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Measuring Ranks via the Complete Laws of Iterated Contraction

Measuring Ranks via the Complete Laws of Iterated Contraction. Wolfgang Spohn Dagstuhl Seminar, Aug. 26 - 30, 2007. 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 :

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Measuring Ranks via the Complete Laws of Iterated Contraction

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  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Athe conditional rank of B given A is defined by (B | A) = (A B) – (A). Spohn, Dagstuhl, 26.-30.8.07

  3. Belief Sets, Ax-Conditionalization A belief setK 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) > 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}. Spohn, Dagstuhl, 26.-30.8.07

  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 > 0. For AA the contraction÷AofbyA 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). Spohn, Dagstuhl, 26.-30.8.07

  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 ). Spohn, Dagstuhl, 26.-30.8.07

  6. 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  = . Spohn, Dagstuhl, 26.-30.8.07

  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 potentialdisbelief comparisonassociated 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. Spohn, Dagstuhl, 26.-30.8.07

  8. Reasons Expressed by Iterated Contractions A is a reason forB or positively relevant to B w.r.t.  iff (¬B | A) > (¬B | ¬A) or (B | A) < (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. Spohn, Dagstuhl, 26.-30.8.07

  9. 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). Spohn, Dagstuhl, 26.-30.8.07

  10. Potential Doxastic Difference Comparisons We need to extend a potential DisDC  from quadruples of mutually disjoint propositions in Ato 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.) Spohn, Dagstuhl, 26.-30.8.07

  11. 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]. Spohn, Dagstuhl, 26.-30.8.07

  12. 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)  (Ai+1 - 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'). Spohn, Dagstuhl, 26.-30.8.07

  13. 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 > 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.] Spohn, Dagstuhl, 26.-30.8.07

  14. 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? Spohn, Dagstuhl, 26.-30.8.07

  15. Iterated Contractions  is an iteratedcontraction (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]. Spohn, Dagstuhl, 26.-30.8.07

  16. 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. Spohn, Dagstuhl, 26.-30.8.07

  17. 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) < (¬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. Spohn, Dagstuhl, 26.-30.8.07

  18. 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. Spohn, Dagstuhl, 26.-30.8.07

  19. 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 > 0 such that ' = x. Spohn, Dagstuhl, 26.-30.8.07

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