Editing Belief revision (section)

==The AGM postulates==
The AGM postulates (named after their proponents Alchourrón, [[Peter Gärdenfors|Gärdenfors]], and [[David Makinson|Makinson]]) are properties that an operator that performs revision should satisfy in order for that operator to be considered rational. The considered setting is that of revision, that is, different pieces of information referring to the same situation. Three operations are considered: expansion (addition of a belief without a consistency check), revision (addition of a belief while maintaining consistency), and contraction (removal of a belief).

The first six postulates are called "the basic AGM postulates". In the settings considered by Alchourrón, Gärdenfors, and Makinson, the current set of beliefs is represented by a [[Deductive closure|deductively closed]] set of logical formulae <math>K</math> called belief set, the new piece of information is a logical formula <math>P</math>, and revision is performed by a binary operator <math>*</math> that takes as its operands the current beliefs and the new information and produces as a result a belief set representing the result of the revision. The <math>+</math> operator denoted expansion: <math>K+P</math> is the deductive closure of <math>K \cup \{P\}</math>. The AGM postulates for revision are:

# Closure: <math>K*P</math> is a belief set (i.e., a deductively closed set of formulae);
# Success: <math>P \in K*P</math>
# Inclusion: <math>K*P \subseteq K+P</math>
# Vacuity: <math>\text{If }(\neg P) \not \in K,\text{ then }K*P=K+P</math>
# Consistency: <math>K*P</math> is [[inconsistent]] only if <math>P</math> is inconsistent
# Extensionality: <math>\text{If }P\text{ and }Q\text{ are logically equivalent, then }K*P=K*Q</math> (see [[logical equivalence]])
# Superexpansion: <math>K*(P \wedge Q) \subseteq (K*P)+Q</math>
# Subexpansion: <math>\text{If }(\neg Q) \not\in K*P\text{ then }(K*P)+Q \subseteq K*(P \wedge Q)</math>

A revision operator that satisfies all eight postulates is the full meet revision, in which <math>K*P</math> is equal to <math>K+P</math> if consistent, and to the deductive closure of <math>P</math> otherwise. While satisfying all AGM postulates, this revision operator has been considered to be too conservative, in that no information from the old knowledge base is maintained if the revising formula is inconsistent with it.<ref name="darwiche-pearl">{{Cite journal| doi = 10.1016/S0004-3702(96)00038-0| issn = 0004-3702| volume = 89| issue = 1| pages = 1–29| last1 = Darwiche| first1 = Adnan| last2 = Pearl| first2 = Judea| title = On the logic of iterated belief revision| journal = Artificial Intelligence| date = 1997-01-01| doi-access = free}}</ref>