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DOXASTIC STATES

DOXASTIC STATES. ISAAC LEVI levi@columbia.edu. CHANGING BELIEFS LEGITIMATELY. For X to change X’s beliefs is for X to change X’s state of belief.

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DOXASTIC STATES

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  1. DOXASTIC STATES ISAAC LEVI levi@columbia.edu

  2. CHANGING BELIEFS LEGITIMATELY • For X to change X’s beliefs is for X to change X’s state of belief. • We need therefore an account of potential states of beliefs in order to characterize the doxastic trajectories of inquiring agents and prescribe warranted changes from state to state.

  3. Components of a Potential State of Belief or Doxastic State • (1) A Potential State of Full Belief K belonging to a set  of potential states of full belief (= set of doxastic propositions). Levi, Fixation of Belief and Its Undoing (1991), CUP ch.2. • (2) Credal State B belonging to a set  of potential credal states. “On Indeterminate Probabilities”, J.Philosophy 71(1974), 391-418; Enterprise of Knowledge,MIT Press (1980), ch.1-4. • (3) A Confirmational Commitment: C:  Enterprise of Knowledge, MIT Press (1980), ch.1-4.

  4. State of Full Belief • The state of full belief K serves several functions: • Agent X’s K is X’s standard for distinguishing serious possibility from impossibility. • X’s K is X’s body of evidence and certainties. • X fully believes that h iff that h is a consequence of K.

  5. Credal State • X’s state of credal probability judgment B is representable by a set of probability distributions over the serious possibilities according to X’s state of full belief K. When the credal state is determinate, the set is a singleton. • When the credal state is determinate and the value of consequences of available options in a decision problem is determinate – i.e., representable by a cardinal utility function unique up to a positive affine transformation, options are evaluated in terms of an expected utility using the uniquely permissible probability and probability pair to determine a uniquely permissible expected utility.

  6. Credal State 2 • In general, assessment of expected utilities of options will be representable by a set permissible expected utilities or by the set of permissible pairs of probability and utility functions. • In this sense, credal probability judgment whether numerically determinate or not is expectation determining.

  7. Evidentialism • X’s credal state B is determined by X’s state of full belief K – i.e, X’s total evidence – in accordance with a rule for adjusting X’s state of subjective, belief or credal probability judgment B to changes in K. • The rule is X’s confirmational commitment and is represented as a function

  8. Confirmational Commitment • X is committed to a procedure for deriving X’s credal state from X’s state of full belief K in accordance with a rule, procedure or method representable by a function: • C:   . • The rule C is X’s confirmational commitment

  9. Evidentialism 2 • Changing the credal state is changing either the state of full belief or the confirmational commitment or both. • Evidentialism, therefore, implies that K and C are the basic components of X’s doxastic state. Justifying change in credal state is parasitic on justifying changes in K and C.

  10. Our topic • This discussion is restricted to the discussion of changes in credal state due to changes in evidence and background information or equivalently to changes in X’s state of full belief K.

  11. Two Kinds of Justification of Changes in Full Belief • Justifying changes in commitment to full belief. • My interest here is in justifying changes in commitment. • This concern is analogous to ‘comparative statics’ in Economics as in John Hicks and Paul Samualson.

  12. Potential States of Full belief • A conceptual framework is a conceptually accessible set  of potential states of full belief available to an inquirer. It has a structure of a Boolean Algebra closed under meets and joins of sets of cardinality less than or equal to . • See Levi The Fixation of Belief ch.2 (1991), Mild Contraction (2004), appendix to ch.1,“Degrees of Belief”, n.1 Journal of Logic and Computation (2008) and “The Logic of Full Belief”, ch.3 of The Covenant of Reason (1997).

  13. Relevant Potential States of Full Belief • Inquiry is typically focused on a far more restricted class of changes in state of full belief than those that are conceptually accessible to the inquirer. • The inquirer in a given context of inquiry focuses on issues that are relevant to the demands of the particular inquiry or class of inquiries. The restriction on changes can be captured by the following:

  14. Minimal Relevant Potential State LK • An inquirer X should regard X’s current state K of full belief as an expansion of a potential state LK whose consequences are in the context of the current inquiry taken to be immune to change. • LK has all the truths of logic and conceptual necessities (if there be such) as consequences. For anyone who thinks that there is a body of conceptually necessary conviction above and beyond logical truth), members of that domain are entailed by LK. But so are many extra logical and extra conceptual truths that X does not accord such status. LK represents the set of beliefs that the inquirer currently takes to be immune to change at least for the time being. Levi, Mild Contraction (2004), OUP, 49-53.

  15. Basic Partition ULK • LK entails that exactly one element of ULK is true and each is consistent with LK. • The members of algebra generated by taking joins of any subset of ULK are those potential states that would be relevant answers to the questions studied in the inquiry were LK the inquirer’s state of full belief. • See Levi, Fixation of Belief and Its Undoing, CUP (1991), pp.123-4 • Levi, Mild Contraction, OUP (2004), 49-53

  16. Inductive Expansion • This discussion focuses on Deliberate or Inductive Expansion. • It ignores Routine Expansion (expansion by observation or consulting the testimony of other agents.) • It ignores contraction.

  17. Cognitive Options in Inductive or Deliberate Expansion • Relevant Potential Expansions K+x where • (1) x is the join of a subset A of the ultimate partition UK which is  if the subset is empty. • (2) A = {UK/R} where R is a subset of UK to be rejected. • The expansion is a new doxastic commitment. The agent chooses to come to believe only in that sense. • This characterization of the cognitive options in deliberate expansion is anticipated in Levi,Gambling with Truth, MIT Press, 1967 and Levi, “Information and Inference”, Decisions and Revisions, ch.5, Levi, Fixation of Belief (1984), For the Sake of the Argument CUP (1997), 6.2, and Levi (2004), Mild Contraction, OUP, 2.2 among many other places.

  18. States and Payoffs in Inductive Expansion • States: Elements of Ultimate Partition. • Payoffs: Two dimensional. • (a) Avoidance of Error T(x,t) = 1, T(x,f) = 0 • (b) Value of Information = Cont(x) = 1 – MK(x) • Levi, “Information and Inference; For the Sake of the Argument (1997),CUP, 6.3-6.5; Mild Contraction, 3.1-3.3

  19. The (undamped) Informational value of a potential answer h in LK • Positive affine transformation of contLK(h) = 1 – MLK(h) • Here the M-function is a probability distribution over ULK., • The M-function does not represent a credal or subjective probability but is used to assess the value of information obtained by inductive expansion.

  20. Undamped informational value of potential answer g given K. • contK(g) = 1-MK(g) where MK(x) = MLK(x)/MLK(UK) and x is a potential expansion of K. (that is to say, x is the join of a set of elements of the ultimate partition UK),

  21. Epistemic Utility • The epistemic utility of the outcome of expanding K by adding h when the aim is to obtain new error free and valuable information is • (T(h,x) + (1-) cont(h) [x = t or f] • or positive affine transformation of this. • Levi, “Information and Inference”: Fixation of Belief 3.2-3.3; For the Sake of the Argument, 6.6; Mild Contraction, 3.2-3.3

  22. Special Assumptions about the Relative Importance of Informational Value and Avoidance of Error • (a)  is positive but less than 1. In this way, both the value of avoiding error and obtaining valuable information are assigned positive weight and cannot be ignored. • q = (1- α)/α • (b) Informational value should not have greater weight that avoidance of error. The epistemic utility of importing an error by adding h should not be greater than the epistemic utility of adding g without error for any potential answers h and g. (See “Information and Inference”.) Hence, α should be greater than 0.5 and 0 < q ≤ 1.

  23. Epistemic Utility 2 • (1) T(h,x) – q(MK(x) • is a positive affine transformation of • (2) (T(h,x) + (1-) contk(h). • Hence, maximizing the expected value of (2) • = QK(x) – qMK(x) • ranks the cognitive options (potential expansions of K) identically as maximizing the expected value of (1) • = QK(x) + (1-)MK(x).

  24. Optimal expansion • An expansion strategy rejects some subset of the ultimate partition. An expansion strategy maximizes expected utility if and only if all elements of the ultimate partition x for which QK – qMK(h) are rejected, none for which this difference is positive are rejected.

  25. E-Admissible Expansion • A potential Expansion by adding h to K is E-admissible if there is a permissible credal probability in the credal state B and a permissible epistemic utility function T(h,x) – qMK(h) according to which the expansion is optimal. • For more discussion of E-admissibility and Cognate ideas, see “On Indeterminate Probability,” Jphil (1974), Enterprise of Knowledge, MIT (1984), Hard Choices CUP (1986);Covenant of Reason CUP (1997), chapters 6-10.

  26. Rule for Ties • If there is no indeterminacy in probability and utility, choose the weakest of the optimal expansions. • This will be the rejection procedure that rejects exactly the members of the ultimate partition for which QK(h) – qMK(h) is not positive. • In the face of indeterminacy in probability or utility, choose the weakest of the E-admissible expansions when there is a weakest.

  27. Deliberate Expansion • If credal state and M-function are determinate, choose among all potential expansion strategies based on expected epistemic utility maximization. When credal state and M-function are indeterminate, restrict choice to E-admissible expansions. • If credal state and M-function are determinate, there will be at least one expansion strategy maximizing expected utility when the set of potential expansions are finite. And also in common contexts where there infinitely many such expansions available .

  28. Deliberate Expansion: Iteration to a Fixed Point. • Define K, e ||~ h as h is in the recommended inductive expansion [K+e]i of K when the evidence the expansion of K by e. • When e is a consequence of K, we have K||~h = Ki • The procedure may be reiterated to obtain [Ki]i and so on to a fixed point. (if there is one.)

  29. Iteration to a fixed point continued • When q = 1 and both credal probability and the M-function are numerically determinate, induction iterated to a fixed point, x in the initial UK is rejected if and only if QK (x)/MK(x) is a maximum among all elements of UK. If the MK – distribution is uniform over UK, only elements of UK carrying maximum credal probability go unrejected. • Such conclusions are excessively strong. If one is predicting the relative frequency of heads in 1000 tosses of a fair coin, the conclusion is that the relative frequency is exactly 0.5. • This argues for q being less than 1.

  30. Iteration to a fixed point 2 • Iteration to a fixed point when q = 1 defines a transformation Ki that satisfies the permutability condition: • (1) For every h, ([K+h]i)  [Ki]+h (Importability of Inductive expansion.) • (2) For every h consistent with Ki, • [Ki]+h  ([K+h]i). • (Exportability of Inductive Expansion.)

  31. Iteration to a fixed point 3 • When Ki is substituted for K in the AGM postulates for revision, and inductive expansion satisfies permutability, the AGM axioms are satisfied. • So iteration to a fixed point with q = 1, guarantees that AGM revision of Ki satisfies AGM revision postulates.

  32. Iteration to a fixed point 4. • But if and only if q =1 does the inference relation validate the ranked models of Lehmann and Magidor. Iteration to a fixed point with q < 1 does not. • Iteration to a fixed point with q < 1 violates cautious monotony but satisfies Cut. Conforms to Reiter’s Requirements.

  33. Shackle measures • q(x) is the minimum value of q for which x fails to be rejected unless x is not rejected for q from 0 to 1. In that case q(x) = 0. This measure has properties of possibility measure in the sense of Dubois and Prade. • d(x) = 1 – q(x) = Shackle’s y(x) • b(x) = y(~x)

  34. Shackle measure 2 • b(x) is a satisficing measure of degree of belief..

  35. Contextual Parameters for Inductive Expansion • Minimal State LK • Basic Partition ULK • Initial State of full belief K Ultimate partition UK by determined basic partition and K • Confirmational Commitment C Credal state B = C(K) The Informational Value determining probability MLK. MK is a function of MLK and K Boldness q

  36. Epistemic States • Two of the contextual features are clearly epistemic or doxastic: K and C. These are relevant targets for justified modification of belief.

  37. Research Agenda • ULK, UK and the algebras generated by them represent demands for information, potential answers and, to this extent, research agendas. But they are not aspects of the epistemic or doxastic state.

  38. Research Agenda • The same is true for MLK, MK. These reflect the comparative assessments of the informational values of potential answers and in this way reflect demands for information.

  39. Boldness • The parameter q – the index of boldness – represents the relative importance attached to risk of error and informational value

  40. Epistemic State • If epistemic state is state change in which is to be justified, the candidates are the state of full belief and credal state. The justification of change in the credal state reduces to change in either the state of full belief or confirmational commitment. • So it seems appropriate to consider K and C to be the two components of the epistemic or doxastic state. • Other contextual factors matter. But none of them are features of a state of belief.

  41. References: Isaac Levi • Gambling with Truth (MIT, 1967) • Enterprise of Knowledge (MIT 1980) • Decisions and Revisions (Cambridge 1984) • Fixation of Belief (Cambridge, 1991) • For the Sake of the Argument (Cambridge, 1996) • The Covenant of Reason (Cambridge, 1997) • Mild Contraction (Oxford, 2004)

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