Conditional Logic and Belief Revisionphd/documents/tesi/XIII/TesiGLIOZZI.pdf · 4 Conditionals and...

Universita degli Studi di Torino

Dottorato di Ricerca in Informatica(XIII ciclo)

PhD Thesis

Conditional Logic and BeliefRevision

Valentina Gliozzi

Advisors: Prof.Laura Giordano, Prof Nicola Olivetti

Dipartimento di Informatica — Universita degli Studi di Torino

C.so Svizzera, 185 — I-10149 Torino (Italy)

http://www.di.unito.it/

1

Contents

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Belief Revision 5

2.1 Alchourron, Gardenfors and Makinson’s Theory . . . . . . . . . . 5

2.1.1 Constructive Models for Revision and Contraction . . . . . 11

2.1.2 Limits of AGM Theory . . . . . . . . . . . . . . . . . . . . 19

2.2 Iterated Belief Revision . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Darwiche and Pearl’s theory . . . . . . . . . . . . . . . . . 21

2.2.2 Lehmann’s Theory . . . . . . . . . . . . . . . . . . . . . . 23

3 Conditional Logics 25

3.1 Limits of classical logic’s formalization of conditional sentences . . 25

3.2 Cotenability theory . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Strict implication . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Stalnaker’s logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Lewis’s logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.6 Gabbay’s logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Conditionals and Belief Revision 35

4.1 Evaluation of Conditional Sentences and Belief Revision . . . . . 35

4.1.1 The conditional logic deriving from the Ramsey Test . . . 37

4.2 Triviality result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Gardenfors’s analysis of the triviality result . . . . . . . . . . . . . 39

4.4 But Gardenfors’s analysis is not the only possible one . . . . . . . 40

4.4.1 Expliciting all the assumptions on which lies the triviality

result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

i

ii CONTENTS

4.4.2 Questioning the assumption according to which belief revi-

sion systems are closed with respect to the expansion operator 42

4.4.3 Questioning the Minimal Change Principle when applied

to conditional beliefs . . . . . . . . . . . . . . . . . . . . . 43

4.5 Solutions Proposed in the Literature . . . . . . . . . . . . . . . . 44

4.5.1 Weakening the Ramsey Test . . . . . . . . . . . . . . . . . 45

4.5.2 Levi’s solution . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5.3 Friedman and Halpern’s solution . . . . . . . . . . . . . . 47

4.5.4 Conditionals and belief update . . . . . . . . . . . . . . . . 48

5 A Logic for Belief Revision 51

5.1 The Conditional Logic BC . . . . . . . . . . . . . . . . . . . . . . 52

5.1.1 The language . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.2 The axiomatization . . . . . . . . . . . . . . . . . . . . . . 53

5.1.3 The semantics . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Soundness of BC . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4 Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . 62

5.5 Decidability of BC . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.5.1 Finite-model property . . . . . . . . . . . . . . . . . . . . 68

6 Extension to Iterated Belief Revision 79

6.1 Iterated Belief Revision . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2 The Conditional Logic IBC . . . . . . . . . . . . . . . . . . . . . 85

6.3 Conditionals and Iterated Revision . . . . . . . . . . . . . . . . . 91

6.3.1 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Weakening the AGM Postulates 101

7.1 Weakened AGM postulates . . . . . . . . . . . . . . . . . . . . . . 102

7.2 Non-triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.3 Strong Ramsey Test . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 The logic BCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8 Conclusions and Future Work 121

8.1 Conclusions and Future work . . . . . . . . . . . . . . . . . . . . 121

8.1.1 Related Works . . . . . . . . . . . . . . . . . . . . . . . . 121

8.1.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 123

CONTENTS iii

Bibliography 124

iv CONTENTS

List of Figures

5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

v

Chapter 1

Introduction

1.1 Introduction

Belief change and conditional logics are two important fields of Artificial Intelli-

gence.

Belief change concerns the study and formalization of a crucial capacity of

intelligent agents, namely the capacity to change their beliefs in face of new

information. It has been widely studied in the last twenty years [1, 15, 8, 30,

28], and there are nowadays several theories of belief change: from the most

known AGM theory of belief revision (proposed by Alchourron, Gardenfors and

Makinson in the Eighties [1, 15]) to the more recent theories of iterated belief

revision (proposed by Darwiche, Pearl and Lehmann [8, 30]), and of belief update

(proposed by Katsuno and Mendelzon in [28]). Given a representation of the belief

state of an agent, the various theories define the properties of a belief change

operator that takes the knowledge structure and a formula and gives back a new

belief state in which the formula has been consistently added. Various theories

of belief revision derive from different assumptions on the representation of belief

states and on the properties of belief change operators.

Conditional logics provide a formal theory of conditional reasoning, which

is a kind of reasoning central in many domains of Artificial Intelligence, such

as knowledge representation, non-monotonic reasoning, reasoning about actions,

planning and natural language analysis. These logics were introduced in the

Sixties by Stalnaker [45] and Lewis [33], and recently rediscovered in Artificial

Intelligence. For a complete review of the application fields for conditional logics

see Ginsberg [16] and Fagin [9].

1

2 CHAPTER 1. INTRODUCTION

In the last fifteen years many authors [15, 24, 32, 40, 34, 10] have considered

the problem of establishing a correspondence between these two domains.

Establishing such a relation is interesting from two points of view.

From the point of view of belief change, this correspondence provides useful

formal tools to study some properties of belief change itself, as we will see in

chapter VI.

From the point of view of conditional reasoning, establishing this correspon-

dence allows to provide a semantics of conditional sentences in terms of belief

change, capturing in this way a seminal idea in the analysis of conditionals dat-

ing back to the philosopher F.P. Ramsey [39, 45, 32, 15, 23]. Indeed, in [39]

Ramsey proposed the following criterion to decide whether to accept a condi-

tional sentence “if A, then B”: hypothetically add the antecedent A to your

stock of beliefs and consider whether the consequent B follows; if B follows, ac-

cept the conditional, otherwise reject it. This is undoubtedly a very intuitive

criterion. The problem is how to formalize it.

The first formalization of a relation between conditional sentences and belief

change has been proposed by Gardenfors in [15]. Gardenfors’s formalization

devises a correspondence between conditional sentences and a specific kind of

belief change, namely belief revision (see chapter 2) which is that kind of belief

change that occurs when an agent acquires new information about a “static ”

world (contraposed to belief update that occurs when an agent changes its stock

of beliefs as a consequence of a change occurred in the world).

Gardenfors’s proposal, known as Ramsey Test, is formulated as follows:

RT: A > B ∈ K if and only if B ∈ K ∗ A,

where > is a conditional operator, K is a belief set (i.e. a deductively closed set

of formulas) and ∗ is the belief revision operator that takes a belief set and a

formula and adds the formula to the belief set by minimally changing it in order

to preserve consistency.

Unfortunately, Gardenfors has shown that his formalization leads to a well

known triviality result according to which there is no interesting belief revision

system compatible with the Ramsey Test.

Gardenfors’s negative result has stimulated a wide debate and literature aim-

ing to reconcile the two sides of the coin: belief revision and conditional logic.

The proposals can be divided in a few categories. Some authors maintain that the

Ramsey Test should link conditionals and another kind of belief change, namely

1.1. INTRODUCTION 3

belief update [24]. Other authors suggest to avoid the triviality result by adding

some preconditions to the Ramsey Test [40, 34]. Others, finally, propose to ex-

clude conditional formulas from belief sets, ascribing them a different espistemic

status [32].

In this thesis, we propose a relation between belief revision and conditional

logics that allows to avoid the triviality result incurred by the Ramsey Test.

To this purpose we first propose a conditional logic to represent belief revision.

Secondly, we propose an extension of the logic BC that allows us to represent

iterated belief revision. Finally, we show how the logic BC can be derived by the

Ramsey Test, once abandoned some counterintuitive assumptions responsible of

the triviality result.

The thesis is thus organized as follows. Chapter 2 is devoted to a survey

of the theories of belief revision. Chapter 3 is devoted to a survey of the most

known conditional logics. Chapter 4 is devoted to the Ramsey Test: we expose

Gardenfors’s formulation; we show how it runs into the triviality result; we ex-

amine the literature devoted to finding a solution to the problem; last, and most

importantly, we show how the explanation of the triviality result provided by

Gardenfors lies on some tacit and counterintuitive assumption, and we show how

the result can be avoided by naturally abandoning these assumptions.

The last three chapters are devoted to our logics. In chapter 5, we introduce

the logic BC; we study the properties of BC and show BC is decidable; we

prove a representation theorem that establishes a significant relation between BC

and belief revision. In chapter 6, we propose an extension of BC, called IBC,

that represents iterated belief revision. A representation theorem establishes a

relation between IBC and iterated belief revisions; the relation established by the

representation theorem is tighter than the relation between BC and simple belief

revision. Some corollaries of the representation theorem show how conditional

logics can be a useful tool to prove some general properties of belief revision

operators, or classes of belief revision operators.

Finally, in chapter 7, we show that the correspondence established by the

Ramsey Test can be safely assumed if we weaken in a very intuitive way some of

the revision postulates. In this way, the triviality result is avoided.

4 CHAPTER 1. INTRODUCTION

Furthermore, our weakened revision postulates are compatible also with a

strong version of the Ramsey Test, called Strong Ramsey Test and formulated as

follows:

(SRT)

• if B ∈ K ∗ A then A > B ∈ K and

• if B 6∈ K ∗ A then ¬(A > B) ∈ K.

The conditional logic deriving from the Strong Ramsey Test and our weakened

revision postulates is very similar to the logic BC studied in chapter 5.

Chapter 2

Belief Revision

2.1 Alchourron, Gardenfors and Makinson’s The-

ory

In [1, 12, 15] Alchourron, Gardenfors and Makinson develop a “theory of rational

change of beliefs”, called AGM theory, in which they characterize three forms

of belief change, namely contraction, expansion and revision. Expansion is that

specific form of belief change that consists in introducing a new belief in a be-

lief state. Contraction is the converse of expansion and consists in abandoning

a sentence previously believed. Revision consists in changing opinion about a

sentence, thus accepting it if it was rejected and rejecting it if it was accepted.

The difference between belief revision and expansion is that in expansion a new

belief is adopted without giving up any of the previously held beliefs, whereas in

revision some of the previously held beliefs are abandoned in order to preserve

consistency.

More formally, Alchourron, Gardenfors and Makinson represent agents’ states

of belief as deductively closed sets of formulas, called belief sets, and characterize

contraction, expansion and revision as functions from belief sets and formulas to

new belief sets. The formulas contained in belief sets belong to a language L that

contains the propositional logic language. Belief sets are deductively closed with

respect to consequence relation Cn that includes classical propositional logic, is

monotonic, compact, and validates the cut rule and the deduction theorem, where

a logic Cn is compact if and only if whenever A ∈ Cn(H), there is a finite subset

H ′ of H such that A ∈ Cn(H ′). It validates the cut rule if and only if A ∈ Cn(H)

5

6 CHAPTER 2. BELIEF REVISION

and B ∈ Cn(H ∪ A), imply B ∈ Cn(H). It validates the deduction theorem if

and only if B ∈ Cn(H ∪ {A}) implies A → B ∈ Cn(H).

In this setting, a formula A is said to be “accepted” in a belief set K just in

case A ∈ K, it is “rejected” in K just in case ¬A ∈ K, and it is “unknown” in

K if both A 6∈ K and ¬A 6∈ K.

The contraction operator, denoted by −, is a function that takes a belief set K

and a formula A ∈ K, and leads to a new belief set K −A that does not contain

A. The expansion operator, denoted by +, is a function that takes a belief set

K and the formula A, and gives back a new belief set K + A that contains A.

The revision operator, denoted by ∗, is a function that, given K and A, leads to

a new belief set K ∗A containing A, that is consistent if so is A, and that differs

as little as possible from K.

All these belief change operators have been characterized both by enumer-

ating a set of rationality postulates they have to satisfy and by providing some

constructive models for them. I examine first the rationality postulates proposed

for the three belief change operators and then some of the constructive models.

Expansion

The expansion operator is the simpler belief change operator. It has to satisfy

the following postulates.

Expansion Postulates

(K + 1): K + A is a belief set;

(K + 2): A ∈ K + A;

(K + 3): K ⊆ K + A;

(K + 4): if A ∈ K, K + A = K;

(K + 5): if K ⊆ H, then K + A ⊆ H + A;

(K +6): K +A is the smallest belief set that satisfies (K +1)− (K +5)

Postulate (K+1) simply requires the result of the expansion of a belief set to be a

belief set itself. Postulate (K+2) requires the expansion to be successful. (K+3)

expresses a characteristic feature that we will find for all belief change operators:

since information is not gratuitous, unnecessary losses of information should be

avoided. Therefore, all the beliefs in K should be preserved in the expanded

belief set. Postulate (K + 4) says that if A is already in K, the expansion of

2.1. ALCHOURRON, GARDENFORS AND MAKINSON’S THEORY 7

K by A should have no effect. Postulate (K + 5) is a monotonicity postulates

for expansion and requires that if two belief sets are included in one another,

also their expansions are included in one another. Finally, (K + 6) requires that

beliefs should not be adopted without any justification: the expanded belief set

should contain only the beliefs required by other postulates.

There is an expansion function that satisfies postulates (K + 1) − (K + 6),

and it is the function defined as: K +A = CnPC(K ∪A). The following theorem

holds:

Theorem 2.1.1 ([15], page 51) The expansion function satisfies (K + 1) −(K + 6) if and only if K + A = CnPC(K ∪ {A})

Revision and Contraction

Following the literature, I will treat revision and contraction together, for the

two operators strictly tight to each other. Indeed, revision can be considered as

a contraction followed by an expansion. The two operators are derivable from

each other through the following identities:

Levi identity K ∗ A = (K − ¬A) + A.

Harper identity K − A = K ∩ (K ∗ ¬A).

Throughout the chapter, all what we say about revision (or contraction) can be

extended to the other operator by means of the two stated identities.

If the expansion operator can be simply defined as: K + A = CnPC(K ∪ A),

things are not so straightforward as far as contraction and revision operators

are concerned. Revision (and contraction) are more problematic than expansion

because in general the result of a revision (or of a contraction) of a belief set

K by a formula A is not uniquely determined by K and A. Rather, this result

depends on some extra information that is not represented in K, nor in A.

Consider belief revision. An example, taken from [14] will illustrate the prob-

lem.

Example 2.1.2 [14] Suppose you have a belief set K containing, among other

things, these informations:

• (A) The bird caught in the trap is a swan

• (B′) The bird caught in the trap comes from Sweden


• (B′ → B) Sweden is part of Europe

• (A ∧B → C) All European swans are white

The following fact is derivable from (K):

• (C) The bird caught in the trap is white

Now, suppose that the bird caught in the trap turns out to be black (¬C). In

order to consistently add the new information ¬C to your belief set, you have

to abandon some of your previous beliefs. The problem is to determine which

of the previously held beliefs should be abandoned, for you can choose between

retracting A or B′ or B′ → B or A ∧ C → B or any combination of these

sentences. 2

Alchourron, Gardenfors and Makinson’s rationality postulates “circumscribe the

set of rational solutions” to this kind of problems. These solutions have to enforce

several constraints, among which the fact that the result of the revision of a belief

set has to be a belief set itself, that it should be independent from the syntactic

form of the inserted formula, that it has to be consistent if so is the incoming

formula.

The most important requirement for belief revision is the Minimal Change

Principle that expresses the “economic side of rationality”. According to this

principle, since information is not gratuitous, unnecessary losses of information

are to be avoided, and when revising a belief set with a new formula, as much as

possible of the old belief set should be preserved. More precisely, the revision of

a belief set by a formula consistent with it is the simple addition of the formula

to the belief set.

The Minimal Change Principle is not only the most important requirement

for belief revision in AGM theory. It is also the most criticized. Some authors

[28] have pointed out that the Minimal Change Principle is not always suitable

and have proposed alternative notions of belief change that do not satisfy the

Minimal Change Principle (like belief update). It is undoubtedly true that the

principle is not always suitable and, indeed, the AGM theory for revision does

not aim to be the only theory of belief change valid for all domains. Two good

examples of domains for which belief revision is well suited are the scientific

domain and the legal domain: the Minimal Change Principle should be respected


when introducing a new information into a scientific theory by changing it as little

as possible. Similarly, the Minimal Change Principle is well suited in the legal

domain, as, when introducing a new rule in a normative system, we want to

change the system as little as possible.

A second point that has to be made is that AGM postulates capture the

Minimal Change Principle in a very weak way and only as far as mild revisions

are concerned, i.e. revisions with formulas consistent with the belief set. In-

deed, there is a revision function satisfying the postulates (full meet contraction)

that, whenever the new information is inconsistent with the belief set, simply

throws away the whole belief set and keeps only the new formula and its logical

consequences. This revision function has the property that if ¬A ∈ K, then

K ∗ A = Cn(A). This problem is solved by stronger theories of revision, for

instance by Darwiche and Pearl’s iterated revision theory (see below), that re-

quire any belief revision to be reversible (K ∗ ¬A ∗A = K ∗A). A belief revision

function that eliminates the whole belief set when making a revision is discarded

by such a theory.

Revision Postulates

(K ∗ 1) (K ∗ A) is a belief set;

(K ∗ 2) A ∈ K ∗ A;

(K ∗ 3) (K ∗ A) ⊆ (K + A);

(K ∗ 4) if ¬A 6∈ K,K + A ⊆ K ∗ A;

(K ∗ 5) (K ∗ A) = K⊥ only if ` ¬A;

(K ∗ 6) if ` A ↔ B then K ∗ A = K ∗B;

(K ∗ 7) K ∗ (A ∧B) ⊆ (K ∗ A) + B;

(K ∗ 8) if ¬B 6∈ (K ∗ A), then (K ∗ A) + B ⊆ K ∗ (A ∧B)

Postulate (K ∗ 1) requires the result of a revision to be a belief set. Postulate

(K ∗ 2) says that revision is always successful. Postulate (K ∗ 3) says that the

revision of a belief set with a formula A does not lead to conclude more than

what can be concluded by the simple expansion of K with A. (K ∗ 4) expresses

one side of the Minimal Change Principle, called Preservation Principle that

says that when we make the revision of K with a formula consistent with it, no

information of K has to be rejected. Taken together, (K ∗3) and (K ∗4) say that


if A is consistent with K, then a revision of K with A is just an expansion of K

with A. This corresponds to the Minimal Change Principle. Postulate (K ∗ 5)

says that the revision is consistent unless the added formula is inconsistent by

itself. Postulate (K ∗ 6) says that the result of revision does not depend on the

syntactic form of the added information. Postulates (K ∗ 7) and (K ∗ 8) can be

regarded as a generalization of (K ∗ 3) and (K ∗ 4) to deal with conjunctions.

Once characterized belief revision operators, we can define belief revision sys-

tems as follows.

Definition 2.1.3 [Belief revision system] A belief revision system is a structure

〈K, ∗〉, where K is a set of belief sets and ∗ a belief revision operator.

We now consider the rationality postulates for contraction.

1. Contraction Postulates

(K − 1) (K − A) is a belief set;

(K − 2) K − A ⊆ K;

(K − 3) if A 6∈ K, then (K − A) = K;

(K − 4) if not ` A, then A 6∈ K − A;

(K − 5) if A ∈ K, then (K − A) + A ⊆ K;

(K − 6) if ` A ↔ B then K − A = K −B;

(K − 7) K − A ∩K −B ⊆ K − (A ∧B);

(K − 8) if A 6∈ K − (A ∧B), then K − (A ∧B) ⊆ K − A

Postulate (K − 1) claims that the result of a contraction must be a belief set.

(K − 2) claims that K −A should be included in K. Postulate (K − 3) requires

that when A does not belong to K, the contraction of by A does not have any

effect. It corresponds to the Minimal Change principle. Postulate (K−4) requires

the contraction to be successful, unless A is a tautology (remember that belief

sets are deductively closed, hence it is not possible to contract a belief set by

a tautology). Postulate (K − 5) requires that any contraction can be recovered

by an expansion. This captures the criterion of informational economy, since it

implies that no belief in K should be given up unjustifiably. Postulate (K − 6)

means that the result of a contraction does not depend on the syntactic form

of the formula retracted. Postulate (K − 7) says that the beliefs that are not


rejected by contracting a belief set by A nor by contracting it by B are not even

rejected by contracting the belief set by A∧B. Finally, postulate (K− 8) claims

that contracting a belief set by A should not lead to abandon more beliefs than

contracting it by A ∧B.

2.1.1 Constructive Models for Revision and Contraction

In this section, we examine some revision and contraction functions. Some au-

thors have focused on contraction, whereas others have focused on revision. The

two approaches are equivalent since a revision can be seen as a contraction fol-

lowed by an expansion and, the other way round, a contraction function can be

expressed in terms of revision function.

Partial meet, Maxichoice and Full meet contractions

Given a belief set K and a formula A, let K ⊥ A be the set of all maximal subsets

of K that fail to imply A. Hence, K ⊥ A = {M : M ⊆ K,A 6∈ CnPC(M) and for

any M ′ : M ⊂ M ′, A ∈ Cn(M ′)}. Let S be a selection function that picks out

some of the elements of K ⊥ A. Intuitively, the elements of K ⊥ A selected by S

are the epistemologically most preferred subsets of K ⊥ A that fail to entail A.

In the specific case in which the selection function S selects one single element of

K ⊥ A, we get a maxichoice contraction defined as

K − A = S(K ⊥ A).

Although maxichoice contraction functions satisfy postulates (K − 1)− (K − 6)

for contraction (but not (K − 7), (K − 8)), they are unsatisfactory because they

lead to belief sets that are “too large”. This can be seen by considering the belief

revision operator obtained from maxichoice contraction through Levi identity. As

shown in [3], such a belief revision operator has the following property:

Proposition 2.1.4 If ¬A ∈ K, then for all B, either B ∈ K ∗A or ¬B ∈ K ∗A.

At the other extreme, when the selection function S selects all the elements

of K⊥A, we get full meet contraction, defined as:

K − A =⋂

K ⊥ A.

Full meet contraction satisfies postulates (K − 1) − (K − 8). Nonetheless, if

belief sets selected by maxichoice contraction functions are too large, belief sets


selected by full meet contraction are “too small”. It was shown in [3] that a

revision operator defined via Levi’s identity from a full meet contraction function

has the following property:

Proposition 2.1.5 If ¬A ∈ K, then K ∗ A = CnPC(A).

For this reason, the revision operator defined through Levi’s identity from a full

meet contraction function is called the trivial revision operator.

Between the two extremes lies partial meet contraction, of which maxichoice

and full meet contraction are specific cases. Partial meet contraction is defined

as :

K − A =⋂

S(K ⊥ A) if K ⊥ A is not empty, K otherwise.

Partial meet contraction functions satisfy the basic postulates (K−1)−(K−6)

for contraction. The following proposition holds:

Proposition 2.1.6 A contraction function satisfies postulates (K−1)− (K−6)

if and only if it is a partial meet selection function.

In order for partial meet contraction functions to satisfy postulates (K − 7),

(K − 8), it is necessary to put more constraints on the selection function S. Let

M(K) be the set of all maximal subtheories of K defined as follows:

M(K) =⋃{K ⊥ A : A ∈ K and 6`PC A}

and let ≤ be a reflexive and transitive ordering relation over the set M(K). The

ordering relation ≤ is defined over the set of all maximal subsets of K and it is

independent on which formula we are retracting.

A partial meet contraction function is said to be transitively relational just

in case the selection function S selects the subsets of K ⊥ A that are “most

preferred” with respect to the ordering relation ≤. A partial meet contraction

function is transitively relational just in case it uses a selection function defined

as follows:

S(K ⊥ A) = {M ∈ K⊥A : M ′ ≤ M for all M ′ ∈ K⊥A}The following theorem holds [1]

Theorem 2.1.7 For any belief set K, the contraction function satisfies postu-

lates (K − 1)− (K − 8) iff it is a transitively relational partial meet contraction

function.


Full meet contraction is a relational partial meet contraction in the specific

case in which for all the elements M and M ′ of M(K), M ≤ M ′ and M ′ ≤ M .

Therefore, full meet contraction satisfies postulates (K− 1)− (K− 8). This does

not hold as far as maxichoice contraction is concerned since this is not in general

a relational partial meet contraction.

Epistemic Entrenchement

In the previous subsection, we have seen some contraction functions that use an

ordering among maximal subsets of a belief set K to determine the result of a

contraction. On the contrary, in the epistemic entrenchment approach [15], it is

used an ordering over the single formulas of the language L (in relation to every

single belief set K) rather than on maximal subsets of K. For any A, B in L, for

any belief set K, let A ≤K B denote the fact that B is at least as epistemically

entrenched as A in relation to K.

A contraction function defined by the equivalence

B 6∈ K − (A ∧B) if and only if B ≤K A

satisfies postulates (K−1)−(K−8) just in case the relation≤K satisfies postulates

(EE1)− (EE5) below.

(EE1) For any A, B, C ∈ L, if A ≤K B and B ≤K C, then A ≤K C (Transitivity)

(EE2) For any A and B, if A ` B, then A ≤K B (Dominance).

(EE3) For all A and B ∈ K, either A ≤K (A ∧ B) or B ≤K (A ∧ B). (Conjunc-

tiveness)

(EE4) When K 6= K⊥, A 6∈ K if and only if A ≤K B for all B (Minimality)

(EE5) If B ≤K A for all B, then ` A

(EE1) simply says that the relation of epistemic entrenchment is transitive. (EE2)

says that the logical consequences of A are at least as epistemically entrenched

as A. As far as contraction is concerned, this means that if A entails B, in order

to retract B from K, then we also have to retract A. The rationale for (EE3) is

that if one wants to retract A∧B from K, then one has to retract either A or B.

(EE4) claims that if a formula A is not contained in K then it is not entrenched


in K and therefore it holds that A ≤K B for all B. (EE5) says that the most

entrenched formulas are those that are logically valid.

For more details, see [15, 14]

Safe Contraction

The Safe Contraction approach has been introduced by Alchourron and Makinson

in [2].

A subset K ′ of K is called entailment set for A just in case it entails A, and

it does not contain any other entailment set for A. Thus, K ′ is an entailment set

for A if and only if K ′ ` A but for all proper subsets K ′′ of K ′, K ′′ 6` A.

Let < be an order relation over K. Intuitively, A < B can be read as “A is

less reliable than B”. Let < be acyclical (i.e. for no A1, . . . An of K it holds that

A1 < . . . < An < A1).

An element B of K is said to be safe with respect to A just in case B is not

a minimal element, with respect to <, of any entailment set for A.

Let K|A be the set of all elements of K that are safe with respect to A.

The safe contraction function is then defined as : K − A = Cn(K|A).

This safe contraction function satisfies postulates (K − 1) − (K − 6). The

revision operator obtained from this contraction function satisfies postulates (K ∗1)− (K ∗ 6). In order to obtain a revision function that also satisfies postulates

(K∗7) and (K∗8), it is necessary to impose more conditions on the order relation

<. More precisely the order relation has to be either “continuing up `” over K

or “continuing down `” over K, where an order relation < “continues up `” over

K just in case for all A, B, C in K, if A < B and B ` C, then A < C. Similarly,

the order relation “continues down `” over K if and only if for all A, B, C in K,

if A ` B and B < C, then A < C.

Furthermore, the order relation < has to be “virtually connected” over K, i.e.

for all A, B, C in K, if A < B, then A < C or C < B.

Proposition 2.1.8 A safe contraction function satisfies postulates (K − 1) −(K − 8) if and only if it is generated by an order relation < that is virtually

connected and continues up or down ` over K.

Several equivalences between the approaches presented until now have been

shown.


Rott [41] has shown that an epistemic entrenchement relation can be used as

an order relation for the construction of safe contraction and, viceversa, given an

order relation, it is possible to define an epistemic entrenchement relation.

Furthermore, it has been shown in [3, 41, 14] that:

Proposition 2.1.9 Let K an epistemic state and − a contraction function.

Then the following conditions are equivalent:

• the contraction function − satisfies (K − 1)− (K − 8)

• the contraction function − is a safe contraction function generated by an

order relation that continues up or down ` over K and is virtually connected

• the contraction function − is a transitively relational partial meet contrac-

tion function over K

• the contraction function − is an epistemic entrenchment contraction func-

tion

Minimal Changes of Models

Up to now, we have seen some syntactical approaches to belief contraction and

belief revision. Some authors [25, 27, 7] have proposed a model-theoretic charac-

terization of these operators, with a specific attention to revision. The common

ingredient of these proposals is that they capture the Minimal Change Principle

by imposing the result of the revision of a belief set K by a formula A to be

the set of formulas true in the minimal models satisfying A. These models are

minimal with respect to a preference relation associated to K.

Grove’s Systems of Spheres

Let M be the set of all maximal consistent subsets of L, called possible worlds

(the possible worlds that can be described using the language L). A belief set

K is represented by the subset [K] of M that consists of all the elements of M

that include K. A Grove system of spheres centered on [K] is a collection S of

subsets of M that satisfies the following conditions:

• S is totally ordered by ⊆, that is if S and S ′ are in S, then either S ⊆ S ′

or S ′ ⊆ S


• [K] is the ⊆-minimum of S, that is for all S ∈ S, [K] ⊆ S

• M ∈ S

• for any formula A, if there is a sphere in S intersecting [A], then there is a

smallest sphere in S intersecting [A] (limit assumption).

Grove’s system of spheres is very similar to Lewis’s system of spheres [33](see

Chapter III). The most important difference is that Grove’s system of spheres

is centered on a set of possible worlds, whereas Lewis’s systems of spheres are

centered on one single possible world.

For any formula A, let SA be the smallest sphere of S intersecting [A]. Fur-

thermore, for any subset S of M, let KS be the set of formulas included in all

elements of S. The revision of K by A is defined by:

Definition 2.1.10 [Grove’s revision operator]

K ∗ A = KSA∩[A]

Therefore, the revision of a belief set K by a formula A is a belief set containing

all the formulas belonging to the most preferred possible worlds (with respect to

K) containing A. The two following propositions [25] hold:

Proposition 2.1.11 The belief revision operator defined as

K ∗ A = KSA∩[A]

satisfies postulates (K ∗ 1)− (K ∗ 8)

Proposition 2.1.12 Let ∗ be any revision function satisfying (K ∗ 1)− (K ∗ 8).

For any fixed belief set K there is a system of spheres S centered on [K] that

satisfies definition 2.1.10.

Katsuno and Mendelzon’s minimal models

Katsuno and Mendelzon [27] propose a reformulation of AGM postulates that can

be adopted when the language L is finite. In this reformulation of the postulates,

belief sets are represented by formulas and therefore the revision operator ∗ takes

two formulas and gives a new formula that represents the new belief set.


• KM formulation of AGM postulates

(R1) K ∗ A implies A

(R2) If K ∧ A is satisfiable, then K ∗ A ≡ K ∧ A

(R3) If A is satisfiable, then K ∗ A is also satisfiable;

(R4) If K1 ≡ K2, and A ≡ B, then K1 ∗ A ≡ K2 ∗B;

(R5) (K ∗ A) ∧B implies K ∗ (A ∧B)

(R6) If (K ∗A)∧B is satisfiable, then K ∗ (A∧B) implies (K ∗A)∧B

Let W be the set of all the interpretations of L. For any formula K of L, let

[[K]] be the set of the elements of W that satisfy K. Consider now an assignment

function that associates to each propositional formula K a total pre-order ≤K

over W i.e. a reflexive and transitive relation over W that has the property that

for every i, j ∈ W i ≤K j or j ≤K i. Let i <K j denote the fact that i ≤K j and

j 6≤K i.

The assignment function is said to be faithful if it satisfies the three following

conditions:

• if i, j ∈ [[K]], then i <K j does not hold

• if i ∈ [[K]] and j 6∈ [[K]], then i <K j

• if K ≡ K ′, then ≤K=≤K′

A belief revision operator can then be defined as follows:

[[K ∗ A]] = min≤K[[A]],

where min≤K[[A]] is the set of worlds satisfying A, minimal with respect to ≤K .

Therefore, the revision of a belief set K by a formula A is a belief set whose

models are the minimal models, with respect to ≤K , that satisfy A.

The following theorem, shown in [27], holds:

Theorem 2.1.13 Revision operator satisfies postulates (R1)− (R6) if and only

if there exists a faithful assignment that maps each belief set ψ to a total pre-order

≤K such that [[K ∗ A]] = min≤K[[A]].


Since we can establish a correspondence between the set of all maximal con-

sistent extensions of the language L and the set of all interpretations of L and

since, moreover, we can establish a correspondence between systems of spheres

and total pre-orders, it follows that Grove’s and Katsuno and Makinson’s models

are equivalent.

Some revision operators that use explicit methods to calculate the distance

among interpretations can be seen as specific cases of Katsuno and Makinson’s

models for revision. This is the case, for instance, of Dalal’s revision. In [7]

Dalal proposes to calculate the distance dist between two interpretations as the

number of literals on which the interpretations differ. It is shown in [27] that this

criterion can be used to assign to each belief set K a total preorder over the set

of all possible interpretations by defining dist([[K]], i) = minj∈[[K]]dist(i, j), for

all interpretations i.

Spohn Systems

A Spohn system is a structure 〈R, ∗s〉, where

- R = {k : W → Ord} is a set of functions from the set W of all classical

interpretations to the set of ordinals. The elements k of R are called rank-

ings, and the elements w of W are called worlds. It is assumed that for all

rankings k ∈ R, there is a world w such that k(w) = 0.

- k(A) = min{k(w) : w |= A}.

- The operator ∗s of type: R× L −→ R is defined as follows 1:

k ∗s A(w) =

k(w)− k(A) if w |= A

k(w) + 1 otherwise

Given a Spohn system to any ranking k we can associate a belief set K that

contains the formulas satisfied in all worlds w such that k(w) = 0. Formally,

K = {A ∈ L such that ∀w ∈ W (k(w) = 0 → w |= A)}.

It can be easily seen that if for any K and for any A K ∗ A is the belief set

associated with the ranking k ∗s A, then ∗ satisfies postulates (K ∗ 1)− (K ∗ 8).

1This is the simplified version of Spohn’s function proposed in [8]

2.2. ITERATED BELIEF REVISION 19

2.1.2 Limits of AGM Theory

The main limit of AGM theory of belief revision is that it does not allow to

explicitly represent all the information that is relevant to determine the result of

a revision. Indeed, in AGM theory agents’ belief states are represented by belief

sets that are deductively closed sets of formulas of an extension L of propositional

language. Thus, belief sets only allow to represent factual knowledge, whereas

they do not allow to represent all the information, as preference orderings, or

revision strategies for short that are used to determine the result of a revision,

as we have seen in example 2.1.2.

This limitation is problematic when iterated revision is concerned. Indeed,

in iterated revision, we are interested not only on how factual beliefs evolve

throughout revision, but also on how the revision strategies (all the preferences

on beliefs that determine the result of a revision) change during the revision

process. AGM framework does not provide such a description.

For this reason, several authors [8, 30, 11, 12, 19, 6] have proposed to replace

belief sets by richer structures that allow to represent all the information (such as

the preferences on the sentences contained in belief sets, the conditional beliefs,

the revision strategies) that influence the result of revision. Some proposals in

this direction are considered in section 2.2. Gardenfors in [15] uses conditional

formulas to represent this information. We shall examine his proposal in chapter

4.

2.2 Iterated Belief Revision

As mentioned above, Several authors [8, 30, 11, 12, 19, 6] have recognized that

AGM postulates are too weak to account for iterated belief revision. Indeed,

in order to deal with iterated belief revision, one has to put some constraints

not only on how propositional beliefs change after a revision, but also on how

conditional beliefs evolve throughout revision. AGM postulates put no such con-

straints. Indeed, in AGM framework, two successive revisions can use two dis-

tinct, unrelated, sets of revision strategies. Recall, for instance, that in Katsuno

and Mendelzon’s semantic model for revision, a total preorder on W is associated

to each belief set. There is no constraint to impose the preorder associated to a

belief set K ∗A to be related in some way to the preorder associated to belief set

K. This permissiveness of AGM postulates leads to counterintuitive results, as


the following example from Darwiche and Pearl [8] illustrates.

Example 2.2.1 We see a strange new animal X at a distance, and it appears

to be barking like a dog, so we conclude that X is not a bird and X does not

fly. Still, in the event that X turns out to be a bird, we are ready to change our

mind and conclude that X flies. Observing the animal closely, we realize that it

actually can fly. The question now is whether we should retain our willingness

to believe that X flies in case X turns out to be a bird after all. It would be

of course counterintuitive to give up the conditional belief merely because we

happened to observe that X can fly. Yet there is an AGM-compatible revision

operator that permits such behavior, formalized as follows:

• K = Cn({¬bird,¬flies})

• K ∗ bird = Cn({bird, flies})

• (K ∗ flies) ∗ bird = Cn({bird})

2

In this example, in belief set K we are disposed to conclude that X flies

after learning that X is a bird. However, we are not disposed to draw the same

conclusions in belief set K ∗ flies. This is possible since the preferences (on

L or on W ) associated to belief set K ∗ flies are unrelated to the preferences

associated to belief set K (semantically, if the most preferred worlds with respect

to the pre-order relation associated to K that satisfy bird also satisfy fly, this

is not true for the most preferred bird-worlds with respect to the total pre-order

associated to K ∗ flies). This is, though counterintuitive, formally possible in

AGM framework.

In order to deal with iterated revision, it is therefore necessary to put some

constraints on the preservation of conditional beliefs. Boutilier [5] has proposed a

belief revision operator, called natural revision, which guarantees that the revision

strategies are preserved as much as the AGM postulates permit. Indeed, Boutilier

proposed to adopt the following postulate for the iterated step:

(CB): if ¬A ∈ K ∗B, then (K ∗B) ∗ A = K ∗ A

This radical strategy is counterintuitive, as the following example from Darwiche

and Pearl [8] shows.


Example 2.2.2 We encounter a strange new animal that appears to be a bird.

So, we believe the animal is a bird. As the animal comes closer to our hiding

place, we see clearly that it is red, so we believe it is red. To learn something

more about the animal, we call in a bird expert who claims that the animal is not

really a bird but some sort of mammal. The question is whether, at this point

of the story, we should still believe that the animal is red. According to natural

revision, we should not. To see why, consider the postulate (CB) above with K

being our starting belief set (in which we believe that the animal is a bird) , B

the information that the animal is red, and ¬A the information that the animal

is a bird.

This is clearly counterintuitive. 2

The problem here is that we have to allow the acquisition of an information to

affect successive revisions. This means that not all the conditional beliefs should

be preserved after a revision but only some of them. In [8] Darwiche and Pearl

adopt a more cautious approach.

2.2.1 Darwiche and Pearl’s theory

In order to deal with iterated belief revision, it is necessary to deal with knowl-

edge structures that are richer than belief sets. These knowledge structures allow

to explicitly represent both objective knowledge and revision strategies. These

knowledge structures are called epistemic states. An epistemic state has an asso-

ciated belief set, but also an associated set of revision strategies that the agent

wishes to employ in that state to accommodate new evidences. By describing

how epistemic states evolve through revision, it is possible to characterize iter-

ated belief revision.

Let Ψ be an epistemic state and Bel a function that associate to each epistemic

state its corresponding belief set (a propositional formula). Notice that there can

be two different epistemic states with the same associated belief set. The revision

operator ∗ takes an epistemic state and a formula and gives back a new epistemic

set.

The fact that two epistemic states Φ and Ψ are equal is denoted by Φ = Ψ,

whereas the fact that they have the same associated belief set is denoted by

Φ ≡ Ψ.

The first six postulates are essentially AGM postulates as formulated by Kat-


suno and Mendelzon, with the exception of postulate (R ∗ 4).

(R ∗ 1) Bel(Ψ ∗ A) implies A

(R ∗ 2) If Bel(Ψ) ∧ A is satisfiable, then Bel(Ψ ∗ A) = Bel(Ψ) ∧ A

(R ∗ 3) If A is satisfiable, then Bel(Ψ ∗ A) is also satisfiable;

(R ∗ 4) If Ψ1 = Ψ2, and A ≡ B, then Ψ ∗ A = Ψ ∗B;

(R ∗ 5) Bel(Ψ ∗ A) ∧B implies Bel(Ψ ∗ (A ∧B))

(R∗6) If Bel(Ψ∗A)∧B is satisfiable, then Bel(Ψ∗(A∧B)) implies Bel(Ψ∗A)∧B

The main difference between these postulates and AGM postulates lies in (R∗4).

According to (R ∗ 4) only if two epistemic states are equal, then the revision of

the first epistemic state coincides with the revision of the second epistemic state.

On the contrary, (R4) says that whenever two epistemic states have the same

associated belief set, the revision of the first belief set coincides with the revision

of the second belief set. Indeed, Katsuno and Mendelzon’s postulate (R4) can

be reformulated in Darwiche and Pearl’s notation as: if Bel(Ψ1) ≡ Bel(Ψ2), and

A ≡ B, then Bel(Ψ ∗ A) ≡ Bel(Ψ ∗ B). Since two different epistemic states

can have the same associated belief set, it follows that (R4) is stronger than

(R ∗ 4). This is a rather important point, highlighted by other authors, like

Lehmann[30], who calls (R4) the non-postulate. By rejecting (R4) Darwiche,

Pearl, and Lehmann recognize that the result of belief revision does not only

depend on belief sets, but on some other elements of epistemic states not captured

by belief sets. We will see in the next section that in Lehmann’s approach, these

elements are the sequences of revisions that lead to that epistemic state.

To these postulates, Darwiche and Pearl add four postulates that rule the

iterated case. The first postulate says that if two informations are successively

learnt, the second being more precise than the first, then the first intermediate

information should be ignored. It can be formulated as:

• (C1) If A |= B, then Bel((Ψ ∗B) ∗ A) ≡ Bel(Ψ ∗ A)

The second postulate says that if two contradictory informations are suc-

cessively learnt, then the false intermediate information is to be ignored. This

postulate requires belief revision to be reversible and it is not satisfied by the


trivial revision operator that whenever the incoming information is inconsistent

with the current epistemic state, throws everything away and simply keeps the

new formula. The postulate is formulated as follows:

• (C2) If A |= ¬B, then Bel((Ψ ∗B) ∗ A) ≡ (Ψ ∗ A)

The third postulate claims that a sentence B should be kept after introducing

a sentence A that implies it given the current epistemic state. It is formulated

as:

• (C3) If Bel(Ψ ∗ A) |= B, then Bel((Ψ ∗B) ∗ A) |= B

The last postulate says that if a sentence B is not contradicted after seeing

A, then it is not contradicted when A is preceded by B itself.

• (C4) If Bel(Ψ ∗ A) 6|= ¬B, then Bel((Ψ ∗B) ∗ A) 6|= ¬B.

These postulates impose that all the conditional beliefs that do not compromise

propositional beliefs should be preserved.

An example of a revision operator that satisfies postulates (R ∗ 1) − (R ∗6), (C1)− (C4) is Spohn revision operator.

2.2.2 Lehmann’s Theory

Lehmann in [30] has proposed a set of rationality postulates for iterated revision.

In his framework he represents the sequence of revisions applied to the initial

belief set by a sequence of formulas σ and denotes by [σ] the resulting belief

set. The belief set [σ.A] represents the result of revising the belief set [σ] by the

formula A. In his framework the revision of [σ] by the formula A depends not

only on the belief set [σ], but also on σ, that is the sequence of revisions that

leads to [σ]. This sequence of revisions plays the role of the epistemic states in

Darwiche and Pearl’s context.

As Lehmann shows, (C1), (C3), and (C4) can be derived from his postulates.

However, only a weaker version of (C2) holds in his framework (namely [σ.¬A.A]⊆[σ]+A), and (C2) cannot be derived from Lehmann’s postulates. On the other

hand, Lehmann’s postulates (I4) and (I6) cannot be derived from Darwiche and

Pearl’s postulates. Let us consider, for instance, his postulate (I4): if A ∈ [σ] then

[σ.τ ] = [σ.A.τ ], where σ and τ are sequences of formulas. This postulate could

be restated in Darwiche and Pearl’s notation as: if A ∈ Bel(Ψ), then Ψ ∗A = Ψ.


It means that when A is believed in Ψ, Ψ∗A and Ψ are the same epistemic state.

This property cannot be derived from Darwiche and Pearl’s postulates from which

a weaker property follows, namely if A ∈ Ψ, then Bel(Ψ ∗A) = Bel(Ψ), which is

a consequence of (R ∗ 2).

Among the consequences that Lehmann proves from his postulates, is the

property that a mild revision (that is a revision with a formula consistent with

the current belief set) essentially fades away at the first severe revision (that is

a revision with a formula which is inconsistent with the current belief set). This

property contrasts with the Principle of Minimal Change and is intuitively un-

wanted. Such a property cannot be proved from Darwiche and Pearl’s postulates.

Chapter 3

Conditional Logics

Conditional logics are concerned with the investigation of the logical properties of

conditional sentences, i.e sentences with the form “if A, then B”. They have been

introduced around the half of the last century [45, 33]. Recently, they have been

rediscovered in Artificial Intelligence for their applicability in several domains,

like knowledge representation, non-monotonic reasoning, reasoning about actions,

planning and natural language analysis [16].

3.1 Limits of classical logic’s formalization of

conditional sentences

The analysis of conditional sentences provided by classical logic is not entirely

adequate for many respects. In classical logic, conditional sentences are formal-

ized as material implications A → B, true whenever the antecedent A is false

or the consequent B is true. This formalization entails that any conditional sen-

tence with a false antecedent, called counterfactual, is considered true. This is

counterintuitive, like the two following sentences show:

• “if I were three meters tall, I would be taller than four meters”

• “if 2× 2 = 5, then New York would be in France”

Indeed, in spite of the fact that they have a false antecedent, both the sentences

are clearly false.

Furthermore, the formalization of conditional sentences in terms of material

implication does not allow to explain the fact that there can be two conditional

25

26 CHAPTER 3. CONDITIONAL LOGICS

sentences, whose antecedents and consequents have respectively the same truth

values, but such that the one is false and the other is true. This argument

extends in general to any formalization of conditional sentences in terms of truth

functional operators. Take for instance the two following sentences:

• “if I were three meters tall, I would be taller than two meters.”

• “if I were three meters tall, I would be taller than four meters”

In spite of the fact that the antecedents and the consequents are both false, the

first sentence is true, whereas the second is false. The same applies to the pair

of sentences:

• “if the electricity had not failed, then the dinner would be ready”

• “if the electricity had not failed, pigs would fly”

These examples show that the truth value of a conditional sentence, and more

precisely of counterfactual conditional sentences, does not (uniquely) depend on

the truth value of its antecedent and consequent. Rather, it depends on some

sort of connection between the antecedent and the consequent of the sentence.

Different approaches to conditional logics derive from different possible for-

malizations of this connection between antecedent and consequent. Thus, in

cotenability theory [22] the connection is expressed by the fact that the an-

tecedent, together with physical laws, entails the consequent. Differently, in

possible worlds approaches, the connection is captured by saying that the truth

value of the antecedent and the truth value of the consequent are related in all,

or in the most similar, possible worlds.

In the rest of the chapter we shall examine different approaches in more detail.

Conditional logics use the operator >, rather than the operator → of mate-

rial implication. A conditional sentence with form “if A, then B” will thus be

formalized as

A > B.

3.2 Cotenability theory

One first proposal in this direction is the cotenability theory of conditionals pro-

posed by Goodman [22]. According to this theory, a conditional “if A, then B”

3.3. STRICT IMPLICATION 27

is true in case A, together with some set of laws and true propositions, entails

B. More precisely, a conditional sentence “ if A, then B” is true just in case A,

together with the set of all physical laws and the set of all propositions cotenable

with A, implies B; where a proposition E is cotenable with a proposition C if

it is consistent with C, and if it does not hold that E > ¬C nor that C > ¬E.

The problem of this proposal is that it is circular, since cotenability is expressed

in terms of conditionals and conditionals are expressed in terms of cotenability

theory. Furthermore, this theory is not well suited for counterlegal conditionals,

since they are trivially true.

3.3 Strict implication

According to the theory of strict implication, the truth value of a conditional

does not uniquely depend on the truth value of its components in the actual

world, since this might be accidental. Rather, it depends on the truth value

of the components in all possible worlds. A conditional is true just in case the

antecedent is true or the consequent is false in all possible worlds. Formally, a

conditional sentence “if A, then B” is expressed in terms of material implication

together with a necessity operator:

Definition 3.3.1 A > B ≡ 2(A → B)

The theory of strict implication overcomes the problems entailed by the classical

theory of conditionals, since not every counterfactual conditional is considered

true. Furthermore, the truth value of a conditional sentence does not uniquely

depend on the truth value of its components.

However, the theory of strict implication enforces some counterintuitive prop-

erties of conditionals.

First of all, it entails the property of contraposition for conditionals, accord-

ing to which from “if A, then B” follows “if ¬B, then ¬A”. This property is

counterintuitive, since from :

• “if the electricity hadn’t failed, dinner would have been on time” ,

we do not want to derive that

• “if dinner had been late, the electricity would have failed”.


Second, the theory of strict implication entails the property of transitivity,

according to which “if A, then B” and “if B, then C” entail “if A, then C”. This

is counterintuitive, since from

• “If Carter had not lost the election in 1980, Reagan would not have been

president in 1981”

and

• “if Carter had died in 1979, he would not have lost the election in 1980”

it does not follow that:

• “if Carter had died in 1979, then Reagan would not have been President in

1981”

Last, the theory of strict implication entails the property of Strengthening

the Antecedent, according to which “if A, then B” entails “if A ∧C then B” for

any C. This is counterintuitive, since from

• “if the engine would fail than the pilot would make an emergency landing”

we do not want to conclude that

• “if the left engine were to fail and the right wing were to shear off, then the

pilot would make an emergency landing”

All these counterintuitive properties derive from the fact that, in order to

evaluate a conditional sentence, we consider the truth value of its components

in all possible worlds. Hence, contraposition derives from the fact that if all the

possible worlds satisfying A also satisfy B, it follows that if a possible world does

not satisfy B, then it does not satisfy A. The two other properties derive from

similar considerations. All these properties are avoided by approaches, as the

following ones, that require to consider only some possible worlds, different for

conditionals with different antecedents.

3.4 Stalnaker’s logic

Differently from the theory of strict implication, Stalnaker claims that in order to

determine the truth value of a conditional formula it is not necessary to consider

3.4. STALNAKER’S LOGIC 29

the truth value of the antecedent and the consequent of the conditional in all

possible worlds. Rather, he maintains that it is sufficient to consider the truth

value of the antecedent and the consequent of the conditional only in one single

possible world, which is the possible world satisfying the antecedent most similar

to the actual world. The conditional formula is true if the consequent holds in

that possible world, and it is false otherwise.

This approach clearly avoids the three counterintuitive properties entailed

by the theory of strict conditionals. Roughly speaking, the three properties are

avoided since, as a difference from strict implication, different sets of worlds

are considered when evaluating different conditional sentences. Contraposition

is avoided since from the fact that the most preferred world satisfying A also

satisfies B, it does not follow that the most preferred world satisfying ¬B satisfies

¬A. Similarly, transitivity is avoided, since the most preferred world satisfying A

might be different from the most preferred world satisfying B. Last, strengthening

of the antecedent is avoided since the most preferred world satisfying A might

satisfy ¬B and therefore be different from the most preferred world satisfying

A ∧ C.

The assumption underlying Stalnaker’s proposal, according to which for each

possible world w and formula A, there is a unique possible world satisfying A

which is more similar to the actual world than any other A world is known as

Stalnaker’s uniqueness assumption. It is the assumption that characterizes his

semantic and it is also the most questioned assumption, criticized, for instance

by Lewis [33], as we will see in the next section.

Formally, a semantic model for Stalnakers’s conditional logic is a quadruple

〈W,R, f, [[ ]]〉, in which W is a set of possible worlds; R is an accessibility relation

over W , f is a selection function that assigns to each formula A and world w a

world f(A,w). This world can be considered as the closest world satisfying A.

[[ ]] is a function that assigns to each formula A the set of worlds satisfying it.

We call it interpretation function.

The quadruple 〈W,R, f, [[ ]]〉 satisfies the following properties:

(S1) f(A,w) ∈ [[A]];

(S2) 〈w, f(A,w) ∈ R〉;

(S3) if f(A,w) is not defined, then for all w′ ∈ W , such that 〈w,w′〉 ∈ R,

w′ 6∈ [[A]];


(S4) if w ∈ [[A]], then f(A,w) = w;

(S5) if f(A,w) ∈ [[B]], and f(B, w) ∈ [[A]], then f(A,w) = f(B,w);

(S6) [[A > B]] = {w : f(A,w) ∈ [[B]]}.

Property (S1) is a success property that says that the A-world selected by the

selection function f does indeed satisfy A. Property (S2) claims that every

world is accessible with respect to itself. Property (S3) says that the selection

function is defined for all satisfiable formulas. (S4) says that the most preferred

world with respect to w is w itself. (S5) says that if the most preferred A

world satisfies B and the most preferred B world satisfies A, then the two worlds

coincide. This property allows to establish a preference relation over the set of

possible worlds which is independent from the formula taken as argument by

the selection function. Finally, property (S6) expresses the truth condition for

conditional sentences: a conditional sentence is true in a world w either if there

is no accessible world satisfying the antecedent, or if the most preferred world

satisfying the antecedent, also satisfies the consequent.

The conditional logic determined by Stalnaker’s model theory, called C2, is

the smallest conditional logic containing the following axioms and closed with

respect to the following inference rules and modus ponens.

(RCEA) From A ↔ B infer A > C ↔ B > C

(RCK) From (A1 ∧A2 ∧ . . . An) → B infer ((C > A1) ∧ (C > A2) . . . (C > An)) →(C > B)

(ID) A > A

(MP) (A > B) → (A → B)

(MOD) (¬A > A) → (B > A)

(CSO) ((A > B) ∧ (B > A)) → ((A > C) ↔ (B > C))

(CV) ((A > B) ∧ ¬(A > ¬C)) → ((A ∧ C) > B)

(CEM) (A > B) ∨ (A > ¬B)

3.5. LEWIS’S LOGIC 31

3.5 Lewis’s logic

As we have already mentioned, Lewis rejects Stalnaker’s uniqueness assumption

according to which for each world w and each formula A, there is always a

unique possible world satisfying A which is more similar to w than any other

possible world. Furthermore, he also rejects a weaker assumption, called the

Limit assumption, according to which for each possible world w and formula A,

there exists at least one possible world satisfying A which as similar to w as any

other possible world.

To criticize the Limit assumption, he uses the following argument: consider a

line printed in a book, and suppose that the line was longer than it is. No matter

which greater length we choose for the line, there is always a shorter length which

is still greater than the actual length of the line. Therefore, the worlds which

differ from the actual world only in the length of the line may be more and more

like the actual world as the length of the line in those worlds comes closer to the

line’s actual length. But none of these world is the possible world closest to the

actual world.

Since Lewis rejects both Stalnaker’s uniqueness assumption and the Limit

assumption, his conditional semantics is not formulated in terms of selection

function models (for there might be no preferred possible world that the selection

function can pick up).

Rather, Lewis formulates his conditional semantics by making use of nested

system of spheres.

Before we consider the system of spheres models, we need some definitions.

Definition 3.5.1 Let S be a set of sets of elements. We say that S is nested

if for any two s and s′in S, either s ⊆ s′ or s′ ⊆ s. We say that S is closed

under union if, whenever S ′ is a subset of S, and⋃

S ′ is the set of all elements

that belong to some member of S ′,⋃

S ′ belongs to S. S is closed under finite

intersection if whenever S ′ is a nonempty subset of S, and⋂

S ′ is the set of the

elements that belong to every member of S ′,⋂

S ′ belongs to S.

With these definitions at hand, we can define a nested system of spheres as a

structure 〈W,S, [[ ]]〉 such that

• W is a set of possible worlds


• S assigns to each w ∈ W a nested set Sw of subsets of W which is closed

under union and finite intersection

• [[ ]] is a function, called interpretation function, that associates to each

formula the set of possible worlds satisfying it.

A system of spheres is said to be centered just in case for each w ∈ W ,

• {w} ∈ Sw

Intuitively, the nested spheres associated to a world indicate the relative sim-

ilarity among worlds: a world w1 is more similar to w than w2 if there is a sphere

associated to w that contains w1 but does not contain w2. The idea is that a

conditional A > B is true in a world w if the worlds satisfying A ∧ B are more

preferred than the worlds satisfying A∧¬B (or if there is no world satisfying A).

Thus, the semantics of conditional formulas in a system of spheres is defined as

follows:

[[A > B]] = {w ∈ W :⋃

Sw ∩ [[A]] = ∅ or there is S ∈ Sw such that

S ∩ [[A]] 6= ∅ and S ∩ [[A]] ⊆ [[B]] }

The conditional logic V C determined by centered systems of spheres is the

smallest conditional logic containing the following axioms and derivation rules

(together with modus ponens):


(RCK) From (A1 ∧ A2 ∧ . . . ∧ An) → B infer ((C > A1) ∧ (C > A2) . . . ∧ (C >

An)) → (C > B)

(ID) A > A

(MP) (A > B) → (A → B)

(MOD) (¬A > A) → (B > A)

(CSO) ((A > B) ∧ (B > A)) → ((A > C) ↔ (B > A))

(CV) ((A > B) ∧ ¬(A > ¬C)) → ((A ∧ C) > B)

(CS) A ∧B → A > B

3.6. GABBAY’S LOGIC 33

3.6 Gabbay’s logic

We have seen that in both Stalnaker’s and Lewis’s conditional logics, in order

to determine the truth value of a conditional formula we have to consider what

happens in the worlds satisfying the antecedent that are as similar as possible to

the actual world.

The idea underlying Gabbay’s proposal is rather different. In Gabbay’s view,

the possible worlds considered to determine the truth value of a conditional for-

mula A > B must not necessarily be as similar as possible to the actual world.

The important thing is that they preserve those features of the actual world that

are relevant to the truth of the consequent, or to the effect the antecedent has on

the truth of the consequent.

Therefore, the possible worlds that should be considered when evaluating a

conditional formula do not only depend on the antecedent of the conditional,

but also on its consequent. As a consequence, different possible worlds may

be considered to determine the truth value of two conditionals with the same

antecedent.

Formally, Gabbay’s semantics is formulated in terms of models that are triples

〈W, f, [[ ]]〉 where W is a set of possible worlds, [[ ]] is an evaluation function

and f a selection function that takes three arguments: a possible world w, the

antecedent of the conditional and the consequent of the conditional. The value

of the selection function is a set of possible worlds. Intuitively, these are the

possible worlds that preserve those features of the actual world that are relevant

to determine the truth value of the consequent.

The model 〈W, f, [[ ]]〉 satisfies the following properties:

• w ∈ f(A,B, w)

• if [[A]] = [[C]], and [[B]] = [[D]], then f(A,B, w) = f(B, D, w)

• g(A,B, w) = g(A,¬B,w) = g(¬A,B, w)

The first condition says that the actual world is one of the possible worlds that has

to be considered when determining the truth value of a conditional formula. The

second condition says that the evaluation of a conditional is independent from

the syntactical formulation of the antecedent and the consequent. The third

condition says that the possible worlds considered when evaluating a conditional


A > B coincide with the possible worlds considered when evaluating A > ¬B

and with the worlds considered when evaluating ¬A > B.

The resulting conditional logic, called G is the smallest conditional logic closed

under the three rules:

(RCEC) From A ↔ B infer C > A ↔ C > B


(RCE) From A → B infer A > B

Chapter 4

Conditionals and Belief Revision

4.1 Evaluation of Conditional Sentences and Be-

lief Revision

We have seen in previous chapters that conditional logics and belief revision are

two distinct areas of research. Nonetheless, there is an intuitive relation between

hypothetical reasoning and belief change. This idea has been pointed out by

the philosopher F.P. Ramsey who proposed in [39] an acceptability criterion for

conditional sentences according to which, in order to decide whether to accept

a conditional sentence “if A, then B” in a given belief state, we should add the

antecedent A of the sentence to our stock of beliefs, and then consider whether the

consequent B of the sentence follows. In the positive case, we should accept the

conditional sentence, otherwise we should reject it. This acceptability criterion

applies both to open conditionals (whose antecedent is consistent with the stock

of beliefs) and to counterfactual conditionals (whose antecedent is inconsistent

with the stock of beliefs). In case the antecedent A is inconsistent with the stock

of beliefs, some adjustments should be made in order to preserve consistency.

Ramsey’s proposal has been very influential in the analysis of conditional sen-

tences. Stalnaker’s conditional logic [45], for instance, stems from this intuition,

even if Stalnaker was interested in analyzing the truth conditions of conditional

sentences, whereas Ramsey maintained that conditional sentences do not have

any truth value and was rather interested in studying the acceptability criteria of

these sentences.

In the context of belief revision, Ramsey’s idea has been formalized by Gardenfors

35

36 CHAPTER 4. CONDITIONALS AND BELIEF REVISION

by the so called Ramsey Test, according to which, given a belief revision system

〈K, ∗〉, for all belief sets K in K, and all conditional formulas A > B:

Ramsey Test (RT): A > B ∈ K if and only if B ∈ K ∗ A

Gardenfors’s Ramsey Test introduces some new elements with respect to Ram-

sey’s informal criterion. First of all, the informal notion of belief change used by

Ramsey is here formalized by a revision operator satisfying rationality postulates

(K ∗ 1) − (K ∗ 8). Second, while in Ramsey’s criterion conditional sentences

were simply accepted in a belief state, in Gardenfors’s formalization conditional

formulas are introduced in belief sets and treated as any other element of belief

sets. This means that conditional beliefs are submitted to the rationality postu-

lates (K ∗ 1)− (K ∗ 8) that rule the revision of non-conditional beliefs and that

were originally thought for non conditional belief sets. We will see in the next

section that this leads to a triviality result, shown by Gardenfors [12].

The Ramsey Test was intended to be the starting point for the developpement

of an epistemic semantics for conditional sentences in general, since the relation

established between belief revision and (counterfactual or open) conditionals can

be extended to other forms of conditionals, like, for instance, conditionals with

the form “B would be the case even if A were the case” and “B might be the case

if A were the case”. An acceptability criterion for a conditional with the form “B

even if A” could be formulated as follows: “B even if A” is accepted in K just

in case B is accepted in K and in K ∗A. Similarly, an acceptability criterion for

a conditional with the form “B might be the case if A were the case” could be

formulated by saying that it is accepted in K just in case ¬B is not accepted in

K ∗ A.

Unfortunately, the whole project failed because of a triviality result entailed

by the Ramsey Test.

Since the Ramsey Test and its triviality result appeared, a lot of literature

devoted to re-establishing a formal relation between evaluation of conditional

sentences and belief revision has appeared [32, 29, 36, 10, 34].

To the triviality result, to the literature devoted to re-establishing a formal

relation between conditionals and belief change without running into the triviality

result.

However, before we examine the triviality result entailed by Ramsey Test and

the proposed solutions, we shall consider a bit closer Gardenfors’s proposal.

4.1. EVALUATION OF CONDITIONAL SENTENCES AND BELIEF REVISION37

4.1.1 The conditional logic deriving from the Ramsey Test

Having formalized the relation between the acceptability of conditionals and belief

revision by the Ramsey Test, Gardenfors investigates which properties of condi-

tionals derive from the properties of belief revision. In particular, he considers

which conditional formulas are valid, given the postulates for belief revision. To

this purpose, he defines the notions of satisfiability and validity of a formula as

follows. A formula is satisfiable in a belief revision system just in case it belongs

to a consistent belief set of the belief revision system; it is valid in a belief re-

vision system if its negation is not satisfiable. Last, a formula is valid if it is

valid in all belief revision systems. Valid are therefore all the formulas whose

negation does not belong to any belief set of any belief revision system. It can

be easily verified that the formulas made valid by this definition, together with

the Ramsey Test and the revision postulates, are some standard axioms of con-

ditional logics considered in chapter 3. For instance, postulate (K ∗ 2), according

to which A ∈ K ∗ A entails, by the Ramsey Test, that A > A ∈ K for all K

of all belief revision systems and that (ID): A > A is valid. Similarly, postu-

late (K ∗ 3), according to which K ∗ A ⊆ K + A, together with (RT), entails

that for all belief revision systems and all belief sets K, if A > B ∈ K, then

A → B ∈ K. This means that there is no consistent belief set K such that

(A > B) ∧ ¬(A → B) ∈ K and (DT): A > B → (A → B) is valid. In the same

way, by (K ∗ 5), according to which K ∗ A = K⊥ only if ` ¬A, it follows that

A > ¬A ∈ K then B > ¬A ∈ K. Therefore, there is no consistent belief set K

such that (A > ¬A) ∧ ¬(B > ¬A) ∈ K and (MOD): (A > ¬A) → (B > ¬A) is

valid.

The same reasoning cannot be repeated for all revision postulates. In partic-

ular, it cannot be repeated, on pain of triviality, for postulates (K ∗4) and (K ∗8)

(we will come back to this point in the next section). Instead of postulate (K ∗4),

Gardenfors has to consider its weaker reformulation (K ∗4w), according to which

if A ∈ K, then K ⊆ K ∗ A. From (K ∗ 4w) and Ramsey Test it follows that for

all belief sets K of all belief revision systems, if (A ∧ B) ∈ K then A > B ∈ K.

Therefore, there is no consistent belief set K such that (A∧B)∧¬(A > B) ∈ K

and A ∧B → A > B is valid.

Similarly, instead of postulate (K ∗ 8) Gardenfors has to consider postulate

(K ∗ L) that says that ¬(A > ¬B) ∈ K, then (K ∗ A) + B ⊆ K ∗ (A ∧ B). The

axiom corresponding to (K ∗ L) is (A > B) ∧ ¬(A > ¬C) → (A ∧ C > B).


In this way, Gardenfors derives a conditional logic equivalent to Lewis’s logic

V C, in the sense that the resulting conditional logic can be derived from V C and

V C can be derived from the resulting conditional logic.

4.2 Triviality result

In spite of the intuitive correspondence between the evaluation of conditional

sentences and belief change, the formalization of this correspondence postulated

by Gardenfors with the Ramsey Test leads to a triviality result [15] according

to which the only belief revision systems compatible with the Ramsey Test are

trivial, i.e. satisfy the following definition.

Definition 4.2.1 [Trivial belief revision system] A belief revision system 〈K, ∗〉is trivial if for any three formulas A, B, C in L, pairwise disjoint (such that

`PC ¬(A ∧ B), `PC ¬(B ∧ C), `PC ¬(A ∧ C)) there is no belief set K ∈ K

consistent with all of them (i.e. such that ¬A 6∈ K, ¬B 6∈ K, and ¬C 6∈ K).

Roughly speaking, trivial belief revision systems can be assimilated to complete

belief revision systems, containing only complete belief sets (such that for any

formula A, either A or ¬A is in the belief set). More precisely, trivial belief

revision systems are a generalization of complete belief revision systems. Like

complete belief revision systems, trivial belief revision systems do not allow to

represent agents’ incomplete knowledge about the world. For this reason, they

are not well suited to represent agents’ belief change in a realistic way.

Unfortunately, the following theorem holds:

Theorem 4.2.2 ([12], page 85) There is no non-trivial belief revision system

which satisfies AGM postulates (K ∗ 1)− (K ∗ 8) and (RT).

Proof.[[12], p.85] Let the formulas A B and C be pairwise disjoint. Let K be a

belief set consistent with A, B, C. Consider the double revision (K ∗A)∗(B∨C).

By (K ∗ 1), (B ∨ C) ∈ K,and therefore either ¬B 6∈ (K ∗ A) ∗ (B ∨ C), or

¬C 6∈ (K ∗A) ∗ (B ∨ C). Assume ¬C 6∈ (K ∗A) ∗ (B ∨ C). By the properties of

the expansion operator, K +(A∨B) ⊆ (K +A), and by (K ∗ 4) K +A ⊆ K ∗A,

it follows that K + (A ∨ B) ⊆ K ∗ A. Consider now a direct consequence of the

Ramsey Test, called the Monotonicity Principle, according to which if K ⊆ K ′,

then K∗A ⊆ K ′∗A. The Monotonicity Principle follows directly from the Ramsey

4.3. GARDENFORS’S ANALYSIS OF THE TRIVIALITY RESULT 39

Test since: if K ⊆ K ′ and B ∈ K ∗ A, then by the Ramsey Test, A > B ∈ K,

thus also A > B ∈ K ′, and B ∈ K ′ ∗ A.

Since K + (A ∨ B) ⊆ K ∗ A, the Monotonicity Principle entails that (K +

(A∨B)) ∗ (B ∨C) ⊆ (K ∗A) ∗ (B ∨C) and hence ¬C 6∈ (K +(A∨B)) ∗ (B ∨C).

Next, consider K+(A∨B). It follows from the assumptions about A, B and C

that ¬(B∨C) 6∈ K+(A∨B). By (K∗4), it follows that (K+(A∨B))+(B∨C) ⊆(K+(A∨B))∗(B∨C). By the properties of expansion, (K+(A∨B))+(B∨C) ⊆K +((A∨B)∧ (B∨C)) = K +B. Hence K +B ⊆ (K +(A∨B))∗ (B∨C). Since

B ∈ K +B, it follows that ¬C ∈ K +B and hence ¬C ∈ (K +(A∧B))∗ (B∨C),

which contradicts what previously concluded. 2

4.3 Gardenfors’s analysis of the triviality result

Gardenfors, and most of the authors that have considered the problem after him,

claim that the triviality result derives from the conflict between the Preservation

Principle that rules belief revision, on the one side, and a direct consequence of

the Ramsey Test, namely the Monotonicity Principle, on the other side. The two

principles can be formulated as follows.

• Preservation Principle: if a sentence B is accepted in a given state of belief

K, and if A is consistent with K, then B is still accepted in the revision of

K by A.

• Monotonicity Principle: if K ⊆ K ′, then K ∗ A ⊆ K ′ ∗ A

According to this analysis, in order to avoid triviality, one of the two principles

must be abandoned.

The Preservation Principle is hardly questionable, since it captures the ra-

tional principle of informational economy. Therefore, in Gardenfors’s view, the

more questionable principle is the Monotonicity Principle. However, his argu-

ments against Monotonicity are not convincing, since they address a form of

Monotonicity which is stronger than the one deriving from the Ramsey Test and

used in the triviality result. Indeed, the form of Monotonicity Gardenfors criti-

cizes claims that whenever two belief sets are included in one another, then the

revision of the first belief set is included in the revision of the second belief set.


This holds also in case the two belief sets do not contain conditionals. Of course

this form of Monotonicity is counterintuitive.

But this is not the Monotonicity deriving from Ramsey Test. Indeed, the

precise formulation of the Monotonicity following form the Ramsey Test is: if the

conditional formulas of a belief set K are included in the conditional formulas

of a belief set K ′, then the revision of K is included in the revision of K ′. This

Monotonicity holds only in belief revision systems satisfying Ramsey Test.

In its correct formulation, the Monotonicity Principle is hardly questionable.

If there can be two belief sets containing the same conditional formulas, but

such that their revision behaves differently, in which sense can we say that there

is a correspondence between the evaluation of conditional sentences and belief

revision?

The good news is that it is not necessary to choose between the Monotonicity

Principle and the Preservation Principle. Indeed, we will see in the next section

that the triviality result lies on some very questionable assumptions. Hence, the

two principles can be safely preserved, and the triviality result avoided, once

abandoned these assumptions.

4.4 But Gardenfors’s analysis is not the only

possible one

4.4.1 Expliciting all the assumptions on which lies the

triviality result

The proof of the triviality result by Gardenfors relies on implicit assumptions.

Let us consider again Gardenfors’s proof by explicitly stating the steps in

which it is articulated.

Let 〈K, ∗〉 be a belief revision system satisfying the Ramsey Test. Let A, B

and C be three formulas pairwise disjoint and K ∈ K a belief set consistent with

A, B and C.

(1) Consider the three belief sets:

– K + A

– K + (A ∨B)

4.4. BUT GARDENFORS’S ANALYSIS IS NOT THE ONLY POSSIBLE ONE41

– K + (A ∨ C)

These belief sets belong to the belief revision system 〈K, ∗〉, since, by def-

inition, belief revision systems are closed with respect to the expansion

operator.

(2) Consider now the two belief sets K +(A∨B) and K +A. By the properties

of expansion, K + (A ∨ B) ⊆ K + A. Therefore, the conditional formulas

in K + (A ∨B) are included in the conditional formulas in K + A and, by

the Monotonicity Principle, the revision of K +(A∨B) (by any formula) is

included in the revision of K + A (by the same formula). The same holds

for belief sets K + (A ∨ C) and K + A: since K + (A ∨ C) ⊆ K + A, then

the revision of K + (A ∨ C) (by any formula) is included in the revision of

K + A (by the same formula). It follows that the revision of K + A by any

formula contains both the revision of K +(A∨B) by the same formula and

the revision of K + (A ∨ C) by the same formula.

(3) Consider then the revision of the three belief sets by the formula (B ∨ C).

– Since (B ∨ C) is consistent with K + (A ∨ B), by the Preservation

Principle (postulate (K ∗4)), it follows that: K +(A∨B)+(B∨C) ⊆K + (A ∨ B) ∗ (B ∨ C), and since K + (A ∨ B) + (A ∨ C) = K + B,

it follows that B ∈ K + (A ∨B) + (B ∨ C) ⊆ K + (A ∨B) ∗ (B ∨ C)

and therefore that B ∈ (K + A) ∗ (B ∨ C)

– By the same reasoning, it follows also that C ∈ (K + A) ∗ (B ∨ C).

(4) By step 2, it follows that both B ∈ K+A∗(B∨C) and C ∈ K+A∗(B∨C).

But B and C are pairwise disjoint, and this contradicts the fact that the

revision of K+A by a consistent formula (like (B∨C)) leads to a consistent

belief set.

(5) It follows that the starting hypothesis cannot hold and the belief revision

system is trivial.


4.4.2 Questioning the assumption according to which be-

lief revision systems are closed with respect to the

expansion operator

In our opinion, the more questionable point of the whole proof is the assumption,

made at step (1), according to which any belief revision systems containing a belief

set K, also contains the three belief sets K + A, K + (A ∨B) and K + (A ∨ C).

More generally, we question the assumption according to which all belief revision

systems (also the ones satisfying the Ramsey Test) are closed with respect to the

expansion operator. Without this assumption, the whole proof would not work

and therefore triviality would be avoided.

Let us assume, for the sake of simplicity, that the expansion of a belief set by

a propositional formula only affects the propositional part of the belief set and

leaves unchanged the conditional beliefs 1.

If a conditional belief revision system is closed with respect to the expan-

sion operator, then it contains several belief sets (obtained by the expansion of

the same belief set) that differ in their propositional beliefs but agree in their

conditional beliefs.

By the Ramsey Test, if two belief sets have the same associated conditional

beliefs, they are revised in the same way. It follows that if a conditional belief

revision system is closed with respect to the expansion operator, then it contains

several belief sets that differ in the propositional beliefs but that are revised in

the same way.

But this contradicts the fact that, by revision postulates, the propositional

formulas of a belief set are relevant to determine the result of revision.

The triviality result derives from this contradiction.

The closure with respect to the expansion operator, on the one side, and

belief revision postulates, on the other side, are compatible only in trivial belief

revision systems because in these belief revision system the closure with respect to

expansion has essentially no effect (since there are not many formulas consistent

with a belief set).

1Of course, this does not hold if we allow in belief sets conditionals like A → B > C whereneither A nor B > C belong to K. In this case, the expansion by a propositional formula (forinstance by A), might lead to the acquisition of new conditional beliefs. In this case, we canonly conclude that by expansion we obtain belief sets whose conditional beliefs are included inone another. Substantially, all what we say can be readapted to this case.

4.4. BUT GARDENFORS’S ANALYSIS IS NOT THE ONLY POSSIBLE ONE43

4.4.3 Questioning the Minimal Change Principle when

applied to conditional beliefs

The same considerations we have done for the assumption according to which

belief revision systems are closed with respect to the expansion operator apply

to the postulates that entail that assumption, namely rationality postulates (K ∗3), (K ∗ 4), (K ∗ 7), (K ∗ 8) encoding the Minimal Change Principle2.

The postulates, and the principle, require that, whenever a new information

is consistent with a belief set being revised, the revision of the belief set by the

new information consists in a simple expansion of the belief set.

Now, the principle, and the postulates, are well suited to deal with the propo-

sitional part of belief sets, because they capture the rational principle of informa-

tional economy according to which information is not gratuitous and therefore

unnecessary loss of information should be avoided.

However, both the principle and the postulates are not well suited to rule the

change of the conditional part of the belief set. First of all, because in this case

they lead to belief sets that contain different propositional beliefs, but the same

conditional formulas. We have seen in the previous subsection that this leads to

triviality.

Second, because they are in contrast with the fact that the acquisition of a

new consistent information may lead to change the conditional beliefs associated

to the belief set.

Consider the following examples:

Examples

Example 4.4.1 A very rich woman has been murdered last night. Her nephews

Spike, Adam and Linda are inquired. Among them, the detective believes that

Spike is most probably the murderer, that Adam is a remote but believable

possibility and that Linda is probably innocent. If he were to discover that Spike

is innocent, he would suspect Adam. His belief set can be represented as follows:

K = {Spike,¬Spike > Adam}2From (K ∗ 3) and (K ∗ 4) we have that for all A and K such that ¬A 6∈ K, K ∗A = K +A,

so that K + A is a belief set contained in the belief revision system.


If he now acquired some evidence proving definitely Adam’s innocence, he would

have to change his conditional belief ¬Spike > Adam by the conditional belief

¬Spike > Linda. Thus, the new belief set obtained by a consistent revision of

K contains different conditional beliefs than the original belief set. 2

Example 4.4.2 A woman has been murdered last night. Mary and John, her

neighbours, are the main suspects. To solve up his doubts about who is the

murder, the detective decides to look for the gun, believing that if the gun is

found in John’s room, then John is the culprit, and if it is found in Mary’s room,

Mary is the culprit. His knowledge base can be formalized as follows:

K = {(John ∨Mary), (gun John > John), (gun Mary > Mary)}

Suppose now that the gun is found in John’s room, but with Mary’s fingerprints.

The detective would conclude that, in spite of the fact that the gun was found

in John’s room, Mary is the culprit and therefore abandon his conditional belief

(gun John > John). 2

The moral we can derive from this discussion is that the triviality result can

be avoided by simply giving up the assumption according to which belief revision

systems are closed with respect to the expansion operator, and by limiting the

action of the Minimal Change Principle to the propositional part of belief sets.

We think that these limitations capture our intuition better than the strong form

does. We will come back to this point in further details in chapter VII.

4.5 Solutions Proposed in the Literature

To conclude the chapter, we examine the proposals aimed to avoid the triviality

result that have appeared up to now. Since the explanation of the triviality result

inherited by the literature is Gardenfors’s explanation in terms of the conflict

between Preservation Principle and Monotonicity Principle, the proposals can be

divided in two groups: those that abandon the triviality result and those that

abandon the Preservation Principle.

To the first group belong the proposals aiming to weaken the Ramsey Test.

The authors belonging to this first group are Gardenfors himself [15], Rott [40],

4.5. SOLUTIONS PROPOSED IN THE LITERATURE 45

Levi [32], Lindstrom and W.Rabinowicz [34]. To the second group belongs Grahne

[23] who studies the relation between conditionals and belief update. He avoids

the triviality result since belief update does not enforce the Preservation Princi-

ple.

4.5.1 Weakening the Ramsey Test

Rott, and Gardenfors in [40, 15] claim that the triviality result can be avoided

by weakening the Ramsey Test. The idea is to add some preconditions to the

Ramsey Test in order to prevent the derivation of the Monotonicity Principle.

Rott in [40] has proposed to adopt a weaker form of Ramsey Test, namely:

A > B ∈ K iff B ∈ K ∗ A and B 6∈ K

This version of the Ramsey Test blocks the derivation of the Monotonicity Prin-

ciple, and the counterintuitive property according to which if A ∈ K and B ∈ K,

then A > B ∈ K.

Nonetheless, it entails a weaker form of monotonicity,

(K ∗WM) if K ⊆ K ′, A ∨C 6∈ K, and C ∈ K ∗A, then C ∈ K ′ ∗A,

sufficient to entail the triviality result.

The same applies to other proposed weakenings of the Ramsey Test, such as:

A > C iff C ∈ K ∗ A and C 6∈ K ∗ ¬A

A > C ∈ K iff C ∈ (K − C) ∗ A;

if A ∨ C 6∈ K, then A > C ∈ K iff C ∈ K ∗ A.

All these weakenings entail (K ∗WM) and thus lead to triviality.

A more promising approach is the one proposed by Lindstrom and Rabinowicz:

Strict RT: For every belief set K, A > B ∈ K iff, for every extension H of K,

B ∈ H ∗ A.

In the Strict RT, a conditional A > B belongs to a belief set K just in case B

belongs to the revision of all the extensions of K.

Strict RT is equivalent to keeping the “only if” part of the Ramsey Test for

all belief sets, while restricting the “if” part of the test only to complete belief


sets. Monotonicity does not follow, since there can be two belief sets K, K ′ such

that K ⊆ K ′, B ∈ K ∗A but A > B 6∈ K, therefore A > B 6∈ K ′ and B ∈ K ′ ∗A.

The questionable point of Strict RT, admitted by Lindstrom and Rabinowicz

themselves, is that it is too demanding. Requiring that every extension of a belief

set must be examined before deciding whether to accept a conditional sentence

is not realistic.

4.5.2 Levi’s solution

According to Levi, the triviality result derives from Gardenfors assumption that

conditionals belong to belief sets. Instead, Levi thinks that conditionals are not

claims about reality, but rather epistemic evaluations: a conditional A > B

expresses the possibility of B relative to a transformation of the current belief

set. For any belief set K, the conditionals that are accepted relative to K are

not themselves in K, but rather in the associated corpus of epistemic appraisals

RL(K). Levi’s formulation of the Ramsey Test can therefore be expressed as

follows:

Levi’s RT: For any propositional formulas A and B, for any belief set K,

A > B is accepted in K if and only if B ∈ K ∗ A,

where a formula A is accepted in K if and only if A belongs to RL(K).

An analogous Ramsey Test is proposed for negated conditionals:

Levi’s Negative RT: For any propositional formulas A and B, for any belief

set K, ¬(A > B) is accepted in K if and only if B 6∈ K ∗ A.

Levi’s approach does not run into the triviality result since the Monotonicity

Principle does never apply to two distinct belief sets K and K ′. Since RL(K)

and RL(K ′) are complete with respect to conditionals, if RL(K) ⊆ RL(K ′),

then RL(K) = RL(K ′). Furthermore, it can be easily proven that if RL(K) =

RL(K ′), then also K = K ′.

Note that, in general, it is not sufficient to eliminate conditionals from belief

sets in order to avoid triviality. Indeed, the triviality result can be extended

to mappings between belief revision and conditional formulas weaker than the

Ramsey Test, that do not insert conditional formulas in belief set.

Let us denote by Γ |=M A the fact that for every possible world w of M , if

w |= B for every B ∈ Γ, then also w |= A (see next chapter). The following

proposition holds


Proposition 4.5.1 Given an AGM belief revision system 〈K, ∗〉, suppose there

is a mapping µ : K → 2L> and there is a model M such that:

1. ∀C, ∀K µ(K) ⊆ µ(K + C),

2. ∀K ∈ K, B ∈ K ∗ A iff µ(K) |=M A > B;

then 〈K, ∗〉 is trivial.

Proof. Consider the three belief sets:

• K + (A ∨B)

• K + (A ∨ C)

• K + A

Since K + A = (K + (A∨B)) + A, it follows that (K + (A∨B)) ∗ (B ∨C) ⊆(K + A) ∗ (B ∨ C). Similarly, since K + A = (K + (A ∨ C)) + A, it follows that

(K + (A ∨ C)) ∗ (B ∨ C) ⊆ (K + A) ∗ (B ∨ C). Then triviality result follows for

the same reasoning made in the triviality proof in [[12], p.85] 2

The main problem with Levi’s formalization of the Ramsey Test is that it

does not allow to give account for iterated conditionals, in the sense that it does

not allow to describe how conditionals themselves change throughout revision.

4.5.3 Friedman and Halpern’s solution

In [10] Friedman and Halpern introduce the notion of belief change system (BCS),

which can be regarded as a generalization of Gardenfors’s belief revision system.

A BCS contains “three components: the set of possible epistemic states in which

the agent can be in, a belief assignment that maps each epistemic state to a

set of beliefs, and a transition function that determines how the agent changes

epistemic states as a result of learning new information” [10]. The notion of

BCS provides a very general framework in which both revision and update can

be characterized. A major difference with Gardenfors’s belief revision systems

is that a BCS does not identify epistemic states with belief sets: two states,

although different, may agree on their belief set. In particular, Friedman and

Halpern investigate a class of BCS’s called preferential BCS’s which embody the


semantic model of revision and update [28, 29, 30]. In a preferential BCS a

state can be identified with a set of possible worlds and a preference ordering

on worlds. Friedman and Halpern show that both revision and update (as well

as some belief change operators which generalize them) can be characterized in

terms of properties on preferential BCS’s, and they also provide an axiomatic

characterization of these properties.

As a difference with standard conditional logic, and also with our approach,

Friedman and Halpern consider epistemic states rather than worlds as their prim-

itive objects. The conditional language L> they define is built up using a con-

ditional operator > and a belief operator B, and it contains only subjective for-

mulas, that is those formulas formed out by boolean combinations of conditional

formulas and belief formulas (prefixed by the operator B). Such language L> is

completely disjoint from the language L containing the objective formulas (that

is formulas which contain neither conditionals, nor belief operators), so that, for

instance, a formula as A ∧ (A > B), with A ∈ L, is not allowed. Moreover, only

objective formulas are allowed in the left hand side of a conditional, that is, in

A > B the antecedent A has to be an objective formula.

The reason why Friedman and Halpern’s approach avoids the triviality result

is that “since they restrict their attention to belief sets in L, they avoid the trivi-

ality problem that occurs when applying AGM postulates to conditional beliefs”

[10].

The drawback of the approach is that the obtained conditional logic has a

non-standard, ad hoc, and complicated semantics. Furthermore, the conditional

logic language has severe limitations.

4.5.4 Conditionals and belief update

A possible solution to the problem of triviality is to consider another form of

belief change: belief update. The triviality result does not apply in this case,

since belief update does not enforce the Preservation Principle involved in the

triviality proof. This is the line of research followed by Grahne [23], for instance.

In [23], Grahne formulates a conditional logic containing a Ramsey Rule that

establishes a relation between the conditional operator and the update operator

without running into the triviality result. Recently Ryan and Schobbens [43] have

established a link between updates and counterfactuals, by regarding them as

existential and universal modalities. The Ramsey rule is an axiomatization of the


inverse relationship between the two sets of modalities. The results by Grahne,

Ryan, Schobbens [23, 43] have induced many people to think that belief update

is the “right” form of belief change to be used when dealing with acceptability

conditions for conditional formulas. This is undoubtedly true as far as some

conditionals are concerned. But it is not true for all conditionals. As there

are different kinds of belief change well suited for different circumstances, in the

same way different belief change operators are well suited to evaluate different

conditional formulas.

To this purpose, Lindstrom and Rabinowicz [34], distinguish two kinds of con-

ditionals: epistemic conditionals and ontic conditionals. Belief revision may be

used to evaluate epistemic conditionals that state “our disposition to change our

beliefs in face of new evidence”. These conditionals differ from ontic conditionals

because these last ones “represent the hypothesis concerning what would be the

case if the world itself were different, and have therefore to do with hypothetical

modifications of the facts rather than with the modification of our beliefs about

the facts”. To consider why belief revision is well-suited to evaluate the first kind

of conditionals and not the second kind of conditionals, consider the following

examples. First of all, an example of an epistemic conditional: “If Oswald did

not kill Kennedy, then someone else did”. Now, if someone accepts this epistemic

conditional and learns that the antecedent is true, for Oswald in fact did not

kill Kennedy, then he would accept the consequent, that Kennedy was killed by

someone else. On the other hand, consider the following ontic conditional: “If

Oswald had not killed Kennedy, then no one else would have”. If someone that

accepts this conditional were to learn that the antecedent is indeed true, he could

hardly accept the consequent, unless he explains why Kennedy was killed if no-

body killed him. Would update work? Yes, because he could probably abandon

the information that Kennedy was killed.


Chapter 5

A Logic for Belief Revision

We start with this chapter the propositive part of the thesis, where we propose a

formal relation between belief revision and conditional logics that holds also for

non trivial belief revision systems.

In this chapter, we introduce a conditional logic BC that represents belief

revision. The relation between the logic BC and belief revision is formally estab-

lished by a representation theorem that shows how to any belief revision system

can be associated a BC structure, and, viceversa, to any BC structure (satisfy-

ing an additional condition, called covering condition) we can associate a belief

revision system.

We thus propose a mapping between belief revision and conditionals which

is weaker than the mapping established by the Ramsey Test. In this sense, our

proposal follows the spirit of Levi’s proposal. We have seen in chapter 4 that Levi

[32] suggests to weaken the Ramsey Test as follows: (RT) A > B is “accepted”

in K iff B ∈ K ∗ A, where the notion of “acceptability” of a conditional A > B

in K is a weaker condition than “A > B ∈ K”. In our proposal, we interpret

the acceptability of a conditional A > B in K as: A > B is true in a world wK

associated to K.

Differently from the Ramsey Test, our approach does not run into the triviality

result.

The logic BC introduced in this chapter will be extended in the next chapter

in order to deal with iterated belief revision. Furthermore, we will see in chapter 7

that this is the conditional logic that derives from the Ramsey Test by weakening

the rationality postulates in such a way that they only apply to the propositional

part of belief sets.

51

52 CHAPTER 5. A LOGIC FOR BELIEF REVISION

5.1 The Conditional Logic BC

We have seen in chapters 2 and 3 that the semantics of conditional logics and

the semantics of belief revision are rather similar, since they can be both formu-

lated in terms of possible worlds structures with a selection function. However,

there are some differences between the two semantics that make it difficult to

reconstruct belief revision semantics in a (standard) conditional logic framework.

Semantically, to each belief set K, there corresponds the set of its models (or

possible worlds). In this respect, a revision of K corresponds to a revision of

the entire set of its models, in a way which globally depends on K. This global

dependency on K is the basic semantic difference with the semantics of condi-

tional logics, in which the selection function is defined for any single world. The

difference between the two semantics can be illustrated by the following figure.

Figure 5.1: Difference between belief revision’s semantic and conditional logic’s

semantic

In order to provide a conditional logic semantics as similar as possible to

belief revision semantics, Friedman and Halpern in [10] introduce another level

of semantic objects, called epistemic states, to account for this dependency. Their

semantic models have therefore two levels composed by heterogeneous elements: a

set of possible words, and a set of epistemic states. Each belief set K corresponds

to an epistemic state and each epistemic state has an associated set of possible

worlds.

In contrast, in our approach, we do not introduce any further semantic object

5.1. THE CONDITIONAL LOGIC BC 53

to account for this global dependency on belief sets. We stay as close as possible

to standard conditional logic semantics. We shall see that our conditional logic

allows us to define a belief operator through the conditional implication itself.

This allows us to represent belief sets and their revisions in a standard conditional

logic framework. Semantically, this is possible since we can define, by means of

the selection function, an equivalence relation over the set of possible worlds. We

thus can represent belief sets by equivalence classes of worlds, and belief revision

operator by the selection function.

5.1.1 The language

Definition 5.1.1 The language L> of logic BC is an extension of the language

L of classical propositional logic obtained by adding the conditional operator >.

Let us define the following modalities:

2A ≡ ¬A > ⊥3A ≡ ¬(A > ⊥).

We define the language of modal formulas L2 as the smallest subset of L> in-

cluding L and closed under ¬,∧,2,3.

5.1.2 The axiomatization

The logic BC contains the following axioms and inference rules. We assume that

the conditional > has higher precedence than the material implication →.

Note that a conditional formula > > A can be regarded as a belief operator

meaning that “A is believed”. Moreover, axioms (REFL), (EUC) and (TRANS)

below (the last two ones for A = >) give this belief operator the property of

being reflexive, transitive and euclidean, the same as an S5 modality.

(G I) (CLASS) All classical axioms and inference rules;

(ID) A > A;

(RCEA) if ` A ↔ B, then ` (A > C) ↔ (B > C);

(RCK) if ` A → B, then ` (C > A) → (C > B);

(G II) (DT ) ((A ∧ C) > B) → (A > (C → B)), for A, B, C ∈ L;

(CV ) ¬(A > ¬C) ∧ (A > B) → ((A ∧ C) > B), for A,B,C ∈ L;


(G III) (BEL) (A > B) → > > (A > B);

(REFL) (> > A) → A;

(EUC) ¬(A > B) → A > ¬(> > B);

(TRANS) (A > B) → A > (> > B);

(G IV) (MOD) 2A → B > A, where A ∈ L2;

(U4) 2A → 22A, where A ∈ L2;

(U5) 3A → 23A, where A ∈ L2.

We have gathered the axioms in different groups. Axioms of (G I) are those

of the basic conditional logic CK+ID. Axioms (DT) and (CV) define essential

properties of the conditional operator; they are part of the axiomatization of

Stalnaker’s logic (C2). We will come back to this point.

Axioms of (G III) are motivated by the introduction of the modal operator

> > A (“A is believed”). The other axioms of this group (the last two for A = >)

give to this belief operator the properties of an S5 modality.

Similarly, axioms of (G IV) define a necessity operator 2 and give it S5-

properties. Axiom (MOD) governs the relation between 2 and the conditional

operator.

It is worth noticing that axioms (ID), (DT), (CV), (MOD) belong to Stal-

naker’s logic C2 (see [38]). However, Stalnaker’s logic contains also other axioms

such as (MP), (CS) and (CEM); these axioms could be derived from the axiom-

atization above if we added the axiom A → (> > A) (“everything true is be-

lieved”), that we clearly do not want. First, from (DT) and (REFL) we can derive

(MP)((A > B) → (A → B)) restricted to the case of A,B ∈ L. Moreover, if we

assume the axiom A → (> > A), from (CV) we derive (CS) ((A∧B) → (A > B))

again restricted to the case in which A,B ∈ L, from (EUC) we derive (CEM)

((A > B) ∨ (A > ¬B)); axioms (TRANS) and (BEL) become tautological.

Moreover, it must be noticed that we have put restrictions on some axioms,

by requiring that they only hold for formulas ranging over L rather than any

conditional formula in L>. For a discussion on these limitations, we refer to the

comments on the semantic counterparts of the axioms.


5.1.3 The semantics

The semantics for the logic BC is a standard Kripke-like semantics for conditional

logics. Our structures are possible world structures equipped with a selection

function (see chapter 3).

Definition 5.1.2 A BC-structure M has the form 〈W, f, [[ ]]〉, where W is a

non-empty set, whose elements are called possible worlds, f is a function of type

L>×W → 2W and is called a selection function, [[ ]] : L> → 2(W ) is a valuation

function satisfying the following conditions:

(⊥) [[⊥]] = ∅

(∧) [[A ∧B]] = [[A]] ∩ [[B]]

(¬) [[¬A]] = W − [[A]]

(>) [[A > B]] = {w : f(A,w) ⊆ [[B]]}.

The above definition is extended to the classical connectives ∨,→, ↔, by the

usual classical equivalences.

Let Prop(S) = {A ∈ L: S ⊆ [[A]]}.We assume that the selection function f satisfies the following properties:

(G I) (S − ID) f(A,w) ⊆ [[A]];

(S −RCEA) if [[A]] = [[B]] then f(A,w) = f(B,w);

(G II) (S −DT ) Prop(f(A ∧ C, w)) ⊆ Prop(f(A,w) ∩ [[C]]), for A,C ∈ L;

(S −CV ) f(A,w) ∩ [[C]] 6= ∅ → Prop(f(A,w)) ⊆ Prop(f(A ∧ C,w)), for

A,C ∈ L ;

(G III) (S −REFL) w ∈ f(>, w);

(S − TRANS) x ∈ f(A,w) ∧ y ∈ f(>, x) → y ∈ f(A,w);

(S − EUC) x, y ∈ f(A,w) → x ∈ f(>, y);

(S −BEL) w ∈ f(>, y) → f(A,w) = f(A, y);

(G IV) (S −MOD) If f(B, w) ∩ [[A]] 6= ∅, then f(A, w) 6= ∅, where A ∈ L2;

(S−UNIV ) if [[A]] 6= ∅, ∃B such that f(B, w)∩ [[A]] 6= ∅, where A ∈ L2;


We say that a formula A is true in an BC-structure M = 〈W, f, [[ ]]〉 if [[A]] =

W . We say that a formula is BC-valid if it is true in every BC-structure. For

readability, we also use the notation x |= A instead of x ∈ [[A]].

Given a BC-structure M = 〈W, f, [[ ]]〉, a set of formulas Γ and a formula A,

we define Γ |=M A if for every w ∈ W , if w |= B for every B ∈ Γ, then also

w |= A i.e.⋂

B∈Γ[[B]] ⊆ [[A]]. We then define the entailment relation Γ |= A iff

for every BC-structure M , Γ |=M A. As expected, ∅ |=M A means that A is true

in M and ∅ |= A that A is valid.

In a BC-structure M , we can define by means of the selection function f

the equivalence relation ≈f on the set of possible worlds W as follows: for all

w,w′ ∈ W ,

w ≈f w′ iff w′ ∈ f(>, w).

The properties of ≈f being reflexive, transitive and symmetric come from the

semantic conditions (REFL), (TRANS) and (EUC) of the selection function f

(and, more precisely, from the last two conditions by taking A = >). We will

refer to the equivalence class containing w as [w]≈f. Thus [w]≈f

= f(>, w).

As a consequence of (S-BEL), all worlds in one equivalence class [w]≈fevaluate

conditional formulas in the same way. Moreover, by (EUC) and (TRANS), the

set f(A,w) is an equivalence class in itself. For a graphic representation of BC−models see figure 5.2 below. The capacity of defining equivalence classes through

Figure 5.2: BC-models

the selection function is central to our logic, and it allows us to represent belief

sets within a standard conditional logic model.


Indeed, we will see in the next section that to each model M we can associate

a belief revision system, by associating belief sets to equivalence classes and the

revision operator ∗ to the canonical extension of f on the equivalence classes.

Notice that, since (A > C) ∨ ¬(A > C) is a tautology, from (EUC) and

(TRANS) we can conclude (A > (> > C))∨ (A > ¬(> > C)), that is, C is either

believed or non believed in the most preferred A-worlds. This is the conditional

excluded middle, (CEM), restricted to belief formulas. While the presence of

(CEM) in Stalnaker’s logic causes the selection function to select a single world

(i.e. f(A,w) = {j} for all A and w), when (CEM) is restricted to belief formulas

(as in our logic), it determines the uniqueness of the belief set associated to

f(A,w). When we evaluate a conditional, we want to move from one world, with

an associated belief set, to other worlds with a different belief set. This ability to

explicitly represent the new belief set obtained from the initial one is especially

important if we want to model iterated revision by nested conditionals. We will

come back to this point in the next chapter.

The other semantic conditions are needed to represent belief revision postu-

lates. Conditions (S−DT ) and (S−CV ) allow us to represent revision postulates

(K ∗ 3) − (K ∗ 4) respectively. The restrictions we have put are motivated by

several considerations. The restrictions on the antecedent are motivated by the

fact that revision postulates only deal with the revision by propositional formu-

las (in L). Without the restrictions on the consequent, from (S − CV ), with

A = >, it would follow that if f(>, w) ∩ [[C]] 6= ∅, then f(C, w) ⊆ f(>, w), i.e.

by the properties of f , f(C,w) = f(>, w) (remember that f(C,w) and f(>, w)

are equivalence classes). This means that everything consistent with a belief set

is believed, i.e. for all C, either > > C or > > ¬C hold. We clearly do not

want this property, since we want to be able to represent incomplete belief sets.

Similar considerations hold for (S −DT ).

From (S-UNIV), which corresponds to (U4) and (U5), and from (S-MOD) we

get the property: if [[A]] 6= ∅, then f(A,w) 6= ∅. This property is needed to

model the revision postulate (K*5). The restrictions we have put on (MOD),

(U4), (U5) are needed since we cannot accept that the above property holds for

all formulas A ∈ L>. Having this property for arbitrary A would correspond to

being able to reach any belief set from any other. This cannot be done by means

of the revision operator. In general, given two belief sets K1 and K2, there may

not exist a formula A such that K2 = K1 ∗ A (for instance when B ∈ K1 but

neither B ∈ K2 nor ¬B ∈ K2).


The axiomatization of BC is sound and complete with respect to semantic

introduced above.

5.2 Soundness of BC

Theorem 5.2.1 (Soundness) If a formula A is a theorem of BC then is BC-

valid.

Proof. One checks each axiom and then shows that rules (RCEA) and (RCK)

preserve validity. Let M = 〈W, f, [[]]M〉 be a BC structure.

(ID) By (S − ID) f(A,w) ⊆ [[A]]M . It follows that w |= A > A.

(DT) Let w ∈ W , A,B, C ∈ L, and w |= A∧C > B. Then B ∈ Prop(f(A∧C, w)).

By (S-DT), we have B ∈ Prop(f(A, w) ∩ [[C]]M). Let y ∈ f(A,w) if y ∈ [[C]]M ,

then y |= B, and hence also y |= C → B; if y 6∈ [[C]]M , then y |= ¬C and hence

also y |= C → B. We have shown that w |= A > (C → B).

(CV) Let w ∈ W A, B, C ∈ L, and let w |= ¬(A > ¬C) and w |= A > B. We

have that f(A,w) ∩ [[C]]M 6= ∅, and B ∈ Prop(f(A,w)). From (S − CV ) it

follows that B ∈ Prop(f(A ∧ C), w) and therefore that w |= (A ∧ C) > B.

(REFL) Let w ∈ W and w |= > > A. Then f(>, w) ⊆ [[A]]M . By (S − REFL),

w ∈ f(>, w). Therefore w ∈ [[A]] and w |= A.

(EUC) Let w ∈ W . Let w |= ¬(A > B). Then, there is y ∈ f(A,w) s.t. y |= ¬B.

By (S−EUC), for all z ∈ f(A,w), y ∈ f(>, z). It follows that for all z ∈ f(A,w),

z |= ¬(> > B), and therefore w |= A > ¬(> > B).

(BEL) Let w ∈ W , and w |= A > B. Then, f(A,w) ⊆ [[B]]M . By (S−BEL), for

all y ∈ f(>, w), f(A, y) = f(A,w). Therefore, f(A, y) ⊆ [[B]]M and y |= A > B.

It follows that w |= > > (A > B).

(TRANS) Let w ∈ W and w |= A > B. Then, f(A,w) ⊆ [[B]]M . By (S −TRANS), for any y ∈ f(A,w) and z ∈ f(>, y), z ∈ f(A,w). Therefore, z ∈[[B]]M and y |= > > B. It follows that w |= A > (> > B).

(MOD) Let w ∈ W and w |= 2A, where A is a modal formula. Then f(¬A,w) =

∅. By (S − MOD), f(B, w) ∩ [[¬A]]M = ∅ and therefore f(B,w) ⊆ [[A]]M . It

follows that w |= B > A

5.3. COMPLETENESS 59

(U4) Let w ∈ W and let w |= 2A, where A is a modal formula. Then, f(¬A,w) =

∅. By (S − MOD) it follows that for all B, f(B,w) ∩ [[¬A]]M = ∅ and by

(S − UNIV ) that [[¬A]]M = ∅. By (S − ID), for all y ∈ W f(¬A,w) = ∅.Therefore, for all y, y |= (¬A >⊥) and [[¬A >⊥]]M = W . It follows that

[[¬(¬A >⊥)]]M = ∅ and by (S − ID), that f(¬(¬A >⊥), w) = ∅ and therefore

w |= ¬(¬A >⊥) >⊥ and thus w |= 22A.

(U5) Let w ∈ W and let (1) w |= 3A, where A is a modal formula. Suppose that

(2) w 6|= 23A. By (1) we have f(A,w) 6= ∅ (we recall that 3A ≡ ¬(A > ⊥)),

whence [[A]]M 6= ∅. Observe that 23A ≡ (A > ⊥) > ⊥, thus by (2) w 6|= (A >

⊥) > ⊥ and there is z ∈ f(A > ⊥, w). We have that z |= A > ⊥, i.e. f(A, z) = ∅,whence [[A]]M = ∅ by (MOD) and (UNIV). We have a contradiction.

(RCK) We show that if A → B is valid, then also (C > A) → (C > B) is

valid. Let w ∈ W , w |= C > A. Then f(C, w) ⊆ [[A]]M . Since A → B is

valid, [[A]]M ⊆ [[B]]M . It follows that f(C, w) ⊆ [[A]]M ⊆ [[B]]M and thus

f(C, w) ⊆ [[B]]M and w |= C > B.

(RCEA) We show that if A ↔ B is valid, then A > C ↔ B > C is valid.

If A ↔ B is valid, then [[A]]M = [[B]]M . For all w ∈ W , by (S − RCEA),

f(A,w) = f(B, w). Therefore w |= A > C if and only if w |= B > C. 2

5.3 Completeness

Theorem 5.3.1 (Completeness) If |= A then ` A.

Proof.

By contraposition, we show that if 6` A then there is a BC structure M in

which A is not true. Let us fix the language L>. As usual we can prove that

if 6` A, then there is a maximal consistent set of formulas X0 which does not

contain A. We assume that the usual properties of maximal consistent sets are

known (e.g. if X is maximally consistent, then D ∈ X or ¬D ∈ X). We define

M = 〈W, f, [[]]M〉, as follows

W = {X | X is maximally consistent},f(B,X) = {Y ∈ W | {C ∈ L> | B > C ∈ X} ⊆ Y },

[[p]]M = {X ∈ W | p ∈ X}


One can prove the following facts.

Fact 1 for every formula B ∈ L> and X ∈ W , B ∈ X iff X ∈ [[B]]M .

Fact 2 The structure M satisfies all conditions of definition 5.1.2, except (pos-

sibly) the condition (S-UNIV), namely it satisfies, (S-ID), (S-RCEA), (S-DT),

(S-CV), (S-REFL), (S-TRANS), (S-EUC),(S-BEL), (S-MOD).

(S-ID) By (ID), A > A ∈ X. Therefore, for all Y ∈ f(A,X), A ∈ Y and by Fact

1, Y ∈ [[A]]M . Hence, f(A,X) ⊆ [[A]]M .

(S-RCEA) Let [[A]]M = [[B]]M . It follows (by completeness and consistency of max-

imal consistent sets) that [[A ↔ B]]M = W . It follows that ` A ↔ B.

By (RCEA), for all X, for all C, if A > C ∈ X, then B > C ∈ X. By

definition of f , it follows that f(A,X) = f(B, X).

(S-DT) Let A, B, C ∈ L. If B ∈ Propf(A∧C,X), then (A∧C) > B ∈ X. By (DT),

also (A > (C → B)) ∈ X. Therefore, for all Y ∈ f(A, X), (C → B) ∈ Y ,

and therefore B ∈ f(A,X)∩ [[C]]. Since B ∈ L, B ∈ Prop(f(A,X)∩ [[C]])

(S-CV) Let f(A, X) ∩ [[C]]M 6= ∅ and B ∈ Prop(f(A,X)), where A,B,C ∈ L. By

hypothesis it cannot be A > ¬C ∈ X, thus we have ¬(A > ¬C) ∈ X;

by hypothesis we also have A > B ∈ X. By (CV) we conclude that

A ∧ C > B ∈ X. Thus B ∈ Prop(f(A ∧ C, X)).

(S-REFL) By (REFL), for all B, if > > B ∈ X, also B ∈ X. By definition of f(>, X),

it follows that X ∈ f(>, X).

(S-TRANS) Let X ∈ f(A,Z) and Y ∈ f(>, X). We must show that Y ∈ f(A,Z),

i.e. we must show that for all B such that: A > B ∈ Z, B ∈ Y . By

(TRANS), for all B such that A > B ∈ Z, also A > (> > B) ∈ Z and thus

(> > B) ∈ X. But then B ∈ Y , since Y ∈ f(>, X).

(EUC) Let X, Y ∈ f(A,Z). We must show that X ∈ f(>, Y ). Therefore, we have

to show that if {B : A > B ∈ Z} ⊆ X and {B : A > B ∈ Z} ⊆ Y , then

{C : > > C ∈ Y } ⊆ X. Suppose C 6∈ X. We prove that > > C 6∈ Y . Since

C 6∈ X, A > C 6∈ Z and ¬(A > C) ∈ Z. By (EUC), A > ¬(> > B) ∈ Z.

Hence, ¬> > B ∈ Y and > > B 6∈ Y .

(S-BEL) let X ∈ f(T, Y ), we show that f(D, X) = f(D, Y ). By definition of f , we

have to show that for any Z ∈ W ,

5.3. COMPLETENESS 61

{B : D > B ∈ X} ⊆ Z iff {B : D > B ∈ Y } ⊆ Z.

We actually prove that in the hypothesis X ∈ f(T, Y ) we have D > B ∈ X

iff D > B ∈ Y , from which the result follows. Let D > B ∈ Y , we have

T > (D > B) ∈ Y by (BEL). Thus D > B ∈ X. Conversely, if D > B 6∈ Y ,

T > (D > B) 6∈ Y by (REFL). This implies that ¬(T > (D > B)) ∈ Y ,

whence T > ¬(T > (D > B)) ∈ Y by (EUC). Thus, ¬(T > (D > B)) ∈ X

and also ¬(D > B) ∈ X by (BEL), whence D > B 6∈ X.

(S-MOD) Suppose f(A,X) = ∅, where A ∈ L2, then there is no Y ∈ W such that

{B : A > B ∈ X} ⊆ Y . By the properties of maximal consistent sets, this

implies that the set {B : A > B ∈ X} is inconsistent, whence A > ⊥ ∈ X,

i.e. 2¬A ∈ X. By (MOD), we get that B > ¬A ∈ X, thus for every

Y ∈ f(B,X), ¬A ∈ Y , whence A 6∈ Y . Thus f(B, X) ∩ [[A]]M = ∅.

The structure M does not necessarily satisfy the condition (UNIV). Our plan is

to define a substructure M0 of M which is still a BC-structure, falsifies A and

satisfies the universality condition. In order to define M0, let X0 ∈ W such that

A 6∈ X0 (whence ¬A ∈ X0). We define a binary relation on W , for X, Y ∈ W , let

RXY ≡ ∀D modal formula (2D ∈ X → D ∈ Y ) and then we let

W0 = {Y ∈ W | RX0Y }.

We first show that W0 is closed with respect to f , i.e.

(i) if Z ∈ W0 and Y ∈ f(B, Z), then Y ∈ W0,

(ii) for all Y, Z ∈ W0, RY Z holds.

For (i) let 2D ∈ X, then 22D ∈ X by (U4), then 2D ∈ Z; by (MOD) we

obtain B > D ∈ Z, whence D ∈ Y .

For (ii), let RX0Y and RX0Z, we show that RY Z holds. Suppose D 6∈ Z,

then 2D 6∈ X0, then ¬2D ∈ X0 then 3¬D ∈ X0, so that 23¬D ∈ X0 by (U5).

Then 3¬D ∈ Y , and this implies that 2D 6∈ Y .

Finally, we can show that

(iii) X0 ∈ W0.

To this regard, if 2D ∈ X0, then T > D ∈ X0 by (Mod) and D ∈ X0 by (REFL).

We can now define a structure M0 = 〈W0, f0, [[]]M0〉, where


f0(B, Z) = f(B, Z) and [[p]]M0 = [[p]]M ∩W0,

in particular the definition of f is correct by virtue of (i).

Fact 3 M0 satisfies all conditions of definition 5.1.2, in particular M0 satisfies

the condition (UNIV). In order to check (UNIV), let D be a modal formula and

suppose that for all formulas B and Z ∈ W0 f(B,Z) ∩ [[D]]M0 = ∅, in particular

we have f(D, Z) ∩ [[D]]M0 = ∅; this implies f(D,Z) = ∅, that is D > ⊥ ∈ Z,

whence 2¬D ∈ Z. By (ii) we have that for every Y ∈ W0, RZY holds, thus

¬D ∈ Y , i.e. [[D]]M0 = ∅.Fact 4 For each formula C, [[C]]M0 = [[C]]M

⋂W0. This is proved by induc-

tion on the form of C.

We can now conclude the completeness proof. If A is not a theorem of BC,

then X0 6∈ [[A]]M . Since [[A]]M0 = [[A]]M⋂

W0 and X0 ∈ W0, we have that

W0 − [[A]]M0 6= ∅, which shows that A is not true in M0. 2

5.4 Representation Theorem

We now provide a Representation Theorem which establishes a precise correspon-

dence between belief revision systems and BC−models.

The first part of the theorem shows how, given a belief revision system,

we can build a BC−model that “represents” the belief revision system in the

sense that for each belief set K of K, there is an equivalence class [w]≈f) in

M∗ that represents K, i.e. it contains the classical interpretations of K (thus,

K = Prop([w]≈f)). The selection function can be seen as representing the revi-

sion operator, since for any world w whose equivalence class is associated to K,

and any formula A ∈ L, it leads to the equivalence class associated to K ∗ A.

The BC−model so built satisfies the properties of definition 5.1.2.

The second part of the theorem shows how, given a BC−structure, we can

build a belief revision system that contains the belief sets represented by some

equivalence class in M , and whose revision operator is defined in terms of the

selection function as:

K ∗ A = Prop(f(A,w)) for some w such that K = Prop([w]≈f).

Recall that, by the properties of BC-models, f(A,w) = [w′]≈f), for some w′ ∈

f(A,w). Thus, also K ∗ A is associated to an equivalence class. The revision

operator so defined satisfies the AGM postulates.

5.4. REPRESENTATION THEOREM 63

As we will see, the Representation Theorem only deals with consistent belief

revision systems, in which all belief sets are consistent and in which revision is

only performed by consistent formulas. This is due to the fact that an inconsistent

belief set K⊥ cannot be represented by a world in a model (due to the presence

of the semantic property (REFL)). As a consequence, we cannot represent the

revision of inconsistent belief sets. This fact is explicitly required in the first part

of the theorem, and it follows from the second part.

We say that a BC-structure M = 〈W, f, [[]]〉 satisfies the covering condition

if, for any formula A ∈ L satisfiable in PC, [[A]] 6= ∅ (i.e., there is some world

satisfying A in M).

Theorem 5.4.1 (Representation Theorem) (1) Given a belief revision sys-

tem 〈K, ∗〉, there is a BC-structure M∗ = 〈W, f, [[ ]]〉 such that for every

consistent K of K, there exists w in W such that:

• Prop([w]≈f) = K, and

• for all A ∈ L, K ∗ A = Prop(f(A,w))

(2) Given a BC-structure M = 〈W, f, [[ ]]〉 satisfying the covering condition,

there is a belief revision system 〈KM, ∗M〉 such that :

• KM = {K : K = Prop([w]≈f), for some w ∈ W }

• and for each K ∈ KM, A ∈ L, K ∗M A = Prop(f(A,w)) for some w

such that K = Prop([w]≈f)

Proof. Part (1). Given a belief revision system 〈K, ∗〉, we define a BC-structure

M∗ = 〈W, f, [[ ]]〉 as follows:

W = {(K,w) : w is a classical interpretation, K ∈ K and w |= K};CK = {(K ′, w) ∈ W : K ′ = K};[[p]] = {(K,w) ∈ W : w |= p} for all propositional letters p ∈ L.

f(A, (K,w)) and [[A]] can be defined by double induction on the structure of the

formula A. At each induction step, for each connective ◦, [[A ◦ B]] is defined

by making use of the valuation of the subformulas ([[A]] and [[B]]) and of the

selection function for subformulas (for instance, f(A,w)); moreover, f(A ◦B,w)

is defined by possibly making use of the valuation of the formula A ◦B itself. In

particular we let:


f(A, (K, w)) = CK∗A, if A ∈ L;

f(A, (K, w)) = CK∗ΦA, if A 6∈ L and there exists a formula ΦA ∈ Lsuch that [[A]] = [[ΦA]];

f(A, (K, w)) = ∅, otherwise.

By making use of the properties of the revision operator ∗, we can show that M∗is a BC-structure. We show it for some of the properties

(S − ID): f(A, (K,w)) ⊆ [[A]].

By definition of f , for all (K ′, w′) ∈ f(A, (K,w)) we have K ′ = K ∗ A and

w′ |= K ∗A. Let A ∈ L, then, by postulate (K ∗ 1) we have that A ∈ K ∗A, and

hence (K ′, w′) ∈ [[A]].

The case of A 6∈ L, but [[A]] = [[φA]] with φA ∈ L, is similar. In the case when

A 6∈ L and there is no φA ∈ L such that [[A]] = [[φA]], then f(A, (K,w)) = ∅ and

the property holds trivially.

(S −RCEA): if [[A]] = [[B]] then f(A, (K,w)) = f(B, (K,w)).

Let A, B ∈ L. If in M∗ [[A]] = [[B]] holds, then it must be that A ≡ B is valid, as

the set {w′ : (K ′, w′) ∈ [[A]]} contains all and only the classical models of A. By

postulate (K ∗6), we have that K ∗A = K ∗B. Hence, if (K ′, w′) ∈ f(A, (K, w)),

then K ′ = K ∗ A = K ∗ B and from w′ |= K ∗ A we can conclude w′ |= K ∗ B.

Therefore, (K ′, w′) ∈ f(B, (K,w)).

In the case A 6∈ L (or B 6∈ L), but there is a formula φA (φB) ∈ L such that

[[A]] = [[φA]] (respectively, [[B]] = [[φB]]), the proof is the same.

If A 6∈ L and there is no formula φA ∈ L such that [[A]] = [[φA]], then f(A, (K, w)) =

∅. Moreover, as [[A]] = [[B]], it must be that B 6∈ L and there is no formula φB

∈ L such that [[B]] = [[φB]]. Hence, f(B, (K, w)) = ∅, which conclude the proof

of the property.

(S −DT ): Prop(f(A ∧ C, w)) ⊆ Prop(f(A, (K, w)) ∩ [[C]]), for A, C ∈ L.

Let B ∈ Prop(f(A∧C, (K, w))). By definition, f(A∧C, (K,w)) contains all the

classical models of K ∗(A∧C). Hence, we have that: B ∈ Prop(f(A∧C, (K, w)))

if and only if for all w′, if w′ |= K ∗ (A ∧ C), then w′ |= B. Hence, B ∈Prop(f(A ∧ C, (K, w))) if and only if B ∈ K ∗ (A ∧ C).

Similarly, it is easy to see that B ∈ Prop(f(A, (K,w)) ∩ [[C]]) if and only if

B ∈ (K ∗ A) + C.

From (K ∗ 7) we have that K ∗ (A ∧ C) ⊆ (K ∗ A) + C, from which, given the

above equivalences, the wanted property trivially follows.

The proof of the semantic property (S − CV ) is very similar to the one for

5.4. REPRESENTATION THEOREM 65

(S−DT ). The proofs of (S−REFL), (S−TRANS), (S−EUC) and (S−BEL)

are easy. Let us consider the semantic properties (S −MOD) and (S −UNIV ).

(S−MOD): If f(B, (K, w))∩[[A]] 6= ∅, then f(A, (K,w)) 6= ∅, where A ∈ L2.

It can be easily seen that, if A ∈ L2, then either A ∈ L or [[A]] = [[φA]] for some

φA ∈ L.

Assume first that A ∈ L. Then, either [[A]] = ∅ (and f(B, (K,w)) ∩ [[A]] = ∅,so that the property holds trivially) or [[A]] 6= ∅. In the last case, 6` ¬A, since

there is a classical interpretation w such that w |= A. Therefore, by (K ∗ 5),

K ∗ A 6=⊥, and hence CK∗A 6= ∅ and also f(A, (K,w)) 6= ∅. by construction,

f(A, (K,w)) 6= ∅.In case [[A]] = [[φA]] for φA ∈ L, the proof is similar.

(S − UNIV ): if [[A]] 6= ∅, ∃B such that f(B, (K, w)) ∩ [[A]] 6= ∅, where

A ∈ L2.

As in the previous case, if A ∈ L2, then either A ∈ L or [[A]] = [[φA]] for some

φA ∈ L. In the first case, from [[A]] 6= ∅, we have by construction (as in the

previous case) that f(A, (K, w)) 6= ∅. As (S − ID) holds, the conclusion follows

by taking B = A. In the second case, the proof is similar, by taking B = φA.

To conclude the proof of part (1), it is not difficult to show that the model M∗satisfies the condition that, for every consistent K of K, there exists w in W such

that:

• Prop([w]≈f) = K, and

• for all A, K ∗ A = Prop(f(A,w)).

For all K ∈ K, consider a world (K, w). Then, Prop([(K, w)]≈f) = Prop(f(>, (K,w))) =

Prop(CK∗>) = Prop(CK) = K.

Furthermore, let B ∈ K ∗ A. Then, by definition of f , f(A, (K, w)) ⊆ [[B]],

and therefore B ∈ Prop(f(A, (K,w))), and K ∗ A ⊆ Prop(f(A, (K, w))). On

the other side, let B ∈ Prop(f(A, (K, w))). Then f(A, (K,w)) ⊆ [[B]] and, by

definition of f , B ∈ K ∗ A. Thus also Prop(f(A, (K,w))) ⊆ K ∗ A.

Part (2). Let us consider a BC-structure M =< W, f, [[ ]] > satisfying the

covering condition. We define a belief revision system 〈KM, ∗M〉 with

• KM = {K : K = Prop([w]≈f )} for some w ∈ W .

• K ∗ A = Prop(f(A,w)), for some w ∈ W such that K = Prop([w]≈f ).


Furthermore, if K = Prop([w]≈f ), we let K + A = Prop([w]≈f ∩ [[A]]). The

expansion operator is well defined, since in this way, K + A = Cn(K ∪ A).

The revision system 〈KM, ∗M〉 satisfies the postulates (K ∗ 1) − (K ∗ 8) We

will prove it by making use of the semantic properties of M .

(K ∗ 1): K ∗ A is deductively closed. It holds by definition of Prop.

(K ∗ 2): A ∈ K ∗A. By (S − ID), for all w ∈ W , f(A,w) ⊆ [[A]]. Therefore,

A ∈ Prop(f(A,w)) and A ∈ K ∗ A.

(K ∗ 3): we have to show that (K ∗ A) ⊆ (K + A). Let us assume that

B ∈ (K ∗ A). Then B ∈ Prop(f(A, w)). By (S −DT ), with A = >, C = A, it

follows that B ∈ Prop(f(>, w) ∩ [[A]]). Hence B ∈ K + A.

(K ∗4): we have to show that if ¬A 6∈ K, then K+A ⊆ K ∗A. From ¬A 6∈ K,

we have that Prop([w]≈f )∩[[A]] 6= ∅, for all w such that K = Prop([w]≈f ). Since,

by definition, [w]≈f = f(>, w), it follows that f(>, w)∩[[A]] 6= ∅. From (S−CV ),

it follows that Prop(f(>, w)) ⊆ Prop(f(A, w)).

Now, let B ∈ K + A. It follows that A → B in K. Thus A → B ∈Prop(f(>, w)) and therefore also A → B ∈ Prop(f(A,w)). Since by (S-ID),

A ∈ Prop(f(A,w)), it follows that B ∈ Prop(f(A,w)), and thus B ∈ K ∗ A.

(K ∗ 5): we have to show that K ∗ A = K⊥ only if ` ¬A. We show the

contrapositive, namely that if 6` ¬A, then K ∗ A 6= K⊥. Since M satisfies the

covering condition, if 6` ¬A, then [[A]] 6= ∅. By (S −UNIV ), and (S −MOD) it

follows that f(A,w) 6= ∅, therefore that ⊥6∈ Prop(f(A,w)) and K ∗ A 6= K⊥.

(K ∗ 6): we have to show that if ` A ↔ B then K ∗A = K ∗B. If ` A ↔ B,

then [[A]] = [[B]]. By (S-RCEA), f(A,w) = f(B, w), thus, by definition of K ∗A

and K ∗ B, K ∗ A = K ∗ B (K ∗ 7) can be shown to hold by making use of the

property (S-DT) in a way similar to case (K*3), while postulate (K*8) can be

proved by making use of the property (S-CV) in a way similar to case (K*4).

To conclude the proof of part (2), we observe that the properties that:

• KM = {K : K = Prop([w]≈f), for some w ∈ W }

• and for each K ∈ KM, A ∈ L, K ∗M A = Prop(f(A,w)) for some w such

that K = Prop([w]≈f)

hold by definition.

2

We have to make one important considerations about the Representation

Theorem.

5.5. DECIDABILITY OF BC 67

Although the correspondence between belief revision systems and BC−models

established by the Representation Theorem is significant, it is rather weak, since

it does not allow to devise a one-to-one correspondence between the equivalence

classes of the model and the belief sets in the belief revision system. Rather, it

allows us to establish a correspondence between some of the equivalence classes

in the model and a belief revision system. This is due to the fact that in the

same BC− model, there can be several equivalence classes that satisfy the same

belief set, but in which conditional formulas are evaluated differently.

We will see in the next chapter that a tighter correspondence can be estab-

lished between iterated belief revision systems and BC−structures. This is due

to the fact that iterated belief revision systems deal with epistemic states, rather

than belief sets (see chapter 2). Since epistemic states are knowledge structures

richer than belief sets, it is possible to establish a one-to-one correspondence

between equivalence classes in the model and epistemic states.

Before we conclude the section, let us say a few words about the question of

triviality. The relation established by the representation theorem holds for all

belief revision systems, and therefore also for non-trivial ones.

We have seen in chapter 4, while discussing Levi’s proposal, that triviality

cannot simply be avoided by taking conditional formulas out of belief sets. In-

deed, we provided an example of a mapping between belief sets and conditionals

that does not introduce conditionals in belief sets but that, nonetheless, entails

triviality.

Intuitively, the reason why the Representation Theorem does not run into the

triviality result is that we associate different conditional beliefs to different belief

sets, even if they are included in one another, or are obtained from one another

by expansion.

Before we consider the extension of BC in order to represent iterated belief

revision, in the next section we show that the logic BC is decidable.

5.5 Decidability of BC

In this section we show that:

Theorem 5.5.1 The logic BC is decidable.

To show that decidability holds, we show that, for any formula F , if there is a

BC−model M such that M 6|= F , then there is also a finite BC−model M∗ such


that M∗ 6|= F .

This property, together with the finiteness of the axiomatization of the logic

entails that the logic is decidable: for any formula F , if F is a theorem of the

logic, it will eventually be derived from the axioms by the derivation rules; if F

is not a theorem of the logic, the exam all the finite BC−models will eventually

lead to a model M∗ such that M∗ 6|= F .

5.5.1 Finite-model property

For any formula F , and any BC−structure M , M 6|= F just in case there is a

world w0 in M such that M, w0 |= ¬F . We show that for any formula F , if

M 6|= F , then there is a finite model M∗ such that M∗ 6|= F by showing that for

any formula F and any model M , if M,w0 |= F , then we can build a finite model

M∗ such that M∗, w0 |= F .

Given the model M = 〈W, f, [[]]〉, a formula F and a possible world w0 ∈ W

such that (M,w0) |= F , we build the model M∗ = 〈Wn, f∗, [[]]∗〉 as illustrated

in section 5.5.1. We then show that M∗ is finite and that (M∗, w0) |= F (by

construction, w0 ∈ Wn).

Some definitions of sets of formulas

Given the formula F , we define the degree d of F as follows:

• d(p) = 0;

• d(¬C) = d(C);

• d(C ∧D) = max{d(C), d(D)};

• d(C > D) = d(C) + d(D) + 1.

We now define the following sets of formulas:

• Γ = the set of the subformulas of F ;

• Γ∗ = the boolean closure of Γ.

• Γ∗0 = {G : G ∈ Γ∗ and d(G) ≤ 0};

• Γ∗1 = {G : G ∈ Γ∗ and d(G) ≤ 1};


• . . .

• Γ∗n = {G : G ∈ Γ∗ and d(G) ≤ n}.

Construction of the finite model

First of all, we define an equivalence relation ≡Γ∗0 over the set of worlds W by

saying that two worlds that evaluate in the same way the formulas of Γ∗0 are

equivalent:

w ≡Γ∗0 w′ just in case, for any formula A of Γ∗0, w |= A if and only if w′ |= A.

Afterwards, we call choice function a function π that, given an equivalence

class, selects one single representing-element of the class.

Some notations: for any set of worlds S ⊆ W , we denote with S/ ≡Γ∗0 the set

of the equivalence classes of S, and with π(S/ ≡Γ∗0) a set of worlds containing

exactly one representing-element for each equivalence class of S. The representing

elements are chosen from the elements of S, thus, π(S/ ≡Γ∗0) ⊆ S.

We build the model M∗ = 〈Wn, f∗, [[]]∗〉 as follows. Remember that w0 is

such that (M, w0) |= F .

• W0 = {w0};

• W1 = W0 ∪ ⋃{π(f(A,w)/ ≡Γ∗0) such that:w ∈ W0 and A ∈ Γ∗n−1};

• . . .

• Wi = Wi−1 ∪ ⋃{π(f(A,w)/ ≡Γ∗0) such that:w ∈ Wi−1 and A ∈ Γ∗n−i}

• . . .

• Wn = Wn−1 ∪ ⋃{π(f(A,w)/ ≡Γ∗0) such that:w ∈ Wn−1 and A ∈ Γ∗0}

The evaluation function [[]]∗ is defined as follows:

[[p]]∗ = [[p]] ∩Wn.

The selection function f ∗ is defined as follows.

If w ∈ Wi with 0 ≤ i ≤ n− 1, then

f ∗(A,w) =

f(A,w) ∩Wn if A ∈ Γ∗n−(i+1)

f ∗(C,w) ifA 6∈ Γ∗n−(i+1) but there is a C ∈ Γ∗n−(i+1)such that [[A]]∗ = [[C]]∗

∅ otherwise


Let w ∈ Wn −Wn−1. Let D ∈ Γ∗0, and w′ ∈ Wn−1 be such that w ∈ f(D,w′)

(by construction of Wn, such D and w′ exist), we define f ∗ as follows:

f ∗(A,w) =

f ∗((A ∧D), w′) if A ∈ Γ∗0and f ∗(D,w′) ∩ [[A]]∗ 6= ∅f ∗(A,w0) if A ∈ Γ∗0and f ∗(D,w′) ∩ [[A]]∗ = ∅f ∗(C, w) if A 6∈ Γ∗0and there is a C ∈ Γ∗0such that [[A]]∗ = [[C]]∗

∅ otherwise

Figure 5.3: Construction of the finite model

Figure 5.4: The resulting model contains only a portion of the original model.

Furthermore, filtration is applied to the sets of worlds selected by the selection

function.


The following theorem illustrates which is the correspondence between the

evaluation of formulas in the original model and their evaluation in the new

model.

Theorem 1 For all B ∈ Γ∗i , for all w ∈ Wn−i, w ∈ [[B]]∗ if and only if w ∈ [[B]].

Proof. By induction on the complexity of B. We show the property only for

¬, ∧ and >. The property extends naturally to the other classical connectives

(since they can be all defined in terms of ¬ and ∧).

If B is an atom, this follows by definition of [[]]∗.

If B = ¬C and the property holds for C, the property follows for B in a

straightforward way.

Let B = C ∧D, with C ∈ Γ∗i , d(D) = i, d(D) = j, j ≤ i. Then, d(C ∧D) = i

and B ∈ Γ∗i . By inductive hypothesis we have that ∀w ∈ Wn−i, w ∈ [[C]]∗ iff

w ∈ [[C]], and ∀w ∈ Wn−j, w ∈ [[D]]∗ iff w ∈ [[D]]. Since Wn−i ⊆ Wn−j, we have

that ∀w ∈ Wn−i, w ∈ [[C]]∗ iff w ∈ [[C]], and w ∈ [[D]]∗ iff w ∈ [[D]]. Therefore,

∀w ∈ Wn−i,w ∈ [[C]]∗ ∩ [[D]]∗ if and only if w ∈ [[C]] ∩ [[D]], and ∀w ∈ Wn−i,

w ∈ [[B]]∗ if and only if w ∈ [[B]].

Let B = C > D. By definition of d, d(C > D) > 0, therefore C > D ∈ Γ∗iwith i > 0. We have to show that ∀w ∈ Wn−1, w ∈ [[B]]∗ if and only if w ∈ [[B]].

If w ∈ [[B]]∗, then ∀y ∈ f ∗(C,w), y ∈ [[D]]∗. Since i > 1, Wn−i ⊆ Wn−1.

Furthermore, if B ∈ Γ∗i , C ∈ Γ∗k ⊆ Γ∗i−1. Therefore, f ∗(C,w) = f(C,w) ∩ Wn.

If f ∗(C,w) ⊆ [[D]]∗, then f(C, w) ∩ Wn ⊆ [[D]]∗. By inductive hypothesis,

since D ∈ Γ∗j with j < i (for d(D) < d(C > D)), it follows that ∀w ∈ Wn−j,

w ∈ [[D]]∗ if and only if w ∈ [[D]]. Since f(C, w)∩Wn ⊆ W(n−i)+1 ⊆ Wn−j, from

f(C, w) ∩Wn ⊆ [[D]]∗, we derive that f(C,w) ∩Wn ⊆ [[D]]. Finally, since Wn

contains one representing element for each equivalence class of f(C, w), we can

conclude that f(C,w) ⊆ [[D]] and therefore that w ∈ [[C > D]].

If w ∈ [[B]], f(C, w) ⊆ [[D]], therefore also f(C, w)∩Wn ⊆ [[D]]. By inductive

hypothesis, and by definition of f ∗, it follows that f ∗(C, w) ⊆ [[D]]∗ and therefore

w ∈ [[C > D]]∗. 2

We now show that the model 〈Wn, f∗, [[]]∗〉 is a BC−structure, i.e. satisfies

all the properties (ID)− (S − UNIV ).

Theorem 2 The model 〈Wn, f∗, [[]]∗〉 is a BC−structure.


Proof. We show that the selection function f ∗ satisfies the conditions (ID) −(S − UNIV ).

• ID

1. Let w ∈ Wi, with 0 ≤ i ≤ n− 1

(a) If A ∈ Γ∗n−(i+1), then f ∗(A,w) = f(A,w) ∩Wn. Since f(A,w) ⊆[[A]], also f(A,w)∩Wn ⊆ [[A]] and f ∗(A,w) ⊆ [[A]]. By theorem

5.5.1, we conclude that f ∗(A,w) ⊆ [[A]]∗ (indeed, f ∗(A,w) ∈ Wi+1

and by theorem 5.5.1, since A ∈ Γ∗n−(i+1), we have that ∀w ∈ Wi+1,

if w ∈ [[A]], then w ∈ [[A]]∗).

(b) If A 6∈ Γ∗n−(i+1), but ∃C ∈ Γ∗n−(i+1), such that [[A]]∗ = [[C]]∗, for

what shown at the previous point, f ∗(C,w) ⊆ [[C]]∗ and therefore

f ∗(A, w) = f ∗(C,w) ⊆ [[C]]∗ = [[A]]∗.

(c) If A 6∈ Γ∗n−(i+1), and 6 ∃C ∈ Γ∗n−(i+1) such that [[A]]∗ = [[C]]∗,

f ∗(A, w) = ∅ ⊆ [[A]]∗.

2. Let w ∈ Wn −Wn−1.

(a) If A ∈ Γ∗0, and f ∗(D, w′)∩[[A]]∗ 6= ∅, then f ∗(A,w) = f ∗(A∧D,w′)

for w′ ∈ Wn−1. Since, for what shown up to now, f ∗(A∧D, w′) ⊆[[A ∧ D]]∗, it follows also that f ∗(A ∧ D, w′) ⊆ [[A]]∗ and also

f ∗(A, w) ⊆ [[A]]∗.

(b) If A ∈ Γ∗0, and f ∗(D, w′) ∩ [[A]]∗ = ∅, then f ∗(A,w) = f ∗(A,w0

and the property holds since f ∗(A,w0) ⊆ [[A]]∗.

(c) If A 6∈ Γ∗0, but ∃C ∈ Γ∗0 : [[A]]∗ = [[C]]∗, we can derive that

f ∗(C, w) ⊆ [[C]]∗ and therefore that f ∗(A,w) = f ∗(C,w) ⊆[[C]]∗ = [[A]]∗

(d) In all the other cases, the property holds trivially, since f ∗(A,w) =

∅ ⊆ [[A]]∗.

• DT

1. Let w ∈ Wi, with 0 ≤ i ≤ n− 1.

Since A,C, (A ∧ C) ∈ L = Γ∗0, we only have to consider the case in

which f ∗((A∧C), w) = f((A∧C), w)∩Wn, and f ∗(A,w) = f(A,w)∩Wn.


If A ∧ C ∈ Γn−(i+1), then f ∗((A ∧ C), w) = f((A ∧ C), w) ∩Wn, and

f ∗(A,w) = f(A,w) ∩Wn. If f((A ∧ C), w)∗ ⊆ [[B]]∗, for B ∈ L = Γ∗0,

we have, by definition of f ∗ and by theorem 5.5.1, that f((A∧C), w)∩Wn ⊆ [[B]]. Since in Wn, there is one representing element per each

equivalence class of f(A∧C), it follows that also f((A∧C), w) ⊆ [[B]]

(otherwise, in Wn there would be a representing element of the class of

worlds contained in f((A ∧ C), w) satisfying ¬B and f((A ∧ C), w) ∩Wn 6⊆ [[B]]). By (DT) in the original model M , we have that also

f(A, w) ∩ [[C]] ⊆ [[B]]. By definition of f ∗ and by theorem 5.5.1, we

conclude that f ∗(A,w) ∩ [[C]]∗ ⊆ [[B]]∗.


Since A,C, (A ∧ C) ∈ L = Γ∗0, we only have to consider three cases:

the case in which f ∗(D, w′) ∩ [[A]]∗ = ∅ (in this case, also f ∗(D, w′) ∩[[A∧C]]∗ = ∅); the case in which f ∗(D,w′)∩ [[A]]∗ 6= ∅ but f ∗(D, w′)∩[[A ∧ C]]∗ = ∅; the case in which f ∗(D, w′) ∩ [[A ∧ C]]∗ 6= ∅.(a) If f ∗(D, w′)∩ [[A]]∗ = ∅ and f ∗(D, w′)∩ [[A∧C]]∗ = ∅, f ∗(A,w) =

f ∗(A,w0) and f ∗((A ∧ C), w) = f ∗((A ∧ C), w0). The property

holds for what shown for w ∈ Wi with 0 ≤ i ≤ n− 1.

(b) If f ∗(D,w′) ∩ [[A]]∗ 6= ∅ but f ∗(D, w′) ∩ [[A ∧ C]]∗ = ∅, then

f ∗(A,w) = f ∗((A∧D), w′) and f ∗((A∧C), w) = f ∗((A∧C), w0).

By the properties ((CV)) of the original model M , we know that

Prop(f(D, w′)) = Prop(f((A ∧ D), w′),i.e. for any formula B :

B ∈ L, if (f(D, w′)) ⊆ [[B]], then also (f(A ∧D, w′)) ⊆ [[B]]. By

definition of f ∗ and by theorem 5.5.1, this property extends in the

new model M∗ to (f ∗(D, w′)) and to (f ∗(A ∧D,w′)) (remember

that f ∗(D, w′) = f(D,w′)∩Wn and f ∗(A∧D, w′) = f(A∧D,w′)∩Wn).

Now, since f ∗(D, w′) ∩ [[A ∧ C]]∗ = ∅, f ∗(D,w′) ⊆ [[¬(A ∧ C)]]∗

and therefore f ∗(A ∧ D,w′) ⊆ [[¬(A ∧ C)]]∗ . Since by (ID),

f ∗(A ∧ D,w′) ⊆ [[A]]∗, this entails that f ∗(A ∧ D,w′) ⊆ [[¬C]]∗

and therefore f ∗(A∧D, w′)∩ [[C]]∗ = ∅. Then the property holds

trivially.

(c) If f ∗(D,w′)∩ [[A∧C]]∗ 6= ∅, then f ∗(A,w) = f ∗((A∧D), w′) and

f ∗(A ∧ C,w) = f ∗(((A ∧ C) ∧ D), w′) = f ∗(((A ∧ D) ∧ C), w′)

(by (RCEA)) . The property follows since (DT) holds for all the


worlds in Wi with 0 ≤ i ≤ n− 1.

• CV

1. Let w ∈ Wi with 0 ≤ i ≤ n− 1. Since A and A∧C ∈ Γ∗0, f ∗(A,w) =

f(A,w) ∩Wn and f ∗(A ∧ C, w) = f((A ∧ C), w) ∩Wn.

If f ∗(A,w) ∩ [[C]]∗ = ∅, then the property holds trivially.

Let f ∗(A,w) ∩ [[C]]∗ 6= ∅. Then by definition of f ∗ and by theorem

5.5.1, f(A,w) ∩ [[C]] 6= ∅.For any B ∈ L, if f ∗(A,w) ⊆ [[B]]∗, then f(A,w) ∩Wn ⊆ [[B]]∗ and

by theorem 5.5.1,f(A,w)∩Wn ⊆ [[B]]. Since Wn contains exactly one

representing element of each equivalence class in f(A,w), it follows

that f(A,w) ⊆ [[B]] (if f(A,w) 6⊆ [[B]], it would follow that ∃y ∈f(A,w) such that y 6∈ [[B]], and since B ∈ L = Γ∗0, there would

be in Wn one representing element of the equivalence class of y and

f(A,w) ∩Wn 6⊆ [[B]]). By (CV) in the original model, it follows that

f(A ∧ C,w) ⊆ [[B]], therefore that f(A ∧ C,w) ∩Wn ⊆ [[B]] and by

definition of f ∗ and theorem 5.5.1, we conclude that f ∗(A ∧ C,w) ⊆[[B]]∗.


Since A, C, (A∧C) ∈ L = Γ∗0, we have to consider three cases: the case

in which f ∗(D,w′)∩[[A]]∗ = ∅ (in this case, also f ∗(D, w′)∩[[A∧C]]∗ =

∅); the case in which f ∗(D, w′)∩[[A]]∗ 6= ∅ but f ∗(D,w′)∩[[A∧C]]∗ = ∅;the case in which f ∗(D, w′) ∩ [[A ∧ C]]∗ 6= ∅.(a) Let f ∗(D, w′) ∩ [[A]]∗ = ∅ and f ∗(D, w′) ∩ [[A ∧ C]]∗ = ∅. In this

case, f ∗(A,w) = f ∗(A,w0) and f ∗((A∧C), w) = f ∗((A∧C), w0).

The property holds for it holds for w ∈ Wi with 0 ≤ i ≤ n− 1.

(b) If f ∗(D, w′) ∩ [[A]]∗ 6= ∅ but f ∗(D,w′) ∩ [[A ∧ C]]∗ = ∅, then

f ∗(A, w) = f ∗((A∧D), w′) and f ∗((A∧C), w) = f ∗((A∧C), w0).

By the same reasoning done in the corresponding case of (DT),

we can conclude that f ∗(A,w)∩ [[C]]∗ = ∅ and the property holds

trivially.

(c) If f ∗(D, w′)∩ [[A∧C]]∗ 6= ∅, then f ∗(A,w) = f ∗((A∧D), w′) and

f ∗(A ∧ C, w) = f ∗(((A ∧ C) ∧ D), w′) = f ∗(((A ∧ D) ∧ C), w′)

(by (RCEA)) . This property holds, since we have already shown

that (CV ) holds for w ∈ Wn−1.


• (S-REFL)

1. Let w ∈ Wi with 0 ≤ i ≤ n − 1 f ∗(>, w) = f(>, w) ∩ Wn. Since

w ∈ f(>, w) (by (S-REFL) in the original model), and w ∈ Wn, it

follows directly that w ∈ f ∗(>, w).

2. Let w ∈ Wn − Wn−1. The result follows, since f ∗(>, w) = f ∗((> ∧D), w′) = f ∗(D, w′) (by RCEA) for D and w′ such that w ∈ f ∗(D, w′).

• (S-TRANS)

1. Let w ∈ Wi with 0 ≤ i ≤ n − 1 Here, we have to consider all the

possible values of f ∗(A,w).

(a) If A ∈ Γ∗n−(i+1), then f ∗(A, w) = f(A,w) ∩Wn. If x ∈ f ∗(A,w),

then x ∈ f(A,w). In the same way, since > ∈ Γ∗i , it follows

that if y ∈ f ∗(>, x), then y ∈ f(>, x). By (S − TRANS) in the

original model, then, it follows that y ∈ f(A,w) and since y ∈ Wn,

y ∈ f(A,w) ∩Wn = f ∗(A,w).

(b) If A 6∈ Γ∗n−(i+1) but there is a C ∈ Γ∗n−(i+1) such that [[A]]∗ = [[C]]∗,

f ∗(A,w) = f ∗(C, w) and the property holds, for what has been

shown.

(c) Otherwise, f ∗(A,w) = ∅ and the property holds trivially.


(a) If A ∈ Γ∗0 and f ∗(D, w′) ∩ [[A]]∗ 6= ∅, then f ∗(A,w) = f ∗((A ∧D), w′). If x ∈ f ∗(A,w), then x ∈ f ∗(A ∧D, w′). If y ∈ f ∗(>, x),

then, there are two possibilities: either x ∈ Wn−1 or x ∈ Wn −Wn−1. In the first case, y ∈ f(>, x) and since f ∗(A ∧ D, w′) =

f(A∧D, w′)∩Wn, x ∈ f(A∧D,w′) and by (S−TRANS) applied

to M , we can conclude that y ∈ f(A ∧ D, w′) and therefore y ∈f ∗(A ∧ D, w′) and y ∈ f ∗(A,w). If x ∈ Wn − Wn−1, then if

y ∈ f ∗(>, x), then y ∈ f ∗(A ∧D, w′) and therefore y ∈ f ∗(A,w).

(b) If A ∈ Γ∗0 and f ∗(D,w′) ∩ [[A]]∗ = ∅, then f ∗(A,w) = f ∗(A,w0).

The property holds, since it holds for f ∗(A,w0).

(c) If A 6∈ Γ∗0 but there exist C ∈ Γ∗0 such that [[A]]∗ = [[C]]∗, the

property holds, since it holds for f ∗(C, w) = f ∗(A, w).

(d) Otherwise, f ∗(A,w) = ∅ and the property follows trivially.


• (S-EUC)


(a) If A ∈ Γ∗n−(i+1), then f ∗(A,w) = f(A,w)∩Wn. If x, y ∈ f ∗(A,w),

then x, y ∈ f(A,w). By (S−EUC) applied to the original model

M , x ∈ f(>, y) and, since x ∈ Wn, x ∈ f ∗(A, w).

(b) If A 6∈ Γ∗n−(i+1) but there is a C ∈ Γ∗n−(i+1) such that [[A]]∗ =

[[C]]∗, f ∗(A, w) = f ∗(C,w) and the property holds for it holds for

f ∗(A, y).

(c) Otherwise, f ∗(A,w) = ∅ and the property holds trivially.


(a) If A ∈ Γ∗0 and f ∗(D,w′) ∩ [[A]]∗ 6= ∅, then f ∗(A, w) = f ∗((A ∧D), w′). The property holds, for it holds for any w ∈ Wn−1.

(b) If A ∈ Γ∗0 and f ∗(D, w′) ∩ [[A]]∗ = ∅, then f ∗(A,w) = f ∗(A,w0).

The property holds for it holds for w0.

(c) If A 6∈ Γ∗0 but there exist C ∈ Γ∗0 such that [[A]]∗ = [[C]]∗, the

property holds, since it holds for f ∗(C, w)

(d) Otherwise, it holds trivially.

• (S-BEL) By construction of Wn, and by the properties of the original

model M , in general we have that if if w ∈ f ∗(>, y), then w ∈ Wi if and

only if also y ∈ Wi. The only possible exception to this property concerns

the case in which n = 1 and either w or y are equal to w0. In showing that

(S-BEL) holds in M∗, therefore, we also have to consider three cases: the

case in which n = 1 and w or y are equal to w0 , and the two usual cases

in which w ∈ Wi with 0 ≤ i ≤ n− 1 and w ∈ Wn −Wn−1.

1. If n = 1 and either w = w0 or y = w0

(a) If w ∈ f ∗(>, w0), then if A ∈ Γ∗0, w ∈ Wn −Wn−1 and f ∗(A,w) =

f ∗(A, w0). If A 6∈ Γ0 but there is a C ∈ Γ∗0, the property holds

for it holds for f ∗(A,w) = f ∗(C, w) = f ∗(C,w0) = f ∗(A,w0, by

(RCEA). Otherwise, f ∗(A,w) = ∅ = f ∗(A,w0)

(b) If w0 ∈ f ∗(>, y), then y ∈ W1 −W0 and by definition of f ∗, we

derive that y ∈ f ∗(>, w0) and we proceed as in the previous case.


2. If n > 1 and w, y ∈ Wi with 0 ≤ i ≤ n−1, then if w ∈ f ∗(>, y), also

w ∈ f(>, y). By (S −BEL) applied to the original model, for any A,

f(A, w) = f(A, y). Therefore, f ∗(A,w) = f ∗(A, y).

3. If n > 1 and w, y ∈ Wn−Wn−1, then if w ∈ f ∗(D, w′) D and w′ such

that y ∈ f ∗(D, w′). By definition of f ∗, it follows that f ∗(A,w) =

f ∗(A, y).

• (S-MOD)+(S-UNIV). (S-MOD) and (S-UNIV) together are equivalent

to the condition:

If[[A]]∗ 6= ∅, thenf ∗(A, w) 6= ∅for any A ∈ L2

1. Let w ∈ Wi with 0 ≤ i ≤ n − 1. If A ∈ L = Γ∗0, then f ∗(A,w) =

f(A, w)∩Wn 6= ∅. This follows from three facts: first of all, by theorem

5.5.1, if [[A]]∗ 6= ∅, also [[A]] 6= ∅, secondly, by (S-MOD)+(S-UNIV)

in M , f(A, w) 6= ∅, third, by construction of Wn, Wn contains exactly

one element per each equivalence class of f(A,w). If A is modal, then

it is equivalent either to > or to ⊥ and the result follows.

• (RCEA) Let if [[A]]∗ = [[B]]∗, for A,B ∈ Γ∗n−1.

Fact1: We show that [[A]] = [[B]]. Suppose by absurd that [[A]] 6= [[B]] and

that therefore [[A∧¬B]] 6= ∅ (or, indifferently, that [[¬A∧B]] 6= ∅). Then,

by the properties of M , it would follow that for all w, f((A∧¬B), w) 6= ∅.Therefore, by construction of W1, in W1 there would be a world w such

that w ∈ π(f(A ∧ ¬B), w0). By theorem 5.5.1, since w ∈ [[A ∧ ¬B]], also

w ∈ [[A ∧ ¬B]]∗, and therefore [[A]]∗ 6= [[B]]∗.

Fact2 From Fact1 and (RCEA) applied to M , it follows that f(A,w) =

f(B,w).

We show that f ∗(A,w) = f ∗(B, w).

1. Let w ∈ Wi with 0 ≤ i ≤ n − 1. If A ∈ Γ∗n−(i+1), then f ∗(A,w) =

f(A, w) ∩ Wn = f(B,w) ∩ Wn = f ∗(B,w). The other cases follow

easily.

2. Let w ∈ Wn −Wn−1. The result follows from the definition of f ∗, for

if [[A]]∗ = [[B]]∗, then also [[A∧D]]∗ = [[B∧D]]∗, and (RCEA) applies

to all the worlds w ∈ Wn−1.


2

Chapter 6

Extension to Iterated Belief

Revision

In the previous chapter, we have devised a correspondence between belief revision

systems and conditional logic that, differently from the Ramsey Test, does not

entail the triviality result. We have established a mapping between belief revision

systems and the conditional logic BC by a representation result showing how each

belief revision system determines a BC−structure, and how each BC−structure1

defines a belief revision system.

We have seen in chapter 2 that iterated belief revision has been widely investi-

gated in recent years [5, 8, 30, 46]. In particular, it has been shown that the AGM

postulates are too weak to ensure the rational preservation of conditional beliefs

during the revision process. For this reason, new postulates have been proposed

which characterize the revision of knowledge structures richer than belief sets,

namely epistemic states. Epistemic states contain a belief set (a set of formulas

ranging over L), and conditional beliefs.

In this chapter, we establish a correspondence between conditional logic and

a specific theory of iterated belief revision, obtained by slightly strengthening

Darwiche and Pearl’s revision theory. The logic IBC presented in this chapter

is an extension of BC, obtained by adding to BC some axioms and properties of

f .

We have seen in the previous chapter that the logic BC is well suited to

represent belief revision, since we can represent belief sets by the equivalence

classes among possible worlds defined by the selection function. Nonetheless,

1Satisfying the covering condition

79

80 CHAPTER 6. EXTENSION TO ITERATED BELIEF REVISION

it is not possible to establish a one-to-one correspondence between belief sets

and equivalence classes, since there can be several equivalence classes with the

same associated belief set. As a difference, we will see in this chapter that it is

possible to establish a one-to-one correspondence between epistemic states and

equivalence classes, showing that the semantics of BC is even better suited to

represent iterated belief revision than simple belief revision.

The relation between IBC and iterated belief revision is established by a

Representation Theorem showing that each belief revision system determines an

IBC−structure, and each IBC−structure (satisfying the covering condition) de-

fines an iterated belief revision system. Like for the logic BC, our representation

theorem does not run into the triviality result.

The chapter is organized as follows. In the next section we introduce our

reformulation of Darwiche and Pearl’s postulates for revision. We show that

Darwiche and Pearl’s postulates can be derived from ours. Moreover, we provide

a concrete operator satisfying our postulates, thus proving their consistency. In

section 3 we define the conditional logic IBC. Finally, in section 4 we prove a

representation theorem, establishing a mapping between iterated belief revision

and conditional models: to each iterated belief revision system corresponds an

IBC−structure and to each IBC−structure (satisfying the covering condition)

corresponds an iterated belief revision system.

6.1 Iterated Belief Revision

An iterated belief revision system is a triple 〈S, ∗, [ ]〉 where S is a non-empty

set whose elements are called epistemic states, ∗ : S × L → S is the revision

operator defined by the postulates below, [ ] : S → P (L) is a function that maps

each epistemic state to a belief set (i.e. a deductively closed set of propositional

formulas). Throughout this chapter we shall consider consistent iterated belief

revision systems, i.e. iterated belief revision systems that contain only consistent

epistemic states, where an epistemic state is consistent if its associated belief set

is consistent. Moreover, we will only consider revisions with consistent formulas.

The first six postulates, with the exception of (R∗4) correspond to postulates

(R1), (R2), (R3), (R5) and (R6) in Katsuno and Mendelzon’s reformulation of

AGM postulates (see chapter 2).

(R ∗ 1) A ∈ [Ψ ∗ A]


(R ∗ 2) If ¬A 6∈ [Ψ] , then [Ψ ∗ A] = [Ψ] + A;

(R ∗ 3) If A is satisfiable, then [Ψ ∗ A] is also satisfiable;

(R ∗ 4) If A1 ≡ A2, then Ψ ∗ A1 = Ψ ∗ A2;

(R ∗ 5) [Ψ ∗ (A ∧B)] ⊆ [Ψ ∗ A] + B

(R ∗ 6) If ¬B 6∈ [Ψ ∗ A], then [Ψ ∗ A] + B ⊆ [Ψ ∗ (A ∧B)]

We can observe that all the above postulates, with the exception of (R ∗ 4), are

only concerned with the belief sets resulting from certain revisions, rather than

with the epistemic states. On the contrary, postulate (R∗4), as well as postulate

(A4) below, deals with epistemic states, rather than with belief sets. (R ∗ 4)

says that, given an epistemic state, its revisions by the formula A1 and by an

equivalent formula A2 lead to the same epistemic state. Therefore, postulate

(R ∗ 4) is stronger than postulate (R4).

In addition to the postulates (R ∗ 1) − (R ∗ 6), we introduce the following

postulates for iterated revision. A part from postulate (A4), the postulates are

a syntactical reformulation of Darwiche and Pearl’s postulates (DP1)− (DP4).

We assume that * associates on the left:

(A1) If B |= ¬A, then [Ψ ∗ A ∗B] = [Ψ ∗B];

(A2) If A ∈ [Ψ ∗B], then [Ψ ∗ A ∗B] = [Ψ ∗B];

(A3) If ¬A 6∈ [Ψ ∗B], then [Ψ ∗ A ∗B] ⊆ [Ψ ∗B] + A;

(A4) Ψ ∗ > = Ψ

Postulate (A1) says that if two contradictory pieces of information, A and B,

are successively learned, then the belief set obtained by the revision of Ψ by A and

then by B does not depend on the false intermediate revision by A. In particular,

(A1) implies that [Ψ ∗ A ∗ ¬A] = [Ψ ∗ ¬A]. It corresponds to postulate (DP2).

Postulate (A2) says that if A is believed after the revision of the epistemic state

Ψ by B, then the belief set obtained by the revision of Ψ by A and then by B is

the same as the one obtained by the revision of Ψ directly by B. It corresponds

to Darwiche and Pearl’s postulates (DP1) and (DP4). Postulate (A3) says that

if A is consistent with the belief set resulting from the revision of Ψ by B, then by

revising Ψ by A and then by B we cannot conclude more than by adding A to the


result of the revision of Ψ by B. It corresponds to postulate (DP4). According

to postulate (A4), the revision of an epistemic state with a tautology > does not

affect the epistemic state. Since we only consider consistent epistemic states, (A4)

is consistent with the other postulates. Note, however, that if we also considered

inconsistent epistemic states, postulate (A4) would conflict with postulate (R*3)

(if Ψ is an inconsistent epistemic state, according to (A4) Ψ ∗ > = Ψ, whereas

according to (R*3) Ψ ∗ > is consistent and hence different from Ψ). We think

that in this case it is not obvious which one of the two postulates should be

discarded. A possible way-out could be to weaken postulate (A4), by adding to

(A4) a precondition requiring Ψ to be consistent.

While postulate (A4) (similarly to (R∗4)) claims that the two epistemic states

Ψ ∗ > and Ψ are the same, postulates (A1), (A2) and (A3) state only something

about the relation between the two belief sets [Ψ ∗ A ∗ B] and [Ψ ∗ B]. But this

does not say anything about the conditional beliefs true in the two states Ψ∗A∗Band Ψ ∗ B. Indeed, for postulate (A2), it might be the case that, although A

holds in the state Ψ ∗ B, we do not want to forget that the revision of Ψ by A

has preceded the revision by B. For instance, we might want to give the pieces

of information a different reliability according to when they have been learnt.

It can be shown that the postulates introduced above are consistent. In

particular it is possible to define a revision operator, which is based on Spohn’s

ordinal conditional functions (see chapter 2) and which satisfies the postulates

above.

We prove that, given the modified AGM postulates (R∗1)−(R∗6), Darwiche

and Pearl’s postulates can be derived from our postulates (A1), (A2), (A3).

Lemma 6.1.1 (DP1), (DP2), (DP3) and (DP4) can be derived from (A1),

(A2),(A3) together with (R ∗ 1)− (R ∗ 6).

Proof.

Postulate (DP1) follows immediately from (A2) since, by (R ∗ 1), B |= A

entails A ∈ [Ψ ∗B].

Postulate (DP2) is identical to (A1).

Postulate (DP3) follows immediately from (A2).

Postulate (DP4) follows from (A3). Assume that ¬A 6∈ [Ψ ∗ B], that is,

[Ψ ∗ B] + A is satisfiable, and, therefore, ¬A 6∈ [Ψ ∗ B] + A. Moreover, by

postulate (A3), from the initial hypothesis we have: [Ψ ∗ A ∗ B] ⊆ [Ψ ∗ B] + A.

Therefore, we can conclude that ¬A 6∈ [Ψ ∗ A ∗B].


2

We cannot prove the converse of Lemma 6.1.1, since (A4) and (R ∗ 4) cannot

be derived from Darwiche and Pearl’s postulates. As a matter of fact, none of

Darwiche and Pearl’s postulates enforces the equality between epistemic states

obtained through different revisions. However, we believe that postulates (A4)

and (R ∗ 4) define natural properties of revision functions. (A4) states that

a revision with a tautology cannot change the epistemic state and its revision

strategies, whence Ψ and Ψ ∗ > will determine the same belief sets under any

sequence of revisions. (R ∗ 4) states that the syntactical form of the revision

formula is irrelevant in determining the resulting epistemic state. The weaker

form of (R ∗ 4) adopted by Darwiche and Pearl only enforces that the syntactical

form of the revision formula is irrelevant in determining the resulting belief set.

We can prove the following:

Lemma 6.1.2 (A1), (A2) and (A3) can be derived from (DP1), (DP2), (DP3)

and (DP4) together with (R ∗ 1)− (R ∗ 6).

Proof. Postulate (A2) follows from (DP1), (DP3), (R∗5) and (R∗6). If [Ψ∗B]

is inconsistent, then B must be inconsistent, by (R ∗ 3). Hence [Ψ ∗A ∗B] must

be inconsistent too. In such a case, [Ψ ∗B] = [Ψ ∗A ∗B] holds and (A2) (as well

as postulates (A1) and (A3)) holds trivially.

Let us now consider the case when [Ψ ∗ B] is satisfiable. Assume that A ∈[Ψ ∗ B]. By (DP3), A ∈ [Ψ ∗ A ∗ B]. From A ∈ [Ψ ∗ B], by the consistency of

[Ψ ∗B], we get ¬A 6∈ [Ψ ∗B]. From these assumptions by (R ∗ 5) and (R ∗ 6) we

have: [Ψ ∗ B] = [Ψ ∗ B] + A = [Ψ ∗ (B ∧ A)]. Similarly, from A ∈ [Ψ ∗ A ∗ B],

(R ∗ 5) and (R ∗ 6) we have: [Ψ ∗ A ∗ B] = [Ψ ∗ A ∗ B] + A = [Ψ ∗ A ∗ (B ∧ A)].

Since from (DP1) we derive [Ψ ∗ (B ∧ A)] = [Ψ ∗ A ∗ (B ∧ A)] we can conclude

[Ψ ∗B] = [Ψ ∗ A ∗B].

Postulate (A3) follows from (DP1), (DP4), (R ∗ 5) and (R ∗ 6). Since ¬A 6∈[Ψ ∗ B], by (R ∗ 5) and (R ∗ 6), [Ψ ∗ B] + A = [Ψ ∗ (B ∧ A)]. By (DP4),

from ¬A 6∈ [Ψ ∗ B] we can conclude ¬A 6∈ [Ψ ∗ A ∗ B]. Again, by (R ∗ 5) and

(R ∗ 6), [Ψ ∗ A ∗ B] + A = [Ψ ∗ A ∗ (B ∧ A)]. Since from (DP1) we derive

[Ψ ∗ (B ∧A)] = [Ψ ∗A ∗ (B ∧A)] we can conclude [Ψ ∗B] + A = [Ψ ∗A ∗B] + A.

Therefore: [Ψ ∗ A ∗B] ⊆ [Ψ ∗ A ∗B] + A = [Ψ ∗B] + A. 2


To conclude this section, we want to show that there is a non-trivial iterated

belief revision system satisfying our postulates. More precisely, we show that

Spohn’s revision operator satisfies our postulates for iterated revision and that it

defines a non-trivial iterated belief revision system.

We say that an iterated belief revision system 〈S, ∗, [ ]〉 is non-trivial if there

is at least one epistemic state Φ ∈ S and three propositions A,B,C which are

pairwise (logically) disjointed, such that [Φ] + A, [Φ] + B, [Φ] + C are consistent.

Theorem 6.1.3 There is at least one non-trivial iterated belief revision system.

Proof. Let 〈K, ∗, [ ]〉 be a Spohn revision system (see chapter 2). As proved

in [8], the operator ∗ satisfies Darwiche and Pearl’s postulates. Theorem 6.1.2

shows that all our postulates (except from postulate (A4) and (R ∗ 4)) can be

derived from Darwiche and Pearl’s ones. It is enough to show that the operator

∗ satisfies (A4) and (R ∗ 4) to conclude that it satisfies all our postulates. As far

as (A4) is concerned, k = k ∗ > since for any w, w |= >, and k(>) = 0.

Concerning (R*4), if A1 ≡ A2 then, for any w, we have w |= A1 iff w |= A2.

Therefore, if w |= A1, then

k ∗ A1(w) = k(w)− k(A1) = k(w)− k(A2) = k ∗ A2(w).

If, on the contrary, w 6|= A1 then k ∗ A1(w) = k(w) + 1 = K ∗ A2(w). We can

therefore conclude that 〈K, ∗, [ ]〉 is an iterated belief revision system.

Now, consider the language L containing only the propositional variables

p1, p2, p3, p4. Let A = ¬p2 ∧ ¬p3 ∧ p4; B = ¬p3 ∧ ¬p4 ∧ p2; C = ¬p2 ∧ ¬p4 ∧ p3.

Clearly, `PC ¬(A ∧B),`PC ¬(A ∧ µ) and `PC ¬(B ∧ µ).

We show that there exists an iterated belief revision system and an epistemic

state k in it such that [k] + A, [k] + B and [k] + C are consistent.

Given the set W of all the possible interpretations of L, consider the iterated

belief revision system 〈K, ∗, [ ]〉 in which K is the set of all the possible rankings

on W . In K there is also the ranking k: k(w) = 0 if and only if w |= p1. Since

[k] = p1, it follows that [k] + A, [k] + B and [k] + C are consistent. 2

6.2. THE CONDITIONAL LOGIC IBC 85

6.2 The Conditional Logic IBC

In this section we introduce the conditional logic IBC. Since IBC is an extension

of the logic BC, the basic features of IBC coincide with the features of BC. As far

as the axiomatization is concerned, IBC contains some additional axioms in order

to represent postulates (A1) − (A4) for the iterated step of revision. Similarly,

as far as the semantic is concerned, the difference between IBC and BC lies in

the fact that in IBC the selection function satisfies some extra conditions.

Definition 6.2.1 The language L> of logic IBC is an extension of the language

L of classical propositional logic obtained by adding the conditional operator >.

Let us define the following modalities:

2A ≡ ¬A > ⊥3A ≡ ¬(A > ⊥).

We define the language of modal formulas L2 as the smallest subset of L> in-

cluding L and closed under ¬,∧,2,32. The logic IBC contains the following

axioms and inference rules:

(BC) (CLASS) All classical axioms and inference rules;

(ID) A > A;

(RCEA) if ` A ↔ B, then ` (A > C) ↔ (B > C);

(RCK) if ` A → B, then ` (C > A) → (C > B);

(DT ) (A ∧ C) > B) → (A > (C → B)), for A,B, C ∈ L;

(CV ) ¬(A > ¬C) ∧ (A > B) → ((A ∧ C) > B), for A,B,C ∈ L;

(BEL) (A > B) → > > (A > B);

(REFL) (> > A) → A;

(EUC) ¬(A > B) → A > ¬(> > B);

(TRANS) (A > B) → A > (> > B);

(MOD) 2A → B > A, where A ∈ L2;

(U4) 2A → 22A, where A ∈ L2;

2We assume that the conditional > has higher precedence than the material implication →.


(U5) 3A → 23A, where A ∈ L2.

(ITER) (C1) 2¬(A∧B)∧3A → [(A > B > C) ↔ (B > C)], where A ∈ L2 and

C ∈ L;

(C2) B > A → [(A > B > C) ↔ (B > C)], where A ∈ L2 and C ∈ L;

(C3) [¬(B > ¬A) ∧ (A > B > C)] → (B > (A → C)), where A ∈ L2 and

C ∈ L.

The axioms of group (BC) are common to (BC), see previous chapter for

their discussion.

Axioms of (ITER) encode our postulates for iterated revision (A1), (A2),(A3).

The semantic of IBC is obtained from the semantic of BC by adding the

properties (S − C1)-(S − C3) below.

There is an intuitive correspondence between iterated belief systems and se-

lection function models satisfying the properties of the next definition. The idea

is that an epistemic state Φ can be represented by any set of equivalent worlds

which evaluate conditional formulas in the same way. Like we did for the logic

BC, in IBC models the selection function can be used to specify a revision op-

erator: the revision of an epistemic state Φ by A is simply the epistemic state

f(A,w), for any w associated to Φ (it does not depend on the choice of w).

Definition 6.2.2 An IBC-structure M has the form 〈W, f, [[ ]]〉, where W is

a non-empty set of possible worlds, f is a function of type L> × W → 2W ,

[[ ]] : L> → 2(W ) is a valuation function satisfying the conditions of definition

5.1.2 of the previous chapter.

We assume that the selection function f satisfies the following properties:

(BC) (S − ID) f(A,w) ⊆ [[A]];

(S −RCEA) if [[A]] = [[B]] then f(A,w) = f(B, w);

(S −DT ) Prop(f(A ∧ C,w)) ⊆ Prop(f(A,w) ∩ [[C]]), for A,C ∈ L 3;

(S − CV ) f(A,w) ∩ [[C]] 6= ∅ → Prop(f(A,w)) ⊆ Prop(f(A ∧C, w)), for

A, C ∈ L ;

(S −REFL) w ∈ f(>, w);

3Recall that Prop(S) = {A ∈ L: S ⊆ [[A]]}


(S − TRANS) x ∈ f(A,w) ∧ y ∈ f(>, x) → y ∈ f(A,w);

(S − EUC) x, y ∈ f(A,w) → x ∈ f(>, y);

(S −BEL) w ∈ f(>, y) → f(A,w) = f(A, y);

(S −MOD) If f(B, w) ∩ [[A]] 6= ∅, then f(A, w) 6= ∅, where A ∈ L2;

(S−UNIV ) if [[A]] 6= ∅, ∃B such that f(B, w)∩ [[A]] 6= ∅, where A ∈ L2;

(ITER) (S − C1) if [[A]] ∩ [[B]] = ∅ and y ∈ f(A, x), then Prop(f(B, x)) =

Prop(f(B, y)), where A ∈ L2;

(S − C2) if f(B, x) ⊆ [[A]] and y ∈ f(A, x) then Prop(f(B, x)) =

Prop(f(B, y)), where A ∈ L2;

(S − C3) if f(B, x) ∩ [[A]] 6= ∅ and y ∈ f(A, x), then Prop(f(B, y)) ⊆Prop(f(B, x) ∩ [[A]]), where A ∈ L2.

Similarly to BC, we say that a formula A is true in an IBC-structure M =

〈W, f, [[ ]]〉 if [[A]] = W . We say that a formula is IBC-valid if it is true in

every IBC-structure. For readability, we also use the notation x |= A instead of

x ∈ [[A]].

Given an IBC-structure M = 〈W, f, [[ ]]〉, a set of formulas Γ and a formula

A, we define Γ |=M A if for every and every w ∈ W , if w |= B for every B ∈ Γ,

then also w |= A i.e.⋂

B∈Γ[[B]] ⊆ [[A]]. We then define the entailment relation

Γ |= A if and only if for every IBC-structure M , Γ |=M A. As expected, ∅ |=M A

means that A is true in M and ∅ |= A that A is valid.

As for BC, in an IBC-structure M , we can define by means of the selection

function f the equivalence relation ≈f on the set of worlds W as follows: for all

w, w′ ∈ W ,

w ≈f w′ iff w′ ∈ f(>, w).

The properties of ≈f being reflexive, transitive and symmetric come from the

semantic conditions (REFL), (TRANS) and (EUC) of the selection function f

(and, more precisely, from the last two conditions by taking A = >).

As a consequence of (S-BEL), for any pair w,w′, if w ≈f w′, then f(A,w) =

f(A,w′). Furthermore, f(A,w) is an equivalence class in itself.

The semantic conditions (S-C1), (S-C2) and (S-C3) are associated with the

axioms (A1), (A2) and (A3) for iterated belief revision.


The axiomatization of IBC is sound and complete with respect to semantic

introduced above.

Theorem 6.2.3 (Soundness) If a formula A is a theorem of IBC then is IBC-

valid.

Proof. For the axioms already contained in BC, see the proof of the soundness

of BC in the previous chapter. Let us consider the axioms not already contained

in BC, i.e. (C1), (C2) and (C3).

(C1) let w ∈ W and w |= 2¬(A∧B)∧3A, i.e. w |= (A∧B) > ⊥∧¬(A > ⊥).

This implies that f(A∧B,w) = ∅. By (S-MOD) and (S-UNIV), this implies

that [[A ∧B]] = [[A]] ∩ [[B]] = ∅.(⇒) let w |= A > B > C, where C ∈ L. By hypothesis w |= 3A, i.e.

w 6|= A > ⊥, that implies f(A,w) 6= ∅. Thus there is y ∈ f(A,w). By

(S-C1), we have Prop(f(B, y)) = Prop(f(B, w)). We get y |= B > C.

Thus C ∈ Prop(f(B, y)) = Prop(f(B,w)). This implies w |= B > C.

(⇐) If w |= B > C, then C ∈ Prop(f(B,w)). By (S-C1) for every y ∈f(A,w), we have Prop(f(B, y)) = Prop(f(B,w)) ⊆ [[C]]. Thus for every

y ∈ f(A,w), f(B, y) ⊆ [[C]], whence y |= B > C. We have shown that

f(A,w) ⊆ [[B > C]] and this means w |= A > B > C.

(C2) Let w ∈ W and w |= B > A, we then have f(B,w) ⊆ [[A]].

(⇒) let w |= A > B > C, where C ∈ L. If f(A,w) = ∅, being A a

modal formula, we obtain (by (S-UNIV) and (S-MOD)) [[A]] = ∅. Thus by

hypothesis also f(B,w) = ∅, and trivially w |= B > C. If f(A,w) 6= ∅, then

let y ∈ f(A,w), by hypothesis we have y |= B > C. By hypothesis and

(S-C2) we have Prop(f(B,w)) = Prop(f(B, y)) from which we conclude

w |= B > C being C ∈ L.

(⇐) Let w |= B > C and y ∈ f(A,w), by hypothesis and (S- C2), we have

Prop(f(B, w)) = Prop(f(B, y)) from which we conclude y |= B > C. We

have shown f(A,w) ⊆ [[B > C]] and this means w |= A > B > C.

(C3) Let w ∈ W assume w |= ¬(B > ¬A) and w |= A > B > C, with C ∈ L.

Then f(B, w)∩ [[A]] 6= ∅. Being A a modal formula, by (S-MOD), we have

f(A,w) 6= ∅. Thus let y ∈ f(A,w). By hypothesis we have y |= B > C

which means f(B, y) ⊆ [[C]], whence C ∈ Prop(f(B, y)). Since f(B, w) ∩


[[A]] 6= ∅, by (S-C3), we know that Prop(f(B, y)) ⊆ Prop(f(B, w)∩ [[A]]),

whence f(B, w) ∩ [[A]] ⊆ [[C]]. This implies f(B,w) ⊆ [[C]] ∪ (W − [[A]]).

But this is equivalent to f(B,w) ⊆ [[A → C]].

2

We come now to completeness.

Theorem 6.2.4 (Completeness) If |= A then ` A.

Proof. The proof follows the line of the completeness of BC of the previous

chapter. We only need to show that, given the axioms (C1) − (C3), both the

structure M and the substructure M0 satisfy the conditions (S−C1)− (S−C3).

As an example, we show that M and M0 satisfy (S − C1)− (S − C3).

- For what concerns M

(S-C1) Let [[A]]M ∩ [[B]]M = [[A ∧ B]]M = ∅. By (S-ID), for any X, we have

f(A ∧B,X) = ∅ = [[⊥]]M , thus X ∈ [[(A ∧B) > ⊥]]M , this means

(*) 2¬(A ∧B) ∈ X.

Now let Y ∈ f(A,X), so that f(A,X) 6= ∅. We have f(A,X) 6⊆[[⊥]]M , and this is equivalent to say

(**) X |= 3¬A.

We must show that

Prop(f(B,X)) = Prop(f(B, Y )).

(⇒) Let C ∈ Prop(f(B, X)), then we have C ∈ Prop(f(B, X)) im-

plies B > C ∈ X so that A > B > C ∈ X by axiom (C1), (*), and

(**). Thus B > C ∈ Y , whence C ∈ Prop(f(B, Y )).

(⇐) Let C ∈ Prop(f(B, Y )), then we have C ∈ Prop(f(B, Y )) im-

plies B > C ∈ Y . Thus T > B > C ∈ Y by (BEL). We have

¬(T > B > C) 6∈ Y . Thus A > ¬(T > B > C) 6∈ X, as by hypoth-

esis Y ∈ f(A,X). We can conclude ¬(A > B > C) 6∈ X by (EUC),

whence A > B > C ∈ X by maximality of X. We get

B > C ∈ X

by axiom (C1), (*) and (**), so that C ∈ Prop(f(B,X)).


(S-C3) Let f(B,X) ∩ [[A]]M 6= ∅ and Y ∈ f(A,X). We must prove that

Prop(f(B, Y )) ⊆ Prop(f(B, X) ∩ [[A]]M).

Let C ∈ Prop(f(B, Y )), similarly to the previous case we have:

C ∈ Prop(f(B, Y )) implies B > C ∈ Y , so that T > B > C ∈ Y by

(BEL). We then have ¬(T > B > C) 6∈ Y and hence A > ¬(T > B >

C) 6∈ X, as by hypothesis Y ∈ f(A,X). Thus ¬(A > B > C) 6∈ X by

(EUC); we conclude A > B > C ∈ X by maximality of X.

By hypothesis we know f(B,X)∩[[A]]M 6= ∅, this means that f(B, X) 6⊆[[¬A]]M , whence B > ¬A 6∈ X. By maximality of X, we obtain

¬(B > ¬A) ∈ X

Since A > B > C ∈ X and ¬(B > ¬A) ∈ X, by axiom (C3) we get

B > (A → C) ∈ X.

This shows that X ∈ [[B > (A → C)]]M , i.e. f(B, X) ⊆ [[A →C]]M = [[C]]M ∪ (W − [[A]]M). Thus f(B, X) ∩ [[A]]M ⊆ [[C]]M , and

this implies C ∈ Prop(f(B, X) ∩ [[A]]M).

- For what concerns M0.

(S-C1) Let f0(B, X) ⊆ [[A]]M0 and Y ∈ f0(A,X), for X, Y ∈ W0. Then we

have

f(B, X) = f0(B,X) ⊆ [[A]]M0 = [[A]]M ∩W0 ⊆ [[A]]M .

Thus f(B,X) ⊆ [[A]]M . Since M satisfy (S-C2), we get Prop(f(B,X)) =

Prop(f(B, Y )) hence also Prop(f0(B, X)) = Prop(f0(B, Y )).

(S-C3) Let X, Y ∈ W0, assume Y ∈ f0(A,X) and f0(B,X) ∩ [[A]]M0 6= ∅. It

is easy to see that we get also

Y ∈ f(A,X) and f(B, X) ∩ [[A]]M 6= ∅.Since M satisfy (C3) (being [[A]]M0 ⊆ [[A]]M) we have

f0(B, X)∩ [[A]]M0 ⊆ f(B, X)∩ [[A]]M ⊆ f(B, X) = f0(B, X),

thus

Prop(f0(B,X)) = Prop(f(B, X)) ⊆ Prop(f(B, X)∩[[A]]M) =

Prop(f0(B,X) ∩ [[A]]M0)

2

6.3. CONDITIONALS AND ITERATED REVISION 91

6.3 Conditionals and Iterated Revision

The correspondence between iterated belief revision systems and IBC-structures

is similar to the correspondence established in the last chapter between the logic

BC and belief revision systems. However, we will see in the second part of the

Representation Theorem below that in this case the correspondence is more tight

than in the previous case.

The Representation Theorem consists of two parts. The first part shows that

we can represent a belief revision system in an IBC-structure such that, for each

epistemic state Ψ there is a set of equivalent worlds WΨ in the structure which

globally represent the epistemic state. The set of worlds WΨ (and, actually, each

world in this set) provides all information concerning the epistemic state Ψ, in-

cluding both the belief set and the revision strategies associated with Ψ. Roughly

speaking, the worlds in the equivalence class WΨ are the classical interpretations

of [Ψ], when we restrict our concern to the valuations of the formulas in L. Thus,

[Ψ] = Prop(WΨ), Furthermore, the revision strategies associated with the epis-

temic state Ψ are represented by all the conditional formulas A1 > A2 > . . . > B

true at the worlds in WΨ (which are the same for all the worlds in WΨ).

In the second part of the Representation Theorem, we consider IBC-structures

satisfying the covering condition i.e. such that, for any A consistent, [[A]] 6= ∅.We show that each IBC-structure 〈W, f, [[ ]]〉 satisfying the covering condition

gives rise to an iterated belief revision system. Indeed, consider the structure

〈W/≈f , ∗M , [ ]〉, where W/≈f is the quotient of W with respect to ≈f , ∗M is the

canonical extension of f with respect to ≈f and [ ] is the function Prop that as-

sociates to each set of worlds the set of propositional formulas true in all worlds

of the set. This structure is a belief revision system where epistemic states are

the equivalence classes of W , the belief sets associated to epistemic states are

the sets of propositional formulas holding in the epistemic states, and ∗M is the

extension of f on the equivalence classes.

Theorem 6.3.1 (Representation Theorem) (1) Given a belief revision sys-

tem 〈S, ∗, [ ]〉, there is an IBC-structure M∗ = 〈W, f, [[ ]]〉 such that:

for every Ψ in S, there exists w in W such that:

B ∈ [Ψ ∗ A1 ∗ . . . ∗ An] iff w |= A1 > . . . > An > B.

(2) Given an IBC-structure M = 〈W, f, [[ ]]〉 which satisfies the covering condi-

tion, there is an iterated belief revision system 〈W/≈f , ∗M , [ ]〉 such that:


W/≈f = {[w]≈f : w ∈ W};[w]≈f ∗M A = f(A,w);

[[w]≈f ] = Prop([w]≈f )

and, for each [w]≈f of W/≈f , and A1 . . . An, B ∈ L (A1 . . . An, B consis-

tent),

B ∈ [[w]≈f ∗M A1 . . . ∗M An] iff w |= A1 > . . . An > B.

Proof. Part (1). Given a belief revision system 〈S, ∗, []〉, we define an IBC-

structure M∗ = 〈W, f, [[ ]]〉 as follows:

W = {(Ψ, w) : w is a classical interpretation, Ψ ∈ S and w |= [Ψ]};[[p]] = {(Ψ, w) ∈ W : w |= p} for all propositional letters p ∈ L.

f(A, (Ψ, w)) and [[A]] can be defined by double induction on the structure of the

formula A. At each induction step, for each connective ◦, [[A ◦ B]] is defined

by making use of the valuation of the subformulas ([[A]] and [[B]]) and of the

selection function for subformulas (for instance, f(A,w)); moreover, f(A ◦B, w)

is defined by possibly making use of the valuation of the formula A ◦B itself. In

particular we let:

f(A, (Ψ, w)) = {(Ψ′, w′) ∈ W : Ψ′ = Ψ ∗ A}, if A ∈ L;

f(A, (Ψ, w)) = {(Ψ′, w′) ∈ W : Ψ′ = Ψ ∗ φA}, if A 6∈ Land there exists a formula φA ∈ L such that [[A]] = [[φA]];

f(A, (Ψ, w)) = ∅, otherwise.

By making use of the properties of the revision operator ∗, we can show that M∗is an IBC-structure, that is, all the semantic properties listed in definition 6.2.2

can be proved. We show it for some of the semantic properties.

(S − ID): f(A, (Ψ, w)) ⊆ [[A]].

By definition of f , for all (Ψ′, w′) ∈ f(A, (Ψ, w)) we have Ψ′ = Ψ ∗ A and

w′ |= [Ψ ∗ A]. Let A ∈ L, then, by postulate (R ∗ 1) we have that A ∈ [Ψ ∗ A],

and hence (Ψ′, w′) ∈ [[A]].

The case of A 6∈ L, but [[A]] = [[φA]] with φA ∈ L, is similar. In the case when

A 6∈ L and there is no φA ∈ L such that [[A]] = [[φA]], then f(A, (Φ, w)) = ∅ and

the property holds trivially.

(S −RCEA): if [[A]] = [[B]] then f(A, (Ψ, w)) = f(B, (Ψ, w)).


Let A,B ∈ L. If in M∗ [[A]] = [[B]] holds, then it must be that A ≡ B is valid, as

the set {w′ : (Ψ′, w′) ∈ [[A]]} contains all and only the classical models of A. By

postulate (R ∗ 4) we have that Φ ∗ A = Φ ∗ B. Hence, if (Ψ′, w′) ∈ f(A, (Ψ, w)),

then Ψ′ = Ψ ∗ A = Ψ ∗ B and from w′ |= [Ψ ∗ A] we can conclude w′ |= [Ψ ∗ B].

Therefore, (Ψ′, w′) ∈ f(B, (Ψ, w)).

In the case A 6∈ L (or B 6∈ L), but there is a formula φA (φB) ∈ L such that

[[A]] = [[φA]] (respectively, [[B]] = [[φB]]), the proof is the same.

If A 6∈ L and there is no formula φA ∈ L such that [[A]] = [[φA]], then f(A, (Ψ, w)) =

∅. Moreover, as [[A]] = [[B]], it must be that B 6∈ L and there is no formula φB

∈ L such that [[B]] = [[φB]]. Hence, f(B, (Ψ, w)) = ∅, which conclude the proof

of the property.

(S −DT ): Prop(f(A ∧ C,w)) ⊆ Prop(f(A, (Ψ, w)) ∩ [[C]]), for A,C ∈ L.

Let B ∈ Prop(f(A ∧ C, (Ψ, w))). For all (Ψ′, w′) ∈ f(A ∧ C, (Ψ, w)), Ψ′ =

Ψ∗(A∧C) and w′ |= [Ψ∗(A∧C)]. Since f(A∧C, (Ψ, w)) contains all the classical

models of [Ψ ∗ (A∧C)], we have that: B ∈ Prop(f(A∧C, (Ψ, w))) if and only if

for all w′, if w′ |= [Ψ ∗ (A∧C)] then w′ |= B. Hence, B ∈ Prop(f(A∧C, (Ψ, w)))

if and only if B ∈ [Ψ ∗ (A ∧ C)].

Similarly, it is easy to see that B ∈ Prop(f(A, (Ψ, w)) ∩ [[C]]) if and only if

B ∈ [Ψ ∗ A] + C.

From (R ∗ 5) we have that [Ψ ∗ (A ∧ C)] ⊆ [Ψ ∗ A] + C, from which, given the

above equivalences, the wanted property trivially follows.

The proof of the semantic property (S − CV ) is very similar to the one for

(S − DT ). The proofs of (S − REFL), (S − TRANS), (S − EUC) and (S −BEL) are easy and left to the reader. Let us consider the semantic properties

(S −MOD) and (S − UNIV ).

(S−MOD): If f(B, (Ψ, w))∩ [[A]] 6= ∅, then f(A, (Ψ, w)) 6= ∅, where A ∈ L2.

It can be easily seen that, if A ∈ L2, then either A ∈ L or [[A]] = [[φA]] for some

φA ∈ L.

Assume first that A ∈ L. Then, either [[A]] = ∅ (and f(B,w) ∩ [[A]] = ∅, so

that the property holds trivially) or [[A]] 6= ∅. In the last case, by construction,

f(A, (Ψ, w)) 6= ∅.In case [[A]] = [[φA]] for φA ∈ L, the proof is similar.

(S − UNIV ): if [[A]] 6= ∅, ∃B such that f(B, (Ψ, w)) ∩ [[A]] 6= ∅, where

A ∈ L2.

As in the previous case, if A ∈ L2, then either A ∈ L or [[A]] = [[φA]] for

some φA ∈ L. In the first case, from [[A]] 6= ∅, we have by construction that


f(A, (Ψ, w)) 6= ∅. As (S − ID) holds, the conclusion follows by taking B = A.

In the second case, the proof is similar, by taking B = φA.

(S − C1): if A is modal, [[A]] ∩ [[B]] = ∅ and (Ψ′, w′) ∈ f(A, (Ψ, w)), then

Prop(f(B, (Ψ, w))) = Prop(f(B, (Ψ′, w′))).

We can distinguish three cases: the case when both A, B ∈ L; the case when

A 6∈ L or B 6∈ L, but there is a φA ∈ L (φB ∈ L) such that [[A]] = [[φA]]

(respectively, [[B]] = [[φB]]); the case when either A 6∈ L and there is no φA ∈ Lsuch that [[A]] = [[φA]], or B 6∈ L and there is no φB ∈ L, such that [[B]] = [[φB]].

In the first case, from the fact that [[A]] ∩ [[B]] = ∅, it follows that A ∧ B

is inconsistent (otherwise, by (R*3), for any Ψ ∈ S, [Ψ ∗ (A ∧ B)] would be

satisfiable, which would imply that (Ψ ∗ (A ∧ B), w) ∈ W , [[A ∧ B]] 6= ∅ and

[[A]]∩ [[B]] 6= ∅). As A∧B is inconsistent, B |= ¬A. By postulate (A1), then, we

conclude that [Ψ∗A∗B] = [Ψ∗B]. By the definition of f , it follows immediately

that the property holds.

The case in which ∃φA (or ∃φB in L) such that [[A]] = [[φA]] (respectively,

[[B]] = [[φB]]) can be treated in a similar way.

In the third case, the property follows easily from the definition of f . If φA

does not exist, f(A, (Ψ, w)) = ∅, i.e. there is no (Ψ′, w′) ∈ f(A, (Ψ, w)) and

the property follows trivially. If φB does not exist, then f(B, (Ψ, w)) = ∅ and

f(B, (Ψ′, w′)) = ∅.The proof of the semantic properties (S − C1) and (S − C3) are similar to

that of (S − C2).

To conclude the proof of part (1), it is not difficult to show that the model

M∗ satisfies the condition that, for each epistemic state Ψ in S, and for each

A1, . . . , An, B ∈ L,

B ∈ [Ψ ∗ A1 ∗ . . . ∗ An] if and only if (Ψ, w) |= A1 > . . . > An > B.

Part (2). Let us consider an IBC-structure M =< W, f, [[ ]] > satisfying the

covering condition. For each world w in W , we introduce an epistemic state [w]≈f ,

and define the belief set associated to the epistemic state by [[w]≈f ] = Prop[w]≈f .

We then define a revision operator ∗M by saying that for each [w]≈f , for each

consistent A, [w]≈f ∗M A = f(A,w).

Notice that for the semantic properties of IBC f(A,w) is an equivalence class,

namely the equivalence class [w′]≈f for any w′ ∈ f(A,w). As far as iterated belief

revision is concerned, then, it follows by construction that [w]≈f ∗M A ∗M B =


f(B, w′) for any w′ ∈ f(A,w).

We will show that the revision system 〈W/≈f , ∗M , P rop〉 satisfies the postu-

lates (R∗1)−(R∗6), (A1)−(A4). We will prove it by making use of the semantic

properties of M .

(R ∗ 1): A ∈ [[w]≈f ∗M A].

Since [w]≈f ∗M A = f(A,w), we have that, for all B ∈ L, B ∈ [[w]≈f ∗M A]

if and only if B ∈ Prop(f(A,w)). By the semantic property (S − ID), we

have f(A, w) ⊆ [[A]], and hence A ∈ Prop(f(A,w)), from which we conclude

A ∈ [[w]≈f ∗M A].

(R ∗ 2): If ¬A 6∈ [[w]≈f ] then [[w]≈f ∗M A] = [[w]≈f ] + A.

We first prove the inclusion “⊆”. Let B ∈ [[w]≈f∗MA]. Then, B ∈ Prop(f(A,w)).

By the semantic property (S − DT ), Prop(f(A,w)) ⊆ Prop(f(>, w) ∩ [[A]]).

Hence, B ∈ Prop(f(>, w)∩ [[A]]). But, Prop(f(>, w)∩ [[A]]) = [[w]≈f ] +A, and

hence B ∈ [[w]≈f ] + A.

We prove the inclusion “⊆”. Assume that ¬A 6∈ [[w]≈f ]. Then, f(>, w)∩ [[A]] 6=∅. If B ∈ [[w]≈f ] + A, then B ∈ Prop(f(>, w) ∩ [[A]]). It is easy to see that

from the semantic property (S − CV ) it follows that Prop(f(>, w) ∩ [[A]]) ⊆Prop(f(A,w)). Hence, B ∈ Prop(f(A,w)) and B ∈ [[w]≈f ∗M A].

(R ∗ 3): If A is satisfiable, then [[w]≈f ∗M A] is also satisfiable.

Assume that A is satisfiable. Then, by the covering condition, [[A]] 6= ∅. As a

consequence of (S − UNIV ) and (S − MOD) we have that f(A,w) 6= ∅. But

[w]≈f ∗M A = f(A,w), whence [w]≈f ∗M A 6= ∅.(R ∗ 4): If A1 ≡ A2, then [w]≈f ∗M A1 = [w]≈f ∗M A2

Assume that A1 ≡ A2. Then [[A1]] = [[A2]] and, by (S − RCEA) f(A1, w) =

f(A2, w). But [w]≈f ∗M A1 =f(A1, w) and [w]≈f ∗M A2 =f(A2, w), so that the

thesis follows.

(R ∗ 5): [[w]≈f ∗M (A ∧B)] ⊆ [[w]≈f ∗M A] + B.

Let C ∈ [[w]≈f ∗M (A ∧ B)]. Then, C ∈ Prop(f(A ∧ B, w)). By the semantic

property (S − DT ), Prop(f(A ∧ B,w)) ⊆ Prop(f(A,w) ∩ [[B]]). Hence, C ∈Prop(f(A,w)∩ [[B]]). But, Prop(f(A,w)∩ [[B]]) = [[w]≈f ∗M A] + B, and hence

C ∈ [[w]≈f ∗M A] + B.

(R ∗ 6): If ¬B 6∈ [[w]≈f ∗M A], then [[w]≈f ∗M A] + B ⊆ [[w]≈f ∗M (A ∧B)].

Assume that ¬B 6∈ [[w]≈f∗MA]. Then, f(A,w)∩[[B]] 6= ∅. If C ∈ [[w]≈f∗MA]+B,

then C ∈ Prop(f(A,w)∩ [[B]]). It is easy to see that from the semantic property

(S − CV ) it follows that Prop(f(A,w) ∩ [[B]]) ⊆ Prop(f(A ∧ B,w)). Hence,


C ∈ Prop(f(A ∧B,w)) and C ∈ [[w]≈f ∗M (A ∧B)].

(A1): If B |= ¬A, then [[w]≈f ∗M A ∗M B] = [[w]≈f ∗B].

Assume that B |= ¬A. Then, [[A]] ∩ [[B]] = ∅. Let C ∈ [[w]≈f ∗M A ∗M B]. As

A is a consistent proposition (we only consider revisions by consistent formulas),

f(A,w) 6= ∅. By construction we have: [w]≈f ∗M A ∗M B = f(B, w′) for any

w′ ∈ f(A,w). Hence, C ∈ Prop(f(B, w′)). By the semantic property (S−C1) we

have that Prop(f(B,w′)) = Prop(f(B,w)) and, therefore, C ∈ Prop(f(B, w)).

Thus, C ∈ [[w]≈f ∗ B]. Similarly, from C ∈ [[w]≈f ∗ B] we conclude that C ∈[[w]≈f ∗M A ∗M B].

(A2): If A ∈ [[w]≈f ∗M B], then [[w]≈f ∗M A ∗M B] = [[w]≈f ∗B] .

Suppose then that A ∈ [[w]≈f ∗M B]. It follows that f(B, w) ⊆ [[A]] and, by

the semantic property (S − C2), we can conclude that for any w′ ∈ f(A,w),

Prop(f(B,w)) = Prop(f(B, w′)). As [w]≈f ∗M A ∗M B = f(B, w′), for w′ ∈f(A,w), and [w]≈f∗MB = f(B, w), we have that [[w]≈f∗MA∗MB] = Prop(f(B, w′))

and [[w]≈f ∗M B] = Prop(f(B,w)). from the semantic property (S−C2) it follows

that [[w]≈f ∗M A ∗M B] = [[w]≈f ∗M B].

(A3): If ¬A 6∈ [[w]≈f ∗M B], then [[w]≈f ∗M A ∗M B] ⊆ [[w]≈f ∗M B] + A.

Suppose then that ¬A 6∈ [[w]≈f ∗M B]. It follows that f(B, w) ∩ [[A]] 6= ∅ and,

by the semantic property (S − C3), we can conclude that for any w′ ∈ f(A,w),

Prop(f(B,w′)) = Prop(f(B, w)∩ [[A]]). As [w]≈f ∗M A ∗M B = f(B, w′), if C ∈[[w]≈f ∗M A ∗M B], then C ∈ Prop(f(B, w′)). Hence, C ∈ Prop(f(B, w)∩ [[A]]).

But then C ∈ [[w]≈f ∗M B] + A.

(A4): [w]≈f ∗M > = [w]≈f

As by the semantic property (REFL) it holds that w ∈ f(A,w), we have that

[w]≈f ∗M > = f(>, w) = [w]≈f .

To conclude the proof of part (2), we observe that the property

B ∈ [Ψ ∗ A1 ∗ . . . ∗ An] if and only if (Ψ, w) |= A1 > . . . > An > B

follows directly from the definition of ∗M .

2

Note that the second part of the theorem is stronger than the second part

of the corresponding theorem 4.1 in the previous chapter. Indeed, in this case

to any IBC−model corresponds a unique belief revision system that contains

an epistemic state for each equivalence class of the model. On the contrary, in

theorem 4.1 of the previous chapter, it is not possible to devise such one-to-one


correspondence between equivalence classes and belief sets, since, in general, there

can be several equivalence classes with the same associated belief set. Hence,

as far as the logic BC is concerned, each BC− model originates several belief

revision systems, depending on the choice of the equivalence class representing a

belief set.

As a final remark, we can say that our Representation Theorem does not entail

the triviality result for the same reasons for which the representation theorem of

the previous chapter did not entail the triviality result. The reason is that to

different belief sets, even if included in one another or obtained through expansion

from each other, we associate different conditional beliefs.

6.3.1 Corollaries

The following corollaries of the representation theorem give a characterization

in the logic IBC of what can be derived by the revision of any epistemic state,

independently from the specific properties of the iterated revision system. The

corollaries thus illustrate, from the belief revision point of view, the utility of

representing belief revision systems within a conditional logic framework.

Corollary 6.3.3 characterizes what can be derived from the revision of any

epistemic state with a given belief set K. As a particular case, corollary 6.3.4

gives a characterization of what can be derived by the revision of an “empty”

epistemic state, that is an epistemic state whose only beliefs are the tautologies.

Corollary 6.3.5, on the other hand, gives a characterization of what can be derived

by the revision of any epistemic state, no matter what is its associated belief set.

For any set K of formulas in L, we define the set Th(K) of formulas in L> as

follows:

Definition 6.3.2

Th(K) = {(> > C) : C ∈ K} ∪ {¬(> > C) : C 6∈ K} ∪ {3A : 6`PC ¬A}.

Notice that Th(K) depends non-monotonically on K, in the sense that from K ⊆K ′ it does not follow Th(K) ⊆ Th(K ′). Thus from Th(K) ` A1 > . . . > An > B

we cannot conclude that Th(K ′) ` A1 > . . . > An > B. This reflects the non-

monotonicity of the belief revision operator with respect to belief sets: if [Ψ] = K,

[Ψ′] = K ′ and [K] ⊆ [K ′] it does not follow that [Ψ ∗A1 . . . An] ⊆ [Ψ′ ∗A1 . . . An].


Corollary 6.3.3 For all A1, . . . , An, B ∈ L we have

Th(K) ` A1 > . . . > An > B

iff

∀〈S, ∗, [ ]〉,∀Ψ ∈ S such that [Ψ] = K, B ∈ [Ψ ∗ A1 . . . ∗ An].

Proof. (⇒) We show the contrapositive, i.e. that if there is a revision system

〈S, ∗, [ ]〉, and Ψ ∈ S such that [Ψ] = K and B 6∈ [Ψ∗A1 . . . An] then Th(K) 6`IBC

A1 > . . . > An > B.

Let 〈S, ∗, [ ]〉 and Ψ ∈ S such that [Ψ] = K; let B 6∈ [Ψ ∗ A1 . . . An]. By the

representation theorem, there exists an IBC−model M , and there is a world w

such that ∀E,Di ∈ L, E ∈ [Ψ ∗D1 . . . ∗Dn] iff w |= D1 > . . . > Dn > E; Thus,

by the hypothesis, we have

(1) w 6|= A1 > . . . > An > B.

Since Ψ = Ψ ∗ >, we have

(2) w |= (> > C) holds for any C ∈ [Ψ],

and w |= ¬(> > C) for any C 6∈ [Ψ].

Furthermore, we can show that

(3) w |= 3A for all A :6`PC ¬A.

To see this, let A be such that 6`PC ¬A, we know (by (R ∗ 1) and (R ∗ 3)) that

there is an epistemic state Ψ ∈ S such that A ∈ [Ψ]; by construction of M (see

the proof of the representation theorem) there is w ∈ W such that w |= [Ψ],

whence [[A]] 6= ∅; but this implies that f(A,w) 6= ∅ (by the universality condi-

tion). Therefore, w |= 3A.

By (2) and (3) we have w |= Th(K) and thus by (1) Th(K) 6|= A1 > . . . > An >

B. By the soundness of IBC, it follows that Th(K) 6` A1 > . . . > An > B.

(⇐) Again, we show the contrapositive. Suppose that Th(K) 6` A1 > . . . >

An > B. By the completeness of IBC, Th(K) 6|= A1 > . . . > An > B and

there is M and w ∈ M such that w |= Th(K) and w 6|= A1 > . . . > An > B.

Since w |= {3A ∈ L :6`PC ¬A}, the model M satisfies the covering condition.


Then, by the second half of the representation theorem, there is an iterated

belief revision system 〈W/≈f , ∗M , [ ]〉 and an epistemic state Ψ ∈ S such that

[Ψ] = Prop([w]≈f ) = {C : w |= (> > C)} = K, and such that B 6∈ [Ψ∗A1 . . . An].

2

To introduce the next corollary, we denote by > the belief set which contains

exactly the classical propositional tautologies. We call empty an epistemic state

Ψ if [Ψ] = >. Notice that if K = >, in any IBC model M = 〈W, ∗, [ ]〉, ∀w ∈ W ,

if w |= {¬(> > B) : 6`PC B} then w |= Th(K), therefore

Th(>) = {¬(> > B) : 6`PC B}.


Th(>) ` A1 > . . . > An > B

iff

∀〈S, ∗, [ ]〉,∀Ψ ∈ S empty B ∈ [Ψ ∗ A1 . . . ∗ An].

The last corollary characterizes what can be derived by the revision of any

epistemic state in any belief revision system.


{3A :6`PC ¬A} ` A1 > . . . > An > B

iff

∀〈S, ∗, [ ]〉,∀Ψ ∈ S B ∈ [Ψ ∗ A1 . . . ∗ An].

The set of assumptions {3A ∈ L 6`PC ¬A} is needed to restrict our consider-

ation to models which satisfy the covering condition.


Chapter 7

Weakening the AGM Postulates

We have seen in chapter 4, section 4 that the triviality result entailed by the

Ramsey Test depends on a questionable assumption: the assumption that condi-

tional belief revision systems are closed with respect to the expansion operator.

In turn, the assumption derives from the closure of belief sets with respect to the

revision operator together with postulates (K ∗ 3), (K ∗ 4), (K ∗ 7), (K ∗ 8) that

encode the Minimal Change Principle. We have argued that both the assumption

and the postulates are counterintuitive when applied to conditional belief revi-

sion systems. Examples 4.1 and 4.2 of section 4 show that the Minimal Change

Principle applied to conditional formulas is counterintuitive. Indeed, learning

new information may change our expectations and plausibility judgements about

the world. Even if consistent, new information may change our revision strategies

and conditional beliefs.

In this chapter, we show that the correspondence postulated by the Ramsey

Test can be safely assumed (and the triviality result avoided) if we abandon

the assumption and weaken the rationality postulates in such a way that they

only apply to the non-conditional part of belief sets. The solution is simple, but

effective, and it offers a solution to a problem remained open for more than fifteen

years.

Furthermore, we show that the conditional logic resulting from our weakened

AGM postulates and a strong version of the Ramsey Test is very similar to the

logic BC introduced in chapter 5.

The chapter is organized as follows. In the next section we state our weakened

AGM postulates. In section 2, we show that the Ramsey Test can be safely

preserved, once the rationality postulates are weakened. In section 3, we show

101

102 CHAPTER 7. WEAKENING THE AGM POSTULATES

that the weakened rationality postulates are consistent also with a stronger form

of the Ramsey Test. In section 4, at last, we show how a conditional logic very

similar to BC derives from this strong version of the Ramsey Test.

7.1 Weakened AGM postulates

Let L be a propositional language. We define the language Lu> as the unnested

conditional extension of L, i.e. it is defined as the least set of formulas such that

• if A ∈ L then A ∈ Lu>;

• if A ∈ L and B ∈ Lu> then A > B ∈ Lu

>;

• if A,B ∈ Lu>, then ¬A ∈ Lu

> and A ∧B ∈ Lu>

We can introduce the other connectives →,∨,↔ by the standard propositional

equivalences. Let S be a set of formulas of Lu>, we define CnPC(S) = {A ∈ Lu

> s.t.

S |=PC A}; the relation S |=PC A means that A is a propositional consequence

of S, where conditional formulas are treated as atoms.

We call epistemic state any set of formulas of Lu> which is deductively closed

with respect to CnPC . Therefore, the epistemic states we consider in this section

are more specific than the epistemic states considered in the previous chapter (and

by most of the theories of iterated belief revision). A belief set is a deductively

closed set of formulas of L. We introduce the belief function [ ] that associates to

each epistemic state K its corresponding belief set

[K] = K ∩ L.

A revision operator ∗ is any function that takes an epistemic state and a formula

in L as input, and gives an epistemic state as output. The expansion of an epis-

temic state K by a formula A is the set K + A = CnPC(K ∪ {A}).

Definition 7.1.1 A conditional belief revision system is a pair 〈K, ∗〉, where K

is a set of epistemic states closed under the revision operator ∗. The operator ∗satisfies the following postulates:

(B ∗ 1) K ∗ A is an epistemic state;

(B ∗ 2) A ∈ K ∗ A;

7.1. WEAKENED AGM POSTULATES 103

(B ∗ 3) [K ∗ A] ⊆ [K + A];

(B ∗ 4) if ¬A 6∈ [K], then [K + A] ⊆ [K ∗ A];

(B ∗ 5) K ∗ A `PC⊥ only if `PC ¬A;

(B ∗ 6) if A ≡ B, then K ∗ A = K ∗B;

(B ∗ 7) [K ∗ (A ∧B)] ⊆ [(K ∗ A) + B];

(B ∗ 8) if ¬B 6∈ [K ∗ A], then [(K ∗ A) + B] ⊆ [K ∗ (A ∧B)]

(B ∗ >) for any K consistent, K ∗ > = K.

(RT ) B ∈ K ∗ A if and only if A > B ∈ K.

Postulates (B*1), (B*2), (B*5), (B*6) are AGM postulates (K*1), (K*2),

(K*5), (K*6). Postulates (B*3), (B*4), (B*7), (B*8) are the restriction of AGM

postulates (K*3), (K*4), (K*7), (K*8) to belief sets. Postulates (K*3),(K*4)

represent the Minimal Change Principle. The restriction of (K*3), (K*4) means

that the Minimal Change Principle is restricted to the non-conditional part of

epistemic states. A similar remark applies to (K*7) and (K*8) that are a gener-

alization of (K*3),(K*4)1. Postulate (B ∗>) expresses a rather intuitive property

of revision (for a discussion of the property, see chapter 6). It comes for free from

the original AGM postulates (K*3),(K*4); we have introduced it as we can no

longer derive it from our corresponding (B*3), (B*4).

The restriction we have put on AGM postulates have an impact on the closure

properties of conditional belief revision systems. AGM postulates entail that

belief revision systems are closed with respect to expansion, i.e. if K is the set

of all the epistemic states of a conditional belief revision system, then for any

epistemic state K, if K ∈ K, then also K + A ∈ K. This property follows

immediately from (K ∗ 3), (K ∗ 4) together with the closure with respect to the

revision operator (see chapter 4). In contrast, this property cannot be derived

from our modified postulates. In our setting, the closure with respect to revision

does not imply the closure with respect to expansion. As we have argued in the

introduction, the revision of an epistemic state with a formula A, even when A

is consistent with the state, may affect the conditional formulas holding in that

state in a way that is not reflected by the simple expansion operation.

1Given (B ∗ >), (K*3),(K*4) derive respectively from (K*7),(K*8) by taking A = >.


7.2 Non-triviality

In this section, we show that our weakened AGM postulates (B ∗ 1) − (B ∗ >)

are compatible with the Ramsey Test. More precisely, we show that there are

non-trivial conditional belief revision systems (satisfying (B ∗ 1) − (B ∗ >) and

(RT )).

The notion of non trivial belief revision system can be extended straightfor-

wardly to conditional belief revision systems.

Definition 7.2.1 A conditional belief revision system 〈K, ∗, 〉 is non-trivial if

there are three formulas A, B, C in L, which are pairwise disjoint (i.e. such that

`PC ¬(A ∧ B), `PC ¬(B ∧ C), `PC ¬(A ∧ C)), and an epistemic state K ∈ K

such that ¬A 6∈ [K], ¬B 6∈ [K], and ¬C 6∈ [K].

We show that to any AGM belief revision system we can associate a conditional

belief revision system that is belief conservative with respect to it, where:

Definition 7.2.2 [Belief Conservativity] A conditional belief revision system 〈K′, ∗′〉is belief conservative with respect to an AGM belief revision system 〈K, ∗〉 just

in case for any K ∈ K there is a K ′ in K′ such that K = [K ′].

Non-triviality follows from the fact that non trivial AGM belief revision sys-

tems give rise to non-trivial revision systems satisfying (B ∗ 1)− (RT ).

In short, to any belief set of an AGM belief revision system 〈K, ∗〉 we can

associate an epistemic state that is obtained by extending the belief set by con-

ditional formulas of Lu>. We then collect all the epistemic states built from the

belief sets of K, and we define a new belief revision operator on epistemic states.

Finally, we show that the structure so obtained is belief conservative with respect

to 〈K, ∗〉 and satisfies (B ∗ 1)− (RT ).

Let (nl) be the nesting level of a formula in Lu>, defined as:

• if A ∈ L, nl(A) = 0

• nl(A ∧B) = max(nl(A), nl(B))

• nl(¬A) = nl(A)

• nl(A > B) = 1 + nl(B)

7.2. NON-TRIVIALITY 105

By using the nesting levels just defined, we define some sublanguages of Lu>

ordered by increasing complexity of the formulas contained: the sublangage L0

will contain only the formulas of Lu> with nesting level 0, the sublanguage L1 will

contain only the formulas of Lu> with nesting level ≤ 1 and so on. Formally,

• L0 = {A ∈ Lu> : nl(A) ≤ 0}

• L1 = {A ∈ Lu> : nl(A) ≤ 1}

• . . .

• Li = {A ∈ Lu> : nl(A) ≤ i}

Clearly, it holds that L0 ⊆ Li ⊆ Ln.

For all belief sets K of K, we define

• φ(K, 0) = K

• φ(K, 1) = CnPC(φ(K, 0) ∪ {A > B : A > B ∈ L1 and B ∈ φ(K ∗ A, 0)}

• φ(K, i+1) = CnPC(φ(K, i)∪{A > B : A > B ∈ Li+1 and B ∈ φ(K ∗A, i)}

Finally, we let:

• φ(K) =⋃

i φ(K, i).

From the set of epistemic states so constructed we define a conditional belief

revision system by defining

• K′ = {φ(K) : K ∈ K}

• φ(K) ∗′ A = φ(K ∗ A)

Proposition 7.2.3 〈K′, ∗′〉 is belief conservative with respect to 〈K, ∗〉

Proof. We show that for all K ∈ K, K = [φ(K)]

⇒ By definition, φ(K, 0) = K. Therefore, K ⊆ φ(K). Furthermore, K ⊆ L,

and K ⊆ [φ(K)].

⇐ We show that for any A ∈ L, if A ∈ φ(K, i + 1), then A ∈ φ(K, i).

Suppose, by absurd, it is not so, and that A ∈ φ(K, i + 1)− φ(K, i). Then, there

are two possibilities. One possibility is that there are E0 . . . Ei ∈ φ(K, i + 1) −φ(K, i) such that E0 . . . Ei `PC A. But this is not possible, since E0 . . . Ei are


conditionals, treated as atoms as far as propositional derivability is concerned,

and no occurrence of them belongs to A.

The second possibility is that there are E0 . . . Ei ∈ φ(K, i + 1)− φ(K, i) and

D0 . . . Dj ∈ φ(K, i) such that E0 . . . Ei, D0 . . . Dj `PC A but D0 . . . Dj 6`PC A.

Again, this is not possible, since E1 . . . Ej can be considered as atoms as far as

propositional derivability is concerned, and they do not occur neither in A nor in

D0 . . . Di.

Therefore, for any A ∈ L, if A ∈ φ(K, i + 1), then A ∈ φ(K, i). Hence,

A ∈ φ(K, 0) = K. 2

Proposition 7.2.4 〈K′, ∗′〉 satisfies postulates (B ∗ 1)− (RT ).

Proof. Let φ(K) + A = CnPC(φ(K), A).

(B ∗ 1) By definition of φ(K), φ(K) ⊆ Lu>. Furthermore, we show that for all

D1 . . . Dn and all A ∈ Lu>, if D1 . . . Dn `PC A, and D1 . . . Dn ∈ φ(K), then

A ∈ φ(K). This follows by the fact that if D1 . . . Dn ∈ φ(K), then there

is an i such that D1 . . . Dn ∈ φ(K, i). By definition, φ(K, i) is deductively

closed.

(B ∗ 2) By the properties of ∗, A ∈ K ∗ A. Therefore, A ∈ φ(K ∗ A, 0) = K ∗ A

and A ∈ φ(K ∗ A). By definition of ∗′, A ∈ φ(K) ∗′ A.

(B ∗ 3) By Proposition 7.2.3 [φ(K ∗ A)] = K ∗ A, and by definition of ∗′, also

[φ(K) ∗′ A] = K ∗A. We now show that [φ(K) + A] = K + A. First of all,

[φ(K) + A] = CnPC(φ(K), A)∩L. Therefore, [φ(K) + A] is the set of non-

conditional formulas that can be obtained by propositional calculus from

φ(K) and A. By considerations similar to the ones made in the proof of

Proposition 7.2.3, we know that these are the formulas that can be obtained

through the propositional calculus from A and the non-conditional part

of φ(K). Since the non-conditional part of φ(K) is K, it follows that

[φ(K) + A] = CnPC(K,A), and [φ(K) + A] = K + A.

(B ∗ 3) follows from the identities just shown and from (K ∗ 3).

(B ∗ 4) If ¬A 6∈ [φ(K)], then ¬A 6∈ K (by Proposition 7.2.3). By (K ∗ 4), K + A ⊆K ∗A. Moreover, we have shown just above that [φ(K) + A] = K + A and

[φ(K) ∗′ A] = K ∗ A, and we conclude that [φ(K) + A] ⊆ [φ(K) ∗′ A].

7.2. NON-TRIVIALITY 107

(B ∗ 5) First, we show that if φ(K, i + 1) is inconsistent, then also φ(K, i) is in-

consistent. Suppose φ(K, i + 1) `PC ⊥ and φ(K, i) 6` ⊥. Then, by con-

struction of φ(K, i + 1), there must be a formula A > B ∈ Li+1 such that

A > B ∈ φ(K, i + 1) and ¬(A > B) ∈ φ(K, i + 1). But this is impossi-

ble, since only positive conditionals are inserted at each step. Therefore,

if φ(K, i + 1) is inconsistent, this can only be because φ(K, i) is inconsis-

tent. We can repeat the reasoning until φ(K, 0) = K. It follows that if

φ(K) `PC⊥, then there is a φ(K, i) such that φ(K, i) `PC⊥ and therefore

K `PC⊥.

Since K = φ(K, 0), this allows us to show that if φ(K) is inconsistent, then

also K is inconsistent. From (K ∗ 5) we conclude that if φ(K) `PC A, then

also K `PC A.

(B ∗ 6) By (K ∗ 6), if A ≡ B, then K ∗A = K ∗B, therefore φ(K ∗A) = φ(K ∗B)

and φ(K) ∗′ A = φ(K) ∗′ B.

(B ∗ 7) From Proposition 7.2.3 [φ(K ∗ (A∧B))] = K ∗ (A∧B) and by the consid-

erations made in the proof of (B ∗ 3), we can derive that [φ(K ∗A) + B] =

(K ∗A)+B. By (K ∗7), K ∗(A∧B) ⊆ (K ∗A)+B from which we conclude

that [φ(K ∗ (A ∧B))] ⊆ [φ(K ∗ A) + B].

(B ∗ 8) If ¬B 6∈ [φ(K ∗A)], then by Fact 1, ¬A 6∈ (K ∗A) and by (K ∗3), (K ∗A)+

B ⊆ K ∗ (A∧B). Since [φ(K ∗A) + B] = (K ∗A) + B (for what shown in

the proof of (B ∗ 3)), and [φ(K ∗ (A ∧ B))] = K ∗ (A ∧ B), it follows that

[φ(K ∗ A) + B] ⊆ [φ(K ∗ (A ∧B))]

(B ∗ >) By (K ∗ 3) and (K ∗ 4), if ¬> 6∈ K (i.e. if K is consistent), then K =

K + > = K ∗ >. It follows, by construction of φ(K), that for any K

consistent, φ(K) = φ(K ∗ >) = φ(K) ∗′ >.

(RT ) If B ∈ φ(K) ∗′ A then B ∈ φ(K ∗A), and B ∈ φ(K ∗A, i). By definition of

φ(K, i), it follows that A > B ∈ φ(K, i) and therefore A > B ∈ φ(K).

If B 6∈ phi(K) ∗′ A, then B 6∈ φ(K ∗ A) and by construction, (A > B) 6∈φ(K).

2

We can therefore conclude that:


Theorem 7.2.5 There is a non-trivial conditional belief revision system .

Proof. If a conditional belief revision system is belief conservative with respect

to a non-trivial AGM belief revision system, then it is non trivial. Therefore, it

is sufficient to show that there is a non trivial AGM revision system. To this

purpose, let the language L contain only the propositional variables p1, p2, p3, p4.

Let A = ¬p2 ∧ ¬p3 ∧ p4; B = ¬p3 ∧ ¬p4 ∧ p2; C = ¬p2 ∧ ¬p4 ∧ p3.

Clearly, `PC ¬(A ∧ B),`PC ¬(A ∧ C) and `PC ¬(B ∧ C). Consider now the

AGM belief revision system 〈K, ∗〉 such that K = CnPC(p1) ∈ K and ∗ is a full

meet revision operator (such a conditional belief revision system exists, since K

can be obtained by successively revising any belief set first by ¬p1 and by p1). It

is non trivial, since K + A,K + B and K + C are consistent. 2

7.3 Strong Ramsey Test

Our weakened postulates for belief revision are also consistent with a stronger

version of the Ramsey Test, namely:

(SRT)

• if B ∈ K ∗ A, then A > B ∈ K and

• if B 6∈ K ∗ A, then ¬(A > B) ∈ K.

Indeed, the construction of φ(K) made in the previous section can be extended

in order to include also negative conditional beliefs as follows.

• φ(K, 0) = K

• φ(K, 1) = CnPC(φ(K, 0) ∪ {A > B : A > B ∈ L1 and B ∈ φ(K ∗ A, 0)} ∪{¬(A > B) : B ∈ L0 and B 6∈ φ(K ∗ A, 0)})

• φ(K, i + 1) = CnPC(φ(K, i) ∪ {A > B : A > B ∈ Li+1 and B ∈ φ(K ∗A, i)} ∪ {¬(A > B) : B ∈ L0 and B 6∈ φ(K ∗ A, 0)})

Finally, we let:

• φ(K) =⋃

i φ(K, i).

7.3. STRONG RAMSEY TEST 109

From the set of epistemic states so constructed, we define a conditional belief

revision system by defining

• K′ = {φ(K) : K ∈ K}

• φ(K) ∗′ A = φ(K ∗ A)

We can show, similarly to what we have done in the previous section, 7.2.3, that:

Proposition 7.3.1 〈K′, ∗′〉 is belief conservative with respect to 〈K, ∗〉

In the following of the chapter we consider the conditional logic deriving from

our weakened rationality postulates and the strong version of the Ramsey Test.

Our choice for a strong version of the Ramsey Test is motivated by the fact

that, when conditionals are introduced in epistemic states, their acceptance in

a state K is determined by the behaviour of the revision operator ∗, which is

completely defined for a fixed belief revision system 〈K, ∗〉. In fact, for a given

K ∈ K and A,B ∈ L, we have that either B ∈ K ∗ A or B 6∈ K ∗ A. There

is not a third choice. On the other hand, if we accept the weaker version of

the Ramsey Test (RT ) given above, it may occur that neither A > B ∈ K nor

¬(A > B) ∈ K, that is, the conditional A > B is undetermined in the epistemic

state K. However, we can notice that this is not a real undeterminacy: given the

fact that A > B 6∈ K, according to (RT ) it must be that B 6∈ K ∗A, so that the

undeterminacy on A > B can only be eliminated by adding ¬(A > B) to K (as

advocated by the strong version of the Ramsey Test), and not by adding A > B

to K. In such a case, adding A > B to K would also violate the weak version

(RT ) of the Ramsey test.

We believe that allowing an incomplete specification of Ramsey conditionals

(so that neither A > B nor ¬(A > B) belong to K) can be meaningful if the

revision operator is incompletely specified, so that we do not know whether B ∈K ∗A or B 6∈ K ∗A, but that it is inappropriate when ∗ is given and completely

determined.

In essence, if we accept the weak version of the Ramsey Rule (RT) we are

faced with a mismatch. On one side of the equivalence, two cases are possible:

either B ∈ K ∗ A or B 6∈ K ∗ A.

On the other side of the equivalence, three cases are possible:


A > B ∈ K and ¬(A > B) 6∈ K,

A > B 6∈ K and ¬(A > B) ∈ K,

A > B 6∈ K and ¬(A > B) 6∈ K.

Of the last three cases, the first one corresponds to the case B ∈ K ∗A. The

second one corresponds to the case B 6∈ K ∗ A. What about the third case? In

the third case, as in the second one it must be that B 6∈ K ∗ A, otherwise (RT)

would be violated. Therefore, it is not clear why one should make a distinction

between three cases by using the conditional formulas, while only two cases are

actually possible for the revision operator.

A strong version of the Ramsey Test, which deals also with negated condi-

tionals, has been proposed by Levi [32], by defining the acceptability of both the

positive and negative conditionals. However, as a difference from the rule (SRT),

conditional sentences are excluded from belief sets in Levi’s proposal.

7.4 The logic BCR

In this section, we consider the conditional logic that derives from the strong

Ramsey Test and from our weakened postulates. Valid formulas are just those

that belong to every epistemic state of every belief revision system.

Definition 7.4.1 A formula A ∈ Lu> is revision-valid if for all conditional revision

systems 〈K, ∗〉, for all K ∈ K we have A ∈ K.

The logic resulting from definition 7.4.1, from the postulates (B ∗1)− (B ∗>)

and from (SRT) is defined as follows.

Definition 7.4.2 [BCR] The logic BCR is the smallest logic containing the fol-

lowing axioms and deduction rules:

• (CLASS) All classical propositional axioms and inference rules;

• (CONS) all formulas ¬(A > ⊥) such that A ∈ L and 6`PC ¬A;

• (ID) A > A;

• (DT) (A ∧ C) > B → A > (C → B), where B ∈ L;

• (CV) (¬(A > C) ∧ (A > B)) → (A ∧ C) > B) where B ∈ L;

7.4. THE LOGIC BCR 111

• (BEL) > B) → > > (A > B);

• (REFL) (> > A) → A;

• (TRANS) (A > B) → A > (> > B);

• (EUC) ¬(A > B) → (A > ¬(> > B));

• (RCEA) if ` A ↔ B, then ` (A > C) ↔ (B > C)

• (RCK) if ` A → B, then ` (C > A) → (C > B).

A part from (CONS), and from the limitations on the language, the axioms

of BCR are the same than the axioms of the logic BC introduced in chapter 5.

For a detailed discussion of the axioms, we therefore refer to chapter 5. Axiom

(CONS) is stronger than any of the axioms of BC, since it says that if a formula

is consistent, then there is at least one most preferred world satisfying it, and

therefore A >⊥ does never hold. In the logic BC we did not have this property,

but the weaker property according to which, for any modal formula A, if there is

a world satisfying it, then there is a most preferred world satisfying it. We then

had to add the additional covering condition in order to obtain the property for

any consistent formula (remember that the covering condition requires that for

any consistent modal formula, there is a possible world satisfying it).

We shall see that the theorems of BCR are exactly the revision valid formulas.

We first show that revision-validity is a conservative extension of the classical

propositional calculus. This property is not entirely evident because of axiom

(CONS).

Lemma 7.4.3 If A ∈ L and A is revision-valid then `PC A.

Proof. It can be easily proven that there exists an AGM belief revision system

〈K, ∗〉 containing a K ∈ K be such that [K] = {C ∈ L :`PC C}.By revision validity, we have that A ∈ [K]. Thus `PC A. 2

Theorem 3 All the theorems of BCR are revision-valid.

Proof. We check that given any revision system 〈K, ∗〉 and any K ∈ K, if A is

a theorem of BCR then A ∈ K. We recall that K is complete with respect to

conditional formulas, i.e. for any A > B ∈ Lu> we have either A > B ∈ K or


¬(A > B) ∈ K. By this fact, when we have to show that F → G ∈ K, where F is

a boolean combination of conditional formulas, we can limit our consideration to

the case when F ∈ K, for, otherwise, we would have ¬F ∈ K and hence trivially

F → G ∈ K by deductive closure. Moreover, always by deductive closure, if we

show that G ∈ K, we have also shown that F → G ∈ K. These steps are tacitly

applied in the following proofs.

• (CLASS) All epistemic states are closed with respect to the propositional

calculus by definition. They therefore include all tautologies of proposi-

tional calculus and they are closed with respect to propositional inference

rules.

• (CONS) Let A ∈ L suppose that 6` ¬A then by (B*5) ⊥ 6∈ K ∗ A. By

(SRT) we have ¬(A > ⊥) ∈ K.

• (ID) By (B*2) A ∈ K ∗ A; thus by (SRT) A > A ∈ K.

• (DT) Let A∧C > B ∈ K, with B ∈ L; we have B ∈ K ∗ (A∧C) by (SRT).

Since B ∈ L, we have B ∈ [K∗(A∧C)]. By (B*7), we have B ∈ [(K∗A)+C].

Thus B ∈ (K ∗ A) + C. But this implies that C → B ∈ K ∗ A, so that by

(SRT) A > (C → B) ∈ K.

• (CV) Assume that ¬(A > ¬C) ∧ (A > B) ∈ K, with B ∈ L; then it must

be ¬(A > ¬C) ∈ K and (A > B) ∈ K. Assume that K is consistent, for

otherwise there is nothing to prove. Then A > ¬C 6∈ K. By (SRT) we have

¬C 6∈ K ∗ A and B ∈ K ∗ A. Since ¬C, B ∈ L, we have ¬C 6∈ [K ∗ A] and

B ∈ [K ∗ A]. By (B*8) [(K ∗ A) + C] ⊆ [K ∗ (A ∧ C)]. Since B ∈ [K ∗ A]

we also have B ∈ [(K ∗ A) + C], thus B ∈ [K ∗ (A ∧ C)]. But this implies

B ∈ K ∗ (A ∧ C), thus by (SRT) we have A ∧ C > B ∈ K.

• (BEL) Let A > B ∈ K. By (B*>), we have K = K ∗ >. Thus A > B ∈K ∗ >. By (SRT) > > A > B ∈ K.

• (REFL) Let > > A ∈ K. By (SRT) and (B*>) we have A ∈ K ∗ > = K.

Thus A ∈ K.

• (EUC) Let ¬(A > B) ∈ K. Assume that K is consistent, for otherwise

there is nothing to prove. Then A > B 6∈ K. By (SRT) we have B 6∈ K ∗A.

By (B*>), we have B 6∈ K ∗A∗>. By (SRT) we obtain ¬(> > B) ∈ K ∗A

and by (SRT) again A > ¬(> > B) ∈ K.


• (TRANS) Let A > B ∈ K. By (SRT) B ∈ K∗A. By (B*>), B ∈ K∗A∗>.

Thus, by (SRT) again we have > > B ∈ K ∗ A and A > > > B ∈ K.

• (RCEA) Suppose that ` A ↔ B is revision-valid, where A, B ∈ L. By the

previous lemma we have, `PC A ↔ B. Let A > C ∈ K; by (SRT) we have

C ∈ K ∗A. By (B*6) we have C ∈ K ∗B; thus B > C ∈ K by (SRT). The

other half is symmetrical.

• (RCK) Let A → B be valid and C > A ∈ K. By validity, we have

A → B ∈ K ∗ C. By (SRT) we have A ∈ K ∗ C. By deductive closure, we

have B ∈ K ∗ C, so that C > B ∈ K by (SRT).

2

Corollary 7.4.4 Let A ∈ L, if A is a theorem of BCR then `PC A.

The semantics of BCR is almost identical, a part from CONS, to the se-

mantics of BC. There are nonetheless two differences between BCR models and

BC models. The first difference is due to a limitation of the language Lu>: since

Lu> only contains conditional formulas with propositional (unnested) antecedent,

the selection function f is only defined for propositional formulas ranging over

L. As a difference, the language L> of BC also contains conditional formulas

with conditional antecedents. Therefore, in BC-models the selection function is

defined for all formulas of L>.

Second, as mentioned above, (CONS) entails that for any consistent formulas,

there is a possible world in the model satisfying it. This does not follow from

the axiomatization of BC. For this reason we had to explicitly add the covering

condition to our models (in order to represent all the properties of belief revision).

We now expose BCR semantics very shortly, because of its strong resemblance

to the semantics of BC.

Definition 7.4.5 A BCR-structure M has the form 〈W, f, [[ ]]〉, where W is a

non-empty set, whose element are called possible worlds, f , called the selection

function, is a function of type L × W → 2W , [[ ]] : L> → 2(W ) is a valuation

function satisfying the conditions of definition 5.1.2.

For S ⊆ W , we define

Form(S) = {A ∈ Lu> | S ⊆ [[A]]}.


Therefore, Prop(S) = Form(S) ∩ L.

We assume that the selection function f satisfies the following properties:

(S-ID) f(A,w) ⊆ [[A]];

(S-RCEA) if [[A]] = [[B]] then f(A,w) = f(B, w)

(S-DT) Prop(f(A ∧ C, w)) ⊆ Prop(f(A,w) ∩ [[C]])

(S-CV) f(A,w) ∩ [[C]] 6= ∅ → Prop(f(A,w)) ⊆ Prop(f(A ∧ C,w)).

(S-REFL) w ∈ f(>, w);

(S-TRANS) x ∈ f(A,w) ∧ y ∈ f(>, x) → y ∈ f(A,w);

(S-EUC) x, y ∈ f(A,w) → x ∈ f(>, y);

(S-BEL) w ∈ f(>, y) → f(A,w) = f(A, y);

(S-CONS) For any A ∈ L if 6`PC ¬A, then f(A,w) 6= ∅.

We say that a formula A is true in a BCR-structure M = 〈W, f, [[]]〉 if [[A]] = W .

We say that a formula is BCR-valid if it is true in every BCR-structure. We also

introduce the following notation S |=M A to say that, given a BCR-structure M,

a set of formulas S and a formula A, for all w ∈ M if w ∈ [[B]] for all B ∈ S,

then w ∈ [[A]].

Theorem 7.4.6 (Soundness) If a formula A is a theorem of BCR then is BCR-

valid.

Proof. (Sketch) One easily checks the validity each axiom and shows that rules

(RCEA) and (RCK) preserve BCR-validity. The proof is similar to BC. 2

Theorem 7.4.7 (Completeness) If A is BCR-valid then it is a theorem of

BCR.

Proof. We fix a language Lu> and we build up a canonical model M = 〈W, f, [[ ]]〉

for Lu>, such that for every Lu

>-formula A we have

A is a theorem of BCR iff [[A]] = W .


The construction of the canonical model is very similar to the construction of the

canonical model for the logic BC(5.3.1). Namely, We define M = 〈W, f, [[]]〉, as

follows

W = {X ⊆ Lu> | X is maximally consistent},

f(B,X) = {Y ∈ W | {C ∈ Lu> | B > C ∈ X} ⊆ Y },

[[p]] = {X ∈ W | p ∈ X}.

One can prove the following facts.

Fact 1By the properties of maximal consistent sets, A is a theorem of BCR iff

A ∈ X for every maximal consistent set X.

Fact 2 for every formula A ∈ Lu> and X ∈ W , A ∈ X iff X ∈ [[A]].

Fact 3 the structure M satisfies all conditions of definition 7.4.5: the proof that

the canonical model satisfies the semantic properties of BCR is similar to the

proof made in chapter 5 for the completeness of BC, except for (CONS). So,

we show that the canonical model satisfies S − CONS:

• (S-CONS) let 6`PC ¬A and let X ∈ W , By (CONS) we have ¬(A > ⊥) ∈ X.

Thus A > ⊥ 6∈ X. We can conclude that f(A,X) 6⊆ [[⊥]]. This means that

f(A,X) 6= ∅.

We have shown that M is a BCR structure. The theorem now follows imme-

diately from Fact 1 and Fact 2: if A is BCR valid, then in particular it is valid in

M , i.e. [[A]] = W ; by Fact 2, we have A ∈ X for every maximal consistent set.

By Fact 1, we conclude that A is a theorem of BCR. 2

We can establish a direct relationship between revision systems and BCR

models. They are essentially the same thing. Differently from what we have

done for the soundness and completeness result, we will expose the representa-

tion theorem below in more details. The reason for this is that it is stronger than

the representation theorem linking the logic BC and belief revision systems. This

is due to the fact that here we deal with conditional belief revision systems, con-

taining epistemic states, rather than with belief sets. We have already explained

in chapter 6 that having richer knowledge structures allows us to establish a more

stringent relation between belief revision systems and conditional logic models.

Indeed, whereas before to different equivalence classes correspond different belief

sets, now every equivalence class determines a different epistemic state.


The representation theorem that we now present resembles the representation

theorem of chapter 6 linking semantic models and iterated belief revision systems.

With a difference, that here we do not have to explicitly require the covering

condition.

The representation theorem then shows a precise correspondence between

revision systems and models of this logic. As a by product we obtain the converse

of the previous theorem, that is to say that only BCR theorems are valid with

respect to conditional belief revision systems.

Consider a structure 〈K, ∗〉, where K is a set of sets of Lu>-formulas and ∗ is

a mapping K ∗ L → K. Then we have:

Theorem 7.4.8 (Representation Theorem) 〈K, ∗〉 is a revision system if and

only if there is a BCR model M = (W, f, [[ ]]), such that for all consistent K ∈ K

there is w ∈ W for which it holds

K = Form(f(>, w)) and K ∗ A = Form(f(A,w)).

Proof. (⇒) Let 〈K, ∗〉 be a revision system, we define a BCR-structure M∗ =

〈W, f, [[]]〉 as follows:

W = {(K, w) : w ∈ 2Prop, K ∈ K and w |= [K]};CK = {(K ′, w) ∈ W : K ′ = K}[[p]] = {(K, w) ∈ W : w |=PC p} for all propositional variables p ∈ L;

f(A, (K, w)) = CK∗A

We observe that if K is consistent, K = K ∗>, whence CK = CK∗>. This implies

f(>, (K,w)) = CK∗> = CK . By making use of the properties of the revision

operator ∗, we can show that M∗ is a BCR-structure. Let K ∈ K be consistent,

so that there exists (K,w) ∈ W . We then prove that

K = Form(f(>, (K, w)) and K ∗ A = Form(f(A, (K, w)).

We first prove the property for formulas of a subset L∗ of of Lu> defined as

follows:

if A ∈ L then A ∈ L∗,if A ∈ L and B ∈ L∗ then A > B ∈ L∗ and ¬(A > B) ∈ L∗.


We prove the above property by induction on the structure of A ∈ L∗. Let A ∈ Land (K, w) ∈ W , first obseve that (K, w) |= A iff w |= A. Now given A ∈ Lsuppose A ∈ K, then A ∈ [K]. Let (K, w′) ∈ f(>, (K,w)) = CK , we have

w′ |= [K], thus w′ |= A, whence (K, w′) |= A. If A 6∈ K then [K] ∪ {¬A} is

propositionally consistent, let w′ such that w′ |= [K]∪{¬A}. We have (K, w′) 6|=A and (K,w′) ∈ CK = f(>, (K, w)).

For the induction step let A = B > C, where B ∈ L and C ∈ L∗. We

have B > C ∈ K iff C ∈ K ∗ B. By the induction hypothesis, this holds iff

C ∈ Form(f(>, (K ∗ B, w1)) for some w1 |= [K ∗ B]. We hence have C ∈Form(f(>, (K ∗ B, w1)) iff C ∈ Form(CK∗B∗>) iff C ∈ Form(CK∗B) iff C ∈Form(f(B, (K, w)) for any w |= [K]. But this holds iff (K, w) |= B > C iff

B > C ∈ Form(f(>, (K,w)), by the semantic properties.

The case of A = ¬(B > C) is reduced to the previous one as K and CK =

f(>, (K, w1)) are complete with respect to conditional formulas.

We have shown that for A ∈ L∗ we have A ∈ K iff A ∈ Form(f(>, (K,w)).

To extend the property to Lu>, we observe that every Lu

>-formula A is equiv-

alent to a set of formulas∨

Ci where Ci ∈ L∗. To see this observe that the

following equivalences are valid:

A > (B ∧ C) ≡ (A > B) ∧ (A > C) and

A > (B ∨ (¬)(C > D)) ≡ (A > B) ∨ (A > (¬)(C > D))

These equivalences can be used to move out conjunction and disjunction on

conditionals. It is easy to extend the property to disjunctions of the above

form. Let C =∨

i Ci where Ci ∈ L∗. We can write C as D ∨ D′ where

D ∈ L and D′ ∈ Lu> − L. Suppose that C ∈ K, but C 6∈ Form(f(>, (K, w)),

thus we have D 6∈ Form(f(>, (K, w)) and D′ 6∈ Form(f(>, (K,w)). Since

Form(f(>, (K, w)) is complete on conditionals, and D′ is a disjunction of con-

ditionals, we have ¬D′ ∈ Form(f(>, (K,w)). By the previous result on L∗formulas, we obtain that D 6∈ K and ¬D′ ∈ K. But since C = D ∨D′ ∈ K and

K is deductively closed, it must be D ∈ K and we get a contradiction. The other

direction is exactly analogous.

For the other identity, we have K∗A = Form(f(>, (K∗A,w′)), for some w′ |=[K ∗ A]. But Form(f(>, (K ∗ A,w′))) = Form(CK∗A) = Form(f(A, (K,w)).

(⇐) Given a model M = (W, f, [[ ]]), let K⊥ = Lu> and

KM = {Form(f(>, w)) : w ∈ W} ∪ {K⊥}


K ∗M A = Form(f(A,w)) for K = f(>, w).

We first observe that ∗ is a function: if Form(f(>, w)) = Form(f(>, w′)) then

also Form(f(A,w)) = Form(f(A,w′)) (because for any w, w |= A > B iff

w |= > > A > B). We then check that 〈KM , ∗M〉 satisfies all the revision postu-

lates including (SRT). Observe that for any S ⊆ W , it holds CnPC(Form(S) ∪{A}) = Form(S ∩ [[A]]). Thus, given K = Form(f(>, w)) and a formula A, we

have K + A = Form(f(>, w) ∩ [[A]]). Moreover [K] = Prop(f(>, w)). Clearly

Form(f(>, w)) is an epistemic state. Let K = f(>, w).

• (B*2) A ∈ K ∗M A.

We have w |= A > A. Thus A ∈ Form(f(A,w)) = K ∗M A.

• (B*3) [K ∗M A] ⊆ [K + A].

We have [K ∗M A] = Prop(f(A,w)) ⊆ Prop(f(>, w) ∩ [A]) = [K + A] by

(S-DT) as A ≡ A ∧ >.

• (B*4) If ¬A 6∈ [K] then [K + A] ⊆ [K ∗M A].

Let ¬A 6∈ K = Form(f(>, w)), then w 6|= > > ¬A, so that f(>, w) ∩[[A]] 6= ∅. By (S-CV), we have Prop(f(>, w)) ⊆ Prop(f(> ∧ A,w)) =

Prop(f(A,w)). Thus also [K+A] = Prop(f(>, w)∩[[A]]) ⊆ Prop(f(A,w)) =

[K ∗M A].

• (B*5) K ∗M A `PC⊥ only if `PC ¬A.

If 6`PC ¬A, we have f(A,w) 6= ∅. Thus ⊥ 6∈ Form(f(A,w)) = K ∗M A.

• (B*6) if A ≡ B, then K ∗M A = K ∗M B.

This follows from (S-RCEA).

• (B*7) [K ∗M (A ∧B)] ⊆ [(K ∗M A) + B].

We have [K ∗M (A ∧ B]] = Prop(f(A ∧ B, w)) ⊆ Prop(f(A, w) ∩ [B]) =

[(K ∗M A) + B] by (S-DT).

• (B*8) if ¬B 6∈ [K ∗ A], then [(K ∗M A) + B] ⊆ [K ∗M (A ∧B)].

If ¬B 6∈ [K ∗M A], with B ∈ L, then ¬B 6∈ K∗M = Form(f(A,w)).

Thus w 6|= A > ¬B, whence f(A,w) ∩ [[B]] 6= ∅. But then by (S-CV) we

have: [(K ∗M A) + B] = Prop(f(A,w) ∩ [[B]]) ⊆ Prop(f(A ∧ B, w)) =

[K ∗M (A ∧B)].


• (B* >) for any K consistent, K ∗M > = K.

It follows by definition of ∗M .

• (SRT) if B ∈ K ∗M A then A > B ∈ K and if B 6∈ K ∗M A then

¬(A > B) ∈ K.

We have A > B ∈ K = Form(f(>, w)) iff w |= > > A > B iff w |= A > B

iff B ∈ Form(A,w) = K ∗M A. The other direction follows then by the

fact that f(>, w) is complete on conditionals.

2

The direction (⇐) of the previous theorem proves also the converse of theorem

3.

Theorem 7.4.9 If A is revision-valid then A is BCR-valid, whence a theorem

of BCR.

Proof. Let A be revision-valid. Let M be a BCR strutcture, then we consider

the revision system 〈KM , ∗M〉 obtained from M as shown in the (⇐)-part of the

previous theorem. We have that A ∈ Form(f(>, w)) for every w ∈ W , but this

implies that [[A]] = W (as w ∈ f(>, w)). Thus A is true in M . We have shown

that A is BCR-valid. 2

Corollary 7.4.10 The following are equivalent.

1. for every revision system 〈K, ∗〉, for every K ∈ K if A ∈ K then B ∈ K.

2. `BCR (> > A) → (> > B).


Chapter 8

Conclusions and Future Work

8.1 Conclusions and Future work

In this thesis we have proposed a conditional logic, called BC, to represent belief

revision. The relation between the logic BC and belief revision is established

by a Representation Theorem that shows how to each belief revision system

corresponds a BC-model, and to each BC−model corresponds a belief revision

system.

We have shown in chapter 6 that the logic BC can be extended in order to

represent iterated belief revision. The relation between the resulting conditional

logic, called IBC, and iterated belief revision is established by a Representation

Theorem. The Theorem shows that in this case the relation is tighter than the

relation established in chapter 5 between the logic BC and simple belief revision.

Some corollaries of this theorem show how the conditional logic IBC allows us

to investigate some interesting properties of iterated belief revision systems.

Finally, we have shown how the problem of triviality raised by the Ramsey

Test [15] can be solved by simply weakening, in an intuitive way, the rationality

postulates that rule belief revision. We have shown that a logic very similar to

the logic BC can be derived from the weakened rationality postulates and strong

Ramsey Test.

8.1.1 Related Works

The approach followed in chapters 5 and 6 has some similarities with the ap-

proaches adopted by Levi [31, 32], and by Friedman and Halpern [10] exposed in

121

122 CHAPTER 8. CONCLUSIONS AND FUTURE WORK

chapter 4.

As we have seen in chapter 4, Levi maintained that, in order to avoid the

triviality result, the Ramsey Test should be weakened and formulated as A > B is

“accepted ” in K if and only if B ∈ K∗A, where the notion of “acceptability” of a

conditional A > B in K is a weaker condition than “A > B ∈ K”. Our approach

follows the line of research indicated by Levi [31]. Indeed, in our proposal, the

relation established by the Representation Theorem between the logic BC and

belief revision systems is weaker than the relation established by the Ramsey

Test, for we interpret the acceptability of a conditional A > B in a belief set K

as: A > B is true in a world associated to K.

As far as Friedman and Halpern’s proposal [10] is concerned, recall from

chapter 4 that, similarly to what we have done, they propose a conditional logic

to represent belief revision.

As a difference with our approach, and with standard conditional logics, Fried-

man and Halpern consider epistemic states, rather than possible worlds, as their

primitive semantic objects. Furthermore, they put some severe restrictions on

the conditional language they define, and explicitly introduce an extra belief op-

erator. In contrast, we do not put any syntactic restriction on the language (for

it is sufficient to put some restrictions on the axioms), and we do not need to

introduce an extra belief operator, for we make use of the conditional operator

itself in order to associate a world with a belief set, and we impose conditions on

the selection function in such a way that the value of the selection function in a

world is determined by the belief set associated to that world.

An approach related to the ours is also the approach presented by Katsuno

and Satoh in [29], where they present a unifying view of belief revision, condi-

tional logic and nonmonotonic reasoning, based on the notion of minimality. More

precisely, they introduce ordered structures and families of ordered structures as

a common ingredient of belief revision, conditional logic and nonmonotonic rea-

soning. Ordered structures are triples (W,≤, V ) containing a set W of possible

worlds, a preorder relation ≤, and a valuation function V . They provide a se-

mantic model to evaluate conditional formulas that contain no nesting of the

conditional operator >. Families of ordered structures are defined as collections

of ordered structures, and their axiomatization corresponds to well known condi-

tional logics, such as VW, VC, and SS. While families of ordered structures are

used to give a semantic characterization of belief update, ordered structures are

used to give a semantic characterization of belief revision. In particular, Katsuno

8.1. CONCLUSIONS AND FUTURE WORK 123

and Satoh show that, given a revision operator ∗, for each belief set K there is an

ordered structure OK such that the formulas in K are true in all minimal worlds

of OK (written OK |= K), and

K ∗ A = {B : OK |= A > B}.

Since an ordered model OK contains a single ordering relation ≤, it can only rep-

resent one single belief set K and its revisions. Moreover, since ordered structures

do not handle nested conditionals, iterated belief revision cannot be captured in

this formalization. As a difference, we are able to represent different belief sets

and their revisions within a single structure, by associating worlds with belief

sets.

Finally, also the approach adopted in chapter 7 has some similarities with

Levi’s approach [32]. Indeed, in chapter 7 we have shown that the triviality re-

sult can be avoided by weakening the rationality postulates in such a way that

they only apply to non-conditional formulas. Also in Levi’s proposal the ratio-

nality postulates for belief revision do not apply to conditional formulas, for the

simple reason that conditional beliefs do not belong to belief sets. As a differ-

ence with Levi’s approach, we only restrict some of the postulates and we allow

the occurrence of conditionals in epistemic states, including iterated conditionals

whose acceptance (whence meaning) is not defined in Levi’s framework.

8.1.2 Future Work

The results obtained in the thesis can be extended in several directions.

We argue that our logic BC is not only well suited for modeling belief revision,

but that it can also provide a suitable framework in which to capture other forms

of belief change, as belief update. To this regard, recall from chapter 4 that

Grahne [23, 24] has proposed a conditional logic which combines update and

counterfactual conditionals. Our logic for revision is related to the logic proposed

by Grahne in the case we make the conditional > > A equivalent to A. In fact,

when the axiom A → (> > A) is added to our axiomatization, from (REFL)

we obtain the equivalence A ↔ (> > A), and the belief set associated with a

world becomes equal to the set of formulas true in that world. In such a case, it

can be easily seen that all the axioms of Grahne’s logic can be derived from ours

(provided we lift the restriction that some formulas have to range over L rather

than over L>).

124 CHAPTER 8. CONCLUSIONS AND FUTURE WORK

Obviously, to deal with belief update some of the axioms of BC, which are

tailored for belief revision, should be dropped, since they are not required for

belief update.

It would be interesting to study how the logic BC could be generalized in

order to deal also with belief update. Thus, we would like to study a general

conditional logic for belief change that could be specified to represent both belief

revision and belief update.

Finally, we have seen in chapter 6 how the relation established in the thesis

between conditional logic and belief revision could be used to prove interesting

properties of belief revision operators, or classes of belief change operators. We

intend to pursue the research in this direction, by studying some proof methods

for our conditional logics that allow us to derive some properties of belief change

systems, or classes of belief change systems.

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