kyiv 15 may 2014 taras shevchenko national university "topical directions in modern logic"

Kyiv 15 May 2014

Taras Shevchenko National University

"Topical Directions in Modern Logic"

Four floors for the theoryof theory change

Hans Rott

University of Regensburg

Overview

1. The basic problem of classical belief change theories

2. The classical AGM constructions: partial meets, hierarchies, entrenchments and possible worlds (systems of spheres)

3. The classical AGM rationality postulates

4. The space between the basic and the full AGM model ("ground floor", "top floor")

5. Identifying four floors in between:i. Internal reasonsii. Disjunctive rationalityiii. Imperfect discrimination

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The basic problem of belief revision:New information contradicting old beliefs

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The basic problem of belief revision:New information contradicting old beliefs

The basic problem of belief revision:Revisions and contractions

The Levi identity defines revisions in terms of contractions:

KA = 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:

KA = K (K¬A)

(Such contractions satisfy recovery!)

There is a close interdependence between the two identities (Gärdenfors 1982, Makinson 1987).

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 changes come from? This is not just logic!

Tentative answer: It is part of the belief state – which must thus be more than a belief set.

The classical AGM paradigm:Constructions

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)

Partial meet contraction

PMC is based on a selection of sentences to be retained. Consider the set KA of maximal subsets of K not implying A

(the A-remainders of K).

The set of all remainders is K = {KA: A is a sentence} A selection function is applied to KA. If is the set of A-remainders of K, then (KA) is the subset of

KA that contains the "best" (most plausible or secure or valuable) remainders.

Definition The partial meet contraction function over K associated with

is the intersection of the selected remainders

KA = (KA)

Partial meet contraction

A partial meet contraction is relational if and only if it the selection function is based on a relation < on K such that

(KA) = { X KA: there is no YKA such that X<Y }.

That is, picks the elements of KA that are optimal w.r.t. <.

If < is acyclic (transitive, modular), then the operation is a acyclicly (transitively, modularly) relational partial meet contraction.

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 may be applied to K ‖ A (Hansson 1994).

We assume that there is a "hierarchy" over K, such that AB 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 CB, then B is selected for removal.

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:

KA = Cn(K/A)

Rationality constraints for hierarchies

A hierarchy in the sense of Alchourrón and Makinson satisfies the conditions:

If Aa`A' and Ba`B', then AB iff A'B' (Intersubstitutivity)

If A1A2…An, then not AnA1 (Acyclicity)

A normal hierarchy is a hierarchy that satisfies

If AB and B`C then AC (Continuing up) If AB and C`A then CB (Continuing down)

A modular (virtually connected) hierarchy satisfies If AB , then AC or CB (Modularity)

(Virtual connectivity)

Modularity (virtual connectedness)

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Modularity (virtual connectedness)

R0 x yz zz

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

KA = K {B: A < A˅B}

for A such that not `A . Otherwise KA is set to K.

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<BC iff (AB<C and AC<B) (Choice)

If not `A, then A<B for some B (Maximality)

A generalized entrenchment relation < is a basic entrenchment relation that satisfiesContinuing 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)

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(AB) and not `A

Thesis: Contraction by conjunction is a choice contraction: Giving up AB is the same as giving up at least one of A and B.

Hence A<BC iff BC K(A(BC))

iff B K((AC)B) and C K((AB)C) iff AC<B and AB<C

Epistemic entrenchment contraction

Why this definition?

… remember the principal case:

KA = 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 KA

iff B KA

Possible worlds semantics for belief: Belief sets

A

Worlds considered possible in K

A accepted

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."

Possible worlds semantics for belief change I: Systems of spheres à la Grove (1988)

A

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The numbers just indicate the

relative positions; they

don't mean anything

beyond that.Don't apply +

and !Low numbers indicate high

plausibility.

A system of spheresis equivalent to a totalpre-ordering ofpossible worlds.

The beliefsare definedby theInnermostsphere.

A accepted

Possible worlds semantics for belief change I: Systems of spheres à la Grove (1988)

¬A

New information ¬Aaccepted

A

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Possible worlds semantics for belief change: Systems of spheres à la Grove (1988)

¬A

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A removed

A

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AThe possible worlds around K are partially ordered


AThe possible worlds around K are partially ordered

The yellow worlds form the new "center" for KA

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.

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A

B

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.

Representing belief states: A question of Gestalt

The classical AGM paradigm:Postulates

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, andchoose 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)

Four ideas for the rationality of revisions

1. The input: success and semantics

2. The posterior belief state: closed and consistent

3. The consistent case: the input does not contradict the prior belief state

4. Revisions by varying inputs: a sense of modularity

2. ‒ 4. can be seen as representing

static, dynamic and dispositional

notions of coherence, respectively.

(K*2) A KA.

KA : The belief set K, revisedby the new piece of information A

(K*6) If Cn(A) = Cn(B), then KA = KB.

Postulates for rational revisions of belief sets (AGM):Constraints concerning the input

(K*2) A KA.


Postulates for rational revisions of belief sets (AGM):Constraints concerning the input

(K*1) If K is logically closed, so is KA.

(K*2) A KA.

(K*5) If A is logically consistent, so is KA.


Postulates for rational revisions of belief sets (AGM):Constraints concerning the revised belief set


(K*2) A KA.

(K*3) KA K+A. K+A =df Cn(K{A})

(K*4) If ¬A K, then K+A KA.



Postulates for rational revisions of belief sets (AGM):Constraints concerning the "consistent case"


(K*2) A KA.

(K*3) KA K+A.




(K*7) K(AB) (KA)+B.

(K*8) If ¬B KA, then (KA)+B K(AB).

Postulates for rational revisions of belief sets (AGM):Constraints concerning various inputs


(K*2) A KA.

(K*3) KA K+A.




(K*7) K(AB) (KA)+B.

(K*8) If ¬B KA, then (KA)+B K(AB).

Postulates for rational revisions of belief sets (AGM):Two levels

basic postulates

supplementary postulates

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.

(K4) If A is no logical truth, then is A KA.

(K6) If Cn(A) = Cn(B), then KA = KB.

Postulates for rational contractions of belief sets (AGM): Constraints concerning the "input"

(K1) If K is logically closed, so is KA.

(K2) KA K.

(K3) If A K, then KA = K.



Postulates for rational contractions of belief sets (AGM):Constraints concerning the revised belief set


(K2) KA K.



(K5) K (KA)+A. (Recovery)


Postulates for rational contractions of belief sets (AGM):Recovery


(K2) KA K.



(K5) K (KA)+A.


(K7) KA KB K(AB).

(K8) If A K(AB), then K(AB) KA.

Postulates for rational contractions of belief sets (AGM):Constraints concerning various "inputs"


(K2) KA K.



(K5) K (KA)+A.


(K7) KA KB K(AB).

(K8) If A K(AB), then K(AB) KA.

Postulates for rational contractions of belief sets (AGM):Two levels

basic postulates

supplementary postulates

very strong!

very weak!

Lessons from non-monotonic reasoning: Full AGM rationality,inclusing modularity, is very strong, probably too strong.

Two variants of (K7).(K7P) KA Cn(A) K(AB) (Partial antitony)(K7p) If A K(AB), then A K(ABC)

(Conjunctive trisection)

A weakening of (K7).(K7c) If B K(AB), then KA K(AB)

Postulates for rational contractions of belief sets: Weakening the dispositional postulates for

Weakenings of (K8)

(K8c) If B K(AB), then K(AB) KA

(K8r) K(AB) Cn (KA KB) (Weak conj. inclusion)

(K8r') If C K(ABC), there are D and E such that

Ca` (DE) and D K(AC) and E K(BC)

(K8wd) If C K(AB), then either B˅C KA or A˅C KB

(K8p) If A K(ABC), then either A K(AB) or A K(AC)

(K8d) K(AB) (KA KB)

(K8d') K(AB) KA or K(AB) KB (Conj. factoring)

Postulates for rational contractions of belief sets: Weakening the dispositional postulates for

Ground floor

Postulates (K1)‒(K6)

Partial meet contraction (possibly non-relational) Basic entrenchment contraction:

Intersubstitutivity, Irreflexivity, Choice, Maximality

Safe contractions satisfy more than just (K1)‒(K6)!… some sort of Acyclicity!

1st floor (the logically finite case)

Relational partial meet contraction

Safe contractions with hierarchies satisfying Continuing up and Continuing down

Entrenchment contraction: … add Continuing down and(EIIcoat) If AB < C and C is a coatom of K, then either A < C or B < C.

Safe contractions satisfy more than (K1)‒(K6)!… some sort of acyclicity

… add postulates

(K7) K(AB) (KA)+B

and

(K8r) K(AB) Cn (KA KB)

2nd floor (the logically finite case)

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 AB < C and C is not a logical truth, then there are D and E such that Ca` (DE) and both A < D and B < E.

… add postulate (K8c) If A K(AB), then K(AB) KB

3rd floor (the logically finite case)

Relational partial meet contraction, with the relation satisfying the Interval condition.

Safe contraction: requirements on < identical with entrenchments

Entrenchment contraction: … add

(EII) If AB < 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.

… add postulate (K8d) K(AB) KA KB

Interval condition

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Interval condition

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4th floor (the logically finite case)

Relational partial meet contraction, with the relation satisfying the Interval condition and Semitransitivity.

Safe contraction: requirements on < identical with entrenchments

Entrenchment contraction: … add

(Semitransitivity) If A < B and B < C, then either

A < D or D < C .

… add postulate (K8s) If A K(AB), then K2(AB) KA

Recursive definition

K1A := KA K1A := KA

Ki+1A := K(A(KiA))

K(i+1)A := (K1A) (Ki+1A)


¬A

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K1A = K1A := KA


¬A

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K2A := K(A(KA))


¬A

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K2A := (KA) (K2A)


¬A

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K3A := K(A(K2A))


¬A

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K3A := (K2A) (K3A)


Semitransitvity

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Semitransitvity

R0 x y u zu u

4th floor: Belief revision with semitransitivity

Peppas, Pavlos, and Mary-Anne Williams, "Belief change and semiorders", Principles of Knowledge Representation and Reasoning: Proceedings of the Twelfth International Conference (KR'2014), Vienna, July 20-24, 2014.

Jamison, Dean, and Lawrence Lau, "Semiorders and the theory of choice", Econometrica 41 (1973), 901–912.

Fishburn, Peter C., "Semiorders and choice functions", Econometrica 43 (1975), 975–977.

Jamison, Dean, and Lawrence Lau, "Semiorders and the theory of choice: A correction", Econometrica 43 (1975), 979–980.

kyiv 15 may 2014 taras shevchenko national university "topical directions in modern logic"

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