Synthese (2009) 171:271–289Knowledge, Rationality & Action 873–891DOI 10.1007/s11229-009-9639-0

The game of inquiry: the interrogative approachto inquiry and belief revision theory

Emmanuel J. Genot

Received: 15 December 2008 / Accepted: 7 July 2009 / Published online: 22 July 2009© Springer Science+Business Media B.V. 2009

Abstract I. Levi has advocated a decision-theoretic account of belief revision. Weargue that the game-theoretic framework of Interrogative Inquiry Games, proposedby J. Hintikka, can extend and clarify this account. We show that some strategic use ofthe game rules (or ‘policies’) generate Expansions, Contractions and Revisions, andwe give representation results. We then extend the framework to represent explicitly(multiple) sources of answers, and apply it to discuss the Recovery Postulate. Weconclude with some remarks about the potential extensions of interrogative games,with respect to some issues in the theory of belief change.

Keywords Interrogative Model of Inquiry · AGM axioms · Belief bases

1 Introduction: a strategic viewpoint on revision

Strategies, in the decision-theoretic sense of planned sequential choices, are relevantto Belief Revision Theory (brt). I. Levi promotes a decision-theoretic account ofbelief revision where the inquirer selects the ‘best’ expansion or contraction strategyamong the set of available contraction strategies. The following summarizes this pointof view (w.r.t. contraction):

[A] decision-theoretic approach [is] one that justifies the selection of a contrac-tion strategy from the set of available contraction strategies as best […] amongthe set of available contraction strategies with respect to the promotion of thegoal of contraction. (Levi 2003, p. 209, author’s emphasis.)

E. J. Genot (B)Université Charles-de-Gaulle Lille III, Lille, Francee-mail: emmanuel.genot@free.fr

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In Levi’s account, options to choose from are strategies as wholes. We are not toldwhat kind of sequential choices theses strategies are made of. Since Levi is interestedin comparing the way strategies—for contraction as well as expansion—promote thegoal of inquiry, his model is primarily aimed at telling us which general constraints(as captured by some postulates) reasonable strategies should comply with.

J. Hintikka argues that we need to study belief change as sequences of choices, andthat changes studied in brt should be examined as the result of ‘interrogative games’of information-seeking by questioning: the Inquirer carries a step-by-step procedurethe outcome of which is actual belief change. These games incorporate deductive aswell as interrogative steps, and rules to deal with uncertain answers, thus generating a(non-monotonic) relation of interrogative consequence between (theoretical) premisesand (accepted) answers on the one hand, and an investigated conclusion on the otherhand.

According to Hintikka, rules of logic are ‘definitory’ and permissive, they do nottell us what to do, only what we can do: use of logic in reasoning is strategic. Yet:

[T]heorists of both deductive and non-deductive reasoning [try] to formulatetheir rules as definitory ones, that is to construct sets of definitory rules of somenew logic that captures certain strategic ideas. (Hintikka 1999, p. 4, emphasisadded.)

Levi focuses on the question whether some properties of contractions—and expan-sions—best promote the goal of inquiry. From the point of view of interrogativegames, this makes what Levi considers a decision problem akin to a strategic game (orsequential decision problem).1 Yet the details of the corresponding game (or sequen-tial decision problem) one has to play (or solve) are not spelled out, only the rationalconstraints on overall strategies one’s choices has to comply with. And since logicplays a role to represent changes in the sentential representation of beliefs (usuallyreferred to as ‘corpus’ or ‘belief set’) attention should be paid to the way logic is used‘strategically’ as a part of the process of change:

[Theories] of belief change (cf. Gärdenfors 1988) […] should be subjected to amuch sharper critical scrutiny from a strategic viewpoint that has been the caseso far. (Hintikka et al. 1999, loc. cit.)

Hintikka’s Interrogative Model of Inquiry (imi) aims precisely at capturing thesesteps an inquiry is made of. Depending on a particular inquiry setting, rules of logic,as well as rules for asking questions, are used strategically in order to investigateone given hypothesis. The ‘game of inquiry’ can be viewed as a game Inquirer playsagainst Nature. Games of this kind are degenerate (Nature is not a ‘player’ in the

1 The strategic or normal form of the game displays choices by players of entire strategies, and is usuallygiven a game-matrix representation. By contrast, in games in extensive form, the successive moves areexplicitly displayed. Extensive representation is especially useful when one wants to represent a player’sknowledge of the state she’s reached: several states may be indiscernible from the player’s point of view(e.g. in some games with imperfect information) so that a player cannot tell at which node of the tree she is.It is also interesting when several choices have a different outcome depending on their order—i.e. wheneverthere is some kind of dependency between moves (see Osborne and Rubinstein 1994, Chap. 6).

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strict sense), and are rather sequential decision problems. The analogy with extensivegames allows to consider sequential moves as steps of increasing depth of a reasoningprocess. When sources of answers are agents, with their own beliefs, preferences, etc.,the framework has to be properly game-theoretic, in order to capture the full richnessof the varieties of inquiries.2

We will show that Hintikka’s idea of submitting the theory of belief change to“critical scrutiny from a strategic viewpoint” is indeed fruitful. Strategic use of therules of interrogative games can generate contractions, or revisions of an initial set ofpremises, provided that some ‘policies’, constraining choices, are followed: choicesnot complying (in the long run) with these constraints would not be, in a very clearsense, rational. This can be expressed through two methodological theses:

Thesis 1 Every operation described in the brt framework should be analyzed as theresult of sequential choices, and should be described within the framework of inter-rogative inquiry games.

Thesis 2 Any strategic application of the rules of interrogative games should be justi-fied from an epistemological viewpoint (i.e. by reference to inquiry-relevant factors).

We will proceed as follow: Sect. 2 introduces the notion of Interrogative Conse-quence, as well as a variant of the semantic trees capturing this notion. Some results of‘interrogative logic’ are briefly exposed. Section3 presents Inquiry Games as (exten-sive) games against Nature, with the additional rules needed to cover the non-mono-tonic aspect of interrogative consequence in semantic trees. Representation resultsfor Expansion, and Contraction (and given an analog of the Levi Identity, for Revi-sion) of corpora (closed sets of sentences representing beliefs) are proved.3 Theserepresentation results support Thesis 1.

Section 4 presents an extension of Hintikka’s framework, in which multiple, andpossibly conflicting, sources of answers are explicitly represented. We do so througha case-study, which supports Thesis 2: the possible failure or success of Recovery, isrelated to inquiry-relevant factors (in this case, sources of answers).4 Although wefocus on the recovery postulate, this is by no means the only reason to introduce

2 Sources may respond differently to questions which are, for the Inquirer, equivalent given her background.Subsequently, her strategy may involve a choice of questions, conditional on the source’s beliefs and pref-erences—a common situation in criminal investigations or sociological surveys. Moreover, Rott (2004) hasshown that messages may convey to a receiver more information than their propositional content, when shehas information about beliefs and preferences of the sender.3 Our framework is closer to belief bases (non-closed sets)—the study of which was initiated by Fuhrmann(1991) and Hansson (1991), but closed corpora do have a natural relatum in a approach motivated by theimi, so we won’t eventually be restricted to bases.4 It is known that withdrawal operators (satisfying the agm postulates with the exception of Recovery) canbe defined for (non closed) bases (see Hansson 1991). Hence it is no surprise that Recovery fails in general ina context where the premise sets are not closed. Still we are able to provide some additional conditions underwhich a contraction operated on a base—as the result of sequential choices in an interrogative game—willindeed satisfy Recovery. Our use of “Contraction” in a slightly more comprehensive than usual: ‘interrog-atively induced’ contractions (of the set of interrogative consequences of a given premise set) should beconsidered as withdrawals. But the very method used to generate them guarantees that Recovery can alwaysbe obtained (besides, terminology is flexible: Levi’s ‘mild contraction’ is Rott’s ‘severe withdrawal’).

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an explicit representation of sources (some others are discussed in conclusion). Theimportant contrast between the way contractions and revisions are obtained in Sects. 3and 4 is that, in the former case, a set of restrictions is imposed to the possible strategiesin a ‘question opening’ game. In the latter, the strategic use of question-opening rulesis motivated entirely by local factors. Hence, while it may seem that our approachimposes a priori constraints on Inquirer’s strategies, this impression is deceptive. Weinsist that, in sensible cases, the strategic use of question-opening rules will complywith the constraints which lead to contractions of the premise set. Yet this complianceis not the result of Inquirer’s aiming at a contraction. Hence, the structural features ofcontractions and revisions ought to be viewed as ‘emerging’ from rationally conducedinquiry, i.e. from strategically sound use of the rules of an interrogative game.

We conclude with some remarks about the possible extension of the frameworkpresented here, and its connexion to other brt-related approaches in which questionsplay a role.

2 Questions and interrogative consequence

Let L be a propositional language, and for some T ⊆ L let Cn(T ) denote the set ofclassical consequences of T .5 We restrict our exposition to propositional questions,due to the focus of brt on propositional representation. Let M, T and C be respec-tively a model, a (finite or finitely axiomatizable) set of premises, and a conclusion. LetAM ⊆ L be a set of available answers in M . A question is a move in an interrogativegame, and calls for an input from M and can be played whenever its presuppositionfollows from T (together with already obtained answers). In the case of a propositionalquestion, the presupposition is simply the disjunction of its potential answers.6

Interrogative consequence is defined as follows (from Hintikka et al. 1999): C isan interrogative consequence of T in M (w.r.t. AM iff C follows deductively fromT together with some answers in AM . Let CnI(M, T ) denote the set of interrogativeconsequences of T in M . Whenever the set of questions needed to derive C is empty,C ∈ CnI(M, T ) iff C ∈ Cn(T ): interrogative consequence reduces then to classicaldeductibility. On the other hand, if the premise set T is empty, and all questions canbe answered, C ∈ CnI(M,∅) iff C is true in M (as long as the answers are true).7

Questions are ‘interrogative steps’ in an otherwise deductive process—and non-monotonicity is dealt with strategically, through sequential use of the rules of theinterrogative game (see Sect. 3). Checking interrogative consequence can be done in

5 For the formal properties of the Cn(·) operator, see in Appendix.6 It suffices, for our present purpose, to identify a question with the set of its potential answers. Hintikka’stheory of questions is more fine-grained, first because it extends to all types of questions (and not onlypropositional, or ‘whether’, questions, and second because is requires the use of explicitly modal notions.Roughly, a question can be identified with a pair: its presupposition (a known disjunction, or a knownexistential statement), and its desideratum (the epistemic situation the question is calculated to bring about,i.e. the knowledge of which answer holds). For a detailed analysis and comparison with other theories, thereader should refer to Wisniewski (1995), who characterizes Hintikka’s theory as ‘imperative-epistemic’,since a question is analyzed as an imperative of the form: “Bring it about that I know …”.7 For the case where not all answers are true, truth has to be relative to one set of epistemic alternatives,tantamount to a treatment in modal semantics (see footnote 13).

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Fig. 1 Interrogative trees I—deductive rules

several ways. Hintikka’s tool of choice is a modified version of Beth Tableaux, whichmirrors certain characteristics of the inquiry games, which is viewed as an attemptto eliminate all scenarios compatible with the truth of the premises and answers,and the falsity of the conclusion (closing all subtableaux corresponds to show thatall those scenarios are impossible). In the variant adopted by Hinitkka, some addi-tional rules are needed, in particular, traffic from the right (Inquirer) side to the left(Oracle/Model) side is avoided, since it would allow Inquirer to ‘import’ IN the modelsome information.8

We will use a simplified representation, with Signed Semantic Trees (see Smullyan1968) i.e. trees with a (possibly empty) set of premises is considered as true, and asingle (tested) conclusion considered as false, where formulas treated as true (false)are signed with labels T (F) respectively. The aim is to prove the falsity of the con-clusion incompatible with the truth of the premises, by closing all the branches (seebelow). Standard Signed Semantic Trees are easily extended to cover interrogativeconsequence, with ‘standard’ rules representing deductive steps, and adding rules tocover interrogative steps (questions themselves are not represented, only their presup-positions).

An interrogative tree is built with premises in T prefixed with T (since they areassumed to be true) and conclusion C prefixed with F (since one tries, hopefullywithout success, to show that the conclusion can be false in models of the premises).

A modification of tree-building rules allows to obtain formulas prefixed with T(F)from premises or answers (conclusion) only, as displayed in Fig. 1.

Any T-formula associated with a branching rule (β-type formula, Smullyan 1968)can serve as the presupposition of a question: information retrieved from M—whenthe answer is available, i.e. in AM —dispenses one from reasoning by cases as shownin Fig. 2 (questions are marked with ?, answers with !).9

Extended Interrogative Trees (ei trees) are obtained when it is possible to add as thesuccessor of any node (the presupposition of) a ‘yes–no’question—i.e. an instance ofthe excluded middle—possibly resulting in a single branch if the answer is available,

8 Standard Beth tableaux can be viewed as model-building games, while interrogative inquiry are closer tomodel-checking games.9 Interrogative rules could in principle apply on the F-formulas, since if the presupposition does not holdin the model, there simply will be no answer available. To avoid this complication, we simply assume thatonly Tprefixed β-formulas can prompt questions.

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Fig. 2 Interrogative trees II—interrogative rules

Fig. 3 Extended interrogativetrees

as shown in Fig. 3. Such trees correspond to extended interrogative logic (eil, Hintikkaet al. 1999).10

Closure rules for ei- trees are as follows. A branch is closed iff for someφ (precededor not with !) one of the following pair of formulas appear on the branch: (i)Tφ andFφ; (i i)Tφ and T¬φ; or (i i) Fφ and F¬φ;⊗ indicates a closed branch. A tree is closediff all its branches are closed. Two results in eil will be useful in the next section. Thefirst shows that any interrogative argument can be reconstructed as a deductive one.

Theorem 1 (Completeness, Hintikka et al. 1999) C ∈ CnI(M, T ) if and only if forsome finite A′

M ⊆ AM , C ∈ Cn(T ∪ A′M ).

The second result shows that any (interrogative) argument can be reconstructed asa (interrogative) argument with only ‘yes–no’questions.

Theorem 2 (Yes-No Theorem, Hintikka et al. 1999) If C ∈ CnI(M, T ) has beenestablished with answers to arbitrary questions, then it can be established with answersto ‘yes–no’questions only.11

3 Games of inquiry and representation results

ei trees can represent Interrogative Inquiry Games (iig) defined as games whereInquirer plays against Nature, and tries to establish that C ∈ CnI(M, T ) by putting

10 From the simplified perspective adopted here (see footnote 6), a ‘yes–no’question is a two-member set{φ,¬φ} (for some φ ∈ L).11 The ‘arbitrary questions’ referred to are those, of arbitrary complexity, of which the presupposition canbe derived from T using the deductive rules. The result states that using ‘yes–no’questions the presupposi-tion of which can be added at any time as additional premises, one can always obtain the same conclusion(close the tree). The question as to which problems are solvable using only ‘yes–no’questions is quite adifferent one, which is out of the scope of the present study, and may be of considerable interest for theextension of brt from propositional to first-order representation. In the case of finite domains, or proposi-tional questions, there is a way to obtain ‘yes-no’ strategies (see e.g. Genot 2009), but there is in generalno mechanical procedure to find the ‘yes-no’ interrogative strategy as soon as some answer to a ‘which’question, over an infinite domain, is needed to complete an argument.

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questions to sources (where sources are oracles generating answers).12 Inquirer winsiff the ei tree with premises T and conclusion C closes, that is, if she manages toeliminate the possibility that C fails to obtain, given her assumptions (T ) and theanswers provided by sources about the model (A′

M ). Saying here that Nature winsmeans that Inquirer did not manage to reduce uncertainty. Hence, the aim of the gameis information-seeking, in the sense of elimination of (not-C) possibilities.

The simplest iig is Pure Discovery: “a type of inquiry in which all answers […]can be treated as being true [and in which] we do not have to worry about justify-ing what we find.” (Hintikka 2007b, p. 98). This is the case when Cn(A′

M ) �= Land Cn(T ∪ A′

M ) �= L, i.e. when no conflict arises between sources—Inquirer beingconsidered as a source for T .13

Some of the most interesting questions relative to iigs regard the (quantificational)complexity of answers: if the answers are atomic, no universal (or negative existen-tial) statement can enter the Inquirer’s beliefs through inquiry, without the help ofsome inductive inference rules. Once again, restriction to the propositional fragmentof eil leaves aside these questions.14 We will only examine the case where answers (orpremises) cannot all be trusted (whatever the reason). Interrogative reasoning becomes,in this case, potentially non-monotonic.

If not all answers are consistent (together or with T ) then the need arises for justifi-cation for keeping some answers and premises, and discarding others. The justificationitself is external to the game, though seeking it may be the subject of a parallel inquiry.Once obtained, some answers are kept, others are discarded (typically those for whichjustification, in some sense, is found wanting). The inquiry process can also lead todiscard premises—a simple form of revision.

On the bookkeeping side, ei trees must be supplemented with a technical deviceto represent formulas that are discarded, and new rules are needed. The device is

12 In standard game-theory, Nature is not properly speaking a player. The result of Nature’s moves is aninitial setting, among a set of possibilities, and player’s uncertainty is then represented by a probabilityfunction over those possibilities. An iig can be qualified as a game against Nature in this sense, since Nature‘chooses’ M (and AM ), which is equivalent to choosing a partial model; while Inquirer tries to show thatdespites this uncertainty as to M , the conclusion can be established (it holds in all the situations indiscerniblefrom M given Nature’s choice).13 An iig of Pure Discovery won by Inquirer I establishes that C ∈ CnI(M

′, T ) where M ′ is an epistemicalternative to M for I (‘interrogatively’ indiscernible from M given T and A′

M ). If all answers are true,the M = M ′. Moreover, since in interrogative logic explicit definitions are creative—extending the rangeof available questions, hence the identifiability of certain individuals (see Hintikka et al. 1999 and Hintikka1999, Essay 14)—‘interrogative indiscernibility’ is a dynamic notion. This topic requires moving to fullfirst-order logic, and its relevance for brt will be left for further research.14 Hintikka presents an analysis of the various logics of questioning w.r.t. to quantificational complexityin Hintikka (1985), and in defends in Hintikka (1988) the thesis that at least some answers (to experimentalquestions) should be viewed as universal (and that inductive inference is needed at a given point). Accountfor the possibility of universal inputs is a key problem for the imi, if it is to serve as a general theory ofinquiry. In that respect, brt faces the same challenge: if no sentence incorporated by expansion or revisionis universal, then no general law can enter a belief set by expansion, and general laws can only be removedthrough revision (by counterexamples). The focus of brt on propositional representation leaves this issuelargely overlooked, but it is important if brt is to be thought of as a general theory of belief and theorychange. One possibility is to combine brt with some inductive methodology. The representation resultsoffered in Sect. 5 offer another potential solution, provided the idea of generalized answers proves viable.

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simply the familiar brackets ‘[’ and ‘]’, a bracketed formula being excepted from theproof. The following rules for using brackets generalize those given by Hintikka et al.(1999)15:

R1 At any time, a T-prefixed formula can be bracketed;R2a If a formula φ is bracketed, and ψ has been obtained by application of adeductive rule to φ, then ψ has to be bracketed too.R2b If a formula φ is bracketed, and φ has been obtained by application of adeductive rule toψ , thenψ has to be bracketed too, until an answer is reached.16

R3 If a node is bracketed, it cannot be used in any application of rules, includingclosure rules.R4 At any time, any node may be unbracketed. Then all the later brackets causedby this node having been bracketed can also be removed.

Strategic use of R1–R4 should be motivated, according to Thesis 2, by inquiry-relevant factors, and according to Thesis, should suffice to generate Expansion, Con-traction and Revision.

Since iig intend to capture investigation of conjectures, the following propositionis stated w.r.t. some conclusion C . Let ‘+’ denote the operation of Expansion, suchthat for a set T and sentence φ, T + φ = Cn(T ∪ {φ}):Proposition 3 C is in the interrogative consequences of T in M iff C is in the Expan-sion of T by A′

M , where A′M ⊆ AM is a set of (unbracketed) answers obtained during

the iig.

Expansion of T by A′M is T + A′

M = Cn(T ∪ A′M ), so Proposition 3 is immediate

from Theorem 1 and Theorem 2 (a full proof is given in Appendix).17 Proposition 3shows that the ‘belief consistent case’ (where Cn(T ∪ A′

M ) is consistent) coincidesexactly with Pure Discovery.18

15 In the rules given by Hintikka, initial application of brackets is always to a premise of answer,‘propa-gating’ downward in the tree. The generalization allows for any formula obtained from a premise or answerto be bracketed, and ‘propagation’ rules guarantee that if a formula obtained from a premise or answer isbracketed, then the premise (answer) it was obtained from will be bracketed, as well as all the formulasobtained from it.16 Without this last condition, bracketing a node which depends on an answer would lead to bracket thepremise which allowed to obtain the presupposition. This is sometimes desirable, but it’s a matter of choice.If no answer is reached upward, then application of the rule will end up bracketing a premise.17 Since A′

M is finite (even if AM is infinite, by Compactness of Cn) and since (T +φ)+ψ = (T +ψ)+φexpansion by A′

M is nothing but expansion by ∧A′M (as long as we disregard the possibility of iterated

revision, and the impact of answers at another level of the belief state, such as entrenchments, rankings,etc.; we leave this as a subject for further research).18 Conjectures, viewed as potential answers to some question, are explicitly studied in variants of brt, as(unions of) elements of Ultimate Partitions (maximally consistent potential expansions of a given corpus,see Levi 2004), or a potential answer to some question in a Research Agenda (see Olsson and Westlund2006).

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So far, the interrogative model seems to provide only some way to implementthe agm model of Expansion (or some variant thereof, using ultimate partitions or‘principal’ research questions in an agenda). Hence such a result would only repre-sent a modest advance (since the imi extends results relevant to expansion to the fullfirst-order case). But the real interest of the model lies in the possibility to cope withnon-monotonicity strategically.

We need first to define the dual of iig, in which the aim is not to prove the conse-quence C with premises T in a model M , but the game aimed at checking whether itis possible for C not to follow from T and M .19 Let us define a Dual iig game (iigd)as game in which Inquirer plays against herself, an tries to show how it is possiblefor a conclusion C not to hold in model M given some finite A′

M ⊆ AM . The gameis played using ei trees with no genuine interrogative moves: there is no request forinformation: implicit questions are used to obtain repetitions of premises. Premisesare formulas in T and A′

M .Obviously, Inquirer can trivially win the iigd by bracketing every premise. Hence,

an given iigd is associated with a bracketing policy aimed at establishing a particulartype of restriction. Ultimately only inquiry-relevant factors can motivate the strategicuse of brackets in an iigd game—as stated by Thesis 2. Hence any ‘bracketing policy’should be motivated by particular inquiry-related interest.

We propose the following policy for iigd, named ‘Contraction by Bracketing’ Pol-icy (cbp). It does not come with a full justification, but we will provide some ratio-nale for it in the next sections. No (genuine) interrogative move is played in iigd, soCnI(M, T ) = Cn(T ) (by definition of interrogative consequence).

Definition 4 (‘Contraction by Bracketing’ Policy) Start an ei tree T with premises Tand conclusion φ. If T closes, then for at least one branch, apply R1 to some subsetof T-formulas used to close the branch, in a way sufficient to reopen it; then applyR2a and R2b.

Let cbp(T |φ) denote the outcome of applying cbp to T with φ. Here is an example ofhow cbp works:

Example 1 Let T = {φ, φ → ψ, χ}, and C = ψ . Trees in Fig. 4 display differentpossible applications of cbp: repetitions of premises are marked with ‘!’; ‘∗’ tags for-mulas used to close branches; subscript indicate a possible sequence of application ofR2a–R2b. Three possible ‘weakening’ of T can be generated: T1 = {φ, [φ → ψ], χ},T2 = {[φ], φ → ψ, χ}, and T3 = {[φ], [φ → ψ], χ}, depending on how one appliesR1 and R2.

Let’s recall the postulates for Contraction of a belief set K by some formula φ,denoted K ÷ φ, as given in Gärdenfors (1988):

19 It is interesting to notice that these ‘dual’ games are very close to the construction of a how possibleexplanation, as exposed in Hintikka (2007a).

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Fig. 4 One example of CBP

(÷1) K ÷ φ = Cn(K ÷ φ) (Closure)

(÷2) K ÷ φ ⊆ K (Inclusion)

(÷3) if φ /∈ K, then K ⊆ K ÷ φ (Vacuity)

(÷4) if φ ∈ K ÷ φ, then φ ∈ Cn(∅) (Success)

(÷5) K ⊆ Cn(K ÷ φ ∪ {φ}) (Recovery)

(÷6) if Cn(φ) = Cn(ψ) then K ÷ φ = K ÷ ψ (Extensionality)

(÷7) K ÷ φ ∩ K ÷ ψ ⊆ K ÷ (φ ∧ ψ) (Conj. Overlap)

(÷8) if φ /∈ K ÷ (φ ∧ ψ), then K ÷ (φ ∧ ψ) ⊆ K ÷ φ (Conj. Inclusion)

(A detailed discussion of the postulates is given in Appendix.)Assuming that when the cbp procedure has to be repeated, choices of premises to

bracket remain the same (see also in Appendix for a detailed proofs), the followingpropositions hold:

Proposition 5 The set of interrogative consequences of cbp(T |φ) satisfies the follow-ing postulates w.r.t. T and φ (and any ψ equivalent to φ): Closure, Inclusion, Vacuity,Success, Extensionality, Conjunctive Overlap, Conjunctive Inclusion.

Proposition 6 If expanding cbp(T |φ) by φ is equivalent to the reintroduction in T ofpremises bracketed in cbp(T |φ), then cbp(T |φ)+ φ satisfies Recovery.

It follows from Proposition 5 that cbp generates a withdrawal of a sentence φ fromthe consequences of a set of premises T (and by Extensionality, w.r.t. the propositionexpressed by φ). If condition of Proposition 6 holds, it is also an agm contraction (ofCn(T ) by φ), even for a non-closed premise set.20 Since cbp depends on T , it cannotinduce every admissible agm contraction of Cn(T ).

Example 2 Let T = {φ ∧ ψ}. Both Cn({ψ}) and Cn({φ ↔ ψ}) are admissible agmcontractions of Cn(T )byφ, but cbp(T |φ) = ∅ since neitherψ ∈ T nor (φ ↔ ψ) ∈ T ,and neither can be left unbracketed in T .

20 Independently of the condition in Proposition 6 holding due to the context of inquiry, Recovery can beenforced by applying R4. Hence cbp is reversible, and can always generate agm contractions. It remains tofind a convincing and natural condition in which the unbracketing, rather than some expansion equivalentto it, may be justified (see below, Sect. 4).

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The possibility of generating every admissible agm contraction of Cn(T ) dependsonly on closure properties of T . The following observation illustrates this:

Proposition 7 If T = Cn(T ), then, for any φ and admissible agm contraction T ÷φ,there is an application of cbp such that Cn(cbp(T |φ)) = T ÷ φ.

Closure of the premise set can be obtained constructively, and interpreted as a limitcase of a process of ‘sub-inquiries’.21

Revision can be reconstructed as a succession of iig and iigd, equivalent to theLevi Identity. We can formulate the Levi Thesis for iig as follows: an InterrogativeInquiry Game of Revision, the aim of which is to revise some set of premises T andsome set of previously obtained answers A′

M ⊆ AM with some new answer φ, can bereconstructed as follows: (1) an iigd with premises T ∪ A′

M and conclusion ¬φ, withbracketing policy π (if π is cbp, one obtains agm revision); (2) an iig with premiseset π(T ∪ A′

M |¬φ) and (unbracketed) answer φ to the question ?T(φ ∨ ¬φ).

4 Recovery as a problem of sources

Let us first review one of the paradigmatic counterexamples to Recovery:

Example 3 (Cleopatra) “Suppose that I have read in a book about Cleopatra that shehad both a son and a daughter. My set of beliefs therefore contains both φ and ψ ,where φ denotes that Cleopatra had a son and ψ that she had a daughter. I then learnfrom a knowledgeable friend that the book is in fact a historical novel. After that Icontract φ ∨ ψ from my set of beliefs, i.e., I do not any longer believe that Cleopatrahad a child. Soon after that, however, I learn from a reliable source that Cleopatra hada child. It seems perfectly reasonable for me to then add φ ∨ ψ to my set of beliefswithout also reintroducing either φ or ψ .” Hansson (1991)

This example is striking, in the light of the model of iig, where all information‘coming in’ is viewed as having been, at one time, an answer to some question from aparticular source. Here there is some admissibility structure related to the reliabilityof sources which is not reflected in the postulates. Hence appeal to source is a wayto enrich the representation of overall epistemic state of an agent and the informationavailable to her, in contrast to the belief set or corpus which in iig consists only in theaccepted premises and answers. Figure 5 describes the evolution of the premise set Tthroughout the Example 3.

A suggested reading of Fig. 5 is as follows: (a) should be read as an iig with nointerrogative (model-checking) move played; (b) as the result of an iigd applying cbp-assuming that there is no T ′ ⊂ T \ φ,ψ s.t. (φ ∨ ψ) ∈ Cn(T ′); and (c) as an iig

21 Bracketing some premises or answers does not prevent from using them in a ‘sub-inquiry’, even forthe sake of the argument. As an example, consider that, after cbp has been run in Example 2, one runs aniig with, as premise set, cbp(T |φ) ∪ {ψ}, and conclusion φ. Since the extension of the premise set doesnot close the tree, one can add ψ to the premise set T . The procedure can be repeated with other possibleconsequences of the bracketed premises—e.g., with ψ ↔ ψ in parallel—checking then whether or not theextensions of the branches of T generated by the addition of those do all close, and consider adding themif it is not the case—and one can then proceed with the union of such premise sets, etc.

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(a) (b) (c)

Fig. 5 Cleopatra formalized

game with no interrogative move played.22 In some sense, (c) can be viewed as anextension of (a). Expansion by (φ ∨ ψ) is not equivalent to unbracketing φ and ψ .Hence Recovery fails.

Yet Fig. 5 does not do justice to the origin of the contraction-like pattern of brac-keting. Adding sources permits this, provided that they can be treated in iig and iigd

as inducing patterns of bracketing, while keeping the object-language as simple aspossible.

To represent sources in ei trees, we only need a set SM = {s0, . . . , sm} of availablesources (in M). Rules of ei trees are stated w.r.t. nodes of the form: 〈φ, s1

i , . . . , snm〉

(with n ≥ 0), representing a sentence of L followed by a (possibly empty) sequenceof ‘source tags’. (If n = 0 for some formula φ, then we can simply write ‘φ’ makingtrees with no source tag a special cases of trees with tagged answers). Elements of Tare not excluded from tagged sentences. We want to have the possibility to ‘ask for’ apremise: we can ask a ‘yes–no’question about it, calling for a new source, or we canhave a deductive move from T representing the use of memory, and an answer taggeds0 (representing Inquirer). The following bookkeeping rules are then added:

S1 At any time, Inquirer can add a source tag sn+1 to 〈φ, si , . . . , sn〉.S2 If a node depends on (by application of a tree rule) 〈φ, si , . . . , sm〉, then it must

be tagged with si ,…, sm as well.

S1 is useful when new sources are available for a premise or answer, while S2 isnot meant to capture any ‘justificational’ pedigree, but rather to keep track of infer-ential relations between tagged sentences. In particular, it is instrumental in definingbracketing rules for sources.

22 The additional premise (φ ∨ ψ) may have been received as the ‘answer’ to a question, but in this case,Theorem 1 applies.

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Not any premise or answer need to be tagged (Inquirer may not recall his sourcesfor premises, and a source of answer can remain unidentified), so new rules will applyin addition of those already given for bracketing. Moreover, application of bracketscan be independent of the sources, so keeping the two sets of rules is convenient.Brackets rules can now be applied to source tags in the following way:

R1′ At any time, any source tag can be bracketed;R2′ If a node depends on a node with a bracketed si by application of a rule, si

has to be bracketed at that node too.23

R3′ If at a node all tags are bracketed, it cannot be used in any application ofrules, including closure rules.24

R4′ At any time, any source tag may be unbracketed. Then all the later bracketscaused by this node having been bracketed can also be removed.

The study of strategic use of R1′–R3′ belongs to epistemology, in the sense that itdepends on the information one has on sources, the way one ranks them, etc. Theserankings need not be represented at the ‘object language’ level. As a result of the brac-keting rules for sources, Proposition 6 can cover use of sources (once reformulated,which is easy given the relation between R3 and R3′, and left to the reader).

Sources being now represented in our model, we are able to give another modelingof Example 3, changing the representation given in Fig. 5 in the following: assume thats1 is the novel, and s2 the reliable source.

As previously, Fig. 6a is analogous to an iigd applying cbp to T1: assuming thatthere is no T ′

1 ⊂ T1 \ {φ,ψ} s.t. (φ∨ψ) ∈ Cn(T ′), no further development of the iigd

is needed. Yet, the application of R1′ and R3′ here is not the result of Inquirer aimingat a contraction, though it induces one (or rather a withdrawal). Only informationabout the source leads to application of rules R1′ and R3′; the result is equivalent toapplying cbp to T with (φ ∨ ψ).

In the tree (b), Expansion by (φ ∨ ψ) is clearly not equivalent to unbracketingφ and ψ , and is represented as an extension of the iigd, in order to show that T3 =T2∪{〈φ∨ψ, s2〉} is the outcome of the procedure (contracting, then asking a re-openedquestion). Hence Recovery fails. Two branches represent two ‘possible worlds’ (orrather, two classes thereof) indiscernible so far given the obtained answer.

Recovery holds when answers (or premises) are ‘recovered’ in a specific way,for example: (1) If all answers (or premises) from a given source (set of sources)is (are) unbracketed as soon as one is; this case corresponds e.g. to some form ofreasoning for the sake of the argument (and applications of rules R4/R4′). (2) If onesource is added which makes available (again) all the premises bracketed at a certainpoint. This case depends on contextual parameters of inquiry (and application of ruleR1).

23 We need not state ‘upward’ propagation rules for source tags, since the initial application of a bracketswill always be to a given answer.24 A different formulation for R3′ would be that if all tags are bracketed at a node, then rule R3 has to beapplied at that node too. But this would cause brackets induced by rule R2 to be conditional on the linesdepending on the bracketed line to have the same tags. We thus keep the two sets of rules as separated.

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(a) (b)

Fig. 6 Cleopatra, adding sources

In any case, the application of the bracketing rules and the rules equivalent tounbracketing has to be—in some sense—uniform (or symmetrical). Only in the firstof the above cases, it is a natural requirement. But we can certainly consider that‘reversal’ of applications of bracketing rules is always possible (rules R4/R4′ guar-antee this possibility), hence guaranteeing the possibility that Recovery be satisfiedeven for bases, when it is a desirable property.

As a last word about sources, it should be noticed that taking them into account—their reliability, and when they are agents, taking into account their beliefs and expec-tations of sources—opens the prospect to be able to treat some further problems, whichhave come unnoticed but to a few. As an example, Rott (2004) shows that when a mes-sage comes from a given source, some patterns of inference, as well as some principlesof belief revision, may seem invalid:

The fact that a piece of information comes from a certain origin or source oris transmitted by a certain medium conveys information of its own. In a shortslogan, there is no message without a medium. […] [In] some cases, the rea-soner should receive not just the content of a message, but take account of themessage-with-the-medium. (Rott 2004, p. 73)

Rott then presents belief revision theorists and formal epistemologists with a chal-lenge: how to represent the association of information with a medium? He dismissesthe extension to intensional operators of the type ‘source si says that’, given that “[the]reasoner needs to detach the message from the medium, in order to be able to utilizeits content for unmediated inferences” (Rott, loc. cit., p. 74), hence the need for anextensional (sentential) formalization. A version of the imi which could represent theset of additional questions answered by an information coming from a given sourcecould be a promising avenue, without moving to a richer language than the extensionalone of first-order logic.

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5 Conclusion

Thesis 1 is supported by the possibility to generate Expansions, Contractions (henceRevisions) of initial premises (together with some answers), by strategic use of therules. Thesis 2, by some feature of this strategic use being explained by inquiry-rele-vant factors (in this case, appeal to sources). Selection Functions widely used in brtcorrespond to choice functions encoding the (final) patterns of brackets, and choicefunctions are used in game-theory to represent strategies.

Within this model, inquiry is question-driven at two levels: (i) Potential answersto a ‘principal’ question serve as the conclusion of an iig; (i i) Instrumental questionsare put to sources as means of information retrieval.25 This model is consonant withthe use of Ultimate Partitions or Research Agendas driving the process of Expansion(and in some cases, of Contraction, in order to reopen questions or alternatives). Atwo-level model has been argued for in Genot (2009), along different lines: operationson the belief set and agenda management are performed sequentially (the agenda fol-lowing changes in the belief set). The present work is tantamount to a unification ofboth tasks at the level of iig.

Comparing sources in terms of reliability, strength, etc. could give rise to severalrepresentation results, including (hopefully): (i) Subjective Probabilities as measuresof risk of acceptation of a source’s testimony; (i i) Ranking Functions as measures of the‘strength’ needed for a new source to make an agent change one’s mind. Let us call thecondition in Proposition 6 a Uniformity Condition. We conjecture that to obtain suchrepresentation results, Uniformity Conditions not holding in the general case wouldhave to be assumed. Such a critical examination is one of the direction of researchfor the interrogative model—see Hintikka (2007b) (Essay 9) for an application to theso-called ‘cognitive fallacies’ in probabilistic reasoning.26 Other issues which can bestudied through the correspondence between brt and the imi include the conditionsneeded to generate well-behaved entrenchments (as defined e.g. in Gärdenfors 1988),or to the closely related issue of functional vs. relational revision (see Lindström andRabinowicz 1991). If, as Levi argues, several (contraction) strategies can be admissi-ble, revision becomes relational. But it is unlikely that an inquiry could genuinely beput to a halt before some decision has been reached. Hence any indecision betweenrevisions should be considered as provisional. And relational revision should be seenas a form of intermediate step in the process of revision, rather than a rival modelingof this process. Finally, we have not touched the problem of Iterated Revision, but theimi could also hopefully bring some light on this problem, too.27

25 But other uses may be possible, such as evaluation of sources by control questions, see Hintikka (2007b),Essay 7.26 Hintikka 1987 develops a extension of the interrogative framework to deal with uncertain answers,and but rather than discussing conditionalization, focuses on strategic use of probabilistic inferences (e.g.consilience of inductions, and the role of repeated ‘questions’, i.e. experiment).27 The standard way to model iteration, as involving a reordering of some sort (a new entrenchment, ora new ranking function) would probably lead to modify the proof of Proposition 3, since in this case theorder of expansions may not longer be indifferent. This problem can be related to ranking functions, whichoffer a solution to iterated revision, and we suspect that the standard use of those ranking necessitates ratherstrong uniformity conditions.

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It should also be noticed that Bracketing is not Contraction, hence it is not in need ofthe same conceptual clarification as ‘pure contraction’. Contraction becomes a struc-tural description of the outcome of a possible (sub)process of inquiry, each step ofwhich is unproblematic, since appraisal of sources is the daily bread of investigatorsof all ilk, and bracketing has clear empirical counterparts.

Choices between possible contractions can also be studied as ‘subinquiries’ aimedat determining which source to trust, etc., and following other ‘strategic principles’—see Hintikka 2007b (Essays 7 and 9). Moreover, information is not lost, since bracket-ing only prevents further applications of the rules, yet the information is still ‘there’and available for unbracketing based on further insights: not using an answer is notloosing it.

Change occurs at the outcome of a procedure, when decision is taken to put inquiryto a halt, when enough has been done to form a belief. In this case, a decision to giveup (contract) or revise will lead to a loss of what has been ‘bracketed’. But this, asHintikka reminds us, can only happen when we do have reached the point where thisdecision is possible or warranted given some standards of inquiry which are, in a verystrong way, tied to the decisions we have to make based on the answers (to ‘principal’questions) we obtained. Let us evoke Georges Simenon’s Inspector Maigret, as doesHintikka, to illustrate this point:

Inspector Maigret is sometimes asked what he believes about the case he is inves-tigating. His typical answer is “I don’t believe anything”. […] [In] one story hesays, “The moment for believing or not believing hasn’t come yet.” […] It is notthat Maigret has not carried his investigation far enough to be in a position toknow something. He does not know enough to form a belief. (Hintikka 2007b,p. 32, emphasis added.)

The very possibility to reverse the hierarchy between knowledge and belief is notthe least interest of a decision-theoretic account of the two notions.

Appendix

Proofs of representation results

For the following proofs, we assume the standard properties of a consequence opera-tor, Inclusion : A ∈ Cn(A); Monotony: if A ⊆ B, then Cn(A) ⊆ Cn(B); and Iteration:Cn(Cn(A)) = Cn(A). We furthermore assume that Cn is Compact, satisfies the Deduc-tion Property and Truth-functionality (i.e. includes the truth-functional consequencesof A). Then, φ ∈ Cn(A) abbreviates A � φ.

We need to consider the deductive closure of T , i.e. Cn(T ), since we are alsointerested in any conclusion that may follow from T without the need of further inter-rogative moves (in which case, by definition, interrogative consequence reduces todeductibility). Since A′

M is finite, and since (T + φ)+ψ = (T +ψ)+ φ, and giventhe properties of Cn, expansion by A′

M is equivalent to expansion by ∧A′M . For the

right-to-left direction, we want to prove that inquiry will establish C in M (givenT ), without assuming that some specific questions have been asked. In this case, AM

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may be infinite, but since Cn is compact, no further difficulty arises. The proof ofProposition 3 is a direct consequence of Theorem 1 and Theorem 2.

Proof of Proposition3 ⇒ Assume that C ∈ CnI(M, T ). By Theorem 1, C ∈ Cn(T ∪A′

M ) for some finite set A′M ⊆ AM of answers (needed to establish that C ∈

CnI(M, T )). Hence, since Cn(T ∪ A′M ) ⊆ Cn(Cn(T ) ∪ A′

M ), by definition of ‘+’,C ∈ T + A′

M .⇐ Assume that C ∈ Cn(T ∪ AM ), since Cn is compact—and T is, by assumption,

finite—C ∈ Cn(T ∪ A′M ) for some finite A′

M ⊆ AM . By Theorem 1, it follows that ifC ∈ Cn(T ∪ A′

M ), then C ∈ CnI(M, T ). (Theorem 2 allows to obtain A′M construc-

tively using moves ?T(φ ∨ ¬φ) for every ±φ ∈ A′M .) ��

The use of iig and iigd suggests a partition in three of the agm postulates. Firstcome the Definitory postulates (d-postulates) needing no assumption about the agent’schoices: postulates (÷1) to (÷4). Then the Restrictive postulates (r-postulates), forwhich some additional assumptions are needed (about the agent’s options and choices):postulates (÷6) to (÷8). Finally, (÷5) to hold, some sort of Uniformity Conditionis needed—i.e. some inquiry-relevant parameters have to be fixed in a certain way.Hence (÷5) could be called a Uniformity Postulate (other uniformity postulates maybe needed in some contexts, see Sect. 5.)

The r-postulates makes the link with strategies and selection functions clearer. Inorder to be stated in their full generality, their proofs indeed need assumptions aboutpatterns of choices generating selections of premises to bracket. Those selection func-tions are interpreted as choice functions, and encode strategies. The r-postulates needthe assumption that in selection functions (strategies) available in one case, are alsoavailable in another, i.e. that “applying the same strategy” is a well defined notion.

Proofs are formulated in terms of an arbitrary but constant (through repetition)choices of brackets, satisfying the conditions for cbp(T |φ) for some T and φ. We willuse the following lemma:

Lemma 8 Assume that we have constructed cbp(T |φ). Then there is a [T ]φ ⊆ T s.t.:[T ]φ = {ti ∈ T : [ti ] ∈ cbp(T |φ)}, and if no interrogative move is played (for anunderlying model M):

if φ ∈ Cn(T), then φ ∈ Cn([T]φ) (1a)

CnI(M, cbp(T |φ)) = Cn(cbp(T |φ)) = Cn(T \ [T ]φ) (1b)

Proof of Proposition3 (1a) follows by construction of cbp(T |φ); (1b) by definitionof interrogative consequence (no interrogative move is played) CnI(M, cbp(T |φ)) =Cn(cbp(T |φ)), and by Bracketing rule R3, Cn(cbp(T |φ)) = Cn(T \ [T ]φ). ��Proof of Proposition5 (1) d-postulates:

Closure: the set of interrogative consequences of cbp(T |φ) is closed by defini-tion, and if no interrogative move is played, CnI(M, cbp(T |φ)) = Cn(cbp(T |φ)), asdesired (if some answers are added, Theorem 1 applies).

Inclusion: By set theory, T \ E ⊆ T for any E . Let E = [T ]φ . By Monotony ofCn, Cn(T \ [T ]φ) ⊆ Cn(T ), and by Lemma 8, Cn(cbp(T |φ)) ⊆ Cn(T ).

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Fig. 7 Trees for cbp for T with φ,ψ and (φ ∧ ψ)

Vacuity: If φ /∈ Cn(T ) then (by Completeness of ei tree) the tree does not close.Hence there is no application of R1–R2, hence [T ]φ = ∅, and T \ [T ]φ = T . ByMonotony of Cn, elementary set theory and Lemma 8, Cn(T ) ⊆ Cn(cbp(T |φ)).

Success: If φ ∈ Cn(T ) and cbp is applied to T , φ ∈ Cn(cbp(T |φ)) when there isno premise to bracket i.e. if φ ∈ Cn(∅) (by Completeness of ei trees).

(2) r-postulates:Extensionality: We prove that if (some) cbp(T |φ) is obtained, then applying the

same bracketing to T will be sufficient to remove ψ as well from the interroga-tive consequences of T if Cn(φ) = Cn(ψ). Assume that neither φ ∈ Cn(∅) norφ /∈ Cn(T ) (in case (÷3) and (÷4) apply resp.), and that Cn(φ) = Cn(ψ). Sinceψ ∈ Cn(φ), by Lemma 8, φ ∈ Cn([T ]φ). Hence (by Monotony and Iteration of Cn),ψ ∈ Cn([T ]φ). Assume (for reductio) that ψ ∈ Cn(cbp(T |φ))—i.e. by Lemma 8again ψ ∈ Cn(T \ [T ]φ). Then, since φ ∈ Cn(ψ), it follows (by Monotony and Iter-ation again) that φ ∈ Cn(cbp(T |φ)), contrary to our assumption (since φ /∈ Cn(∅)).Hence, ψ /∈ Cn(cbp(T |φ)) after all, and by Lemma 8, ψ /∈ Cn(T \ [T ]φ). Providedthat cbp(T |ψ) can be constructed with the same strategy as cbp(T |φ), cbp(T |φ) =cbp(T |ψ). Hence (by Monotony of Cn), Cn(cbp(T |φ)) = Cn(cbp(T |ψ)). (A sym-metric argument yields the same result beginning with cbp(T |ψ).)

Conjunctive Overlap: Let T1 = T \ [T ]φ (T2 = T \ [T ]ψ ). Figure 7 displays theinitial steps of the cbp procedure, as well as the first step of building cbp(T |φ ∧ ψ).If we assume that the choices of brackets for [T ]φ and [T ]ψ correspond to strategiesavailable to the agent when applying cbp to T with (φ ∧ ψ), then, in order to obtainT3 = T \ [T ]φ∧ψ , it suffices to ‘repeat’ (or perform simultaneously) the strategy forobtaining [T ]φ to the Fφ branch (on the leftmost tree), or the strategy for obtaining[T ]ψ to the Fψ branch, or both. In the latter case, T3 = T \ [T ]φ ∪ [T ]ψ = T1 ∩ T2.In the two former cases, either T3 = T \ [T ]φ = T1, or T3 = T \ [T ]ψ = T2. Assumethat it is T1. Then, T1 ⊆ T3, and, by elementary set theory, T1 ∩ T2 ⊆ T3. The sameholds assuming that it is T2. So, in any case, T1 ∩ T2 ⊆ T3, and by monotony of Cn,Cn(T1 ∩ T2) ⊆ Cn(T3). Since Cn distributes over ∩, we have Cn(T1) ∩ Cn(T2) ⊆Cn(T3). Hence (by Lemma 8) Cn(cbp(T |φ)) ∩ Cn(cbp(T |ψ)) ⊆ Cn(cbp(T |φ ∧ ψ)as desired.

Conjunctive Inclusion: Let T1 = T \ [T ]φ and T2 = T \ [T ]φ∧ψ resp, and assumethat φ /∈ Cn(∅)— since φ /∈ Cn(cbp(T |φ∧ψ)). (Figure 7 shows the beginning of thecbp procedure for each. Since we deal with Revision rather than Update, T remainsconstant.) Since ex hypothesis (and Lemma 8)φ /∈ Cn(T2), one of the branches extend-ing Fφ (on the rightmost tree in Fig. 7) must be open after applying rules R1–R3. If we

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assume that the choice of brackets for [T ]φ∧ψ (on the Fφ branch) corresponds to a strat-egy available to the agent when applying cbp to T with φ, then [T ]φ ⊆ [T ]φ∧ψ . HenceT \ [T ]φ∧ψ ⊆ T \ [T ]φ , i.e. T2 ⊆ T1, and (by Monotony of Cn) Cn(T2) ⊆ Cn(T1).By Lemma 8, we have Cn(cbp(T |φ ∧ ψ)) ⊆ Cn(cbp(T |φ), as desired. ��Proof of Proposition6 Assume that cbp has been applied to T with φ, resulting inT ′ = cbp(T |φ). Then, if Expansion of Cn(T ′) byφ reintroduces in cbp(T |φ) premisesin [T ]φ , it yields the same result as applying R4 to cancel every applications of rulesR1 and R2. Then Cn(T ′)+ φ = Cn(T ). ��Proof of Proposition7 Assume that ψ ∈ T ÷ φ for some arbitrary agm contraction,and let’s prove that if for no ‘cbp-complying’ bracketing strategy, ψ ∈ cbp(T |φ),then T �= Cn(T ). By (÷1), ψ ∈ Cn(T ÷ φ), and by (÷2), ψ ∈ Cn(T ). If for nocbp-complying bracketing strategy, ψ ∈ cbp(T |φ), then (by the construction methodof cbp) there is no T \ [T ]φ such that ψ ∈ T \ [T ]φ (i.e. there is no way to leave ψunbracketed in T ). Hence ψ /∈ T , and as a consequence, T �= Cn(T ). Contraposing,we have: if T = Cn(T ), then there is at least one cbp-complying strategy such thatψ ∈ cbp(T |φ). ��

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