Belief Revision

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Belief Revision

• Lecture 1 AGM
• April 1, 2004
• Gregory Wheeler
• greg_at_di.fct.unl.pt

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Outline

• Modeling Belief States
• AGM Rationality Postulates
• Expansion
• Contraction
• Revision
• Entrenchment
• Correspondence Results

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Belief

• A belief is a kind of mental state that
  represents an agents attitude toward a
  proposition
• Washington D.C. is the capital of the U.S.A.
• Sam believes that New York City is the capital of
  the U.S.A.

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Belief

• Propositions are true or false
• An agent S may take one of three attitudes of
  belief toward a proposition p
• S may believe that p is true
• S may believe that p is false
• S may neither believe that p is true nor that p
  is false.

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Two Dimensions of Belief Change

• STATIC DIMENSION
• -internal
• operations of reasoning
• reflective equilibrium

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Two Dimensions of Belief Change
?

• STATIC DIMENSION
• -internal
• operations of reasoning
• reflective equilibrium

. . .
. . .
. . .
t
t
DYNAMIC DIMENSION -external input (e.g.,
?) -learning -absorb new information
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Coherence

• Three senses of coherence/incoherence
• May apply to a single belief state
• Sam believes that 01.
• May apply to a sequence of beliefs
• Sam believes A. Sam believes not A.
• May apply to an agents disposition to choose.
• Sam prefers outcome O to P but chose outcome O.
• Note coherence is used differently in
  epistemology

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Static Constraint Inferential Coherence

• Minimum synchronic conditions for inferential
  coherence of a belief state
• Maxim 1. An agent Ss beliefs should be logically
  consistent.
• Maxim 2 If proposition ? is inferable from Ss
  beliefs, then S should believe ?.

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Diachonic coherence

• Economic constraints
• Maxim 3 The amount of information lost in a
  belief change should be kept minimal.
• Maxim 4 In so far as some beliefs are considered
  more important than others, one should retract
  the least important to restore equilibrium.

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Coherence of choice

• Dynamic Constraint
• Maxim 5 In so far as choices are to be made when
  performing a belief change, these choices should
  be coherent.
• (i.e., preference orderings should be respected)

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Modeling Belief States

• Logical model of rational belief change

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Modeling Belief States

• Logical model of rational belief change
• Let X and Y denote sets of well-formed formulae
  (wffs) in a propositional language, L, and ? and
  ? denote arbitrary formulas in L.
• e.g., X ?, ? ? ?
• Important We will interpret these sets of wffs
  as sets of beliefs held by an ideal agent. This
  is the motivation for considering the
  non-classical extensions and alterations to
  propositional logic.

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Modeling Belief States

• Logical model of rational belief change
• Let X and Y denote sets of well-formed formulae
  (wffs) in a propositional language, L, and ? and
  ? denote arbitrary formulas in L.
• A set X of wffs is inconsistent if there exists a
  wff ? such that X ? ? and X ? ??. If there is
  no such wff, then X is consistent.
• Inference operation Let Cn(X) ? X ? ?.
• Let K and H denote logical theories, e.g., K
  Cn(X), for some set of wffs X.

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Belief Change

• Three values true (t) false (f) undefined (u)
• Belief change may be thought of as a set
  operations that change the value of a wff.
• Expansion u t or u f.
• Contraction t u or f u.
• Revision t f or f t.

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AGM

• Alchourrón, Gärdenfors and Makinson (1985)
  proposed a set of rationality postulates for
  expansion, contraction and revision operators.
• A belief change operator is a 2-place function
  from a logical theory, K, and target formula, ?,
  to a replacement theory
• 2L ? L ? 2L.

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Expansion the operator

• The expression K ? denotes the replacement
  theory resulting from an expansion of K by ?.

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Expansion postulates

• (1) If K is a theory, then K ? is a theory.
• The expansion operator applied to a theory
  yields a theory.
• (2) ? ? (K ?).
• Expansion always succeeds the target formula ?
  is always included in the expanded theory.

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Expansion postulates

• (3) K ? (K ?).
• An expanded theory includes the original
  theory.
• (4) If ? ? K, then (K ?) K.
• Expanding a theory K with a formula that is
  already in K does not change K.

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Expansion postulates

• (5) If K ? H, then (K ?) ? (H ?).
• Expansion by the same formula ? on a theory K
  that is a subset of a theory H preserves the
  set-inclusion relation between K and H.

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Expansion postulates

• (6) (K ?) is the smallest theory satisfying
  (1) to (5).
• The expanded theory is the smallest possible and
  does not include wffs admitted by an operation
  which does not satisfy postulates (1) to (5).
  The set of theories satisfying (1) to (5) is
  closed under set intersection.

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Expansion postulates

• Remarks
• One way to expand a theory K is simply to add
  the target formula and close this set under
  logical consequence, that is to replace K by K
  Cn(K ? ?).
• Thm 3 K ? Cn(K ? ?).
• Note this is the only AGM operation which
  guarantees a unique replacement theory.

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Contraction the - operator

• The expression K - ? denotes the replacement
  theory resulting from a contraction of K by ?.

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Contraction postulates

• (-1) If K is a theory, then K - ? is a theory.
• The contraction operator applied to a theory
  yields a theory.
• (-2) (K - ?) ? K.
• The contracted theory is a subset of the
  original theory.

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Contraction postulates

• (-3) If ? ? (K - ?) then (K - ?) K.
• If the target formula ? to be contracted is not
  in the original theory, then the replacement
  theory is identical to the original theory.
• (-4) ? ? (K - ?) only if ? is not a tautology.
• The target formula ? is always removed from a
  theory by contraction unless ? is a tautology.

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Contraction postulates

• (-5) If ? ? K then K ? ((K-?)?).
• The Recovery Postulate.
• (-6) If ? ? ?, then (K - ?) (K - ?).
• Logically equivalent formulas give rise to
  identical contractions.

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Contraction postulates

• (-7) (K - ?) ? (K - ?) ? (K - ? ? ?).
• The formulas that are in both the theory
  contracted by the target formula ? and the theory
  contracted by the target formula ? are in the
  theory contracted by the target conjunction, ? ?
  ?. It is important to note that contracting by a
  conjunction is not the same as iterative
  contractions by each conjunct. Contracting by a
  conjunction entails removing the joint truth of
  the two formulas, which may be achieved by
  retracting one of the conjuncts.

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Contraction postulates

• (-8) If ? ? (K - ? ? ?), then (K - ? ? ?) ? K -
  ?.
• If the target formula of a contraction operation
  is a conjunction succeeds in removing one of the
  conjuncts, ?, then every formula that is removed
  by a contraction with that conjunct (i.e., ?)
  alone is also removed by the contraction with the
  conjunction.

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Contraction postulates

• Remarks
• While Expansion guarantees a unique replacement
  theory, note that the contraction postulates do
  not determine a unique replacement theory.
• This property will be illustrated with a series
  of examples. Notice that this feature introduces
  the need for extra-logical constraints to guide
  our choice among candidate replacement theories.

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Revision the operator

• The expression K ? denotes the replacement
  theory resulting from an revision of K by ?.

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Revision postulates

• (1) If K is a theory, then K ? is a theory.
• The revision operator applied to a theory
  yields a theory.
• (2) ? ? (K ?).
• Revision always succeeds the target formula ?
  is always included in the expanded theory.

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Revision postulates

• (3) (K ?) ? (K ?).
• A revision never incorporates formulas that are
  not in the expansion of the original theory by
  the same target formula.

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Revision postulates

• (4) If ? ? K, then (K ?) ? (K ?).
• If the negation of a target formula is not in a
  theory, then the result of expanding the theory
  by that target formula is a subset of the result
  of revising the theory by the target formula.
  When taken with (3), it follows that if the
  target formula is consistent with the original
  theory, then a revision is identical with the
  expansion, that is
• (K ?) (K ?).

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Revision postulates

• (5) K ? ? if and only if ? ??.
• Given that a theory is consistent, if we attempt
  to revise the theory by a contradiction the
  replacement theory is inconsistent. This is the
  only case where revision applied to a consistent
  theory yields an inconsistent theory.
• (6) If ? ? ?, then (K ?) (K ?).
• Logically equivalent formulas give rise to
  identical revisions.

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Revision postulates

• (7) (K ? ? ?) ? ((K ?) ?).
• When revising a theory by a target formula that
  is a conjunction we may retain every formula in
  the original theory by (1) first revising the
  original theory by one conjunct and then (2)
  expand the revised theory by the other conjunct.
  Compare
• (K ? ??) ? (K ? ??) ((K ?) ?), by
  (3).
• Since (3) gives us (K ?) ? (K ?), (7)
  gives us a tighter upper-bound on (K ? ??) than
  (3).

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Revision postulates

• (8) If ? ? (K ? ? ?), then ((K ?)?) ? (K
  ? ??).
• So long as a formula ? is consistent with a
  revised theory K by another formula, ?, then the
  resulting theory from applying the two step
  procedure mentioned in (7) is a subset of
  revising K by the conjunction of the two formula
  in question, ? ??. Together, (7) and (8) entail
  that the two step process in (7) is identical to
  the conjunction as a whole, that is
• ((K ?)?) (K ? ??)
• given that ? is consistent with the revised
  theory in the first step.

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Revision postulates

• Remarks
• Like Contraction, the revision postulates do not
  determine a unique replacement theory.
• While we defined the revision operator, , the
  contraction operator, -, and the expansion
  operator, , independently of one another, we may
  nevertheless define these operators in terms of
  one another.

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The Levi Identity

• Thm 8 Given that the contraction function
  satisfies postulates (-1) to (-4) and (-6), and
  the expansion function satisfies (1) to (6),
  the revision function as defined by the Levi
  Identity
• K ? ((K - ?) ?)
• satisfies (1) to (6). Furthermore, if (7) is
  satisfied, then (7) is satisfied if (8) is
  satisfied, then (8) is satisfied.

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The Harper Identity

• Thm 9 Given that the revision function
  satisfies (1) to (6), the contraction function
  - as defined by the Harper Identity
• K - ? K ? (K ?)
• satisfies (-1) to (-6). Furthermore, if (7) is
  satisfied, then (-7) is satisfied and if (8) is
  satisfied, then (-8) is satisfied.

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Entrenchment

• Def. An epistemic entrenchment relation e is
  defined on formula of L, where
• ? e ?
• is interpreted to express that ? is as
  epistemically entrenched as ? and satisfies the 5
  postulates, (EE1) through (EE5). Let ? lte ? stand
  for ? is strictly more entrenched than ?, and ?
  e ? for ? and ? are equally entrenched.

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Entrenchment postulates

• (EE1) If ? e ? and ? e ?, then ? e ?.
• The epistemic entrenchment relation is
  transitive.
• (EE2) If ? ? ? , then ? e ?.
• A formula is at most as entrenched as the
  formulas it logically entails. If ? entails ?
  and we wish to retract ?, we need to retract ?
  also to avoid deriving ? in the replacement
  theory. On the other hand, ? should be at most as
  entrenched as ? so that ? may be retracted
  without necessarily retracting ?.

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Entrenchment postulates

• (EE3) For all ?, ?, either ? e ? ? ? or ? e ?
  ? ?
• Retracting the conjunction ? ?? is achieved by
  either retracting ? or retracting ?. Thus, the
  conjunction is at least as entrenched as one of
  the conjuncts
• From (EE2), we have the opposite relations ? ??
  e ? and ? ?? e ?. From (EE2) and (EE3),
  together, we have ? ?? e ? or ? ?? e ?. In
  other words, a conjunction is as entrenched as
  its least entrenched conjunct.

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Entrenchment postulates

• (EE4) When K ??, then ? ? K iff ? e ? ? ? or ?
  e ? ? ?.
• Formulas not in the theory are the least
  entrenched and, if the theory is consistent, vice
  versa.
• (EE5) If ? e ? for all ?, then ? ?
• The most entrenched formulas are the
  tautologies.

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Correspondence Results

• (C-) ? ? (K - ?) iff either ? e ? ? ? or ? ?.
• (C-) specifies what formulas are retained in a
  contraction given an epistemic entrenchment
  relation. Only formulas that are in the original
  theory K can be included in the contracted
  theory. In addition, if the target formula is a
  tautology, then all formulas are retained.
  Otherwise, if the target formula ? is less
  entrenched than the disjunction
• ? ? ?, then ? is retained.

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Correspondence Results

• (Ce) ? e ? iff ? ? K or ? (? ? ?)
• An epistemic entrenchment relation can be
  constructed from a contraction function by (Ce).
  If a conjunct ? is not retained in the contracted
  theory, then it cannot be more entrenched than
  the other conjunct. Note that both conjuncts can
  be absent from the contracted theory, in which
  case the two conjuncts are equally entrenched,
• ? e ?.

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Correspondence Results

• Thm Given an epistemic entrenchment ordering e
  that satisfies (EE1) to (EE5), condition (C-)
  uniquely determines a contraction function which
  satisfies the AGM contraction postulates (-1) to
  (-8) and condition (Ce).

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Correspondence Results

• Thm Given a contraction function which satisfies
  the AGM contraction postulates (-1) to (-8),
  condition (Ce) uniquely determines an epistemic
  entrenchment ordering e that satisfies (EE1) to
  (EE5) and condition (C-).

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Remarks

• Rationality postulates generate a set of
  candidate theory change functions
• An entrenchment relation allows us to pick a
  specific function among the class.

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Remarks

• The entrenchment ordering provides a constructive
  way to choose a specific contraction operator
  from the set of all possible operators.