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WHICH WORLDS ARE POSSIBLE? A JUDGMENT AGGREGATION PROBLEMAuthor(s): CHRISTIAN LISTReviewed work(s):Source: Journal of Philosophical Logic, Vol. 37, No. 1 (February 2008), pp. 57-65Published by: SpringerStable URL: http://www.jstor.org/stable/41217822 .

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JournalfPhilosophicalogic 2008) 37:57-65 Springer008DOI: 10.1007/sl0992-007-9058-yCHRISTIAN LIST

WHICH WORLDS ARE POSSIBLE? A JUDGMENTAGGREGATION PROBLEMReceived 0 August 006

ABSTRACT. upposethemembers f a group e.g., committee,ury,expertpanel) each form judgment n which worlds n a givenset are possible,subject o theconstrainthat t leastone world s possiblebutnot ll are. Thegroup eeks oaggregatehese ndividualudgmentsnto collectiveudgment,subject o the same constraint. showthatno udgment ggregationule cansolve this problemin accordance with three conditions:"unanimity,""independence"nd"non-dictatorship,"lthoughheresult s a variant f anexistingheoremn"groupdentification"Kasher ndRubinstein,ogiqueetAnalyse 60:385-395, 997), he ggregationf udgmentsnwhichworlds repossible orpermissible,esirable,tc.)appearsnottohave beenstudied et.The result hallenges s to take stance n which f tsconditionso relax.KEY WORDS: judgment ggregation,ossibleworlds, ropositions

1. IntroductionHow can a group of individuals (e.g., committee,ury, expertpanel)make collective udgments true/false) n some propositionsbased onthe groupmembers' individual udgmentson these propositions?Thistask - "judgment aggregation" - becomes non-trivial when thepropositions are interconnected, s shown in a growing literature.Suppose a three-memberroup seeks to make collective udgmentsonp, q, and p q9 whereone member udges all threepropositions o betrue, second udges p tobe truebutq andp q to be false,and a thirdjudges q to be truebutp andp q to be false. Then majorities udge pand q to be true nd yet/? q to be false,an inconsistent etofmajorityjudgments. This problem has been called the "discursive dilemma"(Pettit 2001 extendingKornhauser and Sager 1986) and shown toillustrate moregeneral mpossibility esult List and Pettit 002; 2004).Several extensions nd generalizationshave been given (e.g., Pauly andvan Hees 2006; Dietrich2006, 2007; Nehring nd Puppe 2005; Bovensand Rabinowicz 2006; Pauly 2007; van Hees 2007). Crucially, in the

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58 CHRISTIAN LISTexisting iterature n udgment ggregation, ropositions remodelled assentencesof a formal anguage,not as sets of possible worlds.In this shortpaper, I want to draw attention o a differentudgmentaggregationproblemnot discussed in that iterature, hich arises whenpropositions re modelled as sets of possible worlds. Here each groupmember's judgment consists in the acceptance of a single non-contradictorynd non-tautological roposition, epresented y a set ofworldsdeemedpossible bythat ndividual.The groupthen eeks to makea collective udgment,which also consists n theacceptance of a singlenon-contradictorynd non-tautological roposition, epresented y a setof worlds deemed possible by the group. Thus the problem is toaggregate n /z-tuplef non-contradictorynd non-tautological roposi-tions (across n individuals) into a single non-contradictorynd non-tautologicalpropositionforthegroupas a whole).The theorem want to puton thetable fordiscussionshows that t isimpossible o solve thisproblem naccordancewith ome seeminglymildconditions: unanimity," independence," nd "non-dictatorship." athe-matically, hisresult s a variant f an existing heorem n theso-called"group dentification"roblem yKasher nd Rubinstein1997), but to thebest of my knowledge the interpretation roposed here - i.e., theapplication o udgmentson which worlds are possible - is new. Here Iderivethetheorem rom recent heorem n udgment ggregationn thestandard ense (Dietrichand List 2007a; Dokow and Holzman 2005),which, in turn,has precursors n abstractaggregation heory Wilson1975; Rubinstein nd Fishburn1986; Nehring nd Puppe 2002).As the difficultieswith udgment aggregation re usually thought ostem from the presence of multiple, nterconnected ropositions, t issurprisinghat n impossibility esult an arise evenwhen ndividual ndcollective judgments consist only in the acceptance of a singleproposition. t is also worthnotingthat the resultapplies not only tojudgments n whichworlds arepossible,but also to udgmentson whichworlds are permissible,or desirable, etc. In each of these cases, thetheorem oses interesting uestions.

2. TheoremLet Q = {cc>i,..,uJk}be a finite et of worlds,assuming > 2; and letN = {1,...,} be a finitegroup of individuals,assumingn > 1. Eachindividual GN makes a judgmenton which of theworlds G arepossible (e.g., relative o his/hervidence,but various interpretationsf



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WHICH WORLDS ARE POSSIBLE? 59the formalisman be given).Formally, judgments a subsetJ CO..We saythat is consistentfJ 0 and nformativefJ Q. Makingconsistentnd informativeudgmenthus onsists n accepting non-contradictoryndnon-tautologicalroposition.et J be theset of allconsistentnd informativeudgments.Thegroup equires method faggregatingach-tuple f consistentand informativendividualudgmentsJ',...,Jn)GJn into consistentand informativeollectiveudgment e J. Call such a method naggregationule,defineds a function/Jn - Canwe find naggregationule hatmeets omeminimal onditions?Let me introduce hree conditions.The firstrequiresthat if allindividualsubmit he same udgment,his udgment hould also bethecollective ne. The secondrequireshat hecollectiveudgmentnwhether givenworlduj is possible houlddepend nlyon individualjudgments n whether is possible,not on individualudgments nwhether therworldsJ are possible.The third equires hat hecollectiveudgmenthould ot lwaysbe determinedyanantecedentlyfixed ictator.Unanimity.orallJ G3,f(J, ...,J) = J.Independence.or all (Ju ..,), {J',~.,J'n)GJ" and all GQ,[(V/ n)(lj e J& le j)] = [ujgf(ju ...,) e> oj ef(j'X .., )].

Non-dictatorship.here s no /GN suchthat, or ll (J', ..,) GJn,f(Ju...,Jn)=JhSurprisingly,hese hree onditionsannot e satisfiedimultaneously.

THEOREM 1. There is no aggregation ule/ : J" -> J satisfyingunanimity,ndependencendnon-dictatorship.A proof s given n theAppendix.n thecase n= k, theorem isequivalento a theoremyKasher nd Rubinstein1997) on theproblemof"group dentification."ere then members f a group ach makeajudgmentn whichmembers fthat rouphave a certain ropertye.g.,being true cientistrhaving particulareligiousdentity),ubject othe constrainthat t least one individual as thepropertyut notallindividuals o. The groupthen seeks to aggregate hen individualjudgments n who has thegivenpropertynto a resultingollectivejudgment.Here, too, no aggregationule can simultaneouslyatisfy

unanimity,ndependencendnon-dictatorship.



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60 CHRISTIAN LIST3. Illustration

While the fullproofof theorem1 is somewhattechnical, he intuitionbehind it can be conveyed by considering more restrictive lass ofaggregationrules and showing that this class is empty.Consider anaggregation ule that satisfiesnot only unanimity, ndependence nd astrengthenedersion of non-dictatorship,amely "anonymity" i.e., allindividuals ave equal weight ndetermininghe collectiveudgment), utalso "monotonicity"i.e., ifmore ndividualsudge that givenworld ispossible,this annot urn collective udgmentthat is possible intoonethat is impossible)and "neutrality"i.e., the criterion ordeterminingthecollectiveudgment n whether givenworld is possible s the samefor all worlds a;). (In standardudgment aggregation, arious generalcharacterizationsfaggregation ulessatisfyingheseconditions regivenin Nehring nd Puppe 2005 and Dietrich and List 2007b.) A necessarycondition or n aggregation ule to meet all theseconditions s that t s athreshold rule with some acceptance threshold te {l,2,...,/i}9 defined asfollows. For all (Ji,...,yw)G *7,/(^1,...,) is the set of all worlds judged possible by at least t individuals, ormallyf(J] ,...,y) = {weu: number of ieN with ueJi>t}.

Examples of threshold ules are the unionrulef(J' , ..,) = J' U ... (here i=l), the intersection rule f(J', ...,) =J' ... HJn (heret = ), and simple majority ule (here t is the smallest integergreaterthan/2).Can we find an acceptance threshold uch that the correspondingthreshold ule assigns to every -tuple of consistent nd informativeindividualudgments consistent nd informativeollective udgment? tis easy to see that, o ensureconsistency, he thresholdmust not be toohigh, i.e., it must not happen that no world is deemed possible. Anecessary and sufficient ondition is / ^-' e.g., the intersectionuleguaranteesnformativeness.o theconjunctionfconsistencynd informa-tiveness equireshe cceptance hresholdosatisfy ^p 1 individuals and > 2 possible worlds, as assumed, thisinequality an easilybe seen to have no solution. In thedegenerate aseof = 2 possible worlds and an odd numberof individualsw, simplemajorityulemeets ll conditions.) hereforeheclass ofaggregationulessatisfyinghe conditionsof theorem1 togetherwithmonotonicitynd



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WHICH WORLDS ARE POSSIBLE? 61neutralityand non-dictatorshiptrengthenedoanonymity)s emptywhenn > 1 and > 2. The fullproof hows that his mpossibilityontinues ohold even withoutmonotonicity,eutralityr the strengtheningf non-dictatorshipo anonymity.

4. Avoidingthe ImpossibilityAs with thermpossibilityesults f social choicetheory notably, rrow's(1951) theorem n preferenceggregation thesignificancef theoremlies not primarilyn establishinghe impossibilityf solvinga particularaggregation roblem,but rather n indicatingwhich conditionsmust berelaxed n order ofind solution. he following scape routes re available:Relaxing unanimity.funanimitys dropped, constant ggregationrulesatisfies all other conditions. Such a rule assigns to every -tuple(/i, ..,) Jn the same fixed collective udgmentJ G J. This is notan attractiveolution, s itpays no attention o the udgments ubmittedby the individuals.Relaxing independence. f ndependencesdropped, distance-basedggre-gationrule can be constructedinspired ytheapproach nPigozzi 2006),which satisfies ll other onditions. or each -tuple J' , ..,Jn)G Jn, wehere compare each "candidate" collective udgmentJ e J with eachsubmittedndividual udgmentJi9using the followingmethod.We saythatJ agrees withJi on a givenworld if [u e J ifand only fa; G //].For each individual GN, we now count the total number of worlds G Q on which J agrees withJ and consider the sum-total f thesecounts across individuals GN. The collective udgmentJ e J is thenchosen so as tomaximize thissum-totalwithsome additionalprovisionsforbreaking ies). Under uch an aggregation ule, hecollectiveudgmenton whether given world is possible depends not only on individualjudgments egardinghatworldbut also on individualudgments egardingotherworlds. fthis mplications accepted,a distance-based ggregationrulemaybe a satisfactoryolution o thepresent ggregation roblem.Relaxing non-dictatorship.f non-dictatorships dropped,a dictatorialaggregation rule satisfies all other conditions.Such a rule assigns toeach -tuple (J', ...,Jn)G Jn the udgmentJ of the same antecedentlyfixed individual i e N. For obvious reasons, this is not generallyanattractive olution.



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62 CHRISTIAN LISTRestricting he domain of admissible inputs. f the aggregationrule isrequired o work not foreverypossible w-tuple f individual udgmentsbutonlyfor hose thatmeet some additionalconstraints,henunanimity,independence nd non-dictatorshipan be satisfied imultaneously. orexample, if only -tuples (J',...,Jn) satisfyingJ' ... 0 areadmissible, hen he ntersectionule s defined bove meets ll conditions.Likewise, if only tuples (J',...,Jn) satisfyingJ''J ...UJn^Q. areadmissible, henthe union rule as defined bove meetsthem.A domainrestrictionf thiskind s feasible n cases where hediversityn individualjudgments s limited. f,for xample, udgments epresentheknowledge(as opposedto merebeliefs)of ndividuals, hen tmayseem reasonable oassume that nly-tuples J' , ..,) with non-emptyntersectionccur,and the ntersection ulemay seemplausible.Extendingthe range of admissible outputs. f the aggregationrule ispermitted o generatecollective udgmentsthat violate consistencyorinformativeness, hen it is possible to satisfy all other conditionssimultaneously.For example, if judgments representthe beliefs ofindividuals rather han theirknowledge),thenperhapstherequirementsofconsistency r informativenessre too strong t the collective evel. Asalreadynoted, he union ruleguarantees onsistencybutnot nformative-ness), and the intersection ule guarantees informativenessbut notconsistency),while both rules satisfy nanimity,ndependence nd non-dictatorship.More radically, ne mightpermit ollective udgments hattake the form of continuousprobability ssignments cross worlds asopposedtobinaryudgments fpossibility r mpossibility;utprobabilityaggregation ivesriseto somewell-known roblems fitsown.

In conclusion, have shown that a non-trivialmpossibilityesultofaggregation rises not only for udgmentson multiple, nterconnectedpropositions,s in the much-discussed discursivedilemma,"but also forjudgments hat ach consist n theacceptanceof onlya singleproposition,modelled as a set of possibleworlds.Whether r not we findthis resultcompelling,t hallenges s totake stance n which f ts onditionsorelax.

AcknowledgementsThis work was presented t the Formal EpistemologyWorkshop2006,UC Berkeley,5/2006and at the 2006 Conferenceof thePhilosophyofScience Association,Vancouver,11/2006.Much of thewritingwas doneat the Australian National University, /2006. I am gratefiilfor thehospitality nd discussions enjoyed in all threeplaces. For comments



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WHICH WORLDS ARE POSSIBLE? 63and criticism, thankRichardBradley,FranzDietrich,BrandenFitelson,Robert Goodin, Alan Hayek, Marc Pauly, David Schmeidler andTimothyWilliamson.

AppendixI derive theorem1 from a result in Dietrich and List (2007a), embedding the presentaggregationproblem into the standard udgmentaggregationmodel (in Dietrich's 2007"general logics" version), where propositionsare modelled as sentences of a formallanguage.A notationally ifferent,utmathematically quivalentderivation an be givenfromDokow and Holzman's (2005) theoremor fromRubinstein nd Fishburn's 1986)results,on which Kasher and Rubinstein's (1997) theorem on "group identification" sbased. The presentproof invokes the concept of "total blockedness" introducedbyNehring nd Puppe (2002), whose main theorem s a precursor o the results n Dietrichand List (2007a) and Dokow and Holzman (2005). It is interesting o note that thegeneral result fromwhich theorem 1 follows also has Arrow's (1951) impossibilitytheorem n preference ggregation s a corollary.Let L be a simple sentential language, with atomic sentences a', ..., andconnectives-,, V. To each worldujjG , there orresponds sentence, interpretedomean thatworldUj is possible. For any5L and any p G L, writeS (=p if and only ifthe set S UZ U {-/?} is inconsistent n the standard sense of sentential logic, whereZ = {a' V ... Vfl*,-(i - *)}- Informally, '= p means thatS entails/?relative totheconstraint hat hedisjunctionof a', ..., a* is true consistency)and their onjunctionfalse (informativeness).Further, set S L is called inconsistent in L) if S '= p andS'=-p for some p G L, and consistentotherwise. The logic given by the pair (L, '=)captures the consistency and informativeness onstraintson the present aggregationproblemand satisfiesaxioms L1-L3 in Dietrich's (2007) general logics model.Define theagenda of sentenceson which udgments in the standard ense) are to bemade asX = {a', ...,*,-1,...-*}. A judgment G J as defined bove corresponds oa maximal consistent ubsetA X9 where,for ach uj G , a GA ifand only ifujj GJand -4ij GA if and only if u>jJ. An aggregationrule/ : J" - as defined abovecorrespondsto a judgment aggregationrule F for the agenda X in the standard sense(which, in addition, atisfies universal domain and collective rationality), .e., a functionF thatmaps each -tupleof maximal consistent ubsetsof A' (individualudgment ets nthe standard ense) to a maximal consistent ubset of X (collective judgmentset in thestandard ense). It is easy to see thatF satisfies he standardunanimitynd independenceconditions fand only if/ satisfiesunanimitynd independence s definedhere.One of the two main theorems n Dietrich and List (2007a) states that,when theagendaX satisfies he richness onditionof"strong onnectedness" as definedbyclaims1 to 3 below), a judgment aggregationrule F satisfies independence and unanimity(togetherwith universal domain and collective rationality) fand only if it is dictatorial.In order to see thatthepresenttheorem1 follows from hisresult, t remainsto beshown that the agenda X as defined here is "stronglyconnected." To show this, thefollowingmust be established,where in)consistencymeans (in)consistency n L:Claim 1. There exists a minimal nconsistent ubset Y X with | > 3.Claim 2. There exists a minimal inconsistent subset YX such that(Y'Z) U {-z : z G Z} is consistent or ome subset Z Y of even size.



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64 CHRISTIAN ISTClaim 3. For ny ,q eX, therexist ' ,p2>--,Pk X (with = p' andq= Pk)suchthat ' =*/?2,2^*, >Pk-'t*Pk,where jt*Phis defined o mean that'Pji ~Ph} U is inconsistentor omeY X consistent ith/?,ndwith >/?(claim3 states hat heagendahas thepropertyf "totalblockedness,"sdefined y NehringndPuppe, 002).To prove laims1 and2,notice hat, ince > 3, Y = {a', ...,*} has thepropertiesrequiredyclaims1 and2,with = {a',ai).To proveclaim3, pick any pairof propositions,q eX. Consider hefollowingexhaustiveistof cases:

(a) p = q: trivially,*q (withY = 0);(b) p = uj andq = -*withy h: here ^*-^ (withY = {a : /^,});(c) p = -, ndq = ahwithy h: here */1=*withY = {-a/ / y,A});(d) p = a, and# withy A: hereo,l=*-a/or ome /}, h (byb) and -â^*ah(byc);(e) /? -ndwithy:here âjt*afor ome ]^ (byc) and ^*->(byb);(f) /? / nd/. ere ^* for ome7 byd) anda^*-âj (byb);(g) p = -ndq = a/,here ayâ/, or ome A^y (byc) and1=*byd).Thiscompletesheproof.

ReferencesArrow, .: 1951, ocial Choice and IndividualValues,Wiley,New York.Bovens,L. andRabinowicz,W.: 2006,Democraticnswers o complex uestions:Anepistemic erspective,ynthese50, 131-153.Dietrich,F.: 2006, Judgment ggregation: Im)possibility heorems, ournal ofEconomics heory 26,286-298.Dietrich, .: 2007,A generalizedmodel of udgmentggregation,ocial Choice andWelfare8, 529-565.Dietrich, . and List,C: 2007a, Arrow'stheoremn judgment ggregation,ocialChoice nd Welfare 9, 19-33.Dietrich, . andList,C: 2007b,Majority oting eneralized: udgmentggregationyquotarules, ournal fTheoretical olitics19, npress.Dokow,E. andHolzman,R.: 2005,AggregationfBinary valuations,Working aper,Technion srael nstitutefTechnology.Kasher,A. andRubinstein,.: 1997,On thequestion Who is a J?":A social choiceapproach, ogiqueetAnalyse 60,385-395.Kornhauser,. A. andSager,L. G.: 1986,Unpackinghecourt, ale Law Journal 6,82-117.List,C. and Pettit, .: 2002, Aggregatingets of udgments: n impossibilityesult,Economics ndPhilosophy8,89-1 10.List,C. andPettit, .: 2004,Aggregatingets of udgments: wo impossibilityesultscompared,ynthese40,207-235.Nehring, . and Puppe,C: 2002, Strategy-proofocial Choice on Single-peakedDomains:Possibility,mpossibilitynd the paceBetween,Working aper,University

of Californiat Davis.



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WHICH WORLDS ARE POSSIBLE? 65Nehring, . andPuppe,: 2005,Consistentudgementggregation: Characteriza-tion,Working aper,Universityf Karlsruhe.Pauly,M.: 2007,Axiomatizingollectiveudgmentets na minimalogical anguage,Synthesenpress.Pauly,M. and vanHees,M.: 2006,Logicalconstraintsn udgmentggregation,ournalof Philosophical ogic35, 569-585.Pettit, .: 2001, Deliberative emocracy nd the discursive ilemma, hilosophicalIssues 11,268-299.Pigozzi,G.: 2006, Beliefmergingnd the discursive ilemma:An argument-basedaccount oparadoxes f udgment ggregation,ynthese52,285-298.Rubinstein, . and Fishburn, .: 1986, Algebraic aggregationheory, ournalofEconomicTheory8, 63-77.vanHees,M.: 2007,The limits fepistemic emocracy,ocial Choice nd Welfare8,

649-666.Wilson,R.: 1975,On the heoryfaggregation,ournal fEconomicTheory0,89-99.DepartmentfGovernment,London choolofEconomics,LondonWC2 2AE.UKE-mail: List@lse. c. uk

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