Mathematical Social Sciences ( ) –

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Mathematical Social Sciences

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Unawareness with ‘‘possible’’ possible worlds

Oliver WalkerNERA Economic Consultancy and The Grantham Research Institute on Climate Change and the Environment, London School of Economics and PoliticalScience, United Kingdom

a r t i c l e i n f o

Article history:Received 3 December 2011Received in revised form17 March 2014Accepted 19 March 2014Available online xxxx

a b s t r a c t

Logical structures for modeling agents’ reasoning about unawareness are presented where it can holdsimultaneously that: (i) agents’ beliefs about whether they are fully aware need not be veracious withpartitional information; and (ii) the agent is fully aware if and only if she is aware of a fixed domain offormulae. In light of (ii), all states are deemed ‘‘possible’’. Semantics operate in two stages, with belief inthe second stage determined by truth in the first stage. Characterization theorems show that, without thefirst stage, the structures validate the same conditions as those of Halpern and Rego (2009).

© 2014 Published by Elsevier B.V.

1. Introduction

The most widely used framework for modeling uncertainty indecision and game theory is the standard state space, which consistsof all decision-relevant states of affairs or possible worlds, eachcorresponding to a truth assignment to a set of relevant sentences.A prominent source of uncertainty in many settings is unaware-ness, where being unaware of a sentence means that one cannotconceive of it. However, Dekel et al. (1998) show that the standardstate space cannot accommodate non-trivial degrees of unaware-ness under minimal coherence conditions. This paper introducesa logical structure for describing economically interesting types ofunawareness that remains close in important respects to the stan-dard state-space; it thus aims to provide some foundations for in-corporating unawareness into mainstream economic theory.

Allowing for unawareness in economic models of decisionmakes it possible to explain an array of phenomena that standardframeworks preclude. For example, suppose Donald is unaware ofthe possibility of a bank run and chooses to invest his money inshares in HSBC. If one were to give Donald themessage, ‘‘there willor will not be a bank run on HSBC’’, standard Bayesian models of

I would like to thank Ian Jewitt, Sujoy Mukerji, Aldo Rustichini, BurkhardSchipper, and two anonymous referees for extremely helpful comments andguidance in writing this paper. All remaining errors are my own. This paperis based on Chapter Two of my DPhil thesis at the University of Oxford. Igratefully acknowledge financial support from University College, Oxford, theEconomics Department of the University of Oxford, the Grantham Foundation forthe Protection of the Environment, and the Centre for Climate Change Economicsand Policy, which is funded by the UK’s Economic and Social Research Council andby Munich Re.

E-mail address: oliver.walker@nera.com.

decision would deny any possibility of Donald changing his mindas the message is not ‘‘informative’’ in the sense of ruling out anystate of affairs. However, given Donald’s prior unawareness, themessage does impact upon Donald’s information (put another way,it changes his beliefs) and it therefore seems reasonable to allowDonald to revise his investment strategy in light of receiving it.The question of how unawareness may affect dynamic decision inthis way has been studied by Li (2008), Grant and Quiggin (2013),and Karni and Viero (2013).

Donald’s unawareness of the possibility of a bank run is an ex-ample of what this paper refers to as specific unawareness: un-awareness of a particular statement or event. Though no agentwhowas (specifically) unaware of the sentence φ could ever believeshe was unaware of φ – to do so would imply initial awareness ofφ – such an agent could hold beliefs about the more general pos-sibility that there was something (non-specific) of which she wasunaware. This kind of reasoningmay also pertain to the awarenessof other agents. For example, Donald may believe his financial ad-viser is aware of more relevant features of his investment problemthan he is (even though, of course, he can have no idea what theseadditional features might be), and thus defer to his adviser’s betterjudgment when it comes to his decision.1 In this paper the phe-nomenon of holding beliefs concerning the extent of one’s own orother agents’ awareness is called conscious unawareness.

Conscious unawareness may have a range of interestingeconomic consequences. Tirole (2009) argues that if contractingparties believe they are unaware, it may be efficient for themto allow scope for renegotiation in case they become aware of

1 This echoes an example in Halpern and Rego (2009).

http://dx.doi.org/10.1016/j.mathsocsci.2014.03.0030165-4896/© 2014 Published by Elsevier B.V.

2 O. Walker / Mathematical Social Sciences ( ) –

more relevant facts in future (that is, it may be efficient to writeincomplete contracts). Other applications are interactive: Halpernand Rego (2009) suggest that a local doctor may refer her patientto a specialist because she believes the specialist is aware ofrelevant things that she is not, while Heifetz et al. (2012) arguethat differing degrees of awareness among agents may lead themto speculatively trade against each other. Walker and Dietz (2011)claim that even in static single-agent settings, a decision maker’sconscious unawarenessmay leadher to violate subjective expectedutility.

Frameworks for modeling conscious unawareness should beconsistent with agents satisfying reasonable introspective proper-ties. For instance, as Dekel et al. (1998) argue, it should at least bepossible that an agent who is unaware of φ does not believe she isunaware of φ. Further introspective properties are highlighted bythe examples below:

Example 1: Martha is an inexperienced gambler playing a cardgame with her friends. On the basis of her understandingof the rules of the game, she believes that her currenthand cannot be beaten. This belief (and the conceptionof the rules on which it is founded) is correct. However,when the time comes for her to place a bet, she decidesnot to gamble, reasoning that there may additional rulesof which she is unaware that could cause her to losemoney.

Example 2: Eunice is a social worker visiting a family she hasknown for several years. She interviews the familymembers and observes their behavior, watching for signsof anything that would be of concern to social services,and then writes a report stating her conclusions. Givenher longstanding acquaintance with the family and yearsof professional experience, she is confident that she isaware of all of the relevant concerns and, furthermore,she has sufficient evidence to conclude that none of themapply to the family. She therefore writes that there isnothing to beworried about. This assessment later proveswrong.

While it is true that Martha and Eunice both choose in ways thatturns out to be contrary to their interests, none of the examples isintended to portray unreasonable or irrational beliefs or behavior.

Consider Martha’s case and let r be one of the rules of the game.It is natural to suppose that the fact of Martha’s awareness of r isself-evident to her and thus that she believes she is aware of r . Afterall, the very fact that she uses r to deduce that she has a winninghand should be enough to convince her that she is aware of it. Giventhis, an economist modeling a consciously unaware agent mightwish to require that for any statement φ:

The agent is aware ofφ H⇒ She believes she is aware ofφ. (1)

By contrast, there seems to be little justification for assumingthat the extent of Martha’s awareness is self-evident to her. Foralthough she can have no idea what an additional rule to the gamemight involve, this could also be the case if therewas a rule shewasunaware of. Thus, her inexperience alone gives her good groundsto suspect that there she is not fully aware. Models of consciousunawareness should therefore be consistent with:

The agent is fully aware H⇒ She believes she is fully aware. (2)

Turning to the case of Eunice, similar reasoning applies. She shouldnot be expected to recognize the existence of problems outsideher conception since, by their nature, these problems could notbe self-evident. And given her close knowledge of the family, it isreasonable for her to suppose that theremay be no problems of this

kind. This suggests that models of conscious unawareness shouldsatisfy:

The agent is not fully awareH⇒ She believes she is not fully aware. (3)

However, it turns out that (1) is inconsistent with (2) and (3) in themost straightforward types of model for conscious unawareness.To see this consider the case of Martha again. According to (1)whatever she is aware of, she believes she is aware of, andwhatever she believes must be true in every state of the world sheconsiders possible. This means that in every state she regards aspossible, her level of awareness is at least as great as it is in thetrue state of the world. But this implies that she is fully aware inevery state she considers possible, which means that she believesshe is fully aware, a violation of (2).

This difficulty, which is present in Halpern and Rego (2009),has been recognized in the literature and several authors2 haveproposed alternative approaches that circumvent the problem.Theseworks resolve the inconsistency by allowing the set of thingsthe agent couldbe aware of to vary fromstate to state—thus, thoughMartha is aware of the same things in all states she regards aspossible, it need not be that she is aware of everything there is tobe aware of in all these states so it does not follow that she believesshe is fully aware.

Varying the domain of ‘‘things to be aware of’’ across statesof the world in this way allows one to reconcile (1) with (2) and(3), but the resulting state space is no longer of the kind familiarfromSavage (1954). In the latter type of state space, states stand fora complete and consistent description of reality, where notions of‘‘completeness’’ and ‘‘consistency’’ are fixed across states. Yet if theset of ‘‘things to be aware of’’ differs from state to state, it must bethat the consistency of the sentence ‘‘the agent is fully aware’’ witha given level of awareness is not fixed. Thus, it cannot be that allstates use a single, ‘‘objective’’, criterion of consistency: those thatdepart from this – for instance, any state used to model Example 2where the set of rules is greater than it really is – are described inthis paper as ‘‘impossible’’ states of the world.

There are at least three reasons why it is desirable to modelconscious unawareness using only ‘‘possible’’ states. The first isdecision theoretic. In order to characterize a decision maker’sbeliefs in revealed preference terms,3 the usual approach is tospecify payoffs to different courses of action under every state andendow the decision maker with conditional preferences in everystate of the world.4 But if a state of the world is impossible, it isdifficult to see how payoffs and conditional preferences under thisstate could ever be defined.

Second, working with a framework that includes both possibleand impossible states of the world complicates any reductionfrom a linguistic to a set-theoretic representation of the agent’sreasoning. As was shown by Dekel et al. (1998), set-theoreticmodels for non-trivial specific unawareness must enrich thestandard Savage framework by describing both what the agent’slevel of awareness is andwhat events she regards as possible.5 Set-theoretic models for conscious unawareness that use impossiblestates (for example, Board and Chung, 2009) need to elaborate thisframework further by specifying, for each state, what degree ofawareness amounts to ‘‘full’’ awareness.

2 These include Sillari (2008), Board and Chung (2011), and Halpern and Rego(2012).3 Morris (1996) shows how this may be done in a framework without unaware-

ness. Schipper (2011) and Li (2008) offer revealed-preference characterizations ofbelief in the presence of specific unawareness.4 Though note the contribution of Lipman (1999).5 Heifetz et al. (2006) and Li (2009) offer different approaches to this.

O. Walker / Mathematical Social Sciences ( ) – 3

Finally, a pragmatic but arguably more important reason forworkingwith only possible states is that this has been the approachused in the overwhelming majority of existing economic theory.If the goal is to integrate conscious unawareness with this bodyof work in a manner that is accessible to non-experts, it seemssensible to avoid departures from its existing standards whereverthis is feasible.

This paper presents a logical structure for modeling consciousunawareness where all states are possible and where (1), (2), and(3) are nonetheless mutually consistent. It achieves this by mak-ing the agent’s conscious awareness in any state dependent on herconjecture about her level of awareness in that state rather thanon a comparison between her actual awareness and the domain of‘‘things to be aware of’’. The semantics of these structures works intwo stages. At the first stage, ‘‘conjectured truth’’ is assigned to log-ical formulae in all states for each agent. For any formula that doesnot include strings of the form ‘‘agent i is fully aware’’ or ‘‘agent i isaware of everything agent j is’’, this first-stage truth is determinedas it would be under conventional semantic rules. The formulae ‘‘iis fully aware’’ or ‘‘i is aware of everything j is’’, by contrast, aretrue at a state for an agent if and only if she conjectures them tobe so in that state. The second stage then assigns truth to formulaein a similar way to Fagin and Halpern (1988), with the exceptionthat an agent believes a formula if and only if she is aware of it andconjectures it true in every state she considers possible. Oneway ofinterpreting this two-stage approach is as allowing the agent’s con-scious unawareness to be determined by some process of inductivereasoning (as in Grant and Quiggin, 2013), whereas her other be-liefs are arrived at by a process of deduction fromwhat is true (andshe is aware of) in the states she deems possible.

As well as the two-stage semantics, a notable feature of theframework proposed here is that it follows Agotnes and Alechina(2007) in treating the formulae ‘‘agent i is fully aware’’ and ‘‘agent iis aware of everything agent j is’’ as propositional constants (opera-tors that function as primitive propositions) rather than quantifiedsentences over propositional variables. The disadvantage of doingthis is that it means the language presented in the sequel is less ex-pressive than others in the literature, including that of Halpern andRego (2009),6 though the logic here can express all of the examplesdiscussed in the Introduction. On the other hand, an advantage ofthe approach is that it greatly simplifies the language, obviatingthe need for propositional variables and quantification, which canpresent both notational and conceptual difficulties. A further ad-vantage of this – as shown inWalker (2012) – is that it facilitates arelatively straightforward translation of the logical structures pre-sented here into a set-theoretic framework based on that of Heifetzet al. (2006) for modeling specific unawareness.

The remainder of the paper is organized as follows. Section 2presents the elements of the logical structures including the two-stage semantics, while Section 3 discusses various introspectiveaxioms. Section 4 characterizes the structures axiomatically undera number of well known restrictions, showing that when theagent’s conjecture about her conscious unawareness is constrainedto be veracious, the structures are equivalent to those in Halpernand Rego (2009). Section 5 concludes with a brief review of someof the other related literature.

2. Two-stage semantics for awareness and belief

Consider a formal language, denoted L, for expressing the kindof reasoning that was discussed in the Introduction where there

6 For example, one cannot construct a formula for ‘‘Agent i believes that there isno more than one proposition that j is aware of and k is not’’ here, whereas one canin Halpern and Rego’s logic.

are N agents. L consists of a countably infinite (and no larger) setof primitive propositions, P , the propositional constants Ci and Rijfor i, j = 1, . . . ,N , the logical connectives ¬ and ∨, the brackets[ and ], and the modal operators Ai, Li, and Bi for i = 1, . . . ,N .The primitive propositions are a set of claims about the world thatdo not concern the decision maker’s beliefs or awareness, suchas ‘‘it will rain tomorrow’’ or ‘‘demand for bread will be high’’.The string Ci stands for the claim ‘‘agent i is fully aware’’ and Rijis read ‘‘agent j is aware of everything i is aware of’’. The use ofpropositional constants to express these claims and the notation Ciand Rij is from Agotnes and Alechina (2007). The notation reflectstheir terminology ‘‘i has complete awareness’’ for Ci and ‘‘j hasrelative awareness compared to i’’ for Rij.

Anymember of P , any Ci, and any Rij is a formula of the language,the full set of which is denoted Φ . Φ is then closed under thefollowing formation rules, for i = 1, . . . ,N:

φ ∈ Φ H⇒ ¬φ, Aiφ, Liφ, Biφ ∈ Φ

φ,ψ ∈ Φ H⇒ [φ ∨ ψ] ∈ Φ.

The connectives ¬ and ∨ respectively stand for ‘‘not’’ and ‘‘or’’,and allow formulae expressing negation and disjunction to beconstructed from themembers ofΦ . Thus, ifφ andψ are formulae,then so are¬φ and [φ∨ψ]. Themodal operators Ai, Li, and Bi, read‘‘agent i is aware of the formula’’, ‘‘agent i implicitly believes’’, and‘‘agent i (explicitly) believes’’, allow one to construct statementsabout the agent’s epistemic status (how such statements shouldbe interpreted is discussed below). As is standard: [φ ∧ ψ] (‘‘φand ψ ’’), [φ ⇒ ψ] (‘‘φ implies ψ ’’), and [φ ⇔ ψ] (‘‘φ if andonly if ψ ’’) will be used as shorthand for ¬[¬φ ∨ ¬ψ], [¬φ ∨ ψ]

and [[φ ⇒ ψ] ∧ [ψ ⇒ φ]] respectively throughout this paper,and brackets will generally be suppressed unless doing so leads toambiguity.

A structure, D, in L, the full set of which is denoted D , is a tupleΩ,V,P1, . . . ,PN ,A1, . . . ,AN ,X1, . . . ,XN.Ω is a state spaceconsisting of possible worlds or states of affairs. V : Ω → 2P isa truth assignment to the primitive propositions, listing those thatare true in each state, and Pi : Ω → 2Ω is agent i’s possibilitycorrespondence, where Pi(ω) stands for the set of states that theagent considers possible when the true state is ω. Ai : Ω → 2P isthen i’s awareness correspondence, mapping each state to the setof propositions that the agent is aware of in that state. Finally,where O is the set of preorders, ≽, on N ∪ P that satisfy P ∈

max≽N ∪ P, Xi : Ω → O describes, for each state, theconjecture the agent i has about the relative extent of all agents’awareness, with P corresponding to the level of full awareness.Writing≽

iω forXi(ω), j≽i

ω kmeans i conjectures that j’s awarenessin ω is more extensive than k’s, and j≽i

ω P means i imagines j isfully aware in ω. Note that P ≽

iω j means simply that i conjectures

j to be no more than fully aware: it emphatically does not imply i’sawareness of the members of P .

In what follows, a subformula of any formula φ is anyconsecutive string of characters inφ that is itself a formula ofL. Forexample, if φ = Ai[p∨Bjq], then Ai[p∨Bjq], p∨Bjq, Bjq, p, and q arethe subformulae of φ. Use Sub(φ) for the set of subformulae of anyφ ∈ Φ and define the referents of φ as Ref(φ) := Sub(φ) ∩ P (thatis, Ref(φ) is the set of primitive propositions that are subformulaeof φ).

The semantics of D comprises a set of rules that assign truth toevery formula in Φ in every state inΩ according to V , Pii=1,...,n,Ai=1,...,n, and Xi=1,...,n. The rules presented here are unusual inthat they operate in two stages: first, there is a set that establisheswhat is ‘‘conjectured true’’ or ‘‘stage-one true’’ in each state; andsecond, there are rules for assigning ‘‘stage-two truth’’, or simply‘‘truth’’ to the formulae. The role of stage-one truth is to determinewhat the agent believes and at this level the truth of the formulaeRij and Ci depends – for each agent – on that agent’s conjecture

4 O. Walker / Mathematical Social Sciences ( ) –

rather than the extent of his awareness or any other agent’sawareness. Stage-two truth differs in that Ci is true at this levelif and only if agent i is in fact aware of all propositions, and Rijis true iff j really is aware of everything i is. As will be shown,this divergence between the ‘‘conjectured’’ way in which stage-one truth is assigned to Ci and Rij and the ‘‘objective’’ standardapplied to it for stage-two truth is what allows the members of Dto satisfy the various introspection properties propounded in theIntroduction.

Use the operator V i1 : Ω × Φ → 0, 1 to denote the formulae

that are stage-one true according to i’s conjecture at each state of agiven structure. V i

1(ω, φ) = 1 is read ‘‘φ is stage-one true for agenti in state ω’’ and V i

1(ω, φ) = 0 means φ is first-stage false for i atω. The operator is defined inductively, for all i as follows:

S1. V i1(ω, p) = 1 iff p ∈ V(ω)where p ∈ P;

S2. V i1(ω, Cj) = 1 iff j≽i

ω P;S3. V i

1(ω, Rjk) = 1 iff k≽iω j;

S4. V i1(ω,¬φ) = 1 − V i

1(ω, φ);S5. V i

1(ω, [φ ∨ ψ]) = maxV i1(ω, φ), V

i1(ω,ψ)

;

S6. V i1(ω, Ajφ) = 1 iff Ref(φ) ⊆ Aj(ω);

S7. V i1(ω, Ljφ) = 1 iff V j

1(ω′, φ) = 1 for all ω′

∈ Pj(ω); andS8. V i

1(ω, Bjφ) = 1 iff V i1(ω, Ajφ) = 1 and V i

1(ω, Ljφ) = 1.

S1, S4, and S5 are familiar from propositional calculus. S1 requiresthat p is stage-one true for i in stateω if and only if it is true in thatstate according to V , while S4 ensures the negation of φ is stage-one true for i iff φ is stage-one false for i and S5 makes [φ ∨ ψ]

stage-one true for i iff at least one of its disjuncts is stage-one truefor i.

S6–S8 govern awareness and belief and operate in an anal-ogous fashion to their equivalents in Fagin and Halpern (1988)and Halpern (2001). S6 implies that it is stage-one true for i thatagent j is aware of φ in ω iff j is aware of all of φ’s referents in ω.This is what Halpern (2001) calls ‘‘awareness generated by primi-tive propositions’’. S7 says that it is stage-one true for i that j im-plicitly believes φ in ω iff φ is stage-one true for j in every state jconsiders possible at ω, and S8 then identifies belief with implicitbelief plus awareness.

The rules suggest that implicit belief can be interpreted as acomponent of belief rather than an epistemic property in its ownright: part of believing φ is considering possible only states whereφ is true, but there is no state of mind that corresponds this. Toillustrate, though under restrictions considered later on agent iwillnever be able to believe she is both fully aware and unaware of theformulaφ, itmay still be that, for someω, V i

1(ω, Li(¬Aiφ∧Ci)) = 1.The latter simply means that, in every state i considers possiblein ω, it is (stage-one) true that ¬Aiφ and Ci—no interpretation isoffered in terms of the agent’s thoughts (if she has any) concerningthe formula ¬Aiφ ∧ Ci.

The novel part of the stage-one semantics – and the respect inwhich these semantics are ‘‘conjectured’’7 – comes in S2 and S3.This rule states that Cj is stage-one true for i in ω if and only if i’sconjecture is that j’s awareness is comprehensive in ω (S2), andthat Rjk is stage-one true for i at ω if and only if i conjectures kto have greater awareness than j in ω (S3). The way in which thestage-one truth of Cj and Rjk (and other formulae of which theseare subformulae) in any given state is determined therefore differsfrom other formulae in that it depends on the agent’s conjecturerather than features of the state itself.

7 Clearly not all formulae such that V i1(ω, φ) = 1 are really conjectured by i to be

true in stateω—if, for example,φwere¬Aiψ , iwould be unaware ofφ inω and thusunable to form a conjecture about it. Oneway of thinking of the first-stage semanticvalue assigned to a formula in a state is as its truth as supposed by a hypotheticalagent with full awareness who nonetheless shared i’s conjecture.

Second-stage truth,which determines inwhich states any givenformula does hold true, is assigned according to a second set ofrules. Where V (ω, φ) = 1 means φ is stage-two true (or simply‘‘true’’) in state ω and V (ω, φ) = 0 means φ is false at ω, theserules are as follows:

O1. V (ω, p) = 1 iff p ∈ V(ω)where p ∈ P;O2. V (ω, Ci) = 1 iff Ai(ω) = P;O3. V (ω, Rij) = 1 iff Aj(ω) ⊇ Ai(ω);O4. V (ω,¬φ) = 1 − V (ω, φ);O5. V (ω, [φ ∨ ψ]) = max V (ω, φ), V (ω,ψ);O6. V (ω, Aiφ) = 1 iff Ref(φ) ⊆ Ai(ω);O7. V (ω, Liφ) = 1 iff V i

1(ω′, φ) = 1 for all ω′

∈ Pi(ω); andO8. V (ω, Biφ) = 1 iff V (ω, Aiφ) = 1 and V (ω, Liφ) = 1.

Most of the rules function in essentially the same manner as theirequivalents in S1–S8. The exceptions are O2, O3, andO7. O2 andO3use ‘‘objective’’ criteria in place of the agent’s conjecture to assigntruth to formulae of the form Ci and Rij, stating simply that Ci istrue in ω if and only if the agent is aware of all propositions (andtherefore, by O6, all formulae) in ω, while Rij is true in ω if andonly if agent j is aware of all propositions that agent i is aware of.O7 marks the point of contact between the first- and second-stagesemantics, and implies that i implicitly believes φ in ω if and onlyif φ is stage-one true for i in every state she considers possible atω. Thus, whether or not an agent believes a formula depends on itsstage-one truth in the states she regards as possible and, for thoseformulae in which something of the form of Cj or Rjk appears as asubformula, this can be different from its second-stage truth.

Note that all of the states in any structure inD are ‘‘possible’’ inthe sense outlined in the Introduction as the consistency of Ci andRij with any given level of awareness is the same in all states.

3. Properties of awareness

It remains to be shown what the intrinsic properties of Dare, and whether its structures can be used to describe the sortof reasoning portrayed in the Introduction. A useful concept forinvestigating these matters is validity. A set of formulae ∆ will besaid to be valid in the structure D iff, for every φ ∈ ∆, V (ω, φ) = 1for all ω in D’s state space.∆ is then valid in D∗

⊆ D iff it is validin all D ∈ D∗, and if it is valid in D , it will be described simply asbeing valid.

Many of the sets of formulae described in this section arepresented as schemes, where a scheme is a formula, certainsubformulae of which can be replaced uniformly by any otherformula from a particular domain. For example, the scheme φ ∨

[ψ ∧ ¬φ] for any φ,ψ ∈ Φ contains p ∨ [q ∧ ¬p] but notp∨ [BiCj ∧ ¬Ljr], since φ is not uniformly replaced by any formulain the latter case. A scheme is valid in some D∗

⊆ D wheneverevery instance of it is valid in D∗.

There are certain schemes whose validity is necessary if the Aiand Bi operators are to be plausibly interpreted as representingawareness and belief across all D . Consider the following sets,where φ can be any member ofΦ and i ∈ 0, . . . ,N:

PL ¬Aiφ H⇒ ¬Biφ ∧ ¬Bi¬Biφ; andAU ¬Aiφ H⇒ ¬Ai¬Aiφ.

PL, which stands for ‘‘plausibility’’8, is a formal statement of the factthatwhenever an agent is unaware of something, he cannot believeit and cannot believe that he does not believe it. Similarly, AU (for‘‘awareness–unawareness introspection’’) expresses the fact that

8 This terminology, and that of AU introspection, is due to Dekel et al. (1998).

O. Walker / Mathematical Social Sciences ( ) – 5

an agent who is unaware of φ cannot be aware that he is unawareof φ. Any structure in which either PL or AU were not valid wouldnot be suitable for modeling belief and awareness, but it is easy toshow that both are valid in D .

The validity of other sets of formulae shows what sort ofawareness the structures in D can model. Take the followingformulae, where once again φ and ψ can be any formulae andi, j, k ∈ 1, . . . ,N:

A1 Ai¬φ ⇐⇒ AiφA2 Ai[φ ∧ ψ] ⇐⇒ Aiφ ∧ Aiψ;A3 AiLjφ ⇐⇒ Aiφ;A4 AiAjφ ⇐⇒ Aiφ;A5 AiBjφ ⇐⇒ Aiφ;A6 AiCj;A7 AiRjk;C1 Ci H⇒ Aiφ;C2 Ci H⇒ Rji;C3 [Ci ∧ Rij] H⇒ Cj;R1 Rij H⇒ [Aiφ ⇒ Ajφ];R2 [Rij ∧ Rjk] H⇒ Rik; andR3 Rii.

A1–A5 jointly correspond to ‘‘awareness generated by primitivepropositions’’ – Aiφ iff Aip for all p ∈ Ref(φ) – which seems to bea natural form of awareness to attribute to an economic agent.9One way of interpreting A1–A5 is the assumption that the agentis aware of all aspects of the grammar of L (i.e. the rules forconstructing formulae) but not necessarily all of the vocabulary(i.e. the propositions). Seen from this perspective, A6 and A7 thenimply that i’s vocabulary always encompasses formulae of the typeCj and Rjk. It is straightforward to show that all of A1–A7 are validin D and that A1 and A4 entail AU.

C1–C3 formalize the interpretation of Ci as ‘‘agent i is awareof all formulae’’ while R1–R3 formalize Rij as ‘‘agent j is aware ofeverything i is aware of’’. These axioms all feature in Agotnes andAlechina (2007) and are all valid in D .

A further group of formulae can be regarded as conditions onthe rationality of the agent under consideration. Where φ can beany member ofΦ , none of the following is valid in D:

PI∗ Biφ H⇒ BiBiφ;NI∗ ¬Biφ ∧ Aiφ H⇒ Bi¬Biφ; andBA Aiφ H⇒ BiAiφ.

PI∗ (for ‘‘positive introspection’’) and NI∗ (‘‘negative introspec-tion’’) respectively state that an agent believes she believes what-ever formulae she believes and that she believes she does notbelievewhatever formulae she is aware of but does not believe.BA,which as (1) was endorsed in the Introduction, then says that shebelieves she is aware of whatever she is aware of. Considered to-gether, the three properties amount to an agent knowing the con-tent of her own mind: she knows the formulae she is aware of andknows whether or not she believes each of these formulae. If partof being rational is deducingwhat is self-evident and the content ofthe agent’s mind is self-evident to her, then PI,NI, and BA functionas rationality conditions.

So far much of the discussion has concerned ways in which i’sbeliefs about the Cj and Rjk need not be constrained by the truth ofor her beliefs about other formulae. However, inmany applicationsit is sensible to impose some restrictions in this regard. Considerthe following, where φ may be any formula inΦ:

C4 Aiφ H⇒ Bi¬Ajφ ⇒ ¬Cj

; and

R4 Aiφ H⇒ Bi[Ajφ ∧ ¬Akφ] ⇒ ¬Rjk

.

9 Halpern (2001) describes other types of awareness, which could be accommo-dated in structures very similar to D .

C4 implies that whenever i is aware of some formula and believesj is unaware of it, she may not believe j is fully aware. Similarly,R4 says that whenever i is aware of a formula, she must believethat if j is aware of it but k is not, j is not aware of everything k is.The two axioms thus state minimal conditions on the coherenceof i’s beliefs about the extent of agents’ awareness. Neither is validin D .

In some economic theories, rationality also commits the agentto having only true beliefs. The argument for this is that in order tohold false beliefs, an agent must misinterpret whatever evidenceof the true state of the world she has at his disposal, and such anagent could not be said to deduce beliefs fromevidence in a rationalfashion. However, since in D the agent’s beliefs concerning someformulae about full and relative awareness are determined by aconjecture rather than true features of the states she considerspossible, in this paper a somewhat weaker veracity condition isconsidered that is applied only to a restricted domain. As explainedbelow, this weaker condition is valid across a subset of D ofparticular interest to economists. Define the restricted domain,denotedΦ∀−, inductively as follows:

p ∈ P H⇒ p ∈ Φ∀−

φ ∈ Aiψ, Liψ, Biψ H⇒ φ ∈ Φ∀−

φ ∈ Φ∀−H⇒ ¬φ ∈ Φ∀−

φ, χ ∈ Φ∀−H⇒ [φ ∨ χ ] ∈ Φ∀−

where ψ may be any formula in Φ and i = 1, . . . ,N . Φ∀− is theset of formulae whose stage-two truth in any state in any D ∈ Dis always the same as their stage-one truth in that state.10Therationality condition can now be stated as:

T∗ Biφ H⇒ φ for any φ ∈ Φ∀−;

T∗, which can be interpreted as allowing for delusion concerningthe formulae inΦ \Φ∀− but not for those inΦ∀−, is not valid inD .

Finally, there are some formulae whose validity would entailthe agent’s having overly strong powers of cognizance. Considerthe following:

C5 Ci H⇒ BiCi; andC6 ¬Ci H⇒ Bi¬Ci.

C5 and C6 formalize the notion that an agent should always knowwhether or not he is aware of everything. As was argued in theIntroduction (they are the negations of (2) and (3)), this is not acompelling property in a wide range of economic applications, sostructures inwhich either formula is valid should be avoidedwhenmodeling such environments. Neither formula is valid in D .

4. Characterization results

Characterization theorems identify those formulae that arevalid in a given class of structures in D . They thus show preciselywhat can and cannot be modeled using the structures in question.The purpose of this section is to state a characterization theoremfor the whole of D , as well as for certain subsets of D inwhich the rationality conditions introduced above are validated.A further characterization is provided for the structures in Dthat correspond to those in Agotnes and Alechina (2007), andit is shown that these cannot be reconciled with the rationalityconditions without entailing C5 and C6.

The sets of formulae that characterize various subsets of Dwill be presented as axiom systems. An axiom system consists ofa number of foundational schemes known as axioms and a set of

10 Note that this is not the same as φ : Ci ∈ Sub(φ) and Rij ∈ Sub(φ) foranyi, j =

1 . . .N—for instance AiRjk ∈ Φ∀− .

6 O. Walker / Mathematical Social Sciences ( ) –

deductive rules, which take the form ‘‘from φ1, . . . , φn concludeψ ’’. A proof of a formula, φn, in an axiom system is a finite list offormulae, φ1, . . . , φn, such that for each φi in the list, either φi is aninstance of an axiom scheme, or there is a deductive rule that saysφi can be concluded from some subset ofφ1, . . . , φi−1. A theorem ofa system is any formula for which there exists a proof within thatsystem, and a system characterizes some D∗

⊆ D iff the set oftheorems of the system is identical to the set of formulae that arevalid in D∗.

All of the axiom systems to be considered in this section includethe following:

PC Any tautology in propositional calculus; andK [Liφ ∧ Li[φ ⇒ ψ]] H⇒ Liψ (Distribution)

where φ,ψ ∈ Φ and i ∈ 1, . . . ,N. The tautologies of proposi-tional calculus are those formulae that are valid in virtue of rulesO4 andO5. Distribution states that if an agent implicitly believes animplication and its antecedent, he must implicitly believe its con-sequent. PC and K are common to all ‘‘normal’’ axiom systems inmodal logic, where belief may be characterized using a possibilitycorrespondence (see Hughes and Cresswell (1996) or Fagin et al.(1995) for more details).

A further axiom that is present in all the systems discussed hereis equivalent to the definition of belief given in O8:

A8 Biφ ⇐⇒ Aiφ ∧ Liφ

for any φ ∈ Φ . Note that A8 in combination with A1 and A5 entailsPL.

All axiom systems in this paper include the following set ofdeductive rules, where i ∈ 1, . . . ,N and φ,ψ ∈ Φ:

MP From φ and φ ⇒ ψ infer ψ (Modus Ponens); andN If φ can be obtained without the use of C1 or R1 infer Liφ

(Necessitation).

Modus Ponens and a variant of Necessitation are common to allmainstream axiom systems in modal logic. MP allows one to per-form deductive inference in proving theorems in an axiom system,while N implies that an agent can deduce any theorem obtainedwithout the use of C1 and R1 provided she is aware of it. An un-usual feature of N is the fact that it applies only to formulae thatcan be derived without C1 or R1, reflecting the fact that the agent’sbeliefs concerning full and relative awareness are determined byher conjectures. As will be shown below, strengthening N to theversion that is more common in the literature precludes the exam-ples that were used to motivate this approach.

Finally, stronger versions of some of the rationality conditionsdescribed above will be used in the characterization results:

PI Liφ H⇒ LiLiφ for any φ ∈ Φ;NI ¬Liφ H⇒ Li¬Liφ for any φ ∈ Φ; andT Liφ H⇒ φ for φ ∈ Φ∀−

for any i ∈ 1, . . . ,N. PI, NI, T are difficult to interpret in termsof rationality since the operator L does not represent an epistemicproperty in its own right. The fact that the characterizations belowuse these conditions in place of PI∗, NI∗, and T∗ is thus a weakness,especially from the point of view of economics and game theory,where agents’ interactive reasoning about each others’ states ofmind is of central interest. Nonetheless, it should be noted that,given A1–A8, the three conditions respectively imply PI∗, NI∗,and T∗.

To state the main characterization result, write AB for thesystem comprising the axioms PC, K, A1–A8, C1–C3, and R1–R3,and the deductive rules MP and N. The notation ABX then refersto AB supplemented with all the axioms in X (so, for example,ABT,PI

= AB ∪ T, PI). Let D r⊂ D be the set of structures

in which Pi is reflexive for all i, D t⊂ D be those where Pi is

transitive for all i, De be those where Pi is Euclidean11for all i,Db be those where ω ∈ Pi(ω

′) implies Ai(ω) ⊇ Ai(ω′) for all

i, and let Do be those structures where Ai(ω) ∩ Aj(ω) ⊇ Ai(ω) ∩

Ak(ω) implies j ≽iω k for all i, j, k,12 conditional on i’s awareness, i’s

conjecture is The shorthandDx,y,...,z is used forDx∩Dy

∩· · ·∩Dz .

Theorem 1. The following hold:

i. D is characterized by AB;ii. D t is characterized by ABPI;iii. De is characterized by ABNI;iv. D r,t,e is characterized by ABPI,NI,T;v. Db is characterized by ABBA;vi. D r,t,e,b,o is characterized by ABPI,NI,T,BA,R4;

And if Dz , Dz′ , . . . , Dz′′ are respectively characterized by ABx, ABx′ ,. . . , ABx′′ , then Dz,z′,...,z′′ is characterized by ABx,x′,...,x′′ .

The theorem shows that the rationality conditions can be neatlycharacterized in terms of properties of Pi, Ai, and Xi. Part (iv)is somewhat weaker than standard results of this kind, whichtypically show that a class of models with reflexive probabilitycorrespondences is characterized by a system containing an axiomakin to T. This reflects the fact that T is unusually weak, applyingonly to members of Φ∀−. Further, there are two notable featuresof (vi). First, the rationality condition C4 can be proven inABPI,NI,T,BA,R4 and so is valid in D r,t,e,b,o. Second, in a companionwork, Walker (2012) describes a class of set-theoretic structuresfor modeling conscious unawareness13 that can be interpreted asbeing equivalent to D r,t,e,b,o. The axiomatization of D r,t,e,b,o maythus also apply to these structures.

Now consider a new subset of D , DT , containing all thosestructures where j≽i

ω P iff Aj(ω) = P and k≽iω j iff Ak(ω) ⊇

Aj(ω) for all ω ∈ Ω . The structures in DT are those where theagents’ conjectures about full and relative awareness are alwaysveracious, a feature that renders the first-stage element of thesemantics redundant (that is, it implies V i

1(ω, φ) = V (ω, φ) forall ω ∈ Ω , all i = 1, . . . ,N , and every φ ∈ Φ). The semantics ofthese structures are effectively those of the subclass of structuresin Agotnes and Alechina (2007) where awareness is generated byprimitive propositions, and it is clear that DT ⊂ Do.

A more conventional version of the derivation rule fornecessitation can be expressed as follows:

N′ From φ infer Liφ.

And a more standard rendition of the truth axiom is:

T′ Lφ H⇒ φ.

T′ is just the same as T, except it applies to all formulae instead ofonly those belonging toΦ∀−.

Use ABxT for the axiom system that is identical to ABx in every

respect, except thatN is replaced byN′. The following result, whichlargely parallels Theorem 1, can be proved for DT :

Theorem 2. The following hold:

i. DT is characterized by ABT ;ii. D t

T is characterized by ABPIT ;

iii. DeT is characterized by ABNI

T ;

11 Pi is Euclidean iff ω′, ω′′∈ Pi(ω) implies ω′

∈ Pi(ω′′).

12 One way of interpreting the structures in Do is as those where the followingholds: for any agent i in any state ω, i’s conjecture at ω is consistent with otheragents’ awareness of those formulae i is aware of at ω.13 These structures are themselves adapted from those introduced byHeifetz et al.(2006) for modeling specific unawareness.

O. Walker / Mathematical Social Sciences ( ) – 7

iv. D rT is characterized by ABT′

T ; andv. Db

T is characterized by ABBAT .

And if DxT ,D

yT , . . . ,D

zT are respectively characterized by ABx

T , AByT ,

. . . , ABzT , then Dx,y,...,z is characterized by ABx,y,...,z

T .

The principal advantage of using the structures in DT is thatit allows one to do without the two-stage semantic apparatus— there is no difference between first- and second-stage truthso one may evaluate a formula’s truth in any state using the V i

1operator where i can be chosen arbitrarily. However, as has beennoted by Board and Chung (2011) and Sillari (2008), the structureshave the undesirable property that BA cannot hold without C5 andBA, T,NI, PI do not hold unless C6 does.

Proposition 1. C5 is a theorem of ABBAT and C6 is a theorem of

ABBA,T,NI,PIT .

5. Other literature and concluding remarks

The logical structures defined here draw on the contributionsof many other authors. Early pioneering work in this field in-cludes Fagin and Halpern (1988) andModica and Rustichini (1994,1999), who present structures for modeling specific unaware-ness. Halpern (2001) shows that the latter structures are equiv-alent to the former when awareness is generated by primitivepropositions and T′, PI, and NI hold. Thijsse (1996) describes astructure for specific unawareness that assigns truth using two se-mantic stages, but in other respects his approach is quite differentto that proposed here.

Li (2009) and Heifetz et al. (2006) give set-theoretic structuresfor modeling specific unawareness, the latter with multipleagents. Heinsalu (2012) shows Li’s structures to be equivalentto the class of single-agent logical structures from Fagin andHalpern, while Heifetz et al. (2008) and Halpern and Rego(2008) demonstrate an equivalence between Heifetz et al. (2006)and the multi-agent structures of Fagin and Halpern. A furtherinteractive, set-theoretic approach is provided by Galanis (2013),which Galanis (2011) compares with Heifetz et al. (2006).

Halpern and Rego (2009) extend Fagin and Halpern to allow forconscious unawareness, though as noted earlier (1) is inconsistentwith (2) in this framework. Board and Chung (2011); Sillari(2008), and Halpern and Rego (2012) describe structures in whichthis inconsistency does not arise provided there are impossiblestates. Board and Chung (2009) render the partitional structuresin Board and Chung (2011) in set-theoretic terms; Board et al.(2011) demonstrate that the sub-class of these structures whereall states are possible is equivalent to those of Heifetz et al. (2006).

Agotnes and Alechina’s (2007) logic for conscious unawarenessis perhaps the closest existing work to this one. As in the currentpaper, Agotnes andAlechina use propositional constants to express‘‘agent i is fully aware’’ and ‘‘agent j is aware of everything k isaware of’’, and the axioms C1–C3 and R1–R3 first appear in theearlier work. The critical divergence between their paper and thisone comes in the use of the two-stage semantics here. This meansthat the structures in D are axiomatized using the weak form ofnecessitation, N, while Agotnes and Alechina use N’. As shown inProposition 1, this has the consequence that C5 is valid in anystructure in Agotnes and Alechina that validates BA.

A paper that is formally different to this, though arguablysimilar in spirit, is Grant and Quiggin (2013). Grant and Quigginpresent a dynamic setting in which an agent’s beliefs about thesentence ‘‘i is fully aware’’ are determined by a process of inductionrather than by what is true in the states she entertains. Thisechoes the use here of conjectures – rather than true featuresof the states agents regard as possible – to determine agents’

conscious unawareness, and these conjectures may be understoodas being inductively arrived at. However, as already noted, Grantand Quiggin use a different formal setting to that presented hereand explicitly describe the process of inductive reasoning, which isnot modeled in this work.

Finally, in a companion paper to this one, Walker (2012)adapts Heifetz et al.’s (2006) set-theoretic structures to allow forconscious unawareness and shows an equivalence between theclass of these structures and D r,t,e,b,o.

Appendix. Proofs

A.1. Proof of Theorem 1

Part (i) of the theorem can be rephrased as claiming that thesystemAB is sound and completewith respect toD , where an axiomsystem is sound wrt some class of structures iff all of its theoremsare valid in the class and complete iff every valid formula in theclass is a theorem of the system.

A.1.1. SoundnessThis amounts to showing that the axioms PC, K, A1–A10 are

valid, and that applications of the deductive rules preserve validity.Some of the arguments below follow a very similar course towell known proofs in, for example, Hughes and Cresswell (1996)and Fagin et al. (1995). All of the details are included here becauseof the non-standard semantics.

PC is valid in virtue of rules O1, O4, and O5 as usual. If K werenot valid, there would be some structure such that for state ω andsome i, (a) V (ω, Liφ ∧ Li[φ ⇒ ψ]) = 1 and (b) V (ω, Liψ) = 0.But (a) implies V i

1(ω′, φ) = 1 and V i

1(ω′, φ ⇒ ψ) = 1 for all

ω′∈ Pi(ω). It immediately follows that V i

1(ω′, ψ) = 1 for all

ω′∈ Pi(ω), implying V (ω, Liψ) = 1, contradicting (b). A6 and

A7 are valid because Ref(Ci) = Ref(Rij) = ∅ for all i, j, while A1–A5are valid in light of the fact that Ref(φ) = Ref(¬φ) = Ref(Ljφ) =

Ref(Ajφ) = Ref(Bjφ) and Ref(φ) ∪ Ref(ψ) = Ref(φ ∨ ψ). C1 andR1 follow directly from O2, O3, and O6, and A10 is implied by O8.Finally, C2–C3 and R2–R3 are valid in virtue of O2, O3, and the factthat Ai(ω) ⊆ P for all i and ω in all structures.

Now suppose φ and φ ⇒ ψ are valid formulae. If there werea state ω in any structure such that V (ω,ψ) = 0 then, since φ isvalid, it must be that V (ω, φ ⇒ ψ) = 0, but this contradicts thefact that φ ⇒ ψ is valid. Therefore ψ is valid and MP is validitypreserving.

To show that N is validity preserving, first introduce the notionof stage-one validity. A formula φ is stage-one valid in D wheneverfor every state in the state space of D, V i

1(ω, φ) = 1 for all i. It isstage-one valid in any class of models D∗

⊆ D if it is stage-onevalid for all D ∈ D∗. An axiom system is then stage-one sound wrtD∗ iff any theorem φ of it is stage-one valid in D∗.

Definition A.1. The axiom system ABx1 consists of all of the axioms

and deductive rules in ABx except for the axioms C1 and R1.

Lemma A.1. The axiom system AB1 is stage-one sound wrt D .

Proof. The stage-one validity of the axioms PC, K, A1–A8, C2–C3,R2–R3, andMP can be shown in a parallel manner to their validity,proven above.

For N, if φ is stage-one valid, then it follows immediately thatfor any ω in any structure and any i, V i

1(ω′, φ) = 1 for all ω′

∈

Pi(ω), so Liφ is stage-one valid. Therefore N is stage-one validitypreserving.

8 O. Walker / Mathematical Social Sciences ( ) –

By definition, the set of theorems of ABx1 is identical to the set of

theorems of ABx that can be proven without the use of C1 and R1.Suppose φ is a theorem of AB1, which by Lemma A.1 implies that φis stage-one valid. Hence for every ω and any i in any structure itmust be that V i

1(ω′, φ) = 1 for all ω′

∈ Pi(ω) and thus that Liφ isvalid for i = 1, . . . ,N . N is therefore validity preserving.

Thus AB is valid in all ofD . To complete the soundness proof forparts (ii)–(v) of the theorem, proceed as follows:

PI is valid in D t : If V (ω, Liφ) = 1 for any φ ∈ Φ and in someD ∈ D t , then for every ω′

∈ Pi(ω), V i1(ω

′, φ) = 1. SinceD ∈ D t , if ω′′

∈ Pi(ω′) for ω′

∈ Pi(ω), then ω′′∈ Pi(ω).

Hence if ω′∈ Pi(ω), then for all ω′′

∈ Pi(ω′), V i

1(ω′′, φ) = 1,

implying V i1(ω

′, Liφ) = 1. Since this holds for all ω′∈ Pi(ω), it

follows that V (ω, LiLiφ) = 1.NI is valid inDe:Whenever V (ω,¬Liφ) = 1 for anyφ ∈ Φ andD ∈ De, there is some ω′

∈ Pi(ω) such that V i1(ω

′,¬φ) = 1.Since D ∈ De, if ω′′

∈ Pi(ω) then ω′′∈ Pi(ω

′). Therefore ifω′

∈ Pi(ω), there is some ω′′∈ Pi(ω

′) such that V i1(ω

′′,¬φ) =

1, implying V i1(ω

′,¬Liφ) = 1. It follows that V (ω, Li¬Liφ) = 1.T is valid in D r : First prove that if φ ∈ Φ∀−:

V (ω, φ) = 1 ⇐⇒ V i1(ω, φ) = 1. (4)

Note that for φ ∈ P or φ of the form Aiψ , Liψ , Biψ where ψis any member of Φ , the fact that O1 and O6–O8 respectivelyparallel S1 and S6–S8 ensure that (4) is satisfied by φ. Workinginductively, if ψ satisfies (4), then for φ = ¬ψ , V (ω, φ) = 1iff V (ω,ψ) = 0 and V i

1(ω, φ) = 1 iff V i1(ω,ψ) = 0, hence φ

satisfies (4). And if ψ and χ satisfy (4), where φ = [ψ ∨ χ ],V (ω, φ) = 1 iff at least one of V (ω,ψ) = 1 and V (ω, χ) =

1 and V i1(ω, φ) = 1 iff at least one of V i

1(ω,ψ) = 1 andV i1(ω, χ) = 1, so φ satisfies (4). Thus, (4) holds for all φ ∈ Φ∀−

as required.To show that T is valid in D r , suppose that V (ω, Lφ) = 1

for some φ ∈ Φ∀− and ω in the state space of some D ∈ D r .This requires V i

1(ω′, φ) = 1 for allω′

∈ Pi(ω) and therefore (bythe reflexivity of Pi) that V i

1(ω, φ) = 1. By (4) it follows thatV (ω, φ) = 1 and hence that T is valid in D r .BA is valid in Db: If V (ω, Aiφ) = 1 then Ai(ω) ⊇ Ref(φ). Forωin the state space of someD ∈ Db, this impliesAi(ω

′) ⊇ Ref(φ)for every ω′

∈ Pi(ω) and thus that V (ω, LiAiφ) = 1. SinceRef(φ) = Ref(Aiφ), this implies V (ω, BiAiφ) = 1.R4 is valid in Db,o: Suppose R4 is not valid in Db,o and thusthat is some ω in some D ∈ Db,o such that V

ω, Aiφ ⇒

Bi[Ajφ ∧ ¬Akφ] ⇒ ¬Rjk

= 0. This holds iff V (ω, Aiφ) = 1

and V (ω, Bi[Ajφ ∧ ¬Akφ] ⇒ ¬Rjk

) = 0, and, given the for-

mer, the latter holds iff there exists a ω′∈ Pi(ω) such that

V i1(ω

′, Ajφ) = 1, V i1(ω

′,¬Akφ) = 1, and V i1(ω

′, Rjk) = 1. By thefact thatD ∈ Db,Ai(ω

′) ⊇ Ai(ω), fromwhich V i1(ω

′, Aiφ) = 1.But then Ai(ω

′) ∩ Ak(ω′) ⊇ Ai(ω

′) ∩ Aj(ω′) so since D ∈ Do,

V i1(ω

′, Rjk) = 0, a contradiction.

Finally, if X, Y, . . .Z are respectively valid in Dx, Dy, . . . , Dz , itfollows trivially that X, Y, . . . , Z is valid in Dx,y,...,z .

A.1.2. CompletenessBegin with some preliminary notation and definitions. For any

finite set Γ = φ1, . . . φn, writeΓ for [φ1 ∧[φ2 ∧[. . . φn] . . .]].

If Γ is any set of formulae with the property that there is a finiteΓ ′

⊆ Γ such thatΓ ′

⇒ φ is a theorem of axiom system AX ,write Γ ⊢AX φ. The notation ⊢AX φ then means simply that φ is atheorem of AX .

Say φ is consistent with axiom system AX–φ is AX-consistent– iff 0AX ¬φ and φ is consistent with Γ iff Γ 0AX ¬φ. When Γ isa finite set of formulae, it is AX-consistent iff 0AX ¬

Γ , and if

Γ is infinite it is AX-consistent iff every finite subset of Γ is AX-consistent. A set of formulae, Γ , is a maximal AX-consistent set iffit is consistent and for every φ ∈ Φ \Γ , ⊢AX ¬

(Γ ∪ φ). Fagin

et al. (1995) prove the following lemma:

Lemma A.2. Provided AX is consistent and contains PC andMP, if ∆is an AX-consistent set, there exists a maximal AX-consistent set Γsuch that Γ ⊇ ∆. Furthermore, if Γ is a maximal AX-consistent set,then the following holds:

i. For all φ ∈ Φ one of φ and ¬φ is a member of Γ ;ii. [φ ∨ ψ] ∈ Γ iff at least one of φ ∈ Γ and ψ ∈ Γ ;iii. If φ and φ ⇒ ψ are in Γ , then ψ is a member of Γ ; andiv. ⊢AX φ implies φ ∈ Γ .

Now define an AX-acceptable set of formulae as any ∆ suchthat:

i. Whenever, for any i, ∆⊢AX Aip for all but finitely many p ∈ P ,∆⊢AX Ci; and

ii. Whenever, for any i,j,∆⊢AX Aip ⇒ Ajp for all but finitely manyp ∈ P ,∆⊢AX Rij.

A similar property (also called acceptability) was introducedby Halpern and Rego (2009), and plays the same role in theircompleteness proof as it does here. Lemmas A.3–A.5 parallelanalogous results in Halpern and Rego (2009).

Lemma A.3. If ∆ is finite, then∆ is ABx-acceptable for any x.

Proof. Since P is infinite: there is no finite set of formulae, ∆,such that ∆ 0ABx Ci and ∆⊢ABx Aip for all p in any infinite subsetof P; and, similarly, there is no finite ∆ such that ∆ 0ABx Rij but∆⊢ABx Aip ⇒ Ajp for all p in any infinite subset of P .

Lemma A.4. If ∆ is ABx-acceptable, then ∆ ∪ φ is ABx-acceptablefor any φ ∈ Φ .

Proof. Suppose ∆ is ABx-acceptable but ∆ ∪ φ is not. Thiscan only hold if at least one of conditions (i) and (ii) for ABx-acceptability from the definition above fail: either∆∪φ ⊢ABx Aipfor all but a finite set of p ∈ P and some i but not ∆ ∪ φ ⊢ABx Ci(in violation of condition (i)); or ∆ ∪ φ ⊢ABx Aip ⇒ Ajp for allbut finitely many p ∈ P and some i, j but not ∆ ∪ φ ⊢ABx Rij(contravening condition (ii)).

Suppose condition (i) fails, in which case it follows that∆ 0ABx Ci, and since∆ is ABx-acceptable, thismeans that theremustbe an infinite Q ⊆ P such that∆ 0ABx Aip for all p ∈ Q . But then theonly way∆∪φ ⊢ABx Aip for all but a finite set of p ∈ P would be ifφ ⊢ABx Ci. This, however, means that∆∪φmust satisfy condition(i) of ABx-acceptability. The argument for why ∆ ∪ φ satisfiescondition (ii) is precisely parallel to this.

Lemma A.5. If ∆ is ABx-consistent and -acceptable, then there existssome maximal ABx-consistent and -acceptable∆∗ containing ∆.

Proof. The following proceeds along a similar path to Halpern andRego (Lemma A.6). Enumerate Φ , φ1, φ2 . . . and define ∆0 := ∆

and then construct∆k+1 inductively for k = 0, 1, . . . as follows.First, for every i such that Ci is not ABx-consistent with∆k, take

some p ∈ P such that∆k 0ABx Aip and ¬Aip ∈ ∆k, and denote ¬Aipas φk,i. Provided∆k is ABx-acceptable, it will always be possible tofind some such p. If Ci is ABx-consistent with∆k, set φk,i = ∅.

Second, for every i and every j such that Rij is not ABx-consistentwith ∆k, take some p ∈ P such that ∆k 0ABx Aip ⇒ Ajp, ¬Alp ∈

φk,lNl=1 and ¬[Aip ⇒ Ajp] ∈ ∆k, and denote ¬[Aip ⇒ Ajp] as

φk,i,j. Once again, such a pmay always be found provided∆k is ABx-acceptable, and if Rij is ABx-consistent with∆k let φk,i,j = ∅.

Now define∆k,1 as∆k ∪ φk,1 and∆k,i = ∆k,i−1 ∪ φk,i for i =

2, . . . ,N , then let∆k,1,1 = ∆k,N ∪ φk,1,1,∆k,i,j = ∆k,i,j−1 ∪ φk,i,j

O. Walker / Mathematical Social Sciences ( ) – 9

for i = 1, . . . ,N and j = 2, . . . ,N , and∆k,i,1 = ∆k,i−1,N ∪φk,i,1 fori = 2, . . . ,N .∆k+1 is then∆k,N,N ∪ φk+1 if this is ABx-consistentor ∆k,N,N if it is not. By construction and Lemma A.4, if ∆k is ABx-consistent and -acceptable, ∆k+1 will be. Thus, since ∆0 is ABx-consistent and -acceptable,∆k also has these properties for all k.

Let ∆∗=

∆k, and note that ∆∗ must be maximal ABx-

consistent and clearly∆ ⊆ ∆∗. To show that∆∗ is ABx-acceptable,note that either Ci ∈ ∆∗ or there is an infinite Q ⊆ P such that¬Aip ∈ ∆∗ for all p ∈ Q , and that either Rij ∈ ∆∗ or there is aninfinite Q ⊆ P such that ¬[Aip ⇒ Aj] ∈ ∆∗. Hence, since ∆∗

is maximal ABx-consistent, if ∆∗⊢ABx Aip for all but finitely many

p ∈ P , it must be that ∆∗⊢ABx Ci and if ∆∗

⊢ABx Aip ⇒ Ajp forall but finitely many p ∈ P , it follows that ∆∗

⊢ABx Rij. Thus, ∆∗

is acceptable.

In the next proofs, use C for Ci,¬Ci : i = 1, . . . ,N andR for Rij,¬Rij : i, j = 1, . . . ,N. For any binary relation ≽ oni, . . . ,N, P, C≽ then stands for Ci : i ≽ P ∪ ¬Ci : i ≽ P

and R≽= Rij : j ≽ i ∪ ¬Rij : j ≽ i.

Lemma A.6. ∆ is a maximal ABx1-consistent subset of C ∪ R iff ∆ =

C≽∪ R≽ for some ≽∈ O.

Proof. Suppose∆ ismaximalABx1-consistent subset ofC∪R. Define

≽∆ on 1, . . . ,N, P such that i≽∆ j iff Rji ∈ ∆, i≽∆ P iff Ci ∈ ∆,

and for all i ∈ 1, . . . ,N, P, P ≽∆ i.

By (i) of Lemma A.2, if φ = Ci or φ = Rij for i, j ∈ 1, . . . ,N,φ ∈ ∆ iff ¬φ ∈ ∆ and if φ = ¬ψ where ψ = Ci or ψ = Rij fori, j ∈ 1, . . . ,N, φ ∈ ∆ iff ψ ∈ ∆. Thus, i ≽∆ j iff ¬Rij ∈ ∆ andi ≽∆ P iff ¬Ci ∈ ∆. Therefore∆ = C≽

∆∪ R≽

∆.

To show sufficiency, demonstrate that≽∆ belongs toO. By con-struction, P ∈ max≽∆1, . . . ,N, P. Since P ≽

∆ P by constructionand i≽∆ i by the fact that (given R3) Rii ∈ ∆ for i = 1, . . . ,N ,≽∆ is reflexive. For transitivity there are four relevant cases: if

i, j, k ∈ 1, . . . ,N and i≽∆ j, j≽∆ k, then by R2, i≽∆ k; fori, j ∈ 1, . . . ,N where P ≽

∆ i, i≽∆ j, it holds by construction thatP ≽

∆ j; if i, j ∈ 1, . . . ,N and i≽∆ P , P ≽∆ j, then by C2, i≽∆ j; and

if i, j ∈ 1, . . . ,N and i≽∆ j and j≽∆ P , then by C3, i≽∆ P . Thus,≽∆ is a preorderwith amaximal element P andhence belongs toO.For necessity, take any ≽∈ O and construct a structure D =

Ω,V,P ,A,X whereΩ = ω, X(ω) =≽, and V , P , and A arearbitrary. Let∆(≽) := φ ∈ C ∪ R : V1(ω, φ) = 1. By S2, S3, andS4, V1(ω, φ) = 1 iffφ ∈ C≽

∪R≽ so∆(≽) = C≽∪R≽. Since∆(≽) is

finite, it is ABx1-inconsistent iff⊢ABx1

¬∆(≽). LemmaA.1 requires

that if ∆(≽) is ABx1-inconsistent, it must be that V1(ω,¬

∆

(≽)) = 1. But by definition of ∆(≽), it holds that V1(ω,¬∆

(≽)) = 0, so∆(≽) isABx1-consistent. It is amaximalABx

1-consistentsubset of C ∪ R because if φ ∈ C ∪ R \∆(≽), by PC it must be that∆(≽)⊢ABx1

¬φ.

For each maximal ABx-consistent set, Γ , and any ≽∈ O, definea further set of formulae Γ ≽:

p ∈ Γ ≽⇐⇒ p ∈ Γ (5)

Ci ∈ Γ ≽⇐⇒ i ≽ P (6)

Rij ∈ Γ ≽⇐⇒ j ≽ i (7)

¬φ ∈ Γ ≽⇐⇒ φ ∈ Γ ≽ (8)

[φ ∨ ψ] ∈ Γ ≽⇐⇒ At least one of φ ∈ Γ ≽ and ψ ∈ Γ ≽ (9)

Aiφ ∈ Γ ≽⇐⇒ Aiφ ∈ Γ (10)

Liφ ∈ Γ ≽⇐⇒ Liφ ∈ Γ (11)

Biφ ∈ Γ ≽⇐⇒ Biφ ∈ Γ . (12)

Throughoutwhat follows, whereΓ is amaximal ABx-consistentset, Γ ≽ refers to a set of formulae defined as in (5)–(12) with ≽

understood to belong to O.

Lemma A.7. Where Γ is maximal ABx-consistent and -acceptable,there exists some Γ ≽ that is maximal-ABx

1-consistent.

Proof. For anymaximal ABx-consistent Γ , let ≽ be such that i ≽ Piff Aip ∈ Γ for all p ∈ P , P ≽ i for all i, and i ≽ j iff p : Aip ∈ Γ ⊇

p : Ajp ∈ Γ . Thus, ≽ is a pre-order with P as a maximal element,so ≽∈ O. Furthermore, Ci ∈ Γ iff (by C1 and the ABx-acceptabilityof Γ ) Aip ∈ Γ for all p ∈ P iff i ≽ P iff Ci ∈ Γ ≽, and Rij ∈ Γ iff (byR1 and ABx-acceptability) Aip ⇒ Ajp ∈ Γ for all p ∈ P iff j ≽ i iffRij ∈ Γ ≽. Thus Γ ∩

Φ∀−

∪ C ∪ R

= Γ ≽∩

Φ∀−

∪ C ∪ R. Now

note that Φ is simply the closure of Φ∀−∪ C ∪ R under negation

and disjunction. By Lemma A.2, Γ contains ¬φ iff φ ∈ Γ and[φ ∨ ψ] ∈ Γ iff at least one of φ ∈ Γ and ψ ∈ Γ , and by (8)and (9), Γ ≽ is the closure of Γ ≽

∩Φ∀−

∪ C ∪ Runder precisely

these rules. Thus, it must be that Γ = Γ ≽, which is maximal ABx1-

consistent by assumption.

Lemma A.8. If Γ is a maximal ABx-consistent set, Γ ′ is a maximalABx

1-consistent set containing Γ ∩ Φ∀− only if Γ ′∈ Γ ≽

: ≽∈ O.

Proof. If there exists aφ ∈Γ ′

\ Γ∩Φ∀−, thenby LemmaA.2 and

the definition ofΦ∀−,¬φ ∈ Γ ∩Φ∀−. GivenΓ ′∩Φ∀−

⊇ Γ ∩Φ∀−,this implies φ,¬φ ⊂ Γ ′

∩ Φ∀−, which is impossible since Γ ′ isa maximal AB1

x-consistent set.It follows thatΓ ′

∩Φ∀−= Γ ∩Φ∀−

= Γ ≽∩Φ∀− for any≽∈ O.

And since Γ ′ is ABx1-consistent, by Lemma A.6 it contains C≽

∪ R≽

for some ≽∈ O. So Γ ′∩

Φ∀−

∪ C ∪ R

= Γ ≽∩

Φ∀−

∪ C ∪ R.

From this, the same argument used in the proof of Lemma A.7shows that Γ ′

= Γ ≽ for some ≽∈ O.

For any set of formulae,∆, use Keni(∆) to refer to φ : Liφ ∈ ∆.Any maximal ABx

1-consistent set, Γ≽, is then i-irrelevant iff there

exists some maximal ABx-consistent Γ ′ such that Keni(Γ′) ⊆ Γ ≽

′

for some ≽′∈ O, but there is no maximal ABx-consistent Γ ′ such

that Keni(Γ′) ⊆ Γ ≽. Any maximal ABx

1-consistent Γ≽ that is not

i-irrelevant is i-relevant.

Lemma A.9. For every maximal ABx-consistent Γ and every i =

1, . . . ,N, there is at least one i-relevant maximal ABx1-consistent Γ

≽.

Proof. Immediate by construction.

For any maximal ABx-consistent set Γ , define an ABx-conjectureon Γ , Γ , as an N-vector whose ith element is an i-relevant ABx

1-consistent Γ ≽. Where Γ is an ABx-conjecture on some maximalABx-consistent set, write ΓΓ to refer to the set on which Γ is anABx-conjecture, and write ≽

iΓ to refer to ≽∈ O such that the ith

element of Γ equals Γ ≽. For notational brevity, write Γ iΓ for Γ

≽iΓ

Γ .Now, to show completeness, introduce a canonical model for

ABx, Dxc . The state space of the canonical model for ABx contains

one distinct state corresponding to each ABx-conjecture on everymaximal ABx-consistent set Γ and no other states. Write ωΓ forthe state that corresponds to the ABx-conjecture Γ . The remainingelements of Dx

c are as follows:

p ∈ V(ωΓ ) ⇐⇒ p ∈ ΓΓ

ωΓ ′ ∈ Pi(ωΓ ) ⇐⇒ Keni(ΓΓ ) ⊆ Γ iΓ ′

p ∈ Ai(ωΓ ) ⇐⇒ Aip ∈ ΓΓ

Xi(ωΓ ) = ≽iΓ .

To obtain the result, work in two stages.

Lemma A.10. For any φ ∈ Φ , i = 1, . . . ,N, and any ωΓ in the statespace of Dx

c :

V i1(ωΓ , φ) = 1 ⇐⇒ φ ∈ Γ i

Γ . (13)

10 O. Walker / Mathematical Social Sciences ( ) –

Proof. For φ ∈ P , (13) is immediate from Lemma A.7. Whereφ = Cj for any j = 1, . . . ,N , V i

1(ωΓ , Cj) = 1 iff j≽iΓ P iff (by (6))

Cj ∈ Γ iΓ . For φ = Rjk for any j, k = 1, . . . ,N , V i

1(ωΓ , Rjk) = 1 iffk≽

iΓ j iff (by (7)) Rjk ∈ Γ i

Γ .Proceeding inductively where ψ and χ satisfy (13) for all i =

1, . . . ,N , then forφ = ¬ψ , S4 says V i1(ωΓ , φ) = 1 iff V i

1(ωΓ , ψ) =

0, and part (i) of Lemma A.2 implies that ¬ψ ∈ Γ iΓ iff ψ ∈ Γ i

Γ iffV i1(ωΓ , ψ) = 0. Thus (13) is satisfied for φ.Where φ = [ψ ∨ χ ], S5 says that V i

1(ωΓ , φ) = 1 iff at least oneof V i

1(ωΓ , ψ) = 1 and V i1(ωΓ , χ) = 1 iff at least one ofψ ∈ Γ i

Γ andχ ∈ Γ i

Γ . Part (ii) of Lemma A.2 shows that this holds if and only ifφ ∈ Γ i

Γ , so (13) is satisfied for φ.If φ = Ajψ , V i

1(ωΓ , φ) = 1 iff Aj(ωΓ ) ⊇ Ref(ψ) iff Ajp ∈ Γ iΓ for

all p ∈ Ref(ψ) iff (by A1–A7) φ ∈ Γ iΓ .

Where φ = Ljψ , φ ∈ Γ iΓ iff ψ ∈ Kenj(Γ

iΓ ) iff ψ ∈ Kenj(ΓΓ ).

This implies thatψ ∈ ΓjΓ ′ for all ωΓ ′ ∈ Pj(ωΓ ). Asψ satisfies (13),

it follows that V j1(ωΓ ′ , ψ) = 1 for all ωΓ ′ ∈ Pj(ωΓ ). By S8, this

means V i1(ωΓ , φ) = 1.

Working the other way, if φ ∈ Γ iΓ , then Kenj(ΓΓ ) ∪ ¬ψ

must be ABx1-consistent. To see this, suppose Kenj(ΓΓ ) ∪ ¬ψ

were ABx1-inconsistent; then there would be some finite set

ψ1, ψ2, . . . , ψn ⊆ Kenj(ΓΓ ) such that:

⊢ABx1

ni=1

ψi ⇒ ψ

in which case by N,n

i=1 ψi ⇒ ψ ∈ Kenj(ΓΓ ) and by K, ψ ∈

Kenj(ΓΓ ), a contradiction. Therefore Kenj(ΓΓ ) ∪ ¬ψ is ABx1-

consistent, so by Lemmas A.2 and A.8 there exists some maximalABx

1-consistent Γ′≽ such that Kenj(ΓΓ ) ∪ ¬ψ ⊆ Γ ′≽. Since

Kenj(ΓΓ ) ⊆ Γ ′≽, Γ ′≽ is therefore j-relevant, and hence there issome ωΓ ′ in the state space of Dx

c where Γ ′ is such that Γ ′≽= Γ

jΓ ′ .

Furthermore,ωΓ ′ ∈ Pj(ωΓ ). Sinceψ (and hence¬ψ) satisfies (13),it must be that V j

1(ωΓ ′ ,¬ψ) = 1 and hence that V i1(ωΓ , φ) = 0.

Finally, for φ = Bjψ , V i1(ωΓ , Bjψ) = 1 iff V i

1(ωΓ , Ajψ) = 1 andV i1(ωΓ , Ljψ) = 1 iff Ajψ, Ljψ ∈ Γ i

Γ iff (by A10) Bjψ ∈ Γ iΓ .

Lemma A.11. For every φ ∈ Φ:

V (ωΓ , φ) = 1 ⇐⇒ φ ∈ ΓΓ . (14)

Proof. Where φ is a primitive proposition, (14) is satisfiedstraightforwardly by virtue of the definition of V above. If φ = Aipfor any p ∈ P , then φ ∈ ΓΓ iff p ∈ Ai(ωΓ ), which holds iffV (ωΓ , φ) = 1.

For φ = Ci, if φ ∈ ΓΓ then (by C1) Aip ∈ ΓΓ for all p ∈ P ,implying V (ωΓ , φ) = 1; and if V (ωΓ , φ) = 1 then V (ωΓ , Aip) = 1for all p ∈ P , implying Aip ∈ ΓΓ for all p ∈ P , which, given thatΓΓ is ABx-acceptable, implies φ ∈ ΓΓ . For φ = Rij, φ ∈ ΓΓ implies(by R1) Aip ⇒ Ajp for all p ∈ P , which means Ai(ωΓ ) ⊆ Aj(ωΓ )and thus V (ωΓ , φ) = 1; while if V (ωΓ , φ) = 1, then Ai(ωΓ ) ⊆

Aj(ωΓ ), which means that for all p ∈ P , Aip ⇒ Ajp ∈ ΓΓ , and thusby the acceptability of ΓΓ , φ ∈ ΓΓ .

Most of the remainder of the proof proceeds inductively ina parallel fashion to that for Lemma A.10. The single point ofdeparture is the argument that ifψ satisfies (14) thenφ = Liψ alsosatisfies (14). This is now as follows. We have φ ∈ ΓΓ iff φ ∈ Γ i

Γ

iff (given Lemma A.10) V i1(ωΓ , φ) = 1 iff V i

1(ωΓ ′ , ψ) = 1 for allωΓ ′ ∈ Pi(ωΓ ′) iff V (ωΓ , φ) = 1.

Lemmas A.7 and A.9 imply that every ABx-acceptable andmaximal ABx-consistent set Γ has some state ωΓ in the canonicalmodel such that ΓΓ = Γ . Lemmas A.3 and A.5 imply that everyABx-consistent formula belongs to some maximal ABx-consistent

and -acceptable set, so therefore (14) implies that every ABx-consistent formula is true in some state of the canonical modelfor ABx. Therefore the only formulae that are not true in any stateof Dx

c are those that are inconsistent – that is, the negations oftheorems ABx – so the only formulae that are true in every stateof Dx

c are theorems of ABx. Therefore any formula that is valid in Dis a theorem of AB, so AB is complete with respect to D .

To demonstrate that ABx is complete with respect to Dz , itsuffices to show that Dx

c is a member of Dz . For this, proceed asfollows:

DPIc belongs to D t : If ωΓ ′ ∈ Pi(ωΓ ), then Keni(ΓΓ ) ⊆ Γ i

Γ ′ and,by PI, Keni(ΓΓ ) ⊆ Keni(Γ

iΓ ′) = Ken(ΓΓ ′). Hence, if ωΓ ′′ ∈

Pi(ωΓ ′) and therefore Keni(ΓΓ ′) ⊆ Γ iΓ ′′ , Keni(ΓΓ ) ⊆ Γ i

Γ ′′

implying ωΓ ′′ ∈ Pi(ωΓ ). Thus Pi is transitive for all i in DPIc and

therefore DPIc ∈ D t .

DNIc belongs to De: If ωΓ ′ ∈ Pi(ωΓ ), Keni(ΓΓ ) ⊆ Γ i

Γ ′ and by NIandpart (i) of LemmaA.2 ifφ ∈ Keni(ΓΓ ) thenφ ∈ Keni(Γ

iΓ ′) =

Keni(ΓΓ ′). Thus Keni(ΓΓ ′) ⊆ Keni(ΓΓ ) and hence if ωΓ ′′ ∈

Pi(ωΓ ) then Ken(ΓΓ ′) ⊆ Γ iΓ ′′ , implying ωΓ ′′ ∈ Pi(ωΓ ′). Thus

Pi in DNIc is Euclidean for all i and therefore DNI

c ∈ De.DT,PI,NIc belongs to D r : First show that under T, for any

i, Keni(Γ ) must be ABT1-consistent for any maximal ABT-

consistent Γ . For if there were some φ ∈ Keni(Γ ) such that⊢ABT1

¬φ, then it holds by PC that for someψ ∈ Φ∀−, ⊢ABT1φ ⇒

[ψ∧¬ψ], hence byN,φ ⇒ [ψ∧¬ψ] ∈ Keni(Γ ) and thereforeby K, [ψ ∧¬ψ] ∈ Keni(Γ ). But then since [ψ ∧¬ψ] ∈ Φ∀−, byT [ψ ∧ ¬ψ] ∈ Γ , which is impossible as Γ is ABT-consistent.

Now take any ABT,PI,NI-conjecture Γ such that ωΓ exists inDT,PI,NIc . By T, it must be that

Keni(ΓΓ ) ∩ Φ∀−

⊆

ΓΓ ∩ Φ∀−

.

And since Keni(ΓΓ ) isABT1-consistent, by LemmaA.2 itmust be a

subset of somemaximal ABT1-consistent set that is itself a super-

set of ΓΓ ∩Φ∀−. By Lemma A.8, the only such sets are Γ≽

Γ :≽∈

O. Therefore Keni(ΓΓ ) ⊆ Γ≽

Γ for at least one ≽∈ O. So, forall ωΓ ′ , where Γ ′ is an N-conjecture on ΓΓ , there exists an ωΓ ′′ ,where Γ ′′ is an N-conjecture on ΓΓ , such that ωΓ ′′ ∈ Pi(ωΓ ′).

To show the result, demonstrate that for any Γ , ifKeni(ΓΓ ) ⊆ Γ i

Γ then Γ iΓ is i-irrelevant, and therefore that ωΓ

is not defined, from which it follows that whenever ωΓ is de-fined, ωΓ ∈ Pi(ωΓ ) for all i. Suppose this is not the case andthat Keni(ΓΓ ) ⊆ Γ i

Γ and ωΓ is i-relevant, implying that thereis some ΓΓ ′ such that Keni(ΓΓ ′) ⊆ Γ i

Γ . Then by PI and NI,Keni(ΓΓ ′) = Keni(ΓΓ ), so Keni(ΓΓ ) ⊆ Γ i

Γ , a contradiction.DBAc belongs to Db: Suppose ωΓ ′ ∈ Pi(ωΓ ) and thus that

Keni(ΓΓ ) ⊆ Γ iΓ . If p ∈ Ai(ωΓ ) then Aip ∈ ΓΓ and by AB, Aip ∈

Keni(ΓΓ ) and thus Aip ∈ Γ iΓ ′ , implying p ∈ Ai(ωΓ ′). Therefore

ωΓ ′ ∈ Pi(ωΓ ) implies Ai(ωΓ ) ⊆ Ai(ωΓ ′) so DABc belongs to Db.

DPI,NI,T,BA,R4c belongs to Do: Suppose that DPI,NI,T,BA,R4

c ∈ Do, inwhich case there exists some ωΓ in the state space of Do suchthat Ak(ωΓ )∩Ai(ωΓ ) ⊇ Aj(ωΓ )∩Ai(ωΓ ), but k≽

iΓ j for some

i, j, k. Since this can only hold ifAi(ω) = ∅, theremust be somep ∈ P such that Aip, Ajp,¬Akp ∈ ΓΓ . By BA, BiAip ∈ ΓΓ , thenby R4, N, and K, Bi

[Ajp ∧ ¬Akp] ⇒ ¬Rjk

∈ Keni(ΓΓ ). Since

DPI,NI,T,BA,R4c ∈ D r , Keni(ΓΓ ) ⊆ Γ i

Γ , so [Ajp ∧ ¬Akp] ⇒ ¬Rjk ∈

Γ iΓ . This implies¬Rjk ∈ Γ i

Γ and therefore k ≽iΓ j, a contradiction.

To complete the proof, the arguments above can be used to showthat the canonical model of the system ABx,x′,...,x′′ lies withinDz,z′,...,z′′ , so ABx,x′,...,x′′ is complete with respect to Dz,z′,...,z′′ .

A.2. Proof of Theorem 2

The result follows from Theorem 2 in Agotnes and Alechina(2007) and Fagin and Halpern’s (1988) characterization ofawareness generated by primitive propositions.

O. Walker / Mathematical Social Sciences ( ) – 11

A.3. Proof of Proposition 1

Given Theorem 2, it suffices to show the following:

C5 is valid inDbT (15)

C6 is valid inD r,t,e,bT . (16)

To show (15), consider any D ∈ DbT whose state space includes ω.

V (ω, Ci) = 1 implies Ai(ω) = P and thus since D ∈ Db, Ai(ω′) =

P for all ω′∈ Pi(ω). Thus, V (ω′, Ci) = 1 for all ω′

∈ Pi(ω) and,since for any φ and ω′, V i

1(ω′, φ) = V (ω′, φ), V i

1(ω′, Ci) = 1 for all

ω′∈ Pi(ω). Hence V (ω, BiCi) = 1, so C5 is valid.For (16), take D ∈ D r,t,e,b

T with state ω. Since Pi is partitionaland ω′

∈ Pi(ω) implies Ai(ω) ⊆ Ai(ω′), ω′

∈ Pi(ω) impliesAi(ω) = Ai(ω

′). Therefore V (ω,¬Ci) = 1 implies V (ω′,¬Ci) = 1for all ω′

∈ Pi(ω), and thus V i1(ω

′,¬Ci) = 1 for all ω′∈ Pi(ω).

This then implies V (ω, Bi¬Ci) = 1.

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