Measuring Ranks via the Complete Laws of Iterated Contraction

1
Measuring Ranksvia the Complete Laws of
Iterated Contraction

• Wolfgang Spohn
• Dagstuhl Seminar, Aug. 26 - 30, 2007

2
Ranking Functions

• ? is a negative ranking function for the algebra
  A over W iff ? is a function from A into R R ?
  ? such that for all A, B ? A
• (a) ?(A) 0, ?(W) 0, and ?(A) ? iff A ?,
• (b) ?(A ? B) min ?(A), ?(B).
• (Thus, for simplicity we assume regularity.)
• For A ? A the conditional rank of B given A is
  defined by ?(B A) ?(A ? B) ?(A).

3
Belief Sets, A?x-Conditionalization

• A belief set K is a filter in A, i.e., a subset
  of A not containing ? and closed under
  intersection and the superset relation. The
  belief set K(?) associated with ? is defined as
  K(?) A ? A ?(A) gt 0.
• For A ? A and x ? R, the A?x-conditionalization
  ?A?x of ? is defined by ?A?x(B) min ?(B A),
  ?(B A) x.

4
Revisions and Contractions

• The single (AGM) revision ?? induced by ? is
  defined as the function assigning to each A the
  belief set
• ??(A) K(?A?x) for some x gt 0.
• For A ? A the contraction ?A of ? by A is
  defined by ?A ?, if ? (A) 0, and
• ?A ?A?0, if ? (A) 0.
• The single (AGM) contraction ? induced by ? is
  defin-ed as the function assigning to each A the
  belief set ?(A) K(?A).

5
Iterated Contractions of Ranking Functions

• The iterated contraction ??A1, , An? of ? by
  ?A1, , An? is defined as ((?A1)) An this
  includes the iterated contraction ??? ? by the
  empty sequence ??.
• The iterated contraction ? induced by ? is
  defined as that function which assign to any
  finite sequence ?A1, , An? the belief set
• ??A1, , An? K(??A1, , An? ).

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Potential Iterated Contractions

• Let A denote the set of all finite (possibly
  empty) sequences of propositions from A.
• Then ? is a potential iterated contraction, a
  potential IC, for A iff ? is a function from A
  into the set of belief sets.
• A potential IC ? is an iterated ranking
  contraction, an IRC, for A iff there is a
  negative ranking function ? such that ? ??.

7
Potential Disbelief Comparisons

• For any ranking function ? we have
• ?(A) ?(B) iff A ? ???A ? B?.
• This corresponds to Gärdenfors definition of
  an entrenchment relation.
• Let ? be a potential IC for A. Then the potential
  disbelief comparison associated with ? is the
  binary relation on A such that for all A, B ? A
  A ? B iff A ? ? ?A ? B?. Strict comparison ?
  and equivalence ? are defined analogously.

8
Reasons Expressed by Iterated Contractions

• A is a reason for B or positively relevant to B
  w.r.t. ? iff ?(B A) gt ?(B A) or ?(B A) lt
  ?(B A).
• Analogous notions are defined analogously.
• Then we have for any ranking function ? A is not
  a reason against B, or non-negatively relevant to
  B, given C w.r.t. ?
• iff ?(A ? B ? C) ?(A ? B ? C)
• ?(A ? B ? C) ?(A ? B ? C),
• or iff neither (C ? A) ? B nor (C ? A) ? B is
  a
• member of ???C ? A, C ? A, C ? B, C ? B?.

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Potential Disjoint Difference Comparisons

• Let ? be a potential IC for A. Then the potential
  disjoint difference comparison (potential DisDC)
  associated with ? is the relation defined for all
  quadruples of mu-tually disjoint propositions in
  A such that for all such propositions A, B, C, D
• (A - B) ? ? (C - D) iff A, D ?
• ??A ? B, C ? D, A ? C, B ? D?.
• Similarly strict comparison ? and equivalence
  ? .
• ? ? is the potential DisDC associated with the
  IRC ?? we have (A - B) ? ? (C - D) iff ?(A)
  ?(B) ?(C) ?(D).

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Potential Doxastic Difference Comparisons

• We need to extend a potential DisDC ? ? from
  quadruples of mutually disjoint propositions in A
  to arbitrary quadruples of propositions. Such an
  extension is called the potential doxastic
  difference comparison (potential DoxDC)
  associated with and also denoted by ? ?.
• Such an extension can be produced, e.g., by
  assuming that for each non-empty proposition A
  there are four mutually disjoint propositions A1,
  A2, A3, A4 with A ? Ai (i 1,2,3,4). (But one
  might think of other methods.)

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Doxastic Difference Comparisons

• ? is a doxastic difference comparison (DoxDC) for
  A (with ? being the associated equivalence and ?
  the associated strict comparison) iff ? is a
  quarternary relation on A such that for all A, B,
  C, D, E, F ? A
• (a) ? is a weak order on A ? A weak order,
• (b) if (A - B) ? (C - D), then (D - C) ? (B -
  A) sign reversal,
• (c) if (A - B) ? (D - E) and (B - C) ? (E - F),
  then (A - C) ? (D - F) monotonicity,
• (d) if (A - W) ? (B - W), then (A - W) ? (A ? B
  - W) law of disjunction.

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Supplementary Axioms

• The DoxDC ? is Archimedean iff, moreover, for any
  sequence A1, A2, in A
• if A1, A2, is a strictly bounded standard
  sequence, i.e., if for all i (A1 - A1) ? (A2 -
  A1) ? (Ai1 - Ai) and if there is a D ? A N
  such that for all i (Ai - A1) ? (D - W), then the
  sequence A1, A2, is finite.
• Finally, the DoxDC ? is full iff for all A, B, C,
  D ? A
• (f) if (A - A) ? (A - B) ? (C - D), then there
  exist C', D' ? A such that (A - B) ? (C' - D) ?
  (C - D').

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The Representation Theorem

• Let ? be a full Archimedean DoxDC for A. Then
  there is a regular negative ranking function ?
  for A such that for all A, B, C, D ? A
• (A - B) ? (C - D) iff ?(A) ?(B) ?(C) ?(D).
• If ?' is another negative ranking function with
  these properties, there is an x gt 0 such that ?'
  x ? ?.
• Cf. D.H. Krantz et al., Foundations of
  Measurement, vol I, Academic Press 1971, p151.
• Weak order, sign reversal, monotonicity, law of
  disjunction, and the Archimedean property are
  necessary axioms, fullness is a sufficient
  structural axiom.

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Preliminary Conclusion

• We have seen how potential iterated contractions
  in-duce potential disbelief comparisons and, via
  the in-tuitively accessible reason or relevance
  relation, po-tential doxastic difference
  comparisons (namely just the way iterated ranking
  contractions would do it).
• And now we have seen how such potential DoxDC
  measure ranking functions on a ratio scale,
  pro-vided they satisfy the mentioned six axioms.
• The only remaining question is How must a
  potential iterated contraction behave so that the
  potential DoxDC generated by it satisfies the six
  axioms?

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Iterated Contractions

• ? is an iterated contraction (IC) for A iff ? is
  a potential IC for A such that for all A, B, C ?
  A and S ? A
• (IC1) the function A ? ??A? is a single (AGM)
  contraction single contraction,
• (IC2) if A ? ????, then ??A, S? ??S?
  strong vacuity,
• (IC3) if ?A ? ?B ?, then ??A, B, S? ??B, A,
  S? restricted commutativity,
• (IC4) if A ? B and A? ?B ? ??A?, then ??A? ?B, B,
  S? ??A, B, S? path independence,
• (IC5) if A ? ?C or A, B ? C and A ? B, then A ?
  ?C? B, and if the inequality in the antecedent
  is strict, that of the consequent is strict,
  too order preservation,
• (IC6) ??S? is an IC iterability.

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Some Explanations

• Single contraction (IC1) is clearly required.
• Strong vacuity (IC2) is stronger than AGM
  vacuity, but the intention is the same vacuous
  contraction leaves not only the belief set, but
  even the belief state un-changed.
• Iterability (IC6) is again clearly required (and
  does not make the definition circular).
• (IC3) - (IC5) are the proper laws of iterated
  contractions.
• Order preservation (IC5) could, of course, be
  expressed entirely in terms of iterated
  contractions. It is equi-valent to the
  Darwiche-Pearl postulates.
• Thus, it is precisely (IC4) and (IC5) that go
  beyond the so far known and accepted axioms of
  iterated contraction.

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Restricted Commutativity

• Ranking contractions do not always commute ???A,
  B? ? ???B, A? if and only if A, B ? K(?), ?(B
  A) 0 or ?(A B) 0, and ?(B A) lt ?(B
  A). The latter condition says that A is
  positively relevant to B.
• However, (IC3) claims restricted commutativity
  only under the condition that A logically
  excludes B, i.e., is unrevisably negatively
  relevant to B.
• That is, if two disbeliefs are logically
  incompatible, there can be no interaction between
  giving up these disbe-liefs, and hence it seems
  intuitively convincing that the order in which
  they are given up should not matter.

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Path Independence

• Path independence (IC4) says this in terms of
  disbelief
• Suppose you disbelieve two logically incompatible
  pro-positions, and you have to contract both of
  them. Then you can either contract one after the
  other. Or you can first contract their
  disjunction, and if you still dis-believe one of
  them, you then contract it as well. (IC4) says
  that both ways result in the same doxastic state.

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The Completeness of Iterated Contractions

• Let ? be an IC for A. Then ? is called
  Archimedean iff the DoxDC induced by ? is
  Archimedean. And ? is called full iff the DoxDC
  induced by ? is full.
• Then we have the following completeness theorem
  For any IC ? for A, the potential DoxDC induced
  by ? is a DoxDC for A. And any full Archimedean
  IC ? for A is an IRC for A, i.e., there is a
  ranking function ? for A with ? ??. Moreover,
  for each ranking function ?' with ? ??' there
  is an x gt 0 such that ?' x ? ?.