Logical and Probabilistic Models of Belief Change

Transcription

1 Logical and Probabilistic Models of Belief Change Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org August 7, 2017 Eric Pacuit 1

2 Plan Day 1 Introduction to belief revision, AGM, possible worlds models, Bayesian models (time permitted) Day 2 Possible worlds models (continued), Bayesian models (continued), Updating probabilities Day 3 The value of learning, Lottery Paradox, Preface Paradox, Review Paradox, Iterated belief revision, Context shifts, Becoming aware Day 4 The value of learning, Lottery Paradox, Preface Paradox, Review Paradox, Iterated belief revision, Context shifts, Becoming aware (continued) Day 5 Interactive epistemology (Agreement Theorems), Convergence Theorems Eric Pacuit 2

3 pacuit.org/nls2017/belchange/ Eric Pacuit 3

4 Plan for today Introduction to belief revision AGM Possible worlds model Bayesian models (time permitting) Eric Pacuit 4

5 Belief Change Computer science: updating databases (Doyle 1979 and Fagin et al. 1983) Philosophy (epistemology/philosophy of science): scientific theory change and revisions of probability assignments; belief change (Levi 1977, 1980, Harper 1977) and its rationality. Eric Pacuit 5

6 AGM Carlos Alchourrón, Peter Gärdenfors, and David Makinson. C. Alchourrón, P. Gärdenfors and D. Makinson. On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic, 50, , Eric Pacuit 6

7 Classics I Levi. Subjunctives, Dispositions and Chances. Synthese 34: , W. Spohn. Ordinal conditional functions: A dynamic theory of epistemic states. in W.L. Harper and B. Skyrms, eds., Causation in Decision, Belief Change and Statistics, vol 2, pp , W. Harper. Rational Conceptual Change. PSA 1976, pp Eric Pacuit 7

8 Belief Change Consider the following beliefs of a rational agent: p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: q The bird caught in the trap is white. Now suppose the rational agent for example, You learn that the bird caught in the trap is black ( q). There are several logically consistent ways to incorporate q! Eric Pacuit 8

9 Belief Change Consider the following beliefs of a rational agent: p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: q The bird caught in the trap is white. Now suppose the rational agent for example, You learn that the bird caught in the trap is black ( q). There are several logically consistent ways to incorporate q! Eric Pacuit 8

10 Belief Change Consider the following beliefs of a rational agent: p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: q The bird caught in the trap is white. Question: How should the agent incorporate q into his belief state to obtain a consistent belief state? There are several logically consistent ways to incorporate q! Eric Pacuit 8

11 Belief Change Consider the following beliefs of a rational agent: p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: q The bird caught in the trap is white. Problem: Logical considerations alone are insufficient to answer this question! Why?? There are several logically consistent ways to incorporate q! Eric Pacuit 8

12 Belief Change Consider the following beliefs of a rational agent: p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: q The bird caught in the trap is white. Problem: Logical considerations alone are insufficient to answer this question! Why?? There are several logically consistent ways to incorporate q! Eric Pacuit 8

13 Belief Change What extralogical factors serve to determine what beliefs to give up and what beliefs to retain? Eric Pacuit 9

14 Belief Change p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: q The bird caught in the trap is white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

15 Belief Change p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

16 Belief Change /// p 1 All////////////// Europeans/////// swans///// are//////// white. /// p 2 The///// bird///////// caught/// in///// the////// trap/// is//a//////// swan. /// p 3 The///// bird///////// caught/// in///// the////// trap//////// comes/////// from////////// Sweden. /// p 4 ///////// Sweden/// is///// part//// of////////// Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

17 Belief Change p 1 All Europeans swans are white. /// p 2 The///// bird///////// caught/// in///// the////// trap/// is//a//////// swan. /// p 3 The///// bird///////// caught/// in///// the////// trap//////// comes/////// from////////// Sweden. /// p 4 ///////// Sweden/// is///// part//// of////////// Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

18 Belief Change /// p 1 All////////////// Europeans/////// swans///// are//////// white. p 2 The bird caught in the trap is a swan. /// p 3 The///// bird///////// caught/// in///// the////// trap//////// comes/////// from////////// Sweden. /// p 4 ///////// Sweden/// is///// part//// of////////// Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

19 Belief Change /// p 1 All////////////// Europeans/////// swans///// are//////// white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

20 Belief Change p 1 All Europeans swans are white. /// p 2 The///// bird///////// caught/// in///// the////// trap/// is//a//////// swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

21 Belief Change p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. /// p 3 The///// bird///////// caught/// in///// the////// trap//////// comes/////// from////////// Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

22 Belief Change p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. /// p 4 ///////// Sweden/// is///// part//// of////////// Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

23 Belief Change p 1 All Europeans swans are white. p 2 The bird caught in the trap is a swan. p 3 The bird caught in the trap comes from Sweden. p 4 Sweden is part of Europe. Thus, the agent believes: //q The///// bird///////// caught/// in///// the////// trap/// is//////// white. New belief: q The bird caught in the trap is not white. Eric Pacuit 9

24 Belief Change - Guiding Principles 1. When accepting a new piece of information, an agent should aim at a minimal change of his old beliefs. 2. If there are different ways to effect a belief change, the agent should give up those beliefs which are least entrenched. Eric Pacuit 10

25 Textbooks S. O. Hansson. A Textbook of Belief Dynamics. Theory Change and Database Updating. Dordrecht. Kluwer Academic Publishers, P. Gärdenfors. Knowledge in Flux. Modeling the Dynamics of Epistemic States. The MIT Press, H. Rott. Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning. Oxford University Press, Eric Pacuit 11

26 Epistemic States Belief sets (Ellis s belief systems) Possible worlds models (Doyle s truth maintenance systems) Spohn s generalized possible worlds model Bayesian models Generalized Bayesian models (Johnson-Laird mental models)... Eric Pacuit 12

27 Epistemic States Belief sets (Ellis s belief systems) Possible worlds models (Doyle s truth maintenance systems) Spohn s generalized possible worlds model Bayesian models Generalized Bayesian models (Johnson-Laird mental models)... J. Halpern. Reasoning about uncertainty. The MIT Press, Eric Pacuit 12

28 AGM Language of Beliefs in AGM: propositional logic: atomic propositions p, q, r,... connectives: negation ( ), conjunction ( ), disjunction ( ), implication ( ), and equivalence ( ). Eric Pacuit 13

29 AGM Language of Beliefs in AGM: propositional logic: atomic propositions p, q, r,... connectives: negation ( ), conjunction ( ), disjunction ( ), implication ( ), and equivalence ( ). Rationality constraints:: 1. Belief sets should be consistent 2. Belief sets should be closed under logical consequence Eric Pacuit 13

30 Classical Consequence For any set A of sentences, Cn(A) is the set of logical consequences of A. Cn : (L) (L) satisfying the following three conditions: A Cn(A) (inclusion); If A B, then Cn(A) Cn(B) (monotony); Cn(A) = Cn(Cn(A)) (idempotence) If p can be derived from A by classical propositional logic, then p Cn(A). Write A p when p Cn(A). Eric Pacuit 14

31 K is a belief set just in case K = Cn(K). Eric Pacuit 15

32 K is a belief set just in case K = Cn(K). Logical omniscience; explicit vs. implicit beliefs; A belief set is not what you actually believe, but what you are committed to believe (Levi 1991). Eric Pacuit 15

33 K is a belief set just in case K = Cn(K). Logical omniscience; explicit vs. implicit beliefs; A belief set is not what you actually believe, but what you are committed to believe (Levi 1991). Belief bases vs. belief sets. B 1 = {p, p q}, B 2 = {p, q}. Eric Pacuit 15

34 K is a belief set just in case K = Cn(K). Logical omniscience; explicit vs. implicit beliefs; A belief set is not what you actually believe, but what you are committed to believe (Levi 1991). Belief bases vs. belief sets. B 1 = {p, p q}, B 2 = {p, q}. Cn(B 1 ) = Cn(B 2 ). Eric Pacuit 15

35 K is a belief set just in case K = Cn(K). Logical omniscience; explicit vs. implicit beliefs; A belief set is not what you actually believe, but what you are committed to believe (Levi 1991). Belief bases vs. belief sets. B 1 = {p, p q}, B 2 = {p, q}. Cn(B 1 ) = Cn(B 2 ). What happens when we receive the evidence that p? Eric Pacuit 15

36 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 16

37 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 16

38 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 16

39 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 16

40 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 16

41 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 16

42 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 16

43 Change operator: : B L B K ϕ Initial set of beliefs New evidence ϕ Eric Pacuit 17

44 Revision operator: : B L B K ϕ Initial set of beliefs New evidence ϕ Eric Pacuit 17

45 Revision operator: : B L B K ϕ Initial set of beliefs New evidence ϕ Eric Pacuit 17

46 Belief change operator: : B L B K ϕ Initial set of beliefs New evidence ϕ Eric Pacuit 17

47 Belief Change Minimal change: The criterion of informational economy demands that as few beliefs as possible be given up so that the change is in some sense a minimal change of K to accommodate for A. (Gärdenfors 1988, p. 53). Keep the most entrenched beliefs:...beliefs are only given up when there are no less entrenched candidates... If one of two beliefs must be retracted in order to accommodate some new fact, the less entrenched belief will be relinquished, while the more entrenched persists. (Boutilier 1996, pp ). Eric Pacuit 18

48 Belief Change Suppose that K is the current beliefs. If you give priority to the new information ϕ, then there are three belief change operations: Eric Pacuit 19

49 Belief Change Suppose that K is the current beliefs. If you give priority to the new information ϕ, then there are three belief change operations: 1. Expansion: K + ϕ; ϕ is added to K. Eric Pacuit 19

50 Belief Change Suppose that K is the current beliefs. If you give priority to the new information ϕ, then there are three belief change operations: 1. Expansion: K + ϕ; ϕ is added to K. 2. Contraction: K. ϕ; ϕ is removed from K. Eric Pacuit 19

51 Belief Change Suppose that K is the current beliefs. If you give priority to the new information ϕ, then there are three belief change operations: 1. Expansion: K + ϕ; ϕ is added to K. 2. Contraction: K. ϕ; ϕ is removed from K. 3. Revision: K ϕ; ϕ is added and other things are removed. Eric Pacuit 19

52 Expansion Postulates (E1) (E2) (E3) (E4) (E5) K + α is deductively closed α K + α K K + α If α K, then K + α = K If K K, then K + α K + α (Minimality) For all belief sets K and all sentences α, K + α is the smallest belief set that satisfies (E1), (E2), and (E3). Eric Pacuit 20

53 Expansion Theorem Let + be a function on belief sets and formulas. Then, + satisfies minimality if and only if K + α = Cn(K {α}). Eric Pacuit 21

54 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 22

55 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 22

56 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 22

57 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 22

58 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 22

59 Contraction Postulates (C1) K. α is deductively closed (C2) K. α K (C3) If α K or α then K. α = K (C4) If α, then α K. α (C5) If α β, then K. α = K. β (C6) K Cn((K. α) {α}) Eric Pacuit 23

60 Definition. An operator is a withdrawal if and only if it satisfies (C1-C5). Eric Pacuit 24

61 Definition. An operator is a withdrawal if and only if it satisfies (C1-C5). The following is a withdrawal: K α = { K Cn( ) if α K if α K Eric Pacuit 24

62 Definition. An operator is a withdrawal if and only if it satisfies (C1-C5). The following is a withdrawal: K α = { K Cn( ) if α K if α K Minimal Information Loss (Recovery): K Cn((K. α) {α}) Eric Pacuit 24

63 Let K be a belief set and ϕ a formula. K ϕ is the remainder set of K. A K ϕ iff 1. A K 2. ϕ Cn(A) 3. There is no B such that A B K and ϕ Cn(B). Eric Pacuit 25

64 K α = {K} iff α Cn(K) K α = iff α Cn( ) If K K and α Cn(K ) then there is some T such that K T K α. Eric Pacuit 26

65 A selection function γ for K is a function on K α such that: If K α, then γ(k α) K α and γ(k α) If K α =, then γ(k α) = {K} Eric Pacuit 27

66 Let K be a set of formulas. A function. is a partial meet contraction for K if there is a selection function γ for K such that for all formula α: K. α = γ(k α) Eric Pacuit 28

67 Let K be a set of formulas. A function. is a partial meet contraction for K if there is a selection function γ for K such that for all formula α: K. α = γ(k α) γ selects exactly one element of K α (maxichoice contraction) γ selects the entire set K α (full meet contraction) Eric Pacuit 28

68 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

69 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

70 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

71 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

72 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

73 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

74 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

75 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

76 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

77 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 29

78 Levi Identity K ϕ = (K. ϕ) + ϕ Eric Pacuit 30

79 Let K be a set of formulas. A function. is a partial meet contraction for K if there is a selection function γ for K such that for all formula α: K. α = γ(k α) γ selects exactly one element of K α (maxichoice contraction) γ selects the entire set K α (full meet contraction) Then, K α = Cn( γ(k α) {α}) Eric Pacuit 31

80 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 32

81 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 32

82 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 32

83 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 32

84 p q p q p q p q q p p q p q q p p q p q p q p q Eric Pacuit 32

85 AGM Postulates (AGM1) (AGM2) (AGM3) (AGM4) (AGM5) (AGM6) (AGM7) K ϕ is deductively closed ϕ K ϕ K ϕ Cn(K {ϕ}) If ϕ K then K ϕ = Cn(K {ϕ}) K ϕ is inconsistent only if ϕ is inconsistent If ϕ and ψ are logically equivalent then K ϕ = K ψ K (ϕ ψ) Cn(K ϕ {ψ}) (AGM8) If ψ K ϕ then Cn(K ϕ {ψ}) K (ϕ ψ) Eric Pacuit 33

86 Theorem (AGM 1985). Let K be a belief set and let be a function on L. Then The function is a partial meet revision for K if and only if it satisfies the postulates AGM1 - AGM6 The function is a transitively relational partial meet revision for K if and only if it satisfies AGM1 - AGM8. Eric Pacuit 34

87 There is a transitive relation on K α such that γ(k α) = {K K α K K for all K K α} { {K K α K is -maximal} if α Cn( ) K. α = K otherwise Eric Pacuit 35

88 Evaluating the AGM postulates Eric Pacuit 36

89 Counterexample to AGM 2 (Success) ϕ K ϕ Eric Pacuit 37

90 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Eric Pacuit 37

91 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Is this a counterexample to AGM 2? Eric Pacuit 37

92 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Is this a counterexample to AGM 2? No Eric Pacuit 37

93 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Is this a counterexample to AGM 2? No Two people, Ann and Bob, are reliable sources of information on whether The Netherlands will win the world cup. They are equally reliable. Eric Pacuit 37

94 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Is this a counterexample to AGM 2? No Two people, Ann and Bob, are reliable sources of information on whether The Netherlands will win the world cup. They are equally reliable. AGM assumes that the most recent evidence that you received takes precedent. Ann says yes and a little bit later, Bob says no. Eric Pacuit 37

95 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Is this a counterexample to AGM 2? No Two people, Ann and Bob, are reliable sources of information on whether The Netherlands will win the world cup. They are equally reliable. AGM assumes that the most recent evidence that you received takes precedent. Ann says yes and a little bit later, Bob says no. Why should the, possibly arbitrary, order in which you receive the information give more weight to Bob s announcement? Eric Pacuit 37

96 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Is this a counterexample to AGM 2? No Two people, Ann and Bob, are reliable sources of information on whether The Netherlands will win the world cup. They are equally reliable. AGM assumes that the most recent evidence that you received takes precedent. Ann says yes and a little bit later, Bob says no. Why should the, possibly arbitrary, order in which you receive the information give more weight to Bob s announcement? Is this a counterexample to AGM 2? Eric Pacuit 37

97 Counterexample to AGM 2 (Success) ϕ K ϕ You are walking down a street and see someone holding a sign reading The World will End Tomorrow, but you don t add this add this to your beliefs. Is this a counterexample to AGM 2? No Two people, Ann and Bob, are reliable sources of information on whether The Netherlands will win the world cup. They are equally reliable. AGM assumes that the most recent evidence that you received takes precedent. Ann says yes and a little bit later, Bob says no. Why should the, possibly arbitrary, order in which you receive the information give more weight to Bob s announcement? Is this a counterexample to AGM 2? No Eric Pacuit 37

98 Rott s Counterexample AGM 7: K (ϕ ψ) Cn(K ϕ {ψ}) AGM 8: if ψ K ϕ then Cn(K ϕ {ψ}) K (ϕ ψ) So, if ψ Cn({ϕ}), then K ϕ = Cn(K ϕ {ψ}) Eric Pacuit 38

99 Rott s Counterexample There is an appointment to be made in a philosophy department. The position is a metaphysics position, and there are three main candidates: Andrew, Becker and Cortez. 1. Andrew is clearly the best metaphysician, but is weak in logic. 2. Becker is a very good metaphysician, also good in logic. 3. Cortez is a brilliant logician, but weak in metaphysics. Scenario 1: Paul is told by the dean, that the chosen candidate is either Andrew or Becker. Since Andrew is clearly the better metaphysician of the two, Paul concludes that the winning candidate will be Andrew. A K (A B) Eric Pacuit 38

100 Rott s Counterexample There is an appointment to be made in a philosophy department. The position is a metaphysics position, and there are three main candidates: Andrew, Becker and Cortez. 1. Andrew is clearly the best metaphysician, but is weak in logic. 2. Becker is a very good metaphysician, also good in logic. 3. Cortez is a brilliant logician, but weak in metaphysics. Scenario 1: Paul is told by the dean, that the chosen candidate is either Andrew or Becker. Since Andrew is clearly the better metaphysician of the two, Paul concludes that the winning candidate will be Andrew. A, B, C K (A B) Eric Pacuit 38

101 Rott s Counterexample 1. Andrew is clearly the best metaphysician, but is weak in logic. 2. Becker is a very good metaphysician, also good in logic. 3. Cortez is a brilliant logician, but weak in metaphysics. Scenario 2: Paul is told by the dean that the chosen candidate is either Andrew, Becker or Cortez. Knowing that Cortez is a splendid logician, but that he can hardly be called a metaphysician, Paul comes to realize that his background assumption that expertise in the field advertised is the decisive criterion for the appointment cannot be upheld. Apparently, competence in logic is regarded as a considerable asset by the selection committee. Paul concludes Becker will be hired. A, B, C K (A B C) Eric Pacuit 38

102 Rott s Counterexample 1. Andrew is clearly the best metaphysician, but is weak in logic. 2. Becker is a very good metaphysician, also good in logic. 3. Cortez is a brilliant logician, but weak in metaphysics. Scenario 2: Paul is told by the dean that the chosen candidate is either Andrew, Becker or Cortez. Knowing that Cortez is a splendid logician, but that he can hardly be called a metaphysician, Paul comes to realize that his background assumption that expertise in the field advertised is the decisive criterion for the appointment cannot be upheld. Apparently, competence in logic is regarded as a considerable asset by the selection committee. Paul concludes Becker will be hired. A, B, C K (A B C) Eric Pacuit 38

103 Rott s Counterexample Data: A, B, C K (A B) A, B, C K (A B C) Theory: (AGM7) K (ϕ ψ) Cn(K ϕ {ψ}) (AGM8) If ψ K ϕ then Cn(K ϕ {ψ}) K (ϕ ψ) So, if ψ K ϕ, then K ϕ K (ϕ ψ) Eric Pacuit 38

104 Rott s Counterexample Problem: A, B K (A B C) A B K (A B C) K (A B C) K ((A B C) (A B)) = K (A B) A, B K (A B) Eric Pacuit 38

105 ...Rott seems to take the point about meta-information to explain why the example conflicts with the theoretical principles, Eric Pacuit 39

106 ...Rott seems to take the point about meta-information to explain why the example conflicts with the theoretical principles, whereas I want to conclude that it shows why the example does not conflict with the theoretical principles, since I take the relevance of the meta-information to show that the conditions for applying the principles in question are not met by the example. Eric Pacuit 39

107 ...Rott seems to take the point about meta-information to explain why the example conflicts with the theoretical principles, whereas I want to conclude that it shows why the example does not conflict with the theoretical principles, since I take the relevance of the meta-information to show that the conditions for applying the principles in question are not met by the example... I think proper attention to the relation between concrete examples and the abstract models will allow us to reconcile some of the beautiful properties [of the abstract theory of belief revision] with the complexity of concrete reasoning. asdf dsaf (Stalnaker, 204) Robert Stalnaker. Iterated Belief Revision. Erkenntnis 70, pp , Eric Pacuit 39

108 Counterexamples to Recovery (C6) K Cn((K. α) {α}) While reading a book about Cleopatra I learned that she had both a son and a daughter. I therefore believe both that Cleopatra had a son (s) and Cleopatra had a daughter (d). Eric Pacuit 40

109 Counterexamples to Recovery (C6) K Cn((K. α) {α}) While reading a book about Cleopatra I learned that she had both a son and a daughter. I therefore believe both that Cleopatra had a son (s) and Cleopatra had a daughter (d). Later I learn from a well-informed friend that the book in question is just a historical novel. I accordingly contract my belief that Cleopatra had a child (s d). Eric Pacuit 40

110 Counterexamples to Recovery (C6) K Cn((K. α) {α}) While reading a book about Cleopatra I learned that she had both a son and a daughter. I therefore believe both that Cleopatra had a son (s) and Cleopatra had a daughter (d). Later I learn from a well-informed friend that the book in question is just a historical novel. I accordingly contract my belief that Cleopatra had a child (s d). However, shortly thereafter I learn from a reliable source that in fact Cleopatra had a child. I thereby reintroduce s d to my collection of beliefs without also returning either s or d. S. Hansson. A Textbook of Belief Dynamics, Theory Change and Database Updating. Kluwer Academic Publishers, Eric Pacuit 40

111 Evaluating counterexamples... information about how I learn some of the things I learn, about the sources of my information, or about what I believe about what I believe and don t believe. If the story we tell in an example makes certain information about any of these things relevant, then it needs to be included in a proper model of the story, if it is to play the right role in the evaluation of the abstract principles of the model. Robert Stalnaker. Iterated Belief Revision. Erkenntnis 70, pp , Eric Pacuit 41

112 Belief Revision: The Semantic View A. Grove. Two modelings for theory change. Journal of Philosophical Logic, 17, pgs , EP. Dynamic Epistemic Logic II: Logics of information change. Philosophy Compass, Vol. 8, Iss. 9, pgs , Eric Pacuit 42

113 ... w The set of states, with a distinguished state denoting the actual world The agent s (hard) information (i.e., the states consistent with what the agent knows) Eric Pacuit 43

114 ... w The set of states, with a distinguished state denoting the actual world. The agent s (hard) information (i.e., the states consistent with what the agent knows) Eric Pacuit 43

115 ... w The agent s (hard) information (i.e., the states consistent with what the agent knows) The agent s beliefs (soft information the states consistent with what the agent believes) Eric Pacuit 43

116 ... w The states consistent with what the agent knows with a distinguished state (the actual world ) Each state is associated with a propositional valuation for the underlying propositional language Eric Pacuit 44

117 ... w The agent s beliefs (soft information -the states consistent with what the agent believes) The agent s contingency plan : when the stronger beliefs fail, go with the weaker ones. Eric Pacuit 44

118 ... w The agent s beliefs (soft information the states consistent with what the agent believes) The agent s contingency plan : when the stronger beliefs fail, go with the weaker ones. Eric Pacuit 44

119 ... w The agent s beliefs (soft information the states consistent with what the agent believes) The agent s contingency plan : when the stronger beliefs fail, go with the weaker ones. Eric Pacuit 44

120 Sphere Models Let W be a set of states, A set F (W ) is called a system of spheres provided: For each S, S F, either S S or S S For any P W there is a smallest S F (according to the subset relation) such that P S The spheres are non-empty F and cover the entire information cell F = W Eric Pacuit 45

121 Let F be a system of spheres on W : for w, v W, let w F v iff for all S F, if v S then w S Then, F is reflexive, transitive, and well-founded. w F v means that no matter what the agent learns in the future, as long as world v is still consistent with his beliefs and w is still epistemically possible, then w is also consistent with his beliefs. Eric Pacuit 46

122 Belief Revision via Plausibility W = {w 1, w 2, w 3 } w 1 w 2 and w 2 w 1 (w 1 and w 2 are equi-plausbile) w 1 w 3 (w 1 w 3 and w 3 w 1 ) E D B w 3 A ϕ w 2 w 3 (w 2 w 3 and w 3 w 2 ) {w 1, w 2 } Min ([w i ]) w 1 w 2 Eric Pacuit 47

123 Belief Revision via Plausibility W = {w 1, w 2, w 3 } w 1 w 2 and w 2 w 1 (w 1 and w 2 are equi-plausbile) w 1 w 3 (w 1 w 3 and w 3 w 1 ) E D B w 3 A ϕ w 2 w 3 (w 2 w 3 and w 3 w 2 ) {w 1, w 2 } Min ([w i ]) w 1 w 2 Eric Pacuit 47

124 Belief Revision via Plausibility W = {w 1, w 2, w 3 } w 1 w 2 and w 2 w 1 (w 1 and w 2 are equi-plausbile) w 1 w 3 (w 1 w 3 and w 3 w 1 ) E D B w 3 A ϕ w 2 w 3 (w 2 w 3 and w 3 w 2 ) {w 1, w 2 } Min ([w i ]) w 1 w 2 Eric Pacuit 47

125 Belief Revision via Plausibility E D B A ϕ ϕ Belief: Bϕ Min (W ) [[ϕ]] M Eric Pacuit 47

126 Conservative Upgrade: Information from a trusted source Eric Pacuit ( ϕ): A C D B E 47 Belief Revision via Plausibility ψ E D B A ϕ C Conditional Belief: B ϕ ψ Min ([[ϕ]] M ) [[ψ]] M Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E

127 Conservative Upgrade: Information from a trusted source Eric Pacuit ( ϕ): A C D B E 47 Belief Revision via Plausibility E ψ D ϕ C Conditional Belief: B ϕ ψ Min ([[ϕ]] M ) [[ψ]] M Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E

128 Plausibility Models Epistemic Models: M = W, { i } i A, V Truth: M, w = ϕ is defined as follows: M, w = p iff w V (p) (with p At) M, w = ϕ if M, w = ϕ M, w = ϕ ψ if M, w = ϕ and M, w = ψ M, w = K i ϕ if for each v W, if w i v, then M, v = ϕ Eric Pacuit 48

129 Plausibility Models Epistemic-Plausibility Models: M = W, { i } i A, { i } i A, V Truth: M, w = ϕ is defined as follows: M, w = p iff w V (p) (with p At) M, w = ϕ if M, w = ϕ M, w = ϕ ψ if M, w = ϕ and M, w = ψ M, w = K i ϕ if for each v W, if w i v, then M, v = ϕ Eric Pacuit 48

130 Plausibility Models Epistemic-Plausibility Models: M = W, { i } i A, { i } i A, V Plausibility Relation: i W W where w i v means w is at least as plausible as v. Assumptions: 1. i is reflexive and transitive (and well-founded) 2. plausibility implies possibility: if w i v then w i v. 3. locally-connected: if w i v then either w i v or v i w. Eric Pacuit 48

131 Plausibility Models Epistemic-Plausibility Models: M = W, { i } i A, { i } i A, V Truth: M, w = ϕ is defined as follows: M, w = p iff w V (p) (with p At) M, w = ϕ if M, w = ϕ M, w = ϕ ψ if M, w = ϕ and M, w = ψ M, w = K i ϕ if for each v W, if w i v, then M, v = ϕ M, w = B i ϕ if for each v Min i ([w] i ), M, v = ϕ [w] i = {v w i v} is the agent s information cell. Eric Pacuit 48

132 Example b T 1, H 2 a w 2 a w 1 T 1, T 2 a a, b b a w 4 H 1, H 2 b H 1, T 2 w 3 w 1 = B a (H 1 H 2 ) B b (H 1 H 2 ) w 1 = B T 1 a H 2 w 1 = B T 1 b T 2 Eric Pacuit 49

133 Example b T 1, H 2 a w 2 a w 1 T 1, T 2 a a, b b a w 4 H 1, H 2 b H 1, T 2 w 3 w 1 = B a (H 1 H 2 ) B b (H 1 H 2 ) w 1 = B T 1 a H 2 w 1 = B T 1 b T 2 Eric Pacuit 49

134 Example b T 1, H 2 a w 2 a w 1 T 1, T 2 a a, b b a w 4 H 1, H 2 b H 1, T 2 w 3 w 1 = B a (H 1 H 2 ) B b (H 1 H 2 ) w 1 = B T 1 a H 2 w 1 = B T 1 b T 2 Eric Pacuit 49

135 Example b T 1, H 2 a w 2 a w 1 T 1, T 2 a a, b b a w 4 H 1, H 2 b H 1, T 2 w 3 w 1 = B a (H 1 H 2 ) B b (H 1 H 2 ) w 1 = B T 1 a H 2 w 1 = B T 1 b T 2 Eric Pacuit 49

136 Grades of Doxastic Strength v 0 v 1 w v 2 Suppose that w is the current state. Knowledge (KP) Belief (BP) Safe Belief ( P) Strong Belief (B s P) Eric Pacuit 50

137 Grades of Doxastic Strength v 0 v 1 w v 2 Suppose that w is the current state. Knowledge (KP) Belief (BP) Robust Belief ( P) Strong Belief (B s P) Eric Pacuit 50

138 Grades of Doxastic Strength v 0 P v 1 P w P v 2 P Suppose that w is the current state. Belief (BP) Robust Belief ( P) Strong Belief (B s P) Knowledge (KP) Eric Pacuit 50

139 Grades of Doxastic Strength v 0 P v 1 P w P v 2 P Suppose that w is the current state. Belief (BP) Robust Belief ([ ]P) Strong Belief (B s P) Knowledge (KP) Eric Pacuit 50

140 Grades of Doxastic Strength v 0 P v 1 P w P v 2 P Suppose that w is the current state. Belief (BP) Robust Belief ([ ]P) Strong Belief (B s P) Knowledge (KP) Eric Pacuit 50

141 Grades of Doxastic Strength v 0 P v 1 P w P v 2 P Suppose that w is the current state. Belief (BP) Robust Belief ([ ]P) Strong Belief (B s P) Knowledge (KP) Eric Pacuit 50

142 Is Bϕ B ψ ϕ valid? Eric Pacuit 51

143 Is Bϕ B ψ ϕ valid? Is B α ϕ B α β ϕ valid? Eric Pacuit 51

144 Is Bϕ B ψ ϕ valid? Is B α ϕ B α β ϕ valid? Is Bϕ B ψ ϕ B ψ ϕ valid? Eric Pacuit 51

145 Is Bϕ B ψ ϕ valid? Is B α ϕ B α β ϕ valid? Is Bϕ B ψ ϕ B ψ ϕ valid? Exercise: Prove that B, B ϕ and B s are definable in the language with K and [ ] modalities. Eric Pacuit 51

146 M, w = B ϕ ψ if for each v Min ([w] [[ϕ]]), M, v = ϕ where [[ϕ]] = {w M, w = ϕ} and [w] = {v w v} Eric Pacuit 52

147 M, w = B ϕ ψ if for each v Min ([w] [[ϕ]]), M, v = ϕ where [[ϕ]] = {w M, w = ϕ} and [w] = {v w v} Core Logical Principles: 1. B ϕ ϕ 2. B ϕ ψ B ϕ (ψ χ) 3. (B ϕ ψ 1 B ϕ ψ 2 ) B ϕ (ψ 1 ψ 2 ) 4. (B ϕ 1 ψ B ϕ 2 ψ) B ϕ 1 ϕ 2 ψ 5. (B ϕ ψ B ψ ϕ) (B ϕ χ B ψ χ) J. Burgess. Quick completeness proofs for some logics of conditionals. Notre Dame Journal of Formal Logic 22, 76 84, Eric Pacuit 52

148 Types of Beliefs: Logical Characterizations M, w = K i ϕ iff M, w = B ψ i ϕ for all ψ i knows ϕ iff i continues to believe ϕ given any new information Eric Pacuit 53

149 Types of Beliefs: Logical Characterizations M, w = K i ϕ iff M, w = B ψ i ϕ for all ψ i knows ϕ iff i continues to believe ϕ given any new information M, w = [ i ]ϕ iff M, w = B ψ i ϕ for all ψ with M, w = ψ. i robustly believes ϕ iff i continues to believe ϕ given any true formula. Eric Pacuit 53

150 Types of Beliefs: Logical Characterizations M, w = K i ϕ iff M, w = B ψ i ϕ for all ψ i knows ϕ iff i continues to believe ϕ given any new information M, w = [ i ]ϕ iff M, w = B ψ i ϕ for all ψ with M, w = ψ. i robustly believes ϕ iff i continues to believe ϕ given any true formula. M, w = B s i ϕ iff M, w = B iϕ and M, w = B ψ i ϕ for all ψ with M, w = K i (ψ ϕ). i strongly believes ϕ iff i believes ϕ and continues to believe ϕ given any evidence (truthful or not) that is not known to contradict ϕ. Eric Pacuit 53

151 Dynamic Epistemic Logic The key idea of dynamic epistemic logic is that we can represent changes in agents epistemic states by transforming models. Eric Pacuit 54

152 Dynamic Epistemic Logic The key idea of dynamic epistemic logic is that we can represent changes in agents epistemic states by transforming models. In the simplest case, we model an agent s acquisition of knowledge by the elimination of possibilities from an initial epistemic model. Eric Pacuit 54

153 Finding out that ϕ M = W, { i } i A, { i } i A, V Find out that ϕ M = W, { i } i A, { i } i A, V W Eric Pacuit 55

154 Example: College Park and Amsterdam Recall the College Park agent who doesn t know whether it s raining in Amsterdam, whose epistemic state is represented by the model: b, d r b, d b w 1 w 2 Eric Pacuit 56

155 Example: College Park and Amsterdam Recall the College Park agent who doesn t know whether it s raining in Amsterdam, whose epistemic state is represented by the model: b, d r b, d b w 1 w 2 What happens when the Amsterdam agent calls the College Park agent on the phone and says, It s raining in Amsterdam? Eric Pacuit 56

156 Example: College Park and Amsterdam Recall the College Park agent who doesn t know whether it s raining in Amsterdam, whose epistemic state is represented by the model: b, d r b, d b w 1 w 2 What happens when the Amsterdam agent calls the College Park agent on the phone and says, It s raining in Amsterdam? We model the change in b s epistemic state by eliminating all epistemic possibilities in which it s not raining in Amsterdam. Eric Pacuit 56

157 Example: College Park and Amsterdam Recall the College Park agent who doesn t know whether it s raining in Amsterdam, whose epistemic state is represented by the model: b, d r w 1 What happens when the Amsterdam agent calls the College Park agent on the phone and says, It s raining in Amsterdam? We model the change in b s epistemic state by eliminating all epistemic possibilities in which it s not raining in Amsterdam. Eric Pacuit 57

158 Model Update We can easily give a formal definition that captures the idea of knowledge acquisition as the elimination of possibilities. Eric Pacuit 58

159 Model Update We can easily give a formal definition that captures the idea of knowledge acquisition as the elimination of possibilities. Given M = W, {R a a Agt}, V, the updated model M ϕ is obtained by deleting from M all worlds in which ϕ was false. Eric Pacuit 58

160 Model Update We can easily give a formal definition that captures the idea of knowledge acquisition as the elimination of possibilities. Given M = W, {R a a Agt}, V, the updated model M ϕ is obtained by deleting from M all worlds in which ϕ was false. Formally, M ϕ = W ϕ, {R a ϕ a Agt}, V ϕ is the model s.th.: W ϕ = {v W M, v ϕ}; Eric Pacuit 58

161 Model Update We can easily give a formal definition that captures the idea of knowledge acquisition as the elimination of possibilities. Given M = W, {R a a Agt}, V, the updated model M ϕ is obtained by deleting from M all worlds in which ϕ was false. Formally, M ϕ = W ϕ, {R a ϕ a Agt}, V ϕ is the model s.th.: W ϕ = {v W M, v ϕ}; R a ϕ is the restriction of R a to W ϕ ; Eric Pacuit 58

162 Model Update We can easily give a formal definition that captures the idea of knowledge acquisition as the elimination of possibilities. Given M = W, {R a a Agt}, V, the updated model M ϕ is obtained by deleting from M all worlds in which ϕ was false. Formally, M ϕ = W ϕ, {R a ϕ a Agt}, V ϕ is the model s.th.: W ϕ = {v W M, v ϕ}; R a ϕ is the restriction of R a to W ϕ ; V ϕ (p) is the intersection of V (p) and W ϕ. Eric Pacuit 58

163 Model Update We can easily give a formal definition that captures the idea of knowledge acquisition as the elimination of possibilities. Given M = W, {R a a Agt}, V, the updated model M ϕ is obtained by deleting from M all worlds in which ϕ was false. Formally, M ϕ = W ϕ, {R a ϕ a Agt}, V ϕ is the model s.th.: W ϕ = {v W M, v ϕ}; R a ϕ is the restriction of R a to W ϕ ; V ϕ (p) is the intersection of V (p) and W ϕ. In the single-agent case, this models the agent learning ϕ. In the multi-agent case, this models all agents publicly learning ϕ. Eric Pacuit 58

164 Public Announcement Logic One of the big ideas of dynamic epistemic logic is to add to our formal language operators that can describe the kinds of model updates that we just saw for the College Park and Amsterdam example. Eric Pacuit 59

165 Public Announcement Logic One of the big ideas of dynamic epistemic logic is to add to our formal language operators that can describe the kinds of model updates that we just saw for the College Park and Amsterdam example. The language of Public Announcement Logic (PAL) is given by: ϕ ::= p ϕ (ϕ ϕ) K a ϕ [!ϕ]ϕ Eric Pacuit 59

166 Public Announcement Logic One of the big ideas of dynamic epistemic logic is to add to our formal language operators that can describe the kinds of model updates that we just saw for the College Park and Amsterdam example. The language of Public Announcement Logic (PAL) is given by: ϕ ::= p ϕ (ϕ ϕ) K a ϕ [!ϕ]ϕ Read [!ϕ]ψ as after (every) true announcement of ϕ, ψ. Eric Pacuit 59

167 Public Announcement Logic One of the big ideas of dynamic epistemic logic is to add to our formal language operators that can describe the kinds of model updates that we just saw for the College Park and Amsterdam example. The language of Public Announcement Logic (PAL) is given by: ϕ ::= p ϕ (ϕ ϕ) K a ϕ [!ϕ]ϕ Read [!ϕ]ψ as after (every) true announcement of ϕ, ψ. Read!ϕ ψ := [!ϕ] ψ as after a true announcement of ϕ, ψ. Eric Pacuit 59

168 Public Announcement Logic The truth clause for the dynamic operator [!ϕ] is: M, w [!ϕ]ψ iff M, w ϕ implies M ϕ, w ψ. So if ϕ is false, [!ϕ]ψ is vacuously true. Here is the!ϕ clause: M, w!ϕ ψ iff M, w ϕ and M ϕ, w ψ. Main Idea: we evaluate [!ϕ]ψ and!ϕ ψ not by looking at other worlds in the same model, but rather by looking at a new model. Eric Pacuit 60

169 Public Announcement Logic Suppose M = W, { i } i A, { i } i A, V is a multi-agent Kripke Model M, w = [!ψ]ϕ iff M, w = ψ implies M ψ, w = ϕ where M ψ = W, { i } i A, { i } i A, V with W = W {w M, w = ψ} For each i, i = i (W W ) For each i, i = i (W W ) for all p At, V (p) = V (p) W Eric Pacuit 61

170 Public Announcement Logic [!ψ]p (ψ p) Eric Pacuit 62

171 Public Announcement Logic [!ψ]p (ψ p) [!ψ] ϕ (ψ [!ψ]ϕ) Eric Pacuit 62

172 Public Announcement Logic [!ψ]p (ψ p) [!ψ] ϕ (ψ [!ψ]ϕ) [!ψ](ϕ χ) ([!ψ]ϕ [!ψ]χ) Eric Pacuit 62

173 Public Announcement Logic [!ψ]p (ψ p) [!ψ] ϕ (ψ [!ψ]ϕ) [!ψ](ϕ χ) ([!ψ]ϕ [!ψ]χ) [!ψ][!ϕ]χ [!(ψ [!ψ]ϕ)]χ Eric Pacuit 62

174 Public Announcement Logic [!ψ]p (ψ p) [!ψ] ϕ (ψ [!ψ]ϕ) [!ψ](ϕ χ) ([!ψ]ϕ [!ψ]χ) [!ψ][!ϕ]χ [!(ψ [!ψ]ϕ)]χ [!ψ]k i ϕ (ψ K i (ψ [!ψ]ϕ)) Eric Pacuit 62

175 Public Announcement Logic [!ψ]p (ψ p) [!ψ] ϕ (ψ [!ψ]ϕ) [!ψ](ϕ χ) ([!ψ]ϕ [!ψ]χ) [!ψ][!ϕ]χ [!(ψ [!ψ]ϕ)]χ [!ψ]k i ϕ (ψ K i (ψ [!ψ]ϕ)) Theorem Every formula of Public Announcement Logic is equivalent to a formula of Epistemic Logic. Eric Pacuit 62

176 Public Announcement vs. Conditional Belief Are [!ϕ]bψ and B ϕ ψ different? Eric Pacuit 63

177 Public Announcement vs. Conditional Belief Are [!ϕ]bψ and B ϕ ψ different? Yes! Eric Pacuit 63

178 Public Announcement vs. Conditional Belief Are [!ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q Eric Pacuit 63

179 Public Announcement vs. Conditional Belief Are [!ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q Eric Pacuit 63

180 Public Announcement vs. Conditional Belief Are [!ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q w 1 = B p 1 B 2q Eric Pacuit 63

181 Public Announcement vs. Conditional Belief Are [!ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q w 1 = B p 1 B 2q w 1 = [!p] B 1 B 2 q Eric Pacuit 63

182 Public Announcement vs. Conditional Belief Are [!ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q w 1 = B p 1 B 2q w 1 = [!p] B 1 B 2 q More generally, B p i (p K i p) is satisfiable but [!p]b i (p K i p) is not. Eric Pacuit 63

183 The Logic of Public Observation [!ϕ]kψ (ϕ K(ϕ [!ϕ]ψ)) Eric Pacuit 64

184 The Logic of Public Observation [!ϕ]kψ (ϕ K(ϕ [!ϕ]ψ)) [!ϕ][ ]ψ (ϕ [ ](ϕ [!ϕ]ψ)) Eric Pacuit 64

185 The Logic of Public Observation [!ϕ]kψ (ϕ K(ϕ [!ϕ]ψ)) [!ϕ][ ]ψ (ϕ [ ](ϕ [!ϕ]ψ)) Belief: [!ϕ]bψ (ϕ B(ϕ [!ϕ]ψ)) Eric Pacuit 64

186 The Logic of Public Observation [!ϕ]kψ (ϕ K(ϕ [!ϕ]ψ)) [!ϕ][ ]ψ (ϕ [ ](ϕ [!ϕ]ψ)) Belief: [!ϕ]bψ (ϕ B(ϕ [!ϕ]ψ)) [!ϕ]bψ (ϕ B ϕ [!ϕ]ψ) Eric Pacuit 64

187 The Logic of Public Observation [!ϕ]kψ (ϕ K(ϕ [!ϕ]ψ)) [!ϕ][ ]ψ (ϕ [ ](ϕ [!ϕ]ψ)) Belief: [!ϕ]bψ (ϕ B(ϕ [!ϕ]ψ)) [!ϕ]bψ (ϕ B ϕ [!ϕ]ψ) [!ϕ]b α ψ (ϕ B ϕ [!ϕ]α [!ϕ]ψ) Eric Pacuit 64

188 Belief Revision via Plausibility E D B A ϕ C Incorporate the new information ϕ (!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Eric Pacuit 65

189 Belief Revision via Plausibility E D B A ϕ C Incorporate the new information ϕ (!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Eric Pacuit 65

190 Belief Revision via Plausibility E D B A ϕ C Incorporate the new information ϕ (!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Eric Pacuit 65

191 Belief Revision via Plausibility E D B A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Eric Pacuit 65

192 Belief Revision via Plausibility E D B A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Eric Pacuit 65

193 Belief Revision via Plausibility E D B A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Eric Pacuit 65

194 K 0 Find out that ϕ = K t = K 0 ϕ M 0 Find out that ϕ = M t = M ϕ 0 = M ϕ 0 Eric Pacuit 66

195 K 0 Find out that ϕ = K t = K 0 ϕ M 0 Find out that ϕ = M t = M ϕ 0 = M ϕ 0 Eric Pacuit 66

196 K 0 Find out that ϕ = K t = K 0 ϕ M 0 Find out that ϕ = M t = M ϕ 0 = M ϕ 0 Eric Pacuit 66

197 K 0 Find out that ϕ = K t = K 0 ϕ M 0 Find out that ϕ = M t = M ϕ 0 = M ϕ 0 Eric Pacuit 66

198 K 0 Find out that ϕ = K t = K 0 ϕ M 0 Find out that ϕ = M t = M ϕ 0 = M ϕ 0 Eric Pacuit 66