the similarity approach to counterfactuals: some problems - g. lee bowie

7/28/2019 The Similarity Approach to Counterfactuals: Some Problems - G. Lee Bowie

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The Similarity Approach to Counterfactuals: Some ProblemsAuthor(s): G. Lee BowieSource: Nos, Vol. 13, No. 4, Special Issue on Counterfactuals and Laws (Nov., 1979), pp. 477-498Published by: Wiley

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The SimilaritypproachtoCounterfactuals:SomeProblemsG. LEE BOWIE

MT.HOLYOKE COLLEGE

Although modal logicians have traditionally een interestedin counterfactuals, nlyrecently as theapplication of possi-ble world semantics o theproblemofcounterfactuals egunto yieldformalsemanticswhichappear to capturethe truthconditions fordinary nglishcounterfactuals.1 he advancehas been broughtabout bydefining imilarity tructures nsets of possible worlds. This use of similarity mong possibleworlds n theanalysisofcounterfactuals as been thesubjectof considerabletheoreticaldevelopment.Through the workofLewis,Stalnaker, anFraassen, Thomason,et. l. , thebasicidea has been modifiedand generalized in the directionofhigherorderquantification overmodalities),various index-ings for selection functions, mpossibleworlds as limitsofsequences of possible worlds, counternecessaries, ounter-comparatives, ounterfactual robabilities, nd so on, all ac-companied byelegantmodel-theoretic otions and axioma-tics.Our naturaltendency o preservetheproductsof greatlabor is, n this ase, reinforced ythepowerand elegance ofthe theories themselves.But howeverelegant and powerfulthesimilaritypproach is in its full theoretical evelopment,there re difficultiest itsfoundations-difficultieswhich astdoubt on the use of the notion of comparativesimilarity fpossibleworlds n theanalysisof counterfactuals.This paper, then, sconcernedtopointoutsomeofthesedifficulties,hich urround thebasicassumptions fthesimi-larity pproach in ts pplication o theproblemofcounterfac-tuals.I shall notbe arguingthatthesimilaritypproach doesnot capture the logic of counterfactual nference. n fact tNODS 13 (1979) 477t 1979 byndiana University

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478 NOtSappears that the similaritypproach, as developed by Lewis,does apturethe ogic ofcounterfactualnference. ndeed, thefactthat t does is a very trong rgumenton itsbehalf. Norshall I be arguing thatthe notion of similarity f worlds isvague and imprecise although, to be sure, it is vague andimprecise).What shallbe arguing s thatno plausibleunder-standing fthe notion fsimilarity enerates correct nalysisof the truthconditions of ordinarycounterfactuals.Moreformally, hile theoriesbased on similarityan verify s validall and only those ordinarycounterfactualswhichseem, infact,to be valid,nonethelessno plausible understandingofthe notion of similarity illgive us as true (in the intendedmodel) just those ordinarycounterfactualswhich seem, infact, obetrue.Thus whilethe similaritypproach generatessuccessful ogic,itdoes not generatea successful nalysis.Before proceedingto similarityheories, t willbe usefulto review the classicalapproach to counterfactuals. his willbothhighlight he successes of the similarity pproach, andenable us to connectthe failuresof the similarity pproachwiththe failuresof the classicalapproach. I followLewis inusing ALI->B' to represent hecounterfactualifA were thecase then B would be the case'. In addition, '->' is the materialconditional, A is the negation of A, and an A-world is apossibleworld in whichA is true. Now the simplest ort ofpossibleworldsanalysisof counterfactualswould be this: foreach world there s a classof worlds C i suchthata counter-factualALI->B istrue n iffA-RBistrue neveryworld nC .Lewis callsthis heory strict onditional heory, inceon itwecan understandAL-*B as L(A-*B), where C i is then ust thesetof worldsaccessiblefrom withrespectto themodalityMl.In the case whereCi is ust the setof worlds at which someformulaSi (whose choicemay depend on i) is true,we mayrestate hetheory s follows: counterfactual LJ->B s true ti iffA-RBistrueateveryworld accessiblefrom) at whichSis true [iff Si & A) -RB is necessary].It is clear thatthere s no choice of Si (or of Ci) whichcapturesthecounterfactual onditional.For choose A as anypropositionncompatiblewith i. Then nomatterwhatB is, twill vacuously)be true in everyworld in which(Si & A) istrue. Hence on this theoryALI->B will be vacuously truewheneverA is incompatiblewithSi. More generally, fB istrue teveryworld t which S i & A) istrue, hen fortiori is



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THE SIMILARITY APPROACH SOME PROBLEMS 479true at everyworld at which Si & A & C) is true. Hence on astrict onditional theoryAL->B entails A & C)L->B.But this nference s not sound for counterfactuals. orexample, ifMort were to go to the store tomorrowhe wouldbuy bananas. It does not follow hat fMort were to go to thestore omorrow nd getshoton hiswaytothe checkout tand,he would buy bananas.The arguments bove show thatthere s no set of worldssuchthatforeveryA and B, AL->B istrue ffA-RBistrue ateveryworldin that set. The source of the difficultys clear.When we askwhetherAL->B, we are asking, t would seem,about the truth value of B in relativelyfamiliarworlds inwhichA is true. Interpreting he counterfactual s a strictconditional requires that we decide on a standard of famil-iaritybefore we know whatA is. S encodes the standard offamiliarity. ut havingchosen such a standard,we maythenwishtoask whatwouldhappen ifthat tandardwere notmet.Butwe cannotdo this.The standard has alreadybeen fixed.Thus the first bjection bove. Similarly, avingchosen somefamiliar etofworlds o ookat nwhichMortgoes to the tore,havingchosento gnore as too bizarreworlds nwhichhe getsshotwhile hopping,wemayturn round and ask whatwouldhappen ifMortwere to go to the store nd get shot. Now we gowrong fwe ignore worlds nwhich Mortgets shot, whereasbeforewe wouldgo wrong fwe wereto consider suchworlds.Thus the second objection above.Givena possibleworldsapproach tocounterfactuals,t sclearthat nevaluating he truth f ALJ->Bwe want tolook atthetruth f A->B in some setofworlds.Fromthe discussionabove it is clear that thechoice of the set of worlds must bevariable, dependent on A, and perhaps on B.The classical approach to counterfactuals akes this ntoaccount.Accordingto the classicalaccount,a counterfactualALI->B is trueiff here s some true S such thatB is true ineveryworld n which S & A) is true. In contrast o the strictconditional heory,hechoice ofS maydepend onA (oronB).S ingeneralwillbe a huge conjunctionwhichwill, ntuitively,tellus enough about whatourworld s likeotherthan the fact(if tis a fact)thatA, so thatwhenwe conjoin itwithA, B isentailed. For example, if we are wonderingwhetherMortwould buybananas ifhe wereto go to thestoretomorrow,would containnatural aws, noughinformationodetermine



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480 NOtSthat the storewillhave bananas, to determineMort's proc-livities oward banana-buying, s well as information boutearthquakes, homocidal maniacs and other factors whichwould, fthey btained,bar Mort'sbanana-buyingproclivitiesfromtheirnatural fruition.We have required that S be true, but clearly more isrequired. For wheneverA-RB is true, t will, ogetherwithA,entailB. (WhenA--B isfalse, o isAEI-*B.) So ifwecan alwayschoose A-*B as S, the counterfactual onditionalwillcollapseinto the material conditional. Goodman terms co-tenable'those truths which are appropriateforconjoiningwithAwhenentertaining as a counterfactual ypothesis. fwearewonderingwhatwouldhappen if Mortwere togo to the storetomorrow,then truthsof the sort noted earlier would beco-tenable with the proposition thatMort goes to the storetomorrow.Now, why sA--B (when true)notalways o-tenablewithA? At first lance, thismight eemtobe anotherofthosecasesinwhichwe wouldliketo be able todrawa distinction etween"real" facts like natural laws,the factthat there will be nohomocidal maniacs nthe storetomorrow, tc.) and "phoney"facts like the factthatA--B). Having drawn such a distinc-tion,wewould thensay that truth sco-tenablewithAjust incase itrepresented realfact nd itwas (in some appropriatesense) compatiblewithA.Now quite apartfromthe fact hatattempts o so distin-guish the real factsfrom the phoney ones have not beennotable successes, the attemptwould be misguided in thiscontext. Martha's salary s $100,000, and her property axesare $5,000. Out of an equal desireto donate 10% of hersalaryannuallyand to donate twiceher property ax annually,shegives $10,000 everyyearto the FlatEarth Society.Now,whatwouldhappen ifMarthawerepresident oftheU.S.). The factthatMarthaalways gives 10% ofher salaryand the factthatthe president's salary s $200,000 entail (togetherwith thehypothesis hatMartha spresident) hat he willgive $20,000.But the factthatshe always givestwiceher property axandthefact hatthepresidentpaysno property axon the WhiteHouse entail (togetherwith the hypothesisthat Martha ispresident) hat hewillgive nothing. et all ofthe facts lludedto seem (to me) to be as intuitivelyreal" as factswhichtypi-cally supportcounterfactuals. here are, it would seem, too



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THE SIMILARITY APPROACH SOME PROBLEMS 481many "real" facts around, so we must make some furtherchoice. Unfortunately, mong the factsrelied on above, theones thatwe want re ust those facts hatwould emain acts fMartha were president.So we have come full circle, ince inorderto decide what would be the case if A were thecase wemustfind hose truthswhich re co-tenablewithA. And to dothat,we must findwhich facts are "real" facts. But amongthose, heonlyones inwhichwe are interested re thosewhichwould remain facts f A were the case, so we must gain knowwhat would be thecase if A were the case.If the truths o-tenablewithA are ust thosewhich ntui-tively support"thecounterfactualAEI-*Bthenthere sgoodreason to believe that n general the truths o-tenablewithAwillbe ust those truthswhichwould remain truthsf A werethe case. For the claim that S supports the counterfactualALi->B can alwaysbe defeatedby showing t to be falsethat fAwere thecase thenS would be the case (as reflection n a fewexamples shows).So onlytruthswhichwould remain truthsfA were the case support ALI-RB. And while some propersubsetsofthose truthswouldsufficeo supportall truecoun-terfactuals ith ntecedentA (for xample,the setof alltruthsA->B which would remain truths fA were thecase), exam-inationof cases makes it appear likelythat anytruthwhichwould remain true fA were thecase can be enlisted n sup-portof thecounterfactualALJ->Bwith heproviso,perhaps,that tdoes not tself ntailB). A solidargument s notpossiblehere,due tothe fact hatthe notion of"support" appealed toremains a rough, ntuitive otion. The considerations aisedabove, however,make it appear plausible that itwillbe noeasier to characterize arefully he truthswhichsupportthecounterfactual LI->Bthan twillbe tocharacterize heprop-ositionswhichare counterfactuallymplied byA and to selectfrom hemthose which re true and, perhaps,which do notimplyB). Thus ifthetruths o-tenablewithA are ust thosewhichsupportcounterfactualswith ntecedentA, itappearsthat n order to givean accountofco-tenability,he classicaltheorywill have to give an account of the counterfactualconditional tself.3David Lewis providesan elegantaccount of counterfac-tualswhich, obe understoodas an analysis f the counterfac-tual conditional', equires (in addition to the semanticsofpossible worlds) only the intuitive notion of comparative



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482 NOOSover-all imilarity aken as a 3-place relation among possibleworlds. On Lewis' account, givena possible world , there s aset of spheres associatedwith . The set is nested,and closedunder union and intersection. ach sphere is a set of possibleworlds. ntuitively,fthere s a sphere associated with suchthatj is inside it and k is outside t,then is more similar o ithan k is. We need onlyconsider the usual case where i is amember of every,non-empty phere associated with itself.Thus we will peak ofthespheresassociated with as spheresabout. A sphere sA-permittingff here ssomeworld n t atwhichA is true.Then ALI-*B is trueat i iff itherthere s noA-permitting phere about i, or there is some A-permittingsphere about i such that A-HB is trueat everyworld in thatsphere.Phrased n terms fsimilarity, egetthefollowing orthe non-vacuous case: ALI-*B is trueati ifthere s a world atwhichA istruesuch that tno worldat leastas similar o isAtrueand B false.The characterizationn terms of spheres and the char-acterization n terms of the 3-place similarity elation (j ismore similar o i thank is) are equivalent.Given a system fspheres,we can say thatj is more similar o i thank is iffforsome sphere,S, about i, is inside S and k is not. Similarly,given the comparative similarity elation,we can define asystem f spheres as follows.For each world , S is a sphereabouti iff veryworld nS ismore similar o than any worldnot in S.4 There is then a natural isomorphism betweensphere-systemsnd similarity ystemswhichpreserves truthvalue of all formulasat all worlds. I have described Lewis'semanticsnitiallyn terms f spheres,since the truthdefini-tion s simpler nd sinceit s Lewis' preferredformulation.willrely n thesimilarityormulationnthesequel,since t s amoreconvenientvehicleforexposingthe difficulties hichraise. Naturally, n virtueof the close correspondence be-tweenspherestructuresnd similaritytructures, nydoubtsabout similarity generates corresponding doubts aboutspheres;so no purposeotherthanease ofexposition sservedby the decision to focus on the similarity haracterization.Finally althoughthe accessibility elation enters nto Lewis'account, t s not relevant omydiscussion, nd I haveignoredit. Ignoring t s equivalent osupposing ttoholdbetween llpossible worlds.)



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THE SIMILARITY APPROACH SOME PROBLEMS 483If we assume for any world and propositionA, that fthere s anyworldat whichA is truethen there is a closestworld at whichA is true (that s a world at which A is truewhich smore similar oi than any other world atwhich A istrue),we get as a special case Stalnaker's heory:That AL->Bistrueat iffB is trueat the closest mostsimilar)world to atwhichA is true.Lewis' theoryseems superior to Stalnaker's. t is moregeneral,sinceStalnaker's heory an be obtained as a specialcase ofLewis' by placingsuitablerestrictionsn thesystem fspheres or on the similarity elation). Stalnaker'smodels donot seem to be generatedby our intuitive otionofsimilarity.For example suppose, contrary o fact, hat Mort were stand-ing more than 120 feetaway from the WashingtonMonu-ment. twould seem thatgiven anyworld nwhichMortwerestanding 120 + 8 feetaway,there s a world more similar oours inwhichMort sstanding nly 120 + 8/2feet way. Thusthere seems tobe no closestworld in which Mort s standingmore than 120 feet away from the WashingtonMonument.Furthermore, t would seem as though twodifferentworldscould be equallysimilar o a givenworld.Consider a world nwhich threepool ballsare linedup on a pool table,the 8-ballbetween he 7-ball nd the 9-ball. s a world nwhich he 8-ballis 3 inchescloser tothe7-ballmore or less ike thegivenworldthan one inwhichthe 8-ball s 3 inches closer tothe9-ball? twould seem as though he twoworlds reequallysimilar o thegivenworld. But this annot be so inStalnaker'smodels. ThusLewis' models better aptureour intuitive otionofsimilarity.The more telling onsideration s this: Stalnaker's heoryvalidates the scheme (ALI->B) v (ALIJ>B). But this does notseem correct. n theexample above, whichof the followingcounterfactualss true: i) ifthe8-ballwere3 inches closer toeitherball, twould be 3 inchescloser to the7-ball, ii) ifthe8-ballwere 3 inches closer to eitherball itwould be 3 inchescloser to the 9-ball?Surely fthe 8-ballwere 3 inchescloser toeitherball then twould eitherbe 3 inchescloser to the7-ballor it would be 3 inchescloser to the9-ball. But this does notforceus toaccepteither i) or (ii),and intuition eems tohaveitthatneither i) nor(ii)istrue. But either i) or (ii)mustbe trueon Stalnaker'stheory.Both Stalnaker and Lewis give formal semantics andaxiomaticsfor counterfactual ogic. Structures re defined



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484 NOOSoverarbitrary ets,understood as setsofpossibleworlds, ndarbitrary rderingson those sets (or theirequivalents,as inLewis' spheres),understood as similarityrderings.Both in-tend their ystemsogenerate n analysis fcounterfactualsnthe sense that one gets the truthconditions for ordinarycounterfactuals y looking at the truthconditions of theircorrespondingformulas n the standard nterpretation,hatis, the nterpretation hich akes ts etofpossibleworlds o bethe real set ofpossibleworlds, nd which akes the relationonpossibleworlds tobe thereal similarityelationon theset ofpossible worlds. What I shalltry o show in thesequel is thatthere s no plausibleunderstanding f the real similarityela-tion whichgivesus thecorrecttruth onditionsforordinarycounterfactualsneitherLewis' orStalnaker's heory. or easeofexposition t will be convenient o speak, a a Stalnaker, fthe A-worldwhich s closesttoours,or the A-worldmost ikeours, or the A-worldmost similarto ours (these are inter-changeable).All ofmyremarks an be reformulated o as toavoid assumingthat there s a closestA-world,though gen-erally at thecostofmore complex sentencestructure.Now,when wecomparetwoworldsforover-all imilarity,inwhatrespectsdo we compare them? t followsfromLewis'account that we mustcompare them in all respects.For iftherewererespects nwhich worldmight iffer rom whichdid not count at all in udging itsoverall similarityo i, thentherecould be a worldwhichdiffered rom butonly nthoserespectswhch counted fornothing n udging overall similar-ity.This would entail theexistenceofa worldwhichwas notidenticalwith ,but whichwas ust as similar o as i is to itself.But Lewisadopts the"centering ssumption,"whichrequiresthat no world be as similar o i as i is to itself.Furthermore,Lewis evidently ntends that worlds be compared in theirtemporal otality. hat is,thebasisofcomparison sthe entireworld, nits patialand temporal otality,nd that otalitys tobe compared with the spatialand temporaltotality f otherworlds in every possible respect of comparison. Althoughsome respectsof comparisonmaycount for ittle n udgingoverall similarity, o respectsof comparison can count fornothing.But thismethod ofcomparisondoes notseem togive uscorrecttruthconditions.Consider: I am standingwithmyfingeron the ultimatedoomsday button. The buttonand



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THE SIMILARITY APPROACH SOME PROBLEMS 485associatedmechanismshave been thoroughly ested, nd arereliable. If I were to push it, the entire universe wouldexplode-unless a loose piece ofmetaldrops down as I pushthe button nsuch a way as to am the button.But it won't. t isloose, but not loose enough. Yes, I see that f I pushed thebutton, he universewould explode. Butwhathappens intheworld most ikeoursinwhich pushthebutton?There can beonlyone answer-the pieceofmetal falls nyway nd ams thebutton.For surely world n whichthe buttongets ammed isfar more like ours than a world in which it doesn't and theentireuniverseexplodes. Even if the piece of nH~eTalad tocome from ll thewayacross theroom,or if the buttonhad todisintegrate,heresultingworldwould be more ikeoursthana world whichexplodes. In factwe can saywithsome confi-dence that in the world most like ours where I push thebutton,the nearest relativelyunattached small object fliesunder the button and jams it,for surely a temporary ocalbreakdown nthe aws of mechanicswouldpreserve imilarityfarmorethan worldcataclysm. o, theexamplecontinues,fpush the button, he world would not explode. But we knowthat if I push the button,the world will explode. We caninspect the mechanism, do simulated trial runs, etc. So thetheory s wrong. It is wrong because it makes change toodifficult.f there is some simple way to keep the world ontrack, t willhappen. If thingscan go right, heywill.If the objection stands,the theories we are consideringwould be clearly unacceptable, and the similarity pproachwould have togo. Now itmightbe thought hattheobjectioncan be met by placing more weighton laws of nature indetermining imilarity. his approach failsfor several rea-sons. First,no matterhow muchweightwe place on laws,wemust stillplace someweight n particularmatters f fact.Todo otherwisewouldbe to do injustice omore mundane coun-terfactuals nd to run afoul of the centeringprinciple.Butwhateverreasonable distribution fweightswe decide upon,by makingthe results fpreserving he awssufficientlypec-tacular,and by makingthe breakdown in laws sufficientlyunexciting as in the example), we will findcases where asmall, ocalviolation f the aws preserves imilaritymorethanstrict dherence to the laws.Secondly, n entertaining counterfactualwithfalse an-tecedent, heonlyworlds nwhich the antecedent s truebut



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486 NOOSwhere the laws of this world are preserved mightbe worldswhich are unacceptablydifferent romours. As Lewis5says,

Suppose thatthe laws prevailing t a world are deterministic,s weused to think the laws of our own world were. Suppose a certainroulettewheel in thisdeterministic orld stops on blackat a timet,and consider the counterfactual ntecedent that t stopped on red.Whatsortofantecedentworldsare closettoi? On the one hand, wehave antecedentworldswhere the deterministicaws of holdwithoutexception, but where the wheel is determined to stop on red byparticularfacts different romthose of i. Since the laws are deter-ministic, he particular actsmustbe different t all timesbefore ,nomatterhow far back.... On the other hand, we have antecedent-worlds hat areexactly ike until or shortly efore;where the laws of hold almostwithout xception;but where a small, ocalized, inconspicuousmira-cle attorjustbeforepermits hewheeltostoponred nviolation f thelaws. Laws are very mportant,but greatmasses of particularfactcount for somethingtoo; and a localized violation s not the mostserious sortof difference f law.Thirdly,in testing scientifichypotheses we must takesome of thelaws rather ightly. uppose for example thatwe

are testing artofthespecial theory frelativityor the firsttime,bychecking n theapparent positionofMercury uringa solar eclipse. Classical mechanicshas it thatMercurywillappear to be where t s,atpointC. Special relativityas itthatMercurywill appear to be at pointS, distinct romC. Inter-estingly,Mercury ppears to be atS,and we infer hat lassicalmechanicswas wrong.But let us entertain hecounterfactualhypothesis hatMercuryhad appeared tobe at pointC. Thereare twohypotheses at least): HI-that althoughthe photonsweredisturbedbythemassof thesun,randomdisplacementof the photons compensated for this disturbance, makingMercury ppear atpointC. On H I, specialrelativity asrighteventhough Mercury ppeared to be atC, because therewerecompensatingrandom influences.H2 has it that the mass ofthe sun did not disturb hepassage of thephotons, nd class-ical mechanicswas right.Now the point is that n the actualcase, we clearly reject HI as an explanation of ourhypothesizedfailure to confirm pecial relativity,nd assertinstead thatif Mercuryhad appeared at point C, then re-lativisticmechanicswould have been incorrect.Our accept-ance of H2 in thiscounterfactual ituationrequires thatwecount as closera world nwhichrelativisticmechanics s ncor-



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THE SIMILARITY APPROACH SOME PROBLEMS 487rect, world nwhichthe laws of nature are violated,than aworldwhichis fully ccounted for withoutviolationof anylaws of nature, but which is so improbable as to be disre-garded. We conclude with Lewis that"the preeminenceoflaws of nature among cotenable factualpremises is only amatterof degree."6Fortunately,here s an easierway tomeet the objection.It canbe metbymaking lear that he worldwe are comparingwithours is not being compared in virtueof its temporaltotality.We must require only that its historyup to (andperhaps ncluding) he time t whichA is true for ounterfac-tual AEI-*B) is most like the historyof this world. In theexample, we are to imaginestanding n theroom,finger nthe button; hestage sset, nd everythingo far s as muchaspossible ikethings re here. Atthispoint-I havejust pushedthe button-we stopworryingbouthow closetheworlds re;we ust sitback to wait and see whathappens. In thiscase weneedn'twait ong-the world explodes. It no longermatters,though,thattheworldhas ust become verydifferent romours. We just stopped comparing. We made the worlds asclose as possibleup to and includingthe time of the button-push,and then et naturetake tscourse.Sincethatworldwaslikeours,themetalwas loose butnot too loose,the mechanismworked, tc.And because all thatwas thecase, the worldblewup, ust as thisworld would have if I had pushed the button.Once the basis of comparison is made clear, the coun-terexample sblocked. Unfortunately,he solutionraisesnewdifficulties.Paradigm counterfactuals re forward-directed. heyconjecturewhatwould happen in the future f thingsweredifferentnthepastor present, r whatwouldhappen inthefuture fthingsweredifferent t thesame timeor earlier nthe future.But counterfactuals an also be past-directed.There is an alarminginstabilityn the structure f theuniversewhichmakes t iabletocataclysm. ecause we find tcomforting, e haveinvented n earlywarningdevice EWD)which ends out a signal n advance of theuniversaldestruc-tion.The device,forproperoperation, ssituated t thecenterofmass ofthe universe.There are twosocieties-one, SocietyNear,so close to thecenterof the universethat f t receivessignal, herewillbe 3 yearsbeforethecataclysm egins.Theother, ociety ar, ssofarfrom hecenteroftheuniverse hat



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488 NOOSby thetime treceivesthewarningsignal,2/3 f the universewill lreadyhavebeendestroyed, nd theremaining 3willgosoon. Now,entertain hetwocounterfactual ypotheses: So-cietyNear receivesa signalfromtheEWD' and 'SocietyFarreceivesa signalfromthe EWD'. In the first ase, sincethedevice has been tested and is highlyreliable,all workswell.Sincewe are onlycomparingthe histories f the universesupto thetimewhenSocietyNearreceives he signal, universe nwhichthesignal s received and is genuinewillhave a historymore ikethat fouruniverse hana universe nwhich signalis receivedthrougha malfunction f thishighlyreliablede-vice.We may say,then,that fSocietyNear wereto receive asignal,thecataclysmwould begin 3 yearsthence.On the otherhand,suppose SocietyFar receivesa signal(even thesame signal).Now, anyuniverse n which a signal sreceivedbySociety ar, and isgenuine, s a universewhichhasalreadybeen 2/3destroyed.Since we are comparingthe his-toriesof suchuniverses o the history f our universe, uchauniverse,being2/3 estroyed,wouldbe grossly ifferentromour intact ne. We are in the same position s beforewith hedoomsdaybutton.For anyuniverse n which SocietyFar re-ceives a signalwhichis genuine (hence a universewhichisalready /3destroyed) here s a universewhosehistorysmoresimilar o thehistoryfour intact niverse,n which hesignalresulted froma malfunction.Hence the counterfactualifSocietyFar receives signal, hecataclysmhas begun' isfalse,on the similarityheory, ven if we suppose SocietyFar toreceivethe ame ignalwhichwe suppose SocietyNear to havereceived,fromthe same device.So inchoosinghistories s our basisofcomparison,wearestillcomparing too much. Perhaps what we are to do is tocompare someworldwith ursonlywithrespecttoitstempo-ralslice tthe time twhichA istrue, gnoring, orpurposesofcomparion, not only its future from that time (as we didabove), butalso itspastbeforethattime.Thus, forthecoun-terfactualAEJ->B,we picktheA-worldwhich s such that tstemporal lice smost ikeours,and see whatwill did) happenin thatworld.Notefirst hatunlesswe assumea strong orm fdeterminism,his uggestionwillrequirethatwe abandon thecentering ssumption.For thecentering ssumptionrequiresthatno world s as similar o as i is to tself. ut on the currentsuggestionwe are comparingonlytemporalslicesofworlds;



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THE SIMILARITY APPROACH SOME PROBLEMS 489and unless some strongformof determinism olds, two dis-tinctworldswill have identical licesat a given time,differingonly at some earlier or latertime. Since we are comparingonlytemporal lices, wosuchworldswillbe as similar oeachother as each is to itself.But this is not permittedby thecentering principle. However, since we could adopt whatLewis calls "the weak centering principle,"8whichrequiresonlythat no world be more similarto i than i is to itself,nointractableproblemis created for the similarity pproach.When we compare an A-sliceof another world withourworld,what rewecomparing twith?When we werecompar-ing temporally ompleteworlds, hiswas no problem.Butnowthatwe are looking onlyat an A-slice of a possible world,wemust presumably ompare it forsimilarity ith sliceof ourworld; and thequestionis,which slice?Evidentlywe do notcomparetheA-slicewith current liceofthisworld.Supposethat t sDecember, and myfoot sina cast whichwill ome offnextweek. f weare entertaining he counterfactualif wereto go skiingnFebruary, would need a newpairofskiboots',we wouldgowrong fwe compared slices nwhich go skiingin Februarywith urrent lices, ince nthe slice most ike thecurrent lice of ourworld, have a caston myfoot. Hence wewould be askingwhatwillhappen in a world in whichI goskiingnFebruarywith castonmyfoot, nd this sclearlynotgoing to give us theright nswer.Do wecompareA-sliceswith sliceof our world which s,as itwere, cotemporaneouswiththe A-slice?9That is,in theexample above, do we compare A-slices with a slice of thisworldtaken n February? uppose, as a matter ffact, hat nJanuary decide nottogo skiing, ut to cut off he efthalfofmyfoot nstead.Thus inFebruary have onlyhalf a foot.Wewouldgo wrong fwe were tocompareA-sliceswith slice ofthisworld taken nFebruary, or nthat lice haveonlyhalffoot and would need new skiboots). But in thereal counter-factual ituation,f had gone skiingnFebruary, would nothave decided to mutilatemyfoot nJanuary, nd would nothave needed new boots.It is not as easy as it should have been to see where wewentwrong.For in deciding thatwe had to compare slices,and notworlds,we tacitly ssumed thatwe would compareA-sliceswith slicesof our world. But in the skiingexamples,theintent s clearly hatamong all of theA-worlds,we chose



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490 NOOSthe one whosecurrent lice smost ike thecurrent liceof ourworld, and then see what happens there. Thus in the firstskiingexample above, look among the worlds n which goskiing nFebruaryfor hatworldwhose Decemberismost ikeDecember in thisworld,and then see whether need newbootsin thatworld. In thatworldI do not have a caston myfootin February,and I do not need new boots. The sameprocedureapplied to thesecond example givesus a world nwhich go skiing n February, o not mutilatemyfoot, nd donotneed new boots. Now thatthe basis for comparisonhasbeen made clear, things eem to workright.Unfortunately, ehave stillnotgot t right.Return otheexample oftheearlywarningdevice.On thepresentsugges-tion, nevaluating he counterfactualifSocietyFar receivessignal,the cataclysmhas begun',we are to look among all ofthoseworlds nwhichSocietyFar receives signalforthe onewhose current liceresembles our world's currentslice. Butthe currentslice most of any world in which the signal isgenuinewillbe 2/3destroyed, nd willbe less like our currentslice thanone inwhichthesignalresultedfrommalfunction.Thus the counterfactual till urnsout false on the similaritytheory,when it should turnout to be true.Two more examples will help illuminatethe problem.Consider the falsecounterfactualif the Norman Conquesthad failed, he worldwould be prettymuch the way t snow'.The present uggestionhas us look amongall of the worlds nwhich he Norman Conquestfailsfor heone(s) whose currentslice smost ike our current lice,that s,fortheone which smost ike oursnow. But giventhepossibilityfcompensatingdifferenceswhichcounteract he failure f the NormanCon-quest,there s at leastone worldin whichtheConquest failswhichtoday is prettymuch like ours. Hence the world inwhichtheConquest failswhichtodayis most ike our worldtoday,willbe a world whichis prettymuch like our worldtoday.Thus on thepresent uggestion, he counterfactualifthe NormanConquest had failed,theworld would be prettymuchtheway t s now'comesout true.To gettheright nswerin this example we have to compare A-slices slices n whichthe NormanConquest has ust failed)withco-temporaneousslicesof our world,or alternatively,ompare historiesup tothe time of the failureof the Norman Conquest with co-temporaneoushistories f our world.Whiletheseproceduresgiveustheright nswer n this ase, they ailed n earlier ases.



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THE SIMILARITY APPROACH SOME PROBLEMS 491Now consider one last example. Mort is sitting t thedinnertable with dish of magic yogurtn front f him. Mortis 6 feet all.The yogurtworks s follows: fhe eats exactly neteaspoon or less,he will remain 6 feet tall. As he eats morethanone teaspoon, he willgrow as a linear functionof theamount, until, f he eats itall,he willgrowto 7 feet tall. As amatter f fact,Mort abominates yogurt,magic or not, and isnot going to eat anyof it,although he maydip his spoon in itand playwith t. Now, consider the counterfactualif Mortwereto eat some of thisyogurt,he would remain6 feet tall'.My intuition s thatneither this counterfactual,nor the as-sociated counterfactual,if Mort were to eat some of thisyogurt,he would grow', is true.10The antecedent can besatisfiedn too manyways, nd there s no wayofprojectingwhichwayitwouldbe satisfied f itwere satisfied.But thesimilaritypproach saysthat fMort were to eatsome of theyogurt, e would remain6 feet all. For a world nwhich eterisaribus e iseatingonly tiny itofyogurts moresimilar to our world, n which he eats no yogurt, han anyworld in which he eats more. Hence in the closestworld(s),Mort eats only a tiny bit of yogurt and remains the sameheight.It might be thought that there will be featuresof thesituationwhichwouldgiveus a better nswer. f,for xample,Mort were an inveterate glutton, given to extremes,thenperhapswe could saythathe would either at none or eat t ll.If, on the other hand,he were a man of delicate sensibilities,giventocaution n newventures, hen we could say that fhewere to try ome, he would tryonly a tinybit,and wouldremainthe same height.This would amount to sayingthatthere re features f the situationincludingMort's haracter)which re more mportantndetermining imilarity,hanthemere quantity fyogurtwhich Mort eats.While such considerationsmightsettle the issue morefavorablyn this ase,thegeneral problemremains.For how-ever we construe imilarity,f n a counterfactualALJ-RB,heantecedent s false but can be satisfied n a number ofways,and ifthereis no wayof projectingwhichway it would besatisfied,f t were satisfied, henthesimilarity pproach re-quires that t be satisfiednwhateverwayis closestto itsnotbeing satisfied t all. So forany antecedentA, we have thefollowingcounterfactual rue: if A were the case, then A



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492 NOOSwouldbarelybe thecase. Thus, ifMortweighedbetween 300and 400 pounds, he would weigh between 300 and 310, be-tween 300 and 301, etc.Here there snothing n Mort'schar-acter or constitutionwhich suggests any particularrangewithinwhichhisweightwouldfall fheweretoweighbetween300 and 400 (except,ofcourse,therange300-400). We haveno reasonableway ofsaying,forexample,whetherhe wouldweighover 350 or under 350. But the similarity pproachrequires that he weigh between 300 and 301 if he were toweighbetween300 and 400. This is because worlds nwhichhe weighsbetween300 and 301 are, ceterisaribus,more likeour world (in which Mortweighs, et us say, 180) than areworlds n whichhe weighsover 301.11Let us reviewour results husfar. n evaluatingthe truthvalue of the counterfactualAL->B at a possibleworld (wesupposeherethatA is false n ),thesimilaritypproach hasuslook atworldsj, inwhichA istrue, nd whichare similar o .We have asked what thingsone is comparingwhen one isevaluatingoverallsimilarity,nd haveinvestigatedhefollow-ing alternatives:

(1) compare i andj in theirtemporaltotalities,(2) compare the history f up to the timethatA be-comes truewiththecorrespondinghistory fi,(3) compare an A-sliceofj witha current lice of i,(4) comparean A-sliceofj with co-temporaneous liceof i, and(5) compare a current lice of with contemporaneous(current) liceofj.

Although ach ofthese lternativesworkedfor omecounter-factuals,none of themworksfor all of the counterfactualswhich we have considered. Furthermore, there are somecounterfactuals orwhichnone ofthealternativesworks.Theearlywarning device example and the example of Mort'sweight re suchcases. As is clearfrom helist bove, we havenot exhausted the possible alternatives.For example, wemight ry omparing hetwoclosest licesof andj, whenevertheyhappen to take place, or (if this is different) he twoclosestcontemporaneous lices.We have selectedfordiscus-



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THE SIMILARITY APPROACH SOME PROBLEMS 493sion only the most naturalcandidates, but none of the un-naturalcandidateswhich haveinvestigated oes any better.Could itbe thatwhilethere sno singlenotion of similar-itywhich gives us a uniform nalysisofcounterfactuals, hatthere is a range of notions of similarity uch that (a) eachcounterfactualscorrectly nalysedbyone notionof similarityin thisrange, and (b) the contextof assertion ndicateswhichinthisrange s the ntendednotionof similarity?n discussingQuine's examples,namely

IfCaesar had been in command[in Korea]he wouldhaveused the atom bombversusIf Caesar had been in command he would have usedcatapults,Lewis says,

In one context,we may attach great mportanceto similarities nddifferencesn respectof Caesar's character and in respectof regu-larities concerning the knowledge of weapons common to com-manders nKorea. In anothercontextwemay attach ess mportanceto these similaritiesand differences, and more importance tosimilarities nd differencesn respectofCaesar's ownknowledgeofweapons. The first ontext resolves the vagueness of comparativesimilaritynsucha waythat omeworldswith modernizedCaesar incommand come out closer to our world thananywithan unmoder-nized Caesar. It therebymakes the first ounterfactual rue. Thesecondcontext esolves hevagueness ntheoppositedirection,mak-ing the second counterfactual rue. Other contextsmight esolve thevagueness notherways.A third ontext, or nstance,mightproducea tie between the closestworlds with modernized Caesars and theclosestworldswithunmodernized Caesars. That contextmakes bothcounterfactuals alse.12

Could itbe thatourproblem ssimilar, nd that he context fassertion an be calledon to ndicate heappropriatenotionofsimilarity?Note first hat fA isfalse, nd if A & B) and (A & B) arebothpossible,bychoosing appropriatesimilarityelationswecan alwaysmakeALI-*Btrue at ,or makeACl->B true at ,ormakeneither rue.To make ALI-*Btrue,merely ivide the setof worlds nto twosubsets,W1and W2.Let W1be the set ofworlds t whichA->B istrue, nd letW2be the set of worlds twhichA->B is false.Order the worlds so thateveryworld n



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494 NOOSW1 is more similarto i than anyworldin W2. Order worldswithinW1and W2 nanywayyoulike. Then among A-worlds,any in whichB is truewill be more similarto i than any inwhichB is false, o ALI->B will be true ati. To makeAC-> Btrue at i, merely etW1be theset of worlds at whichA-> B istrueandW2 he setofworlds t whichA-> B isfalse.Finally, omake neitherAL->B norACl->B true, etallworldsbeequallysimilar o . Thus, while t s true thatforeach counterfactualthereis some measure of similaritywhichgives the correcttruthvalue, this s trivially rue; there is also a measure ofsimilarity hichgivesus the incorrect ruthvalue.Secondly, he choiceamongthe five or more)alternativeconstruals fsimilaritybove does not seemtobe a choice as tohow to resolvevagueness. tdoes not seem tobe a question, sinQuine's example,of how toweighvariousrespectsof simi-larity. n thatexample, in the contextof a discussion aboutmilitary sychology nd the relative tendencies of militaryleaders towardoffense rdefense,wemight aythat fCaesarwere nKorea, hewould have used theatombomb.However,indiscussing he ways hatgeneralsadapted their ttack o thevagaries of terrain,contrastingCaesar's methods in Alesiawithhis methods nBritain,we might aythat fCaesar were nKorea, hewould have used catapults.Note that n the formercase we would put emphasison theword Caesar', whereas nthe latter ase we wouldplace emphasison theword Korea'.The problem we face here, however,is not a problem ofcontext,forwe mayneed to choose two different ases ofcomparison n the same context.Change thedoomsdaybut-tonexample slightly,upposing the buttonto be an ancient,butstill perational, rtifacteftbya lost cultureof aliens. Inshowing he button o a group ofjournalists, general mightsay, "If I pushed this button,the universe would be de-stroyed."As we have seen, to getthecorrecttruthvalue forthis ounterfactual e seewhathappens inthe world nwhichthegeneral pushesthe buttonwhosehistory p to the timeofthebuttonpush is most ike our cotemporaneous history. utsuppose the generalnow adds, "and ifAttillathe Hun hadpushed thisbutton, he universewouldhavebeendestroyed."Nowwego wrong fwecomparehistories p to thepresent; ogetthecorrect ruth aluewemust omparehistories p to thetimeof Attilla.The choice ofcomparison basis has changed.But thecontext f utterancehas remained thesame. Hence it



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THE SIMILARITY APPROACH SOME PROBLEMS 495cannotbe contextwhichdetermines he choice of comparisonbasis.13Thirdly, n some of our examples there is no plausiblechoice of comparisonbasis which seems to give the correcttruthvalue. In the example of Mort's weight, ny plausiblechoice ofcomparisonbasis forces itherALI-*B or AD}-+B tobe true,whereaswe wantbothto be false. n theearlywarningdevice example, to make the counterfactual if SocietyFarwere to receive a signal,theuniverse would be 2/3destroyed'true nthegivenworld , wemustfindworlds nwhich signalis receivedand compare themwith in respectto theirhis-toriesup tothe time hesignal semitted,otup tothe time hesignal s received.But change theexample slightlykeep thecontextof utterancethe same ifyou like) so that nstead ofemitting he signal 3 yearsbefore the cataclysmbegins,thedevice emitsa signalonlyafter?2 of the universehas beendestroyed, nd incomparinghistories p to the time hesignalis emittedwe will make the counterfactual alse.To make ittruewe have to comparehistories nlyup to someearliertime,some time before thecataclysms sufficientlynderway.Forin the changed example, any world in which the signal isemitted and is genuine will be a world which is already '/2destroyed nd which s therefore ufficientlyifferent romthe givenworldi that worldsin which the signal is emittedthroughmalfunctionwillmore similar.Notefinallyhat hereis no obviousrelationship etweenthecounterfactualifSoci-ety Far were to receive a signal, the universewould be 2/3destroyed' including ts ontext f utterance) nd the timewemust topcomparinghistories.n the unmodified xample westopped comparinghistories t the timeof emissionof thesignal; in the modifiedexample we stopped comparing atsome earlier time beforethe cataclysmwas sufficientlyn-derway.The choiceof timesdoes not seem tobe determinedbythe counterfactual tself rbythe context futterance thecounterfactualnd contextwere the same in bothcases), butrather ythe detailsoftheworking f the device. Thus there sno way n whichthecounterfactual long with ts contextofutterancedetermines n appropriatebasis forcomparison.There is noquestionthat, or counterfactual ypothesisA,there re respects orwhich hepropositions hatwould bethe case ifA were the case are ust thepropositionswhich rethe case in those A-worldsmost similarto ours in the given



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496 NOOSrespects.ALI->B willalwaysbe true ust in case B is true inthoseA-worldsmostsimilar o ours withrespectto the truthswhichwould emain ruefA were he ase.This muchwas clearfromour discussion of the classical approach to counterfac-tuals. Unfortunately t is not generally the case that theA-worldsmost ikeours with espect othetruthswhichwouldremaintruths f A were thecase coincidewiththe A-worldsmost like ours. In manycases the A-worlds which wouldbeactual ifA were truewillbe less similar to ours than otherA-worldswhich ouldbe actual fA were true. Whenthis s thecase, theanalysisof counterfactualsn termsof over-all imi-laritywillgive us thewrongtruth-value. hus in the EWDexample,the worldwhichwould have been actual had a signalbeen received was less like thegivenworld than other worldsin which the signal was received. Given the counterfactualhypothesis ,there remanyways oadjustthetruth aluesofotherpropositionsnorder to accommodateA. In many ases,the adjustmentwhichwouldbe made if A were thecase wouldbringabout a world much less like ours than otherworldswhich ould be brought boutby ess drastic djustments. t isperhaps a brute factthatsome changes would bringaboutmore drastic over-allchanges than they need bring about.There iscertainly o reason to expect thatthechanges whichwouldresult from a change in the truth-value f A are theminimum changes compatible with logical necessity. Butwhen the changeswhich would be broughtabout byA aremore drastic overall then theyneed be, the similarity p-proach willgive us thewronganswers.Since there s some notionofsimilarity,amely imilaritywithrespectto thosetruthswhich would remain truths fAwere thecase,whichgivesus theright nswerfor he counter-factualAL-4B, itis no surprisethat the similarity pproachseems to provide an acceptable logic for the counterfactualconditional.Unfortunately,here eems to be no non-circularwayofexplicating herequirednotionofsimilarity hichdoesnot tself nvolve ounterfactual iscourse. The problem s nodifferent romthatfacedbythe classicalapproach, whereinthere seemed to be no non-circularway of explicatingtherequirednotion of co-tenability.The similaritypproach,understood as an analysis f thetruthconditions or ordinarycounterfactuals, equires thatwhen we specifyfor the formal theoryan interpretation



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THE SIMILARITY APPROACH SOME PROBLEMS 497whose set of possible worlds s the set of real possible worlds,and whose similarityelation s the real (but possiblyvague)similarity elationon possible worlds,the resulting nterpre-tationmakes trueustthosecounterfactuals hich re, nfact,true.14Although for each counterfactual, here s trivially,similarityelationwhichworksfor thatcounterfactual,hereappears to be no similarityrelation specifiable in non-counterfactual erms,which atisfieshe constraint bove. Weconclude thatthesimilaritypproach failstogivean analysisof the truth onditionsof ordinarycounterfactuals.

REFERENCES[1] David Lewis, Completeness ndDecidability fThree LogicsofCounterfactualConditionals,"Theoria 7(1971): 74-85.[2] , Counterfactuals,Harvard University ress, 1973).[3] Donald Nute, "Counterfactuals,"Notre DameJournal of FormalLogic XVI,4(October 1975): 476-482.[4] , "Counterfactuals nd theSimilarityfWords [sic],"JournalfPhilosophyLXXII, 21(1975): 773-8.[5] Robert Stalnaker, A TheoryofConditionals," n N. Rescher, tudiesnLogicalTheory,Oxford: Blackwell, 1968).[6] and R. Thomason,"A SemanticAnalysis f ConditionalLogic,"Theoria36(1970): 23-42.[7] Richmond Thomason, "A Fitch-StyleFormulation of Conditional Logic,"Logique tAnalyse 2(1970): 397-412.[8] Bas van Fraassen, The Logicof ConditionalObligation,"JournalfPhilosophicalLogic I(1972): 417-38.

NOTES'Earlierversions f thispaper have been read to colloquia at the University fMichigan and the University f Massachusetts. am grateful o theparticipants fthosegroups, nd especially oJohnG. Bennett, or heirmanyhelpful uggestions.2See David Lewis, 1, 2]; RobertStalnaker, 5]; RobertStalnaker nd RichmondThomason, [6]; RichmondThomason, [7]; Bas van Fraassen, [8]. All subsequentreferences o Lewiswillbe toCounterfactuals,nd all subsequent references o Stal-nakerwillbe to "A Theoryof Conditionals."3Lewis observes the equivalence of the Classical theory with his under theassumption hat he truthso-tenablewithA arejust thosewhich accordingtoLewis'theory)would remain truthsf A were the case. Cf. Lewis, p. 69-70.4Cf. Lewis, p. 48.5Lewis,p. 75.6Lewis,p. 7570f course thesituationwillvarydepending on our individuation rinciplefor

temporalslices. Consider, forexample, the factthat the paper whichyou are cur-rently eadingwas written yme. Is that fact boutthe current liceof theworld? fso, then no world could have an identical urrent liceunless in thatworld,at sometime nthepast, wrote hispaper. Ifnot,then world nwhich omebody lse wrote



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498 NOSthispaper could have an identicalcurrentslice. I shall ignore such problems, l-though o theextent hatwe taketemporal lices eriously,hey re seriousproblems.8This s not a possibilityn Stalnaker'saccount.

9I shall gnorethe considerableproblems nvolved n cross-worlddentificationof times.?Remember hat n Lewis' accountACEJ-RBnd AEIR-B re not contradictories."See Donald Nute, [3, 4]. This particularproblemcan be avoided within heframework roposedbyNute. n hisformulation, CEJ-RBs true ti iffA-HB s true tallworlds ufficientlyimilar o to warrant onsideration. n this xample, t can beclaimed thatworlds n whichMort'sweight s toward he topofthe 300-400 rangearealso sufficientlyimilar o the givenworldto warrant onsideration.12Lewis, p. 67.13Lewis uses an analogousargument o showthat he counterfactual onditionalcannotbe a strict onditionalwhose strength epends on contextof utterance.Cf.Lewis,p. 13."4Naturally we must also specifyvalues for the non-logicalconstantsof thelanguage.

TheBritishournalor hePhilosophyfScienceContents, olume30 Number September 979ArticlesDAVID ZARET Absolute pace andConventionalismALAN CHALMERS Towards n Objectivist ccount fTheoryChangeFREDERICK SCHICK Self-Knowledge,ncertainty,nd ChoiceDiscussionsCOLIN HOWSON andGRAHAM ODDIE Miller'sSo-Called Paradoxof nformationGORDON FISHER Cauchy'sVariables nd Orders fthe nfinitelymallJOHN P. CLEAVE The Concept f Variable' n Nineteenth entury nalysisReviewArticleISAAC LEVI Support nd Surprise: . J.Cohen'sviewof nductive robabilityreview fL. J.Cohen: TheProbable nd theProvable)ReviewsL. JONATHAN COHEN on HerbertA. Simon: Modelsof Discovery; ARRY M. BRACKEN onG. Pitcher: erkeley; EIL COOPER on W. Ehrenberg: ice oftheGods;D. M. MACKAY onA. Sloman: The Computer evolutionn Philosophy; . J. DALE on D. Holcroft ed.):Papers nLogicandLanguage; OHN BELL onJonBarwiseed.): Handbook f MathematicalLogic;VERNON PRATT on W. H. Thorpe:Purposena World fChanceAnnouncementsPublishedfor he Britishocietyfor he Philosophyf Science byAberdeenUniversity ress,Farmers all,AberdeenAB9 2XT, ScotlandEditors:JohnWatkinsndJohn WorrallPrice ?2.50 ($8.00) Subscription Price ?7.00 ($22.00)

the similarity approach to counterfactuals: some problems - g. lee bowie

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