Kyiv 15 May 2014 Taras Shevchenko National University "Topical Directions in Modern Logic"

1. Kyiv 15 May 2014 Taras Shevchenko National University "Topical Directions in Modern Logic" Four floors for the theory of theory change Hans Rott University of Regensburg

2. Overview • The basic problem of classical belief change theories • The classical AGM constructions: partial meets, hierarchies, entrenchments and possible worlds (systems of spheres) • The classical AGM rationality postulates • The space between the basic and the full AGM model ("ground floor", "top floor") • Identifying four floors in between: • Internal reasons • Disjunctive rationality • Imperfect discrimination

3. r p q q&r  ¬p s t  q The basic problem of belief revision:New information contradicting old beliefs

4. r p q q&r  ¬p s t  q The basic problem of belief revision:New information contradicting old beliefs

5. The basic problem of belief revision:Revisions and contractions • The Levi identity defines revisions in terms of contractions: • KA = Cn((K¬A)  {A}) • Rationale: A sentence can be added to a set without giving rise to inconsistency iff that set does not imply ¬A. • Conversely, the Harper identity defines contractions in terms of revisions: • KA = K  (K¬A) • (Such contractionssatisfyrecovery!) • There is a close interdependence between the two identities (Gärdenfors 1982, Makinson 1987).

6. The basic problem of belief revision:Belief sets and belief states • Belief sets are sets of sentences expressing an agent’s beliefs. • Belief sets should obey some normative requirements. • They should be consistent. • They should be logically closed. • Today we focus on logically closed belief sets. • Belief sets are subject to change. • Belief changes should obey some normative requirements. • Question: Where does the information governing rational belief changescomefrom? Thisis not just logic! • Tentative answer: It is part of the belief state – which must thus be more than a belief set.

7. The classical AGM paradigm: Constructions

8. Belief revision theory: AGM models • AGM — Carlos Alchourrón (1931-1996), Peter Gärdenfors (*1949), and David Makinson (*1941) Constructions for belief changes: • Partial meet contraction and revision ("PMC", AGM 1985) • Safe contraction and revision ("SC", AM 1985) • Epistemic entrenchment contraction and revision ("EEC", G 1988, GM 1988) • System of spheres contraction and revision (Grove 1988); contractions and revisions based on partial orderings of possible worlds (Katsuno/Mendelzon 1991, Rott 1993)

9. Partial meet contraction • PMC is based on a selection of sentences to be retained. • Consider the set KA of maximal subsets of K not implying A (the A-remainders of K). • The set of all remainders is K = {KA: A is a sentence} • A selection function g is applied to KA. • If is the set of A-remainders of K, then g(KA) is the subset of KA that contains the "best" (most plausible or secure or valuable) remainders. Definition • The partial meet contraction function over K associated with g is the intersection of the selected remainders KA = g(KA)

10. Partial meet contraction • A partial meet contraction is relational if and only if it the selection function g is based on a relation < on K such that g(KA) = { X KA: there is no YKA such that X<Y }. That is, g picks the elements of KA that are optimal w.r.t. <. • If < is acyclic (transitive, modular), then the operation  is a acyclicly (transitively, modularly) relational partial meet contraction.

11. Safe contraction • SC is based on selection of sentences to be removed. • Consider the set K‖A of minimal subsets of K implying A (the A-kernels of K). • A selection function s may be applied to K‖A (Hansson 1994). • We assume that there is a "hierarchy" over K, such that AB means that A is "worse" (more exposed or vulnerable, less secure or plausible) than B. • Idea: From each A-kernel, if B is minimal in it, i.e., if there is no C such that CB, then B is selected for removal.

12. Safe contraction • A sentence B in K is safe with respect to A (briefly, A-safe) if and only if it is not selected for removal in any of the A-kernels. • Let K/A be the set of A-safe sentences in K. Definition • The safe contraction function over K associated with  is the set of consequences of the A-safe elements: KA = Cn(K/A)

13. Rationality constraints for hierarchies • A hierarchy  in the sense of AlchourrónandMakinson satisfies the conditions: • If Aa`A' and Ba`B', then AB iff A'B' (Intersubstitutivity) • If A1A2…An, then not AnA1 (Acyclicity) • A normal hierarchyis a hierarchy thatsatisfies • IfABandB`CthenAC (Continuing up) • IfABandC`AthenCB (Continuing down) • A modular (virtually connected) hierarchy  satisfies • IfAB , thenAC orCB (Modularity) (Virtual connectivity)

14. Modularity (virtual connectedness) y z x

15. Modularity (virtual connectedness) R y z z z 0 x

16. Epistemic entrenchment contraction • EEC is based on the entrenchment of the sentences in the belief state. • Epistemic entrenchment relations are similar to hierarchies, but are applied in a "more direct" way. • Thus they need to satisfy some (one?) special rationality constraints. Definition • The entrenchment-based contraction function over K associated with < is KA = K  {B: A < A˅B} for A such that not`A . Otherwise KA is set to K.

17. Rationality constraints for epistemic entrenchments • A basic entrenchment relation < satisfies the conditions • If Aa`A' and Ba`B', then A<B iff A'<B' (Intersubstitutivity) • Not A<A (Irreflexivity) • A<BC iff (AB<C and AC<B) (Choice) • If not `A, then A<B for some B (Maximality) • A generalized entrenchment relation < is a basic entrenchment relation that satisfies Continuing up and Continuing down for <. • A GM entrenchment relation < is a generalized entrenchment relation the satisfies Virtual connectivity for < and • If not K ` , then A is in K iff B<A for some B (Minimality)

18. Rationality constraints for epistemic entrenchments • Basic entrenchment relations need not be acyclic, but generalized entrenchment relations are (Rott 2003). • Why is "Choice" basic? • A re-constructive view of the concept of entrenchment: A<B if and only if B  K(AB) and not `A • Thesis: Contraction by conjunction is a choice contraction: Giving up AB is the same as giving up at least one of A and B. • Hence A<BC iff BC K(A(BC)) iff B K((AC)B) and C K((AB)C) iff AC<B and AB<C

19. Epistemic entrenchment contraction • Why this definition? … remember the principal case: KA = K  {B: A < A˅B} • Because it works perfectly together with the concept of entrenchment (the re-constructive view), even in very "basic" settings. Principal case A < A˅B iff A˅B  K(A(A˅B)) iff A˅B  KA iff B  KA

20. Possible worlds semantics for belief: Belief sets A Worlds considered possible in K A accepted

21. Possible worlds semantics for belief:Maximal non-implying subsets A Worlds considered possible in a maximal subset of K that does not imply A "The spirit of AGM is semantic."

22. Possible worlds semantics for belief change I: Systems of spheres à la Grove (1988) A system of spheres is equivalent to a total pre-ordering of possible worlds. The numbers just indicate the relative positions; they don't mean anything beyond that. Don't apply + and  ! Low numbers indicate high plausibility. A  6 5 4 The beliefs are defined by the Innermost sphere. 3 2 1 A accepted

23. Possible worlds semantics for belief change I: Systems of spheres à la Grove (1988) A ¬A 1 New information ¬A accepted

24. Possible worlds semantics for belief change: Systems of spheres à la Grove (1988) A ¬A 1 1 A removed

25. Possible worlds semantics for belief:Maximal non-implying subsets The possible worlds around K are partially ordered A

26. Possible worlds semantics for belief:Maximal non-implying subsets The possible worlds around K are partially ordered A The yellow worlds form the new "center" for KA

27. Epistemic entrenchment and systems of spheres A is more entrenched than B if and only if the set of A-worlds covers more spheres than the set of B-worlds. A  6 5 4 3 2 1 B

28. Representing belief states: A questionofGestalt • Semantical vs. syntactical representations Horizontal (coherentist) vs. vertical ("data-driven") mode • Systems of spheres (SOSs) vs. prioritized data bases (PDBs) Which representation is to be preferred? • SOSs seem to give the best intuitive understanding of methods for revising belief-states. • However, for the intuitive understanding of the contents of belief states, PDBs seem much more appropriate.

29. The classical AGM paradigm: Postulates

30. Eliminating inconsistency • "Success" idealisation: Accept the new piece of information! • Give up some old belief(s)! • Equivalently: Consider some situation(s) or world(s) that don't satisfy all old beliefs! • A problem of rational choice: Choose carefully which beliefs you want to retain, and choose carefully which ones you want to give up! • Rational choice is relational choice – use a preference relation to determine the relevant decisions which is independent of the input (the sentences to be added or retracted)

31. Four ideas for the rationality of revisions • The input: successandsemantics • The posterior belief state: closedandconsistent • The consistentcase: theinputdoes not contradicttheprior belief state • Revisionsbyvaryinginputs: a sense ofmodularity • 2. ‒ 4. canbeseenasrepresenting • static, • dynamicand • dispositional notionsofcoherence, respectively.

32. Postulates for rational revisions of belief sets (AGM):Constraints concerning the input (K*2) A  KA. KA : The belief set K, revised by the new piece of information A (K*6) If Cn(A) = Cn(B), then KA = KB.

33. Postulates for rational revisions of belief sets (AGM):Constraints concerning the input (K*2) A  KA. (K*6) If Cn(A) = Cn(B), then KA = KB.

34. Postulates for rational revisions of belief sets (AGM):Constraints concerning the revised belief set (K*1) If K is logically closed, so is KA. (K*2) A  KA. (K*5) If A is logically consistent, so is KA. (K*6) If Cn(A) = Cn(B), then KA = KB.

35. Postulates for rational revisions of belief sets (AGM):Constraints concerning the "consistent case" (K*1) If K is logically closed, so is KA. (K*2) A  KA. (K*3) KA  K+A. K+A =dfCn(K{A}) (K*4) If ¬A  K, then K+A  KA. (K*5) If A is logically consistent, so is KA. (K*6) If Cn(A) = Cn(B), then KA = KB.

36. Postulates for rational revisions of belief sets (AGM):Constraints concerning various inputs (K*1) If K is logically closed, so is KA. (K*2) A  KA. (K*3) KA  K+A. (K*4) If ¬A  K, then K+A  KA. (K*5) If A is logically consistent, so is KA. (K*6) If Cn(A) = Cn(B), then KA = KB. (K*7) K(AB)  (KA)+B. (K*8) If ¬B  KA, then (KA)+B K(AB).

37. Postulates for rational revisions of belief sets (AGM):Two levels (K*1) If K is logically closed, so is KA. (K*2) A  KA. (K*3) KA  K+A. (K*4) If ¬A  K, then K+A  KA. (K*5) If A is logically consistent, so is KA. (K*6) If Cn(A) = Cn(B), then KA = KB. (K*7) K(AB)  (KA)+B. (K*8) If ¬B  KA, then (KA)+B K(AB). basic postulates supplementary postulates

38. The central result: Belief change and preferences • The most characteristic postulates for the classical theory of belief change were the postulates concerning various inputs, aka the postulates of dispositional coherence. • These postulates, (K*7) and (K*8), are equivalent with the following claim: (The dispositions for) AGM belief changes are structured as if they were governed by modular doxastic preferences that are independent of the information that actually comes in. • This idea of rationality is in accordance with rational choice theory. • The theoretical problem of belief formation is considered as a practical problem of rational choice.

39. Postulates for rational contractions of belief sets (AGM): Constraints concerning the "input" (K4) If A is no logical truth, then is A  KA. (K6) If Cn(A) = Cn(B), then KA = KB.

40. Postulates for rational contractions of belief sets (AGM):Constraints concerning the revised belief set (K1) If K is logically closed, so is KA. (K2) KA  K. (K3) If A  K, then KA = K. (K4) If A is no logical truth, then is A  KA. (K6) If Cn(A) = Cn(B), then KA = KB.

41. Postulates for rational contractions of belief sets (AGM):Recovery (K1) If K is logically closed, so is KA. (K2) KA  K. (K3) If A  K, then KA = K. (K4) If A is no logical truth, then is A  KA. (K5) K  (KA)+A. (Recovery) (K6) If Cn(A) = Cn(B), then KA = KB.

42. Postulates for rational contractions of belief sets (AGM):Constraints concerning various "inputs" (K1) If K is logically closed, so is KA. (K2) KA  K. (K3) If A  K, then KA = K. (K4) If A is no logical truth, then is A  KA. (K5) K  (KA)+A. (K6) If Cn(A) = Cn(B), then KA = KB. (K7) KA  KB  K(AB). (K8) If A  K(AB), then K(AB) KA.

43. Postulates for rational contractions of belief sets (AGM):Two levels (K1) If K is logically closed, so is KA. (K2) KA  K. (K3) If A  K, then KA = K. (K4) If A is no logical truth, then is A  KA. (K5) K  (KA)+A. (K6) If Cn(A) = Cn(B), then KA = KB. (K7) KA  KB  K(AB). (K8) If A  K(AB), then K(AB) KA. basic postulates very weak! supplementary postulates very strong!

44. Postulates for rational contractions of belief sets: Weakening the dispositional postulates for  Lessons from non-monotonic reasoning: Full AGM rationality, inclusing modularity, is very strong, probably too strong. Two variants of (K7). (K7P) KA  Cn(A)  K(AB) (Partial antitony) (K7p) If A K(AB), then A K(ABC) (Conjunctive trisection) A weakening of (K7). (K7c) If B K(AB), then KAK(AB)

45. Postulates for rational contractions of belief sets: Weakening the dispositional postulates for  Weakenings of (K8) (K8c) If B K(AB), then K(AB) KA (K8r) K(AB)  Cn (KA KB) (Weak conj. inclusion) (K8r') If C K(ABC), there are D and E such that Ca`(DE) and D K(AC) and E K(BC) (K8wd) If C K(AB), then either B˅C KA or A˅C KB (K8p) If A K(ABC), then either A K(AB) or A K(AC) (K8d) K(AB)  (KA KB) (K8d') K(AB)  KA or K(AB)  KB (Conj. factoring)

46. Ground floor • Postulates (K1)‒(K6) • Partial meet contraction (possibly non-relational) • Basic entrenchment contraction: Intersubstitutivity, Irreflexivity, Choice, Maximality • Safe contractions satisfy more than just (K1)‒(K6)! … some sort of Acyclicity!

47. 1st floor (the logically finite case) … add postulates (K7) K(AB)  (KA)+B and (K8r) K(AB)  Cn (KA KB) • Relational partial meet contraction • Safe contractions with hierarchies satisfying Continuing up and Continuing down • Entrenchment contraction: … add Continuing down and (EIIcoat) If AB < C and C is a coatom of K, then either A < C or B < C. • Safe contractions satisfy more than (K1)‒(K6)! … some sort of acyclicity

48. 2nd floor (the logically finite case) … add postulate (K8c) If A  K(AB), then K(AB)  KB • Transitively relational partial meet contraction • Safe contractions with transitive hierarchies satisfying Continuing up and Continuing down • Entrenchment contraction: … add Continuing up‒ and strengthen (EIIcoat) to (EII-o) If AB < C and C is not a logical truth, then there are D and E such that Ca`(DE) and both A < D and B < E.

49. 3rd floor (the logically finite case) … add postulate (K8d) K(AB)  KA  KB • Relational partial meet contraction, with the relation satisfying the Interval condition. • Safe contraction: requirements on < identical with entrenchments • Entrenchment contraction: … add (EII) If AB < C, then either A < C or B < C, or, alternatively and equivalently, the Interval condition (Int) If A < B and C < D, then either A < D or C < B.

50. Interval condition y v x u