Interrogative Belief Revision Based on Epistemic Strategies

Studia Logica (2012) 100: 453–479DOI: 10.1007/s11225-012-9410-2 © Springer 2012

Sebastian Enqvist Interrogative Belief RevisionBased on EpistemicStrategies

Abstract. I develop a dynamic logic for reasoning about “interrogative belief revision”,

a new branch of belief revision theory that has been developed in a small number of papers,

beginning with E. J. Olsson and D. Westlund’s paper “On the role of the research agenda

in epistemic change” [12]. In interrogative belief revision, epistemic states are taken to

include a research agenda, consisting of questions the agent seeks to answer. I present

a logic for revision of such epistemic states based on the notion of an epistemic strategy,

a stable plan of action that determines changes in the agent’s research agenda. This idea

is a further development of an idea put forward in [6], that changes in the research agenda

of an agent should be determined by stable, “long term” research interests. I provide

complete axioms and a decidability result for the logic.

Keywords: Belief revision, Research agenda, Dynamic logic, Questions.

Introduction

“Interrogative belief revision” is a relatively new branch of belief revisiontheory that has been developed in a small number of papers, beginning withE. J. Olsson and D. Westlund’s paper “On the role of the research agendain epistemic change” [12]. Later, the theory was developed within modallogic, in [5]. What separates interrogative belief revision from other branchesof belief revision is that epistemic states are taken to include, besides thebeliefs of an agent plus some representation of their “deep structure” (usuallyan entrenchment order, see [8]), a representation of the agent’s researchinterests, or his research agenda.

For Olsson and Westlund, an epistemic state is a triple 〈K,�, A〉 whereK is a belief set in the usual sense, i.e. a logically closed set of sentences ina propositional language (with the Boolean operators and a classical conse-quence relation), � is an entrenchment order and A is a socalled K-agenda,defined as follows.

Given a belief set K, a finite and non-empty set of sentences {α1, . . . , αn}is said to be a K-question if

Presented by Heinrich Wansing; Received March 2, 2010; Revised January 5, 2011

454 S. Enqvist

1. α1 ∨ . . . ∨ αn ∈ K

2. for i,j such that 1 ≤ i < j ≤ n, we have ¬ (αi ∧ αj) ∈ K

3. for all i such that 1 ≤ i ≤ n we have ¬αi /∈ K

The idea here is to represent questions by the set of their potential an-swers. Conditions 1 – 3 give constraints on when we are allowed to askcertain questions. Condition 1 says that, if we ask a question, then we arecommitted to believing that at least one of its answers is true. The secondcondition states that we are committed to believing that the answers to aquestion are mutually exclusive, in the sense that no two of them can betrue together. The third condition states that all the possible answers tothe question must be regarded as open possibilities, in the sense of beingconsistent with the agent’s belief set. A K-agenda then is simply a set ofK-questions.

In this paper, I will be using DEL-style dynamic logic as a formal frame-work for interrogative belief revision. The system will be an extension of thelogic for belief revision developed by J. van Benthem in [19] (I will assumethat the reader is familiar with the basic concepts of belief revision theory,as well as with standard techniques in modal logic). My main goal is todevelop formally an idea presented in [6] in order to deal with contraction,although I am here rather concerned with the general context of revision.The idea is that changes in the research agenda of an agent, as an effectof changes in the beliefs of the agent, should be determined by some “deepstructure” of the agenda, much like changes in belief are often taken to bedetermined by an entrenchment order. In [6], this deep structure is taken tobe the stable, long term research interests of the agent. Here, we will takethis interpretation a bit further, and see the deep structure of the agendaas a “global” plan of action for reaching the long term goals of inquiry —we will call it an epistemic strategy. Changes in the agent’s “local” researchagenda, in response to changes of belief, are then determined by this globalepistemic strategy, which is held fixed throughout inquiry.

1. Preparation

1.1. Representing questions

In order to reason about interrogative belief revision in a modal object lan-guage we need to be able to express beliefs of an agent, belief changes (i.e.revisions) and questions on the agent’s agenda. There exists a variety ofmodal languages for reasoning about belief change in the literature. In [5]

Interrogative Belief Revision Based on Epistemic Strategies 455

two such languages are extended with means to reason about questions onthe agenda, in order to capture interrogative belief revision. One is an exten-sion of K. Segerberg’s system “DDL” (Dynamic Doxastic Logic, see [16, 17]),and the other a similar extension of G. Bonanno’s temporal logic of beliefchange in [3].

The way questions are expressed in these languages, which is essentiallythe same method that we shall use here, is based on J. Hintikka’s theory ofquestions, in which questions are thought of as pairs consisting of a desider-atum and a presupposition (see [9]). The desideratum of a question is thedemand for information that we express with the question, a specificationof what beliefs we would need to have to regard the question as answered.In the simplest case, a question can be taken to be a finite set of potentialanswers {α1, . . . , αn}, as in Olsson & Westlund’s model. Hintikka calls thisa propositional question. The desideratum of such a question is simply thatwe come to believe one of the potential answers α1, . . . , αn. The presupposi-tion of a question is a specification of what beliefs we must have to ask thequestion meaningfully. In this case, the presupposition is that we believe inthe disjunction of the answers α1, . . . , αn. The two other preconditions forquestions required by Olsson & Westlund — that answers to questions areconsidered as mutually exclusive, and that answers to questions are consis-tent with what we believe — are not in general required for Hintikka, norwill they be required here.

With this theory in the background, one can think of the agenda as a setof desiderata, and express desiderata by introducing a one-place operator“des” such that desα should be read “α is a desideratum on the agent’sagenda”. A propositional question {α1, . . . , αn} can then be expressed as

des(Bα1 ∨ . . . ∨ Bαn),

where “Bα” should be read “the agent believes α”. This sentence then saysthat it is a desideratum for the agent to believe one of α1, . . . , αn. We shallget back to presuppositions a bit later, in section 2.3.

A point of clarification: the sentence desα does not mean that the agentutters the question with desideratum α, but that it is on his research agenda.Just as I can have a belief without stating that I have it, I can be deter-mined to resolve some research question without uttering the question. Thequestions on the agent’s research agenda should be thought of as the setof questions he is currently interested in, or he is currently determined toresolve, whether he has uttered them or not.

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1.2. Introducing epistemic strategies

A central problem of interrogative belief revision is to model how the agendachanges in response to changes of belief. The problem is raised in connectionwith expansion in [12], and again for contraction in [6]. We shall get backto the treatment in [6] shortly.

In belief revision theory, revision of beliefs is usually taken to be deter-mined by some “deep structure” of the agent’s belief state (entrenchment,choice functions, etc.). In van Benthem’s logic for belief revision, the deepstructure of the agent’s belief state is expressed in terms of conditional be-liefs. A two-place operator is introduced for this purpose, such that B(α | β)is to be read “the agent believes α conditionally upon β”. The new beliefsof an agent after revision will then be obtained by “conditionalization”, i.e.they will be the beliefs conditional on the input proposition prior to revi-sion. Non-conditional beliefs can be defined from this operator by lettingBα abbreviate B(α | T), where T is an arbitrary tautology. Semantically,conditional beliefs can be represented by a plausibility order or a spheresystem — we shall get back to this soon.

Here, we shall use the same approach for agenda change: changes in theresearch agenda will be taken to be determined by some deep structure. Weshall treat the deep structure of agendas similarly to van Benthem’s approachwith conditional beliefs, by introducing conditional desiderata. We introduceinto the language a binary operator such that des(α | β) should be read “αis a desideratum conditional on β”. Similar to the case of beliefs, we obtainthe one-place operator by letting desα abbreviate des(α | T) (so the readingof desα should be “α is a desideratum on the agent’s agenda”). The ideais that the new agenda after revision by a proposition P should, like in thecase of beliefs, be obtained by conditionalization, i.e. the new agenda shouldconsist of the desiderata conditional on P prior to revision.

These conditional desiderata need to be interpreted. My proposal for aninterpretation is a further development of an idea from [6]. In this paper,it is suggested that we should think of the deep structure of the agenda ascorresponding to the long term research interests of the agent. This idea hassome connection with J.Hintikka’s socalled interrogative model of inquiry, inwhich one distinguishes between principal and operational questions (see [9]).The principal questions are the end goals of some inquiry, and operationalquestions are questions we ask to eventually get answers to the principalquestions. The “long term research interests” of [6] then correspond roughlyto Hintikka’s “principal questions”.


My suggestion for an interpretation of the conditional desiderata is,rather, that they correspond to a strategy or plan that the agent followsin order to eventually obtain answers to the long term research questions,or principal questions. This “global” strategy is kept fixed while the “lo-cal” research agenda changes, in response to new information, in accordancewith the strategy. We shall call it the epistemic strategy of an agent. (Thisfurther strengthens the connection with the interrogative model of inquiry,which is formulated in game theoretic terms and in which questioning strate-gies play a central role; for some recent work on the connection between theinterrogative model of inquiry and belief revision, see [7]).

Let us take a simple example to illustrate this concept. Consider aninvestigator who has two main suspects in a murder case, John and Paul. Hislong term research interests simply consist of the question “which one of Johnand Paul killed the victim?”. In order to answer this question, accordingto my suggestion he follows a fixed epistemic strategy that determines howhis research agenda changes throughout the inquiry, and at each step ofthe inquiry, the strategy gives rise to a set of conditional questions. Forinstance, the investigator may initially suspect that the murderer was lefthanded, and so begins by seeking further evidence for this. His strategyto resolve the principal question, at the initial stage of inquiry, gives riseto the question “was the killer left-handed?”, together with the conditionalquestions of whether John is left-handed and whether Paul is left handed,which he will proceed to investigate given that he learns the killer was indeedleft-handed.

2. The static base logic IBR

2.1. A logic of conditional beliefs

Following the usual DEL method, we shall start by constructing a static baselogic and then provide a dynamic extension. Before introducing desideratawe define a language used to reason about the conditional beliefs of an agentas follows:

LBR : p | ¬α | α ∨ α | B(α |α) | Aα

A is the global necessity operator, a useful operator that allows us to quantifyover all the worlds of a model1. A unary belief operator is defined from

1As pointed out to me by an anonymous referee this operator is strictly speakingredundant, since it will be definable as

Aα =df. B(⊥| ¬α)

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the conditional belief operator by Bα =df. B(α | T), T being any tautology.Throughout the paper, by a purely Boolean formula I mean a formula thatdoes not contain any occurrences of modal operators. To construct thesemantics for the language LBR, we begin with a definition.

Definition 2.1. A BR-structure is a sequence

〈W, {Su}u∈ W , V 〉

where, for each u ∈ W , Su is a set of subsets of W , called a sphere system,such that W ∈ Su and Su is nested w.r.t. set inclusion (i.e. X ⊆ Y or Y ⊆ Xfor all X, Y ∈ Su). V is a valuation in the usual sense.

The sphere systems represent the belief state of the agent at each world,and can be thought of as representing a plausibility order over worlds. Forany X ⊆ W we denote by bel(Su, X) the set M ∩ X, where M is thesmallest member of Su that intersects X if such a set exists, and we setbel(Su, X) = ∅ otherwise. The smallest member of Su, if it exists, will bedenoted by min(Su). Of course then min(Su) = bel(Su,W ). Subsets of Wwill sometimes be referred to as propositions.

Given a BR-structure M and a world u in its universe, we shall writeM, u � α to say that α is true at u in M, and we let ‖α‖M denote the truth-set of α in M in the usual sense, dropping the index when no confusionis likely to occur. With this notation, we define truth conditions for LBR-formulas inductively as follows:

1. for a propositional variable p, M, u � p iff u ∈ V (p)

2. standard clauses for ¬,∨3. M, u � Aα iff M, v � α for all v ∈ W

4. M, u � B(α | β) iff bel(Su, ‖β‖M) ⊆ ‖α‖M

Informally, B(α | β) is true if and only if α is true at every member of theset of most plausible β-worlds. This interpretation warrants the followingdefinition:

Definition 2.2. A BR-structure M will be called a BR-model if, for anyformula α, ‖α‖M = ∅ implies bel(Su, ‖α‖M) = ∅.

For the sake of proving completeness of the static base logic I prefer to work with A as aprimitive operator of the language, but this is a matter of taste; in principle we could dowithout it.


In particular, in a model, the set min(Su) always exists. With validityin a BR-structure defined in the usual way, the logic BR is defined as theset of LBR-sentences that are valid in all BR-models, rather than all BR-structures. We thus write �BR α to say that α is valid in every BR-model.We shall sometimes speak of a pair M, u consisting of a model together witha world from its universe W as a pointed model.

We immediately turn to the task of providing a complete proof systemfor the logic BR of conditional beliefs. This will then be used as a basis fora proof system for the static base logic IBR in which we can reason aboutthe agenda. Although the logic BR is very similar to the logic presented byO. Board in [4], I will use a somewhat different completeness proof here.

To axiomatize BR, we take as axiom schemata the following:

BR1: All propositional tautologies.

BR2: S5 schemata for A.

BR3: B(α | α)

BR4: B(⊥| α) → A¬α

BR5: B(α → β | χ) → (B(α | χ) → B(β | χ))

BR6: ¬B(¬α | β) → (B(χ | α ∧ β) ↔ B(α → χ | β))

BR7: Aα → B(α | β)

BR8: A(α ↔ β) → (B(χ | α) ↔ B(χ | β))

As our two rules of inference, we take Modus Ponens and Necessitationfor A; I leave it to the reader to check that these axioms and rules are soundfor BR.

Theorem 2.3. The axiom schemata BR1 −BR8, with Modus Ponens andNecessitation for A as rules, provide a strongly complete proof system forBR.

The proof of this theorem proceeds by constructing a canonical modelwith a non-standard interpretation of the universal modality A, and thenform a generated submodel in which A has its intended interpretation (astandard way of handling the global modality). The axioms I have presentedare similar to some of the axioms used by D. Lewis to obtain a completeproof system for the system VC of counterfactual conditionals (see [11]), andthe most important parts of the completeness argument are very similar toparts of Lewis’s proof. For this reason I shall only sketch the proof, leavingout most of the details.

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Let WC denote the set of maximal consistent sets of sentences in LBR,and for any formula α let α denote the members of WC that contain α.Given S ∈ WC we define, for any sentence α the set

σ(S, α) =df. {T ∈ WC : ∀β[B(β | α) ∈ S ⇒ β ∈ T ]}We then define S

CS as the collection of all subsets X of WC for which it holds

that if X ∩ α = ∅ then σ(S, α) ⊆ X, and that if X = WC and T ∈ X,then there is some α such that T ∈ σ(S, α). A relation RC

A is defined overWC by setting SRC

AT if and only if, for each sentence α, Aα ∈ S impliesα ∈ T . Finally, the valuation V C is defined by setting V C(p) = p for eachpropositional variable p. The canonical model MC is then defined as thequadruple

〈WC , {SCS }S∈W C , RC

A, V C〉Evaluation clauses are as usual, except that A has the non-standard clause:

MC , S � Aα iff MC , T � α for all T such that SRCAT

One can then prove the standard “truth lemma” for this canonical model(i.e. MC , S � α iff α ∈ S).

Given T ∈ WC we define the generated submodel

MT = 〈W T , {STS}S∈W T , V T 〉

of MC by setting:

• W T = {S ∈ WC : TRCAS}

• STS = {X ∈ S

CS : X ⊆ W T } ∪ {W T }

• V T (p) = V C(p) ∩ W T

If we let A have its standard interpretation in this model, we can showthat MT is indeed a BR-model, and that the pointed model MT , T satisfiesprecisely the same formulas as MC , T . This is shown using the followingresult:

Lemma 2.1. Suppose S ∈ W T and β ∩ W T = ∅. Then M ⊆ W T , where Mis the smallest member of S

CS intersecting β.

Proof. Let X be the union of all sets of the form σ(S, δ) such that β ⊆ δ.It can be shown that X ∈ S

CS , and BR3 gives that β intersects X. So to

show that M ⊆ W T it suffices to show X ⊆ W T . By the definition of X wecan show this using the axiom BR7.

Hence we have, for every consistent set Γ of sentences, a model witha world where each member of Γ is true, which proves the completenesstheorem.


2.2. Base language for epistemic states with research agendas

We now extend LBR to form the language LIBR that has the capacity tospeak about the agent’s epistemic strategy as well as his conditional beliefs.For this purpose, we add the binary operator des with the restriction that, inan expression of the form des(α | β), α must be an LBR-formula. Formally,the BNF definition of LIBR is as follows, where χ is any LBR-formula:

LIBR : p | ¬α | α ∨ α | B(α |α) | des(χ | α) | Aα

As mentioned earlier, we shall let desα abbreviate des(α | T). Thus, desαmeans that α is a desideratum on the agent’s agenda, while des(α | β) meansthat α is a desideratum conditional on β.

Let us illustrate how the language works by going back to the simpleexample of the investigator and the murder case. Let us describe the sce-nario using the formal language LIBR; first, on the actual agenda of theinvestigator we find the question “did John or Paul kill the victim?”. Ifwe denote the proposition “John killed the victim” with j and “Paul killedthe victim” with p, then the presence of the aforementioned question on theinvestigator’s agenda is expressed by the sentence

des(Bj ∨ Bp)

Also on his agenda is the “instrumental”‘question “is the killer left-handed?”.If we let lk stand for the proposition “the killer is left-handed”, we have

des(Blk ∨ B¬lk)

We also have two merely conditional questions that are involved in the in-vestigator’s epistemic strategy: given that he finds out that the killer isleft-handed, he will add to his agenda two questions, one asking whetherJohn is left-handed and one asking whether Paul is left-handed. Lettinglj stand for “John is left-handed” and lp for “Paul is left-handed”, we canexpress the situation with the following sentence:

des(Blj ∨ B¬lj | lk) ∧ des(Blp ∨ B¬lp | lk)

Formal semantics for LIBR is defined as follows. An IBR-structure is aquadruple

〈W, {Su}u∈W , {AGu}u∈W , V 〉where the Su are as before, and for each u ∈ W , AGu is a function frompropositions (subsets of W ) to sets of LBR-formulas. Informally, the set

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AGu(X) is the set of desiderata conditional on the proposition X. V is,again, a valuation in the usual sense.

The function AGu in a model can be said to represent the epistemicstrategy of the agent, in the sense in which we used the term in section 1.2.In fact, the function AGu is formally reminiscent of a strategy in the sense theterm is used in the theory of extensive games (see [13]): it is a function frompossible events (input propositions) to actions of the agent, in the form of aset of questions asked by the agent. We shall let AGu abbreviate AGu(W )— intuitively, this is the “local” agenda of the agent at world u. Evaluationclauses for LIBR-formulas in an IBR-structure M are exactly as before,except that we add the following evaluation clause for the des-operator:

M, u � des(α | β) iff α ∈ AGu(‖β‖M)

To define the static base logic IBR, which will serve as the logic of epistemicstates extended with epistemic strategies, we still have to do a bit more work.

2.3. Two semantic postulates

As in the case of BR we distinguish between IBR-structures and IBR-models. One of the properties we shall require of IBR-models is meant toensure that the agent always believes the presuppositions of all the questionson his agenda. We have now come to a stage where it is necessary to treatpresuppositions formally.

Definition 2.4. Given an LBR-formula α, a presupposition of α is a purelyBoolean formula β such that �BR α → Bβ.

For instance, if p1, . . . , pn are propositional variables, then p1 ∨ . . . ∨ pn

is a presupposition of the formula Bp1 ∨ . . . ∨ Bpn (while, of course, noneof p1, . . . , pn are presuppositions of that formula). The motivation for thisdefinition of presuppositions is that, if Bβ is a logical consequence of thedesideratum α of some question Q, then no matter what answer we receiveto Q we will end up believing β. But I should only ask the question Q if Ithink that I can obtain an answer to it without coming to believe somethingfalse, and since obtaining any answer to Q entails that I come to believe β,this means I should only ask the question Q if I believe β to be true. (Ofcourse, this does not mean that β has to be actually true in order for meto be able to ask the question legitimately — what is required is only thebelief that β is true.)

Definition 2.5. An IBR-structure M is said to be an IBR-model if, forevery LIBR-formula α and every world u the set bel(Su, ‖α‖M) is non-empty


given that ‖α‖M is, and in addition the following two postulates are fulfilledfor all formulas α, β, χ:

[pres]: if α ∈ AGu(‖β‖M) and χ is a presupposition of α then we havebel(Su, ‖β‖M) ⊆ ‖χ‖M.

[func]: if bel(Su, ‖β‖M) = bel(Su, ‖χ‖M) then AGu(‖β‖M) = AGu(‖χ‖M).

The interpretation of [pres] is that, if α is a desideratum conditional on β,then the agent ought to believe any presupposition of α conditionally on β.The second postulate, [func], says that if the agent’s beliefs conditional onβ and χ are the same, then his agenda conditional on β is the same as thatconditional on χ. This means that changes in the agenda are determinedentirely by changes of belief, which is desirable given our interpretation ofthe conditional desiderata.

Remark 1. The postulate [pres] is the reason I have chosen to restrictpresuppositions to be purely Boolean formulas. Nothing has been said hereabout what postulates the defined belief operator should satisfy among thosethat are discussed in the literature, such as the “positive introspection”postulate (Bα → BBα). The reason is that I want to remain agnostic hereconcerning the status of such postulates. On the one hand I do not want toassume them to hold, but on the other hand, it should ideally be possible toextend the logic with these postulates without any direct conflict. However,if we allowed all LBR-formulas to serve as presuppositions of desiderata, thenadding the positive introspection postulate would lead to trouble: say thatwe have M, u � des(Bj ∨ Bp), expressing that the propositional question“was the murder committed by John or Paul?” is on the agenda. Normalityof the defined belief operator B together with positive introspection givesthat Bj∨Bp entails B(Bj∨Bp) 2, thus making Bj∨Bp a presupposition ofitself. The postulate [pres] then entails that M, u � B(Bj∨Bp) — informallyput, asking the propositional question of whether it was John or Paul whocommitted the murder requires that we already believe its desideratum tobe fulfilled!

Thus, if we did not impose the aforementioned restriction on presuppo-sitions, [pres] together with positive introspection would lead to paradoxicalconsequences. Furthermore, the restriction is not just an ad hoc device to

2To derive this: positive introspection gives Bj → BBj and Bp → BBp. We haveBj → Bj ∨ Bp and Bp → Bj ∨ Bp by propositional logic, and normality of B thus givesBBj → B(Bj ∨ Bp) and BBp → B(Bj ∨ Bp). By propositional logic we now get thevalidity of Bj ∨ Bp → B(Bj ∨ Bp), q.e.d.

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get around the paradoxes, but there is a natural explanation for it: the mo-tivation for the notion of a presupposition was that, if obtaining any answerto a question Q will cause us to believe β, then asking the question Q shouldcommit us to the belief that β is true. This is reasonable if β expresses aproposition about the world, which presumably does not change when werevise our beliefs. But this is not the case for sentences that express propo-sitions about our beliefs. For instance, when the detective asks whether itwas John or Paul who murdered the victim, while certainly Bj ∨ Bp willbe true after the question has been answered (no matter which answer isreceived), there is no reason for the detective to believe this sentence to bealready true when he asks the question.

With the notion of an IBR-model in place, we define the logic IBR aswe did with BR before, and we write �IBR α to say that the formula α isvalid in all IBR-models.

2.4. A proof system for IBR

The proof system already provided for BR can easily be extended to a com-plete proof system for IBR. Since presuppositions of questions are defined byreference to the sublogic BR, we will have use of the following proposition:

Proposition 2.1. IBR is a conservative extension of BR.

Proof. Given formulas Γ ∪ {α} ⊆ LBR and a BR-model M, u such thatM, u � Γ, M, u � α, extend M to an IBR-structure M∗ by adding, foreach world v, an agenda function AGv defined by AGv(X) = ∅ for eachproposition X. This structure will trivially satisfy both [pres] and [func].Furthermore, using the fact that for every sentence of the form des(δ | χ)we have ‖des(δ | χ)‖M∗ = ∅ we can show by an easy induction on the lengthof formulas that for every LIBR-formula β there is a LBR-formula β′ suchthat

‖β‖M∗ =∥∥β′∥∥

M

From this in turn follows easily that for each β ∈ LIBR and each world v,‖β‖M∗ = ∅ implies bel(Sv, ‖β‖M∗) = ∅. This shows that M∗ is an IBR-model. Furthermore, it is trivial to show that M∗, u � Γ and M∗, u � α.

With this simple fact in place we proceed to construct the proof system.As axioms we take all axiom schemata for BR plus the following singleaxiom schema (χ ∈ LBR):

Func: (B(α | β) ∧ B(β | α)) → (des(χ | α) ↔ des(χ | β))


As rules of inference, we take Modus Ponens, Necessitation for A and thefollowing rule schema with a restricted scope:

PRES:α → Bβ

des(α | χ) → B(β | χ)where α ∈ LBR and β is purely Boolean

The soundness of the resulting proof system is achieved by the followingproposition:

Proposition 2.2. Func is valid in IBR, and the rule PRES preservesvalidity.

Proof. To see that Func is valid, suppose M, u � B(α | β)∧B(β | α). LetX be the smallest member of Su intersecting ‖β‖, and let Y be the smallestmember intersecting ‖α‖. We have X ∩ ‖β‖ ⊆ ‖α‖ and Y ∩ ‖α‖ ⊆ ‖β‖, soboth X and Y intersect ‖α ∧ β‖; clearly both X and Y must be minimalamong the spheres that intersect ‖α ∧ β‖ and so X = Y since Su is nested.It now easily follows that bel(Su, ‖α‖) = bel(Su, ‖β‖) and so, by [func],AGu(‖α‖) = AGu(‖β‖). From this in turn follows that we haveM, u � des(χ | α) ↔ des(χ | β) for any LBR-formula χ.

To see that PRES preserves validity: suppose that �IBR α → Bβ, whereβ s purely Boolean, and suppose M, u � des(α | χ). By propoposition 2.1we have �BR α → Bβ, so β is a presupposition of α and by [pres] we easilyget M, u � B(β | χ).

We shall now show that the proof system is complete.

Theorem 2.6. The axioms and rules presented above provide a stronglycomplete proof system for IBR.

To prove this theorem we again construct a canonical model

MC = 〈WC , {SCS }S∈W C , {AGC

S }S∈W C , RCA, V C〉

where A has a non-standard interpretation. Here, the set WC is of coursethe set of maximal consistent sets of LIBR-formulas in the sense of the proofsystem described above. Since the details of the proof are easy to fill in, Ionce more only sketch the proof.

The construction of MC is the same as before, except for the new com-ponent AGC

S which is defined, for each maximal consistent set S, by set-ting α ∈ AGC

S (X) iff there is some sentence β such that X = β anddes(α | β) ∈ S. As before all operators have their standard interpreta-tions, except for A which has the same non-standard evaluation clause asbefore. As before, we can obtain a truth lemma for MC :

466 S. Enqvist

Lemma 2.2. For every S ∈ WC and formula α, we have MC , � α iff α ∈ S.

Proof. We prove this by induction. I do only the step for des(α | β).Suppose des(α | β) ∈ S. Then α ∈ AGC

S (β), and the desired result followsdirectly from the induction hypothesis.

Conversely, suppose MC , S � des(α | β). Then α ∈ AGCS (‖β‖MC ),

so by the inductive hypothesis α ∈ AGCS (β). This means that there is

some formula χ such that β = χ and des(α | χ) ∈ S. We can show thatA(β ↔ χ) ∈ S and using BR7 and Func we can derive that des(α | β) ↔des(α | χ) ∈ S, hence des(α | β) ∈ S as required.

Given T ∈ WC the construction of a generated submodel

MT = 〈W T , {STS}S∈W T , {AGT

S}S∈W T , V T 〉where A has its standard interpretation proceeds as before, except that wedefine the new component AGT

S , for S ∈ W T , by setting α ∈ AGTS (X) iff

there is some Y ⊆ WC such that X = Y ∩ W T and α ∈ AGCS (Y ).

Lemma 2.3. For any maximal consistent set of sentences S such that TRCAS

and any formula α, we have MT , S � α iff MC , S � α.

Proof. By induction. Again I do only the case for des(α | β). SupposeMT , S � des(α | β). Then α ∈ AGC

S (‖β‖MC ), so α ∈ AGTS (‖β‖MC ∩W T ) =

AGTS (‖β‖MT ) by the induction hypothesis.Conversely, suppose MT , S � des(α | β). Then there is some formula χ

such that ‖β‖MT = χ ∩ W T and des(α | χ) ∈ S. But using the inductivehypothesis and the truth lemma for MC we now get χ ∩ W T = β ∩ W T

and from this we get A(β ↔ χ) ∈ S, from which we can show using BR7and Func that des(α | β) ↔ des(α | χ) ∈ S, hence des(α | β) ∈ S. Thisshows that α ∈ AGC

S (β), and the desired result follows directly from thetruth lemma.

We clinch the completeness argument by proving the following lemma:

Lemma 2.4. For any maximal consistent set of sentences T , the generatedsubmodel MT satisfies [pres] and [func].

Proof. For [pres], suppose that β is a presupposition of α and that α ∈AGT

S (‖χ‖MT ). The truth lemma for MC gives des(α | χ) ∈ S. By theorem2.3 we have �BR α → Bβ and so, obviously, �IBR α → Bβ. By PRES thisgives

�IBR des(α | χ) → B(β | χ)


so B(β | χ) ∈ S. The truth lemma for MC gives MC , S � B(α | χ)and lemma 2.3 gives MT , S � B(α | χ), from which the desired conclusionobviously follows.

The easy argument for [func] is omitted.

This concludes the proof of theorem 2.3.

3. A dynamic extension of IBR

3.1. Some requirements

We now come to the point where it is time to turn the static base logicIBR into a dynamic logic for revision. The aim of this section is to definea revision operation that captures the idea developed before, that changesin the research agenda due to belief change should be determined by theunderlying epistemic strategy of the agent. There are three main desideratathat we want any acceptable revision operation to satisfy.

First, the revision operation should, when applied to a model, yield a newmodel — this means that the structure obtained by applying the operationto a model should satisfy all the constraints we have imposed on models. Inparticular, the postulates [func] and [pres] should be satisfied. We call thisrequirement categorical matching (the term is well known in belief revision,see for instance [18]). Second, the deep structure of the agenda in the initialmodel should determine the new “local” agenda in the expected manner,i.e. the agenda of the agent after revision by χ should equal the agendaconditional on χ before the revision. Let us call this requirement determi-nation. Lastly, and importantly, the epistemic strategy, which is interpretedas a stable plan of action that the agent follows in inquiry, should in somesense be preserved in the revised model. The requirement will henceforth bereferred to as stability.

3.2. Lexicographic upgrade

We shall build our revision operation on van Benthem’s operation of lexi-cographic upgrade, which he uses to model belief revision. In “sphere lan-guage”, lexicographic upgrade can be defined as follows: given a spheresystem Su corresponding to a world u in a model M, and given a propo-sition Y , we we denote by Su ⇑ Y the lexicographic upgrade of Su, givenby:

Su⇑Y =df. {X ∩ Y : X ∈ Su & X ∩ Y = ∅} ∪ {X ∪ Y : X ∈ Su}

468 S. Enqvist

The operation has a very natural interpretation: if we think of the spheresystems as corresponding to an ordering over worlds representing how plau-sible the agent considers them (the worlds closer to the center of the spheresystem being more plausible than those further out), then we can think oflexicographic upgrade by Y as changing the plausibility order so that allY -worlds become more plausible than all non-Y -worlds, while the orderingamong Y -worlds as well as among non-Y -worlds remains the same as before.

We shall extend this to a full revision operation by introducing the fol-lowing upgrade of conditional desiderata:

Definition 3.1. AGu⇑Y (X) =df.

{AGu(X) if X ∩ Y = ∅

AGu(X ∩ Y ) if X ∩ Y = ∅Given a model M =

⟨W, {Su}u∈W , {AGu}u∈W , V

⟩, let M⇑Y be defined

as the structure 〈W, {Su ⇑ Y }u∈W , {AGu ⇑ Y }u∈W , V 〉. This gives a fulloperation ⇑ defined on pairs of models and propositions, which outputs anIBR-structure.

We now introduce a dynamic operator corresponding to our extendedversion of lexicographic upgrade. Formally we define the extended languageLIBR⇑ as follows (χ any LBR-formula):

LIBR⇑ : p | ¬α | α ∨ α | B(α |α) | des(χ |α) | Aα | [⇑ α] α

The new operator has the following evaluation clause:

M, u � [⇑χ] α iff M⇑‖χ‖M , u � α

The logic determined by this semantics will be denoted IBR⇑.To axiomatize IBR⇑, we first define the rule of substitution of equivalents

as follows: if α is any formula, then let α(β/χ) denote the result of uniformlysubstituting χ for β in α. Then substitution of equivalents has the form:from � β ↔ χ, infer � α ↔ α(β/χ). Below, the dual of the operator A,defined as ¬A¬, is abbreviated as E.

Theorem 3.2. IBR ⇑ is completely axiomatized by all rules and axiomschemata of IBR, substitution of equivalents plus the following schemata:

⇑1: [⇑χ] q ↔ q, q a propositional atom

⇑2: [⇑χ]¬α ↔ ¬ [⇑χ] α

⇑3: [⇑χ] (α ∨ β) ↔ ([⇑χ]α ∨ [⇑χ] β)

⇑4: [⇑χ] Aα ↔ A [⇑χ]α


⇑5: [⇑χ] B(α | β) ↔((E(χ ∧ [⇑χ] β) ∧ B([⇑χ] α | χ ∧ [⇑χ] β)) ∨(¬E(χ ∧ [⇑χ] β) ∧ B([⇑χ] α | [⇑χ] β)))

⇑6: [⇑χ]des(α | β) ↔((E(χ ∧ [⇑χ] β) ∧ des(α | χ ∧ [⇑χ] β)) ∨(¬E(χ ∧ [⇑χ] β) ∧ des(α | [⇑χ] β)))

Here, the axiom ⇑ 5 is adapted from van Benthem’s axiom system forlexicographic upgrade. We should notice what a natural extension of vanBenthem’s axioms the axiomatization for IBR⇑ provided above is. Theaxiom ⇑6, which we have added to capture revision of conditional desiderata,is precisely the same as ⇑5, van Benthem’s axiom for revision of conditionalbeliefs, only with “B” replaced with “des” throughout, and with “[⇑χ] α”replaced by “α” everywhere on the right-hand side.

The proof of the completeness theorem consists in checking that thereduction axioms ⇑ 1− ⇑ 6 are all sound, and verifying that they provide atranslation τ of every formula in LIBR⇑ into a provably equivalent formulain LIBR. This translation is defined recursively by the following clauses:

1. τ(q) = q

2. τ(¬α) = ¬τ(α)3. τ(α ∨ β) = τ(α) ∨ τ(β)4. τA(α) = Aτ(α)5. τ(B(α | β)) = B(τ(α) | τ(β))6. τ(des(α | β)) = des(α | τ(β))7. τ([⇑χ] q) = q

8. τ([⇑χ]¬α) = τ(¬ [⇑χ] α)9. τ([⇑χ] (α ∨ β)) = τ([⇑χ] α ∨ [⇑χ] β)

10. τ([⇑χ]Aα) = τ(A [⇑χ] α)11. τ([⇑χ]B(α | β)) =

τ((E(χ ∧ [⇑χ] β) ∧ B([⇑χ] α | χ ∧ [⇑χ]β)) ∨(¬E(χ ∧ [⇑χ] β) ∧ B([⇑χ] α | [⇑χ] β)))

12. τ([⇑χ]des(α | β)) =τ((E(χ ∧ [⇑χ] β) ∧ des(α | χ ∧ [⇑χ] β)) ∨(¬E(χ ∧ [⇑χ] β) ∧ des(α | [⇑χ] β)))

To show that τ is a well defined translation that sends every LIBR⇑-formulato an equivalent LIBR-formula, we need to define a suitable measure ofcomplexity of formulas. The method is explained in detail in [21]. Theargument is standard and left to the reader.

470 S. Enqvist

3.3. Properties of ⇑We now ask whether the operation ⇑, as an extension of van Benthem’s oper-ation of lexicographic upgrade, is appropriate as a revision operation. First,we have to check that ⇑ satisfies categorical matching. This is expressed byproposition 3.1 below.

Lemma 3.1. For any X, Y ⊆ W , we have:

1. bel(Su⇑Y, X) = bel(Su, X), if X ∩ Y = ∅, and

2. bel(Su⇑Y, X) = bel(Su, X ∩ Y ), if X ∩ Y = ∅.Proof. Straightforward.

Let us simplify notation a bit. Given a proposition of the form ‖χ‖M forsome formula χ, we shall write the updated model as M ⇑ χ rather thanM ⇑ ‖χ‖M. Similarly, for a world u in M we shall write Su ⇑ χ insteadof Su ⇑ ‖χ‖M and AGu ⇑ χ instead of AGu ⇑ ‖χ‖M. Using this notationalconvention, we have:

Proposition 3.1. Let M be an IBR-model and let χ be any LIBR⇑-formula.Then M⇑χ is an IBR-model.

Proof. I leave it to the reader to check that Su ⇑ χ is nested. To seethat bel(Su⇑ χ, ‖α‖M⇑χ) is non-empty for every LIBR-formula α such that‖α‖M⇑χ = ∅, let α be such a formula. We recall that by the argumentfor theorem 3.2 the translation τ : LIBR⇑ → LIBR defined earlier sendsevery LIBR⇑-formula to an equivalent LIBR formula. We consider two cases:either ‖χ‖M∩‖α‖M⇑χ = ∅ or ‖χ‖M∩‖α‖M⇑χ = ∅. In the first case we havebel(Su⇑χ, ‖α‖M⇑χ) = bel(Su, ‖α‖M⇑χ) by lemma 3.1. But

‖α‖M⇑χ = ‖[⇑ χ] α‖M = ‖τ([⇑ χ] α)‖M

Since τ([⇑ χ] α) is an LIBR-formula and since M is an IBR-model it followsthat bel(Su, ‖τ([⇑ χ] α)‖M) = ∅, hence bel(Su⇑χ, ‖α‖M⇑χ) = ∅. The secondcase is not harder, and I leave it to the reader.

What remains is to check that M⇑χ satisfies [func] and [pres]. I givethe argument for [pres], leaving [func] to the reader. Suppose α ∈ AGu ⇑χ(‖δ‖M⇑χ) and let β be a presupposition of α. We need to show thatbel(Su ⇑ χ, ‖δ‖M⇑χ) ⊆ ‖β‖M⇑χ; we use the obvious fact that, since β ispurely Boolean, ‖β‖M⇑χ = ‖β‖M. Also, we have ‖δ‖M⇑χ = ‖τ([⇑ χ] δ)‖M

where τ is the translation defined in section 3.2. We distinguish two cases.


Case 1: ‖τ([⇑ χ] δ)‖M ∩ ‖χ‖M = ∅. Then by definition 3.1, we have α ∈AGu(‖τ([⇑ χ] δ)‖M). So since M satisfies [pres] and τ([⇑ χ] δ) ∈ LIBR,we get

bel(Su, ‖[τ(⇑ χ] δ)‖M) ⊆ ‖β‖M

But by clause 1 of lemma 3.1, since ‖[τ(⇑ χ] δ)‖M ∩ ‖χ‖M = ∅ we havebel(Su ⇑ χ, ‖τ([⇑ χ] δ)‖M) = bel(Su, ‖τ([⇑ χ] δ)‖M), and the conclusionimmediately follows.

Case 2: ‖τ([⇑ χ] δ)‖ ∩ ‖χ‖M = ∅. Then by definition 3.1, we have α ∈AGu(‖τ([⇑ χ] δ)‖ ∩ ‖χ‖M). So since M satisfies [pres] and τ([⇑ χ] δ) ∈LIBR, we have

bel(Su, ‖τ([⇑ χ] δ)‖ ∩ ‖χ‖M) ⊆ ‖β‖M

But by clause 2 of lemma 3.1, since ‖τ([⇑ χ] δ)‖ ∩ ‖χ‖M = ∅ we have

bel(Su⇑χ, ‖τ([⇑ χ] δ)‖) = bel(Su, ‖τ([⇑ χ] δ)‖ ∩ ‖χ‖M)

and the conclusion follows.

This ends the proof.

Next comes the question whether the operation satisfies determination.Recalling that we defined the local agenda at a world u as the agenda con-ditional upon the trivial proposition W , we write AGu⇑Y for AGu⇑Y (W ),extending earlier notation. The following proposition then gives us whatwe want:

Proposition 3.2. AGu⇑Y = AGu(Y ), provided Y is non-empty.

Proof. As Y is non-empty, W ∩ Y = ∅. So by definition 3.1, AGu ⇑ Y =AGu⇑Y (W ) = AGu(W ∩ Y ). But of course, W ∩ Y = Y , and we have theresult.

We can view this fact from a syntactic perspective as well. Recall thatwe defined desα (read “α is a desideratum on the current agenda”) as theformula des(α | T), where T is an arbitrary tautology. Putting T for β in⇑ 6, and exploiting some obvious equivalences, we can derive the following:

� [⇑χ]desα ↔ (Eχ ∧ des(α |χ)) ∨ (¬Eχ ∧ desα)

If we apply simple propositional logic to this theorem, we get

� Eχ → ([⇑χ]desα ↔ des(α |χ))

472 S. Enqvist

This is nothing more than an object language expression of proposition 3.2.Finally, it remains to argue that the operation satisfies stability. This

is not an entirely exact question, but I think the reader will agree thatthe operation does satisfy this desideratum as far as one could wish. Thedesiderata conditional on a proposition X after revision correspond to thedesiderata before revision conditional on either X, if X is inconsistent withthe input proposition, or the conjunction (intersection) of X and the inputproposition, if this is consistent. In each case, by lemma 3.1, the agendaconditional on X after revision corresponds to the agenda conditional ona proposition Y prior to revision, such that the agent’s beliefs conditionalon Y prior to revision are the same as his beliefs conditional on X afterrevision. In this sense, it is reasonable to say that the epistemic strategy ofthe agent is preserved through lexicographic upgrade as defined here.

4. Decidability

The purpose of this section is to demonstrate the decidability of the logicIBR and, as a corollary, of its dynamic extension IBR ⇑. We shall provethe decidability of IBR by a filtration argument. Before we start, we needto prove a simple lemma: let us say that a purely Boolean formula β is aprime presupposition of the LBR-formula α if it is a presupposition of α,and every presupposition of α is classically entailed by it.

Lemma 4.1. Every LBR-formula has a prime presupposition.

Proof. We first prove the following statement:

(†) If β is a presupposition of α that classically entails every presupposi-tion χ of α with the property that all propositional variables in χ aresubformulas of α, then β is a prime presupposition of α.

Suppose β is such a presupposition of α, and let δ be any presupposition ofα. Suppose it were not the case that β classically entails δ. Then there isa truth-value assignment t : var → {1, 0} in the usual sense such that β istrue but δ false under t. Since δ is a presupposition of α we have α �BR Bδ.So let M, u be any pointed BR-structure such that M, u � α, and such thatthere is a v ∈ min(Su) for which, for any p ∈ var(α), we have v ∈ V (p) ifft(p) = 1. Such a model exists, for otherwise let

ψ =∨

{¬p : p ∈ var(α) & t(p) = 1} ∨∨

{p ∈ var(α) : t(p) = 0}If we could not find a model of the required sort, then we would have α � Bψ,and so ψ would be a presupposition of α. Since var(ψ) is contained in var(α),


by assumption β classically entails ψ — but since clearly ψ is false underthe truth-value assignment t, this is a contradiction, since β is true under t.

Given that M = 〈W, {Su}u∈ W , V 〉, let V ∗ be the valuation defined by

• w ∈ V ∗(p) iff w ∈ V (p), for w = v

• v ∈ V ∗(p) iff t(p) = 1.

Setting M∗ = 〈W, {Su}u∈ W , V ∗〉, we have M∗, u � α — this is easy to show,using the fact that only valuations of propositional variables not occurringin α are different in M∗ (details left to the reader). Furthermore, sincev ∈ min(Su) and δ is false under t, we get M∗, u � Bδ — this contradicts ourassumption that δ was a presupposition of α, and we are done proving (†).

It now suffices to note that there are, up to classical equivalence, onlyfinitely many presuppositions of α with all propositional variables containedin α. If we let β be the conjunction of one member of each equivalence class,then clearly β is a presupposition of α that entails every presupposition ofα with all propositional variables contained in α. Therefore, by (†), β is aprime presupposition of α.

We say that a set Σ of IBR-formulas is presupposition closed if, giventhat Σ contains some formula of the form des(α |β), Σ also contains at leastone prime presupposition of α.

Lemma 4.2. Every formula α is contained in a finite, subformula closed andpresupposition closed set of formulas.

Proof. Straightforward consequence of lemma 4.1, keeping in mind thatpresuppositions are always purely Boolean.

Let us fix a model

M = 〈W, {Su}u∈ W , {AGu}u∈ W , V 〉and a fixed finite, subformula closed and presupposition closed set of sen-tences Σ. We define the set W f as the quotient of W modulo the usualequivalence relation ∼Σ obtained from Σ, i.e. v ∼Σ w if and only if wehave M, v � β ⇔ M, w � β for each β ∈ Σ. The equivalence class of agiven world u is denoted |u|. We let r be a fixed choice function picking arepresentative r(|u|) for each equivalence class |u|. Given a set X ⊆ W , wedenote by |X| the set {|u| : u ∈ X}. Given a set |X| ⊆ W f , we denote by|X|−1 the set {u ∈ W : |u| ∈ |X|}. Clearly, for any subset |X| of W f , wehave ||X|−1| = |X|. We define the filtration of M through Σ as

Mf =df.

⟨W f , {Sf

|u|}|u|∈ W f , {AGf|u|}|u|∈ W f , V f

⟩

474 S. Enqvist

given by the following definitions:

• Sf|u| =df. {|X| : X ∈ Sr(|u|)}

• α ∈ AGf|u|(|Y |) iff there is some β for which we have des(α | β) ∈ Σ,

α ∈ AGr(|u|)(‖β‖M) and and bel(Sf|u|, |Y |) = bel(Sf

|u|, | ‖β‖M |).• for a propositional variable p,

V f (p) =df.

{|u| ∈ W f : u′ ∈ V (p) for all u′ ∈ |u|

}.

Lemma 4.3. For all u ∈ W and α ∈ Σ, we have M, u � α iff Mf , |u| � α

Proof. By induction; I do the step for des(α | β). If M, u � des(α | β)and des(α | β) ∈ Σ then one easily shows that Mf , |u| � des(α | β), sowe focus on the converse direction. Suppose Mf , |u| � des(α | β), wheredes(α | β) ∈ Σ. We aim to show that M, r(|u|) � des(α | β), from which theresult follows. We have α ∈ AGf

|u|(‖β‖Mf ). Hence there is some sentence

χ such that des(α | χ) ∈ Σ, α ∈ AGr(|u|)(‖χ‖M) and bel(Sf|u|, ‖β‖Mf ) =

bel(Sf|u|, | ‖χ‖M |). Since M satisfies [func], it now suffices to show that

bel(Sr(|u|), ‖χ‖M) = bel(Sr(|u|), ‖β‖M).But using the fact that β ∈ Σ we get | ‖β‖M | = ‖β‖Mf by the induction

hypothesis. Hence bel(Sf|u|, | ‖β‖M |) = bel(Sf

|u|, | ‖χ‖M |), and from this weget

bel(Sr(|u|), | ‖χ‖M |−1) = bel(Sr(|u|), | ‖β‖M |−1)

by a straightforward argument. But since χ, β ∈ Σ the induction hypothesisgives | ‖β‖M |−1 = ‖β‖M and | ‖χ‖M |−1 = ‖χ‖M, and we are done.

It may be worth mentioning that in some places in the proof, we needto make use of the obvious fact that r(|u|) and u satisfy precisely the samemembers of the set Σ.

Lemma 4.4. Mf is an IBR-model.

Proof. One can easily show that each sphere system Sf|u| is nested, and

the fact that bel(Sf|u|, ‖α‖Mf ) is always non-empty provided that ‖α‖Mf is

non-empty is an easy consequence of the fact that W f is finite (which meansthat S

f|u| must be well-ordered w.r.t. set inclusion). The crucial part of the

proof is to show that Mf satisfies [pres] and [func].


For [pres], suppose α ∈ AGf|u|(‖χ‖Mf ). By the definition of AGf there is

some δ such that des(α | δ) ∈ Σ, α ∈ AGr(|u|)(‖δ‖M) and bel(Sf|u|, ‖χ‖Mf ) =

bel(Sf|u|, | ‖δ‖M |). Since Σ is presupposition closed, Σ contains a prime pre-

supposition β of α. Since M satisfies [pres], we have bel(Sr(|u|), ‖δ‖M) ⊆‖β‖M. Using the fact that β, δ ∈ Σ it is now easy to show thatbel(Sf

|u|, | ‖δ‖M |) ⊆ | ‖β‖M | and hence bel(Sf|u|, ‖χ‖Mf ) ⊆ | ‖β‖M |.

Since β ∈ Σ, lemma 4.3 entails that | ‖β‖M | = ‖β‖Mf . Furthermore,since β is a prime presupposition of α it classically entails any presuppositionof α, and from this clearly follows that ‖β‖Mf ⊆ ‖γ‖Mf for any presupposi-tion γ of α. Thus we have bel(Sf

|u|, ‖χ‖Mf ) ⊆ ‖γ‖Mf for any presuppositionγ of α, and Mf satisfies [pres].

The argument for [func] is straightforward.

Theorem 4.1. IBR is decidable.

Since the translation provided by the reduction axioms given for IBR ⇑in section 3.2 is clearly effectively computable, by the decidability of IBRwe immediately get:

Corollary 4.2. IBR⇑ is decidable.

5. Concluding remarks

5.1. Comparisons with related work

The current framework is most closely related to that of [5]. The most im-portant difference between the logics in [5] and the current framework isthat we are now able to capture agenda change in revision, rather than justexpansion, which is achieved by the introduction of the “conditional desider-ata” that we have interpreted as representing the epistemic strategy of theagent. Proper revision is allowed in both logics in [5], but the postulatesgoverning agenda change are concerned only with the special case of expan-sion (this is acknowledged in the concluding remarks section, and the taskof handling proper revision is left as a problem for future research).

As mentioned earlier, the idea that agenda changes should be determinedby stable, long term research interests comes from [6]. However, the notionof epistemic strategies used here is a rather different approach to modellingthe way the long term research interests determine changes of the researchagenda compared to the solution used in [6]. There, the idea is simply to letthe “local” agenda of an epistemic state S consist of those questions that

476 S. Enqvist

are obtained by removing, from some fixed set of “long term” questions,those answers that are inconsistent with the agent’s beliefs in the epistemicstate S. In this way, keeping the long term questions fixed, agenda changesas a result of belief revision are completely determined.

It should further be mentioned that there is some connection between in-terrogative belief revision and the work of A. Wisniewski concerning “eroteticarguments”, understood as inferences involving both declarative sentencesand questions (see [22, 23, 24]); establishing the exact nature of this connec-tion will have to be left for future research, although it could probably bequite enlightening. Another quite recent connection can be found in J. vanBenthem and S. Minica’s paper [20], in which some actions such as raisingand answering a question are examined in a DEL-style setting. Also, theirnotion of a “DELQ protocol”, which is essentially a branching tree-structuredescribing various sequences of actions performed in some specific order, issomewhat reminiscent of the notion of an epistemic strategy developed here,although again the relationship between these two notions is not clear atthis point and needs to be further investigated.

5.2. Future research

A natural way to proceed from here would be to consider dynamic exten-sions of BR based on other revision operations than lexicographic revision.Obviously, the core property of lexicographic revision that allows for thenatural extension to the case of interrogative belief revision is captured bylemma 3.1, which in effect shows us that all conditional belief systems in arevised sphere system correspond to some conditional belief system in theprior sphere system. There are other revision operators for which this holds;one example is the operation that H. Rott calls “natural revision”, whichcoincides with the operation that van Benthem calls “elite change”. In thelanguage of sphere systems, the operation (denoted here by “↑”) can bedefined as follows:

Su ↑ X =df. {bel(Su, X)} ∪ {bel(Su, X) ∪ Y : Y ∈ Su}One can easily establish a modified version of lemma 3.1 for this operation3,and it is rather straightforward to extend ↑ to a revision operation that

3The result would in this case be that, for any X, Y ⊆ W , we have:

1. bel(Su ↑X, Y ) = bel(Su, Y ) if bel(Su, Y ) ∩ X = ∅, and

2. bel(Su ↑X, Y ) = bel(Su, Y ∩ X) if bel(Su, Y ) ∩ X = ∅.


includes update of agenda functions following roughly the same lines as theconstruction for lexicographic revision.

The problematic cases are the ones where no counterpart of lemma 3.1can be established; one example is an operation called irrevocable revision(see [16, 15]). This operation, denoted here by “∗”, is defined by the followingclause for revising a given sphere system Su with some proposition X:

Su∗Y =df. {X ∩ Y : X ∈ Su & X ∩ Y = ∅} ∪ {W}To see that we cannot obtain any variant of lemma 3.1 for this operation,

consider the following example: set

1. W = {a, b, c}2. Sa = {{a}, {a, b}, {a, b, c}}3. X = {c}, Y = {a, b}Then bel(Sa ∗ X, Y ) = Y , but there is no Z ⊆ W such that bel(Sa, Z) = Y— the only candidates would be the supersets of Y , i.e. Y and W , and wehave bel(Sa, Y ) = bel(Sa,W ) = {a}. So we cannot proceed in the way wehave done here with irrevocable revision. A problem for further research istherefore to find a way of handling revision of agenda functions for operationslike irrevocable revision.

Secondly, the fact that I have here used van Benthem’s logic for beliefrevision as a starting point rather than K.Segerberg’s DDL or Bonanno’ssystem could itself provide directions for future research by connecting in-terrogative belief revision with the wide variety of work that is being donein the DEL tradition. For instance, it would be interesting to considermulti-agent versions of the languages developed here. Although the DEL-method of “static base logic + dynamic extension” is generally an elegantand powerful approach to modelling dynamic phenomena, its most commonapplication is indeed within the dynamics of various sorts of multi-agentsystems. Although I have focused exclusively on the single agent case here,with a DEL-style treatment of interrogative belief revision in place the way ispaved to make use of this feature of the DEL-method to study interrogativebelief revision with several agents.

A third possible direction for future research is to study the structuralproperties of epistemic strategies. Here there is a possible connection with[14], in which various properties of socalled goal systems (for example, co-herence) are investigated. One interesting feature to study would be thecomplexity of epistemic strategies; if we see epistemic strategies as research

478 S. Enqvist

strategies that an agent uses in the hope to eventually resolve some “princi-pal” question, then since real agents have limited resources in terms of cog-nitive capacities and time, less complex strategies should ideally be favoredover more complex ones. In order to study this idea formally, we would needto define a precise measure of the complexity of an epistemic strategy. Howto properly explicate this notion could itself be a philosophically interestingproblem.

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Sebastian EnqvistDepartment of PhilosophyLund UniversityKungshuset, Lundagard222 22 Lund, [email protected]

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Interrogative Belief Revision Based on Epistemic Strategies