditionalization: P'(φ) = P(φ|E) upon learning evidence E. AGM theory models belief states as sets of accepted propositions, with change governed by the rationality postulates. The Bayesian operates with graded credences; the AGM theorist with full beliefs.

One might expect these frameworks to be intertranslatable—perhaps AGM belief sets correspond to propositions above some credence threshold, with AGM revision corresponding to conditionalization on threshold-crossing propositions. This Lockean thesis fails in general. The lottery paradox demonstrates why: I may have high credence that ticket 1 will lose, high credence that ticket 2 will lose, and so on for each ticket individually, while having low credence that all tickets will lose. Threshold-based belief sets are not deductively closed.

A deeper divergence concerns commutativity. Bayesian conditionalization is order-independent: conditionalizing on E₁ then E₂ yields the same result as conditionalizing on E₂ then E₁ (given appropriate independence assumptions). AGM revision is not commutative in general. Revising first by φ then by ψ may differ from revising first by ψ then by φ. This reflects that AGM revision involves not just updating on new information but selecting which prior beliefs to retain—a selection that depends on the order of operations.

The frameworks complement rather than compete. AGM theory excels at modeling categorical acceptance and rejection—the kind of belief change involved in scientific theory change, legal reasoning, and database update. Bayesianism excels at modeling graded uncertainty—the kind of belief change involved in prediction, decision-making under uncertainty, and evidence accumulation. Which framework applies depends on whether the domain admits meaningful probability assignments and whether reasoning requires tracking degrees of confidence.

Recent work explores bridges between the frameworks. Ranking theory (Spohn) assigns ordinal degrees of belief that behave like AGM revision while admitting conditional structure. Probability kinematics (Jeffrey) generalizes conditionalization to handle uncertain evidence, approaching AGM's flexibility. The research program of relating these frameworks illuminates what is essential to rational belief change across all formal representations—the invariants that any adequate theory must respect.

Takeaway

AGM theory and Bayesian epistemology model different aspects of rational belief—categorical acceptance versus graded credence—and neither reduces to the other; choosing between them depends on whether your reasoning domain requires tracking degrees of confidence or simply accepted commitments.

AGM theory demonstrates that the logic of belief change is not merely a psychological description but admits rigorous formal characterization. The rationality postulates constrain permissible revisions as tightly as logical laws constrain valid inference. This is a substantive philosophical achievement: showing that norms of belief revision are amenable to mathematical treatment.

The relationship between AGM theory and Bayesian approaches remains an active research frontier. Neither framework subsumes the other; each illuminates aspects of rational belief management that the other obscures. The ongoing dialogue between logical and probabilistic epistemology drives formal epistemology forward.

For practitioners—whether in artificial intelligence, legal theory, or database systems—AGM theory provides concrete computational methods for implementing rational belief change. The possible worlds semantics yields implementable algorithms; the rationality postulates provide correctness criteria. Theory and practice converge in systems that actually work.