a response to orzack and sober

8/13/2019 A Response to Orzack and Sober

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VOLUME 68, No. 4 THE QUARTERL Y REVIEW OF BIOLOGY DECEMBER 1993~~~~~~~~~~~~~~~~~~~~~~~~~~

COMMENTARYA RESPONSE TO ORZACK AND SOBER:FORMAL ANALYSIS AND THE FLUIDITY OF SCIENCE

RICHARD LEVINSDepartmentf Population ciences ndInternationalealth,Harvard chool fPublicHealthBoston,Massachusetts2115 USA

MODELS OR MODEL BUILDING?JNHE BOOKSTORES ofmy youth twas commontofind itles hat ncludedtheterm foundations fscience. These werenotbooks about science at all but about logic,heirs of the Russell-Whitehead program toderive mathematics from ogic, and sciencefrommathematics.The hope persisted thatobjectivitycould be achieved by analyticalmethods:cleardefinitions, nambiguous cat-egories,sharpmeasurement, and thediscov-ery ofalgorithms hatcould substitute or hecaprice ofhuman udgment.But theprogramas a wholehas been a failure, s indeed it hadto be. The foundations of science are to befound in historyand sociology, not formalanalysis. Orzack and Sober's critiqueofmy1966paper fallswithin hetradition fformalanalysis,and ourdisagreements all long theaxisfrom ormal nalytical odialecticalviewsofthe scientific rocess (Levins and Lewon-tin, 1985).Formal nalysisfreezesomentsf processntothings.My essay was concerned withmodelbuilding as a process,embedded inthe argerprocessesofscientificnvestigation. t lookedat thedecisionsthatpopulationbiologistsweremakingatthetime themiddle1960s) inorderto solve different inds of problems. It wasconcernedwithtwodifficulties: ne was that

attemptsooptimize ifferentriteriaor at-isfactorymodels nterfere itheach other,and theotherwas that ll models repartlyfalse. identifiedhree endenciesn modelbuildingn population iology,achofwhichinvolvedhe acrificef neof hedesiderata:generality,ealismndprecision. ther rite-riawerementionedmanageabilitynd un-derstandability)utnotpursued.Nor did Iincludenthe istingtrategieshat acrificedtwocriteria or hethird.t is clearfromhecontext hat he istingf hree trategies asnot ntendedo be exhaustivef all possiblepopulation iology ractice r of cience s awhole. Whatwas important as thenotionof trade-offsn modelbuilding.Orzackand Sober reifyhediscussion fmodelbuildingntoa discussion f models,and turn he rends notednto Levins's ax-onomy f models r Levins's richotomy.Formalnalysisrefersfixedefinitionsf bjects,freef heirontext,n rdero llowfornambiguousmeasurementndranking.ut nscience efini-tionsevolve with heproblem.The formalmode of thought refersharply eparateddisjunct ategories nd has indeedencour-aged the longlistoffalsedichotomieshathaveplagued urbiology: eredityersus n-vironment, hysical ersusbioticcontrol fpopulationbundance,nternalersus xter-nal determination,andom ersus etermin-

The Quarterlyeview ofBiology,December1993,Vol. 68, No. 4Copyright 1993by The University fChicago.All rights eserved.0033-5770/93/6804-0003$100

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548 THE QUARTERLY REVIEW OF BIOLOGY VOLUME 68isticprocesses, nd equilibrium ersusnon-equilibriumystems.Formalnalysisreaksheworld,nd he cien-tificrocesseshattudyheworld,ntomutuallyexclusiveategories,lthoughheworldsfluid ndcategoriesform,issolve,verlap,ndnterpenetrate.OrzackandSoberdo notdefine uantityndquality,buttheydo treat hem s mutuallyexclusivelternativesnd decide hat here reno purely ualitativemathematical odels.They concludefrom histhat TypeII andTypeIII woulddistinguishifferentinds fmodelsfType II models ntailed oquanti-tativepredictionst all p. 538). Since themodel ypeswere riginallyfferedo ndicatedifferentirections f model development,theywerenot ntended obedisjunct atego-ries.Neitherre quantitynd quality. couldnothave suggestedhatTypeIII models repure ualitativemodels.Even hemost urelyuantitativef bjects,numbershemselves,avequalitativeroper-ties (Engels, 1940). In ordinary ifferenceequationsperiodthree mplies haos. Fouristhehighest-orderolynomialquation hatcan be solved nterms f tscoefficients.ixis the um f tsprime actors. here retopo-logical theorems pplicable to dimensionsthat are multiples f 7 ? 2. In nature ndsociety ualitative ifferencesften mergeabruptly rom hresholdransitionsr accu-mulategraduallywithquantitativehange.Quantitynd quality re aspects fthe ameunitary rocesses iewedfrom ifferenter-spectives nd at differentimes.This pre-cludes thatdistinctionrom eparatingnydisjunct lassesofmodels.

CRITERIA FOR CHOOSING MODELSOrzackand Sober offer efinitionsfthecriteria or hoosingmodels: If one modelappliesto morerealworld ystemshan n-other, t is more eneralp. 534). There aretwodifficultiesith his.First, cientificen-eralitys not he ame s ogical rmathemati-cal generality. mathematical ropositionderived or ome setofobjects s generalizedif t can be shown o apply to all membersof a more nclusive et. Whenwe apply hatgeneralizationoanother bject n that nclu-sive setwe specify hat objectmore com-pletely,nd all of ts elevant ropertiesollow

oncewehave done thespecifying.A scientificgeneralization s quite different.tmayapplyto most tropicalforests, r the majorityofbeetles, or to planktoniccommunities mostofthetime. Such statementsmake scientificsense, but it would be of no mathematicalinterest o prove thata class ofequations hasa real root often r even mostof thetime.Second, the term applies must be qualifiedto take relevance into account. In science amodel applies to or fits situation f t iscapable ofanalysingthoseproperties hat reof interest.Formal nalysis ftenumpsdistincthenomenaunder single eadingecausefformalimilarities.Orzack and Sober definerealismas follows:If one model takesaccountof more indepen-dent variables known to have an effect hananother model, it is more ealistic p. 534).This isfar oonarrow:The addingof ndepen-dentvariables is onlyone way ofattemptingto increaserealism.Their definition eemstobe derivedfrom egressionmodelswherethedistinctionbetween independent and depen-dent variables is central to the conceptualframework.Then taking account meanssimplyadding independent variables to themodel,eachwith coefficienthat s eft nde-termined n the uninstantiatedcase and as-signed numericalvalues when instantiated.But we can attempt o make models morerealistic n otherways as well:(1) We can add newvariablesthatmutuallyaffectach other, uchas thepredators rfoodspecies affectinghe species of interest, ela-tive preferencesforone or anotherhabitat,and physiologicalstates.These are not inde-pendentvariablesbutcovariables.The effectson generality nd precisionare ambiguous.(2) We can add a new link between vari-ables already present. For example, in athree-tiered rophicmodel with a plant, anherbivoreand a predator, the predatorcon-sumes the herbivore, which consumes theplant. This is thecore ofthemodel, thepartthat is there by definition.But the plant'sabundance may serveas a signalthatattractsthe predator and therefore ncreases preda-tion.We take account ofthisby recognizinga direct positive link fromthe plant to thepredator.This increase n realismwillreducegenerality.The effectn precisionwillbe am-biguous.



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DECEMBER 1993 RESPONSE TO ORZACKAND SOBER 549(3) We can relax simplifyingssumptions,such as symmetry, r the ogisticformof thegrowth quation or constantselection.Real-ism and generalitywill be gained at the ex-pense ofprecision.(4) We can restrict he domain ofapplica-tion. Thus the growth quationdN/dt = rN

is veryunrealisticas a generaldescriptionofpopulation growth,but bacteriologistshaveused it for differentiating pecies by re-stricting he model to the logarithmicphaseofgrowth,where tapplies bydefinition.Re-alism and precision will be gained at the ex-pense ofgenerality.To make the addition ofa coefficientr ndependentvariable stand forallmodifications f models is itself o sacrificegeneralitynd realismfor puriousprecision.Whetheran additional factor ncreases re-alism depends on the state of the science atthetime. In theoriginalessay I discussed thisaspect as thedistinctionetween implificationand oversimplification:n the early years ofpopulationgenetics,he ssumption f onstantselectioncoefficients as realistic nough forthe question: Can weak selection changepopulations? The addition ofvariable envi-ronmentswould have added only complica-tionto this model. But after hequestion Inaddition to geneticdrift, an a variable envi-ronment ause loss ofgeneticheterogeneity?was posed, models withselectioncoefficientsrepresentedas stochastic whitenoise wereappropriate.The answerwas, yes; a varyingselectioncoefficientmay result n the loss ofheterogeneity.This model in turnbecame unrealistic forconsideringtheproblem, Can the responseto selectionimprovefitness? hen a white-noise environment s completelyunrealistic,and onlywhen an autocorrelatednvironmentwas introducedcould populations be seen totrack heir nvironments.The processcontin-ues, modelingfrequency-dependentelection,demographic covariables, multiple oci, andso on. At each stage a new question is askedand theassumptions,realistic or hepreviousquestion, become unrealistic n the new con-text.The fourproceduresnoted above often n-creaserealism, butnotalways.A paradoxical

situationoccurs when adding specification oa model actually reduces realism. This hap-penswhentheadded variables orconnectionsamong variables change the level of abstrac-tionof a model. It therefore pens up a wholenew domainofvariables forpossible nclusionin the model and is a kind of silence aboutthose variables at that same level that arenotincluded, an implicit decision that they arenot as importantas the ones we included.In thisway, the addition ofvariables canreduce realism, not logically and necessarilybut historically nd practically. To say thatplant growthneeds nutrientss a general andrealistic tatement, nd it wouldbe legitimatetorepresent nutrients s a variable n a model.But ifwe go on to specify nlyzinc and mag-nesium explicitly s variables in the model,thesituationchanges. We have moved to thelevel of ndividualnutrients. he silence boutnitrogen,phosphorus, potassium, and othernutrients as become a seriousmisrepresenta-tion. Thus a changeinthe evelof abstractionchanges the realism of a model.Therefore, the more closely the assump-tions of a model correspondto the processesand level of abstractionbeing studied, themore realistic the model is; and the moreclosely the characteristicsof interestcorre-spond to theoutcomesofthemodel, themorerealistic hemodel is. The factor frelevancemeans that the same model will differ nrealism for investigatorsstudyingdifferentproblemsand for he same investigator t dif-ferent times as an investigationproceeds.This makes it impossible to rank models assuch fortheirrealism.Formal nalysis referso workwith ropertiesthat elong o the bjectn itself,ndependentf tscontext. rzack and Sober find unsatisfac-tory what I finddelightful: hatthe level ofgeneralityof the Hardy-WeinbergLaw andofother aws and modelsdepends on howtheyare used. They understandthis n a very im-ited sense. They only distinguishnstantiatedcases from ninstantiatedmodels. But the de-pendence of a model on how it is applied ismorebroadlyuseful: A theoretical esultmaybe applied as a specific laim about a particu-lar object. It maybe invokedmore generallyand realistically but less preciselyas an ifnothing else interferes laim. This may be



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550 THE QUARTERL YREVIEW OF BIOLOGY VOLUME 68tied to an argumentthat nothingelse doesinterfere ufficientlyo change the outcome.Or it may be used still more generallyas aclaimabout process rather hanresult, denti-fying hedirection nwhich theprocessbeingstudied acts on the variable without pre-dictingthe outcome. All these uses dependon the agenda ofthe science at the time,thegoal of the researcher,and the stage of theinvestigation.The Hardy-Weinberg Law, the heterotictheory fpolymorphism,he MacArthur-Wil-sontheory f slandbiogeography, nd otherscan all be used in these differentways andwillbe general, realistic nd preciseaccordingto how theyare used.Orzack and Sober considerprecisionto bedichotomous. This allows them to make asharp distinction between uninstantiatedmodelsthatdo notgivepoint predictions ndinstantiated nes thatdo. This istoorigidandnarrow. Precision is comparative. Models inwhichequations are specified ymbolicallynterms of their coefficients re more precisethan models thatonly offer graph of flowbetween compartments,but are less precisethan numericalequations. Mixed models as-signnumerical values to somecoefficientsndleave others s symbols.Numerical solutionsalso differn precisiondepending on the effi-cacy ofmeasurementand thedegrees of free-dom. Therefore see theprecisionof a modelas a degreeofspecification,with nstantiationas a finalstage along a continuum.Nowhere istheunsuitability fformal nal-ysismoreapparent than nthestudyofscien-tific rocesses whether nthe arge thetrajec-toryof a scientific ieldor problem area), orin the small (in studyingthe unfolding of aparticular ine ofresearch).

INCREASING OR MAXIMIZING?In the original article I made two claimsthat at the time seemed obvious to thepointoftriviality,nd therefore did notgivethemsupporting rgument.The firstwas that t isnot possible to maximize generality,realismand precision imultaneously.This is a partic-ular case ofthe more general truth hatmax-ima withconstraints annot be greaterthanmaxima withoutconstraints, nd except forveryspecial circumstanceswill be less. The

(a) (b)Cl Bt/ A C1 A

BV(2 (2

FIG. 1. A MODEL FOR MODEL BUILDINGThe axes Cl and C2representwo riteria,uchas generalitynd realism.The enclosed pace isthe etofpossiblemodels t a particularime.Fig.l(a) The initialmodelB can be improvedwithregard o bothcriteria y shiftingt upward ndto theright, s shown y the rrows. ' is superiortomodelB. It s closer o theboundary.Hereonlysmall mprovementsre possiblewithregard oboth riteriaogether,ut ither necan be greatlyimproved t theexpense ftheother.At pointAontheboundary ny mprovement ith egard oone criterionmustreduce heother.Fig. l(b) Inthismodel uniquepointAis optimalwith espectto bothcriteria.

second claim was about robustness:The relia-bilityof an inference s increasedwhen it isthe oint inference fmultiplemodels.My first ssertionwas that these three de-siderata could not be maximizedimultane-ously. I did not claim thattheycould neverbe increasedogether. ndeed thatclaimwouldbe absurd: We could always change a modelin a direction hatwouldmake it ess general,lessrealistic, nd less precise,so thatundoingthat procedure would increase all three atonce. And indeed adding a term to an equa-tion may sometimesincrease all three. Butifadding higher-order erms to an equationalwaysincreasedgenerality, ealismand pre-cision, why would anyone ever stop addingterms?Figure 1 representsthe process of modelimprovement n a space whose axes are mea-sures of two criteria. The available modelsare pointsin thisspace. Point B representspoormodel neartheorigin.Almost ll changesmove B upward and to the right mprovingboth measures. As the model improves to-ward B', the range ofoptions that increases



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DECEMBER 1993 RESPONSE TO ORZACK AND SOBER 551both criteriabecomes more limited, and onlysmall mprovements re possible. But optionsare still vailable thatmake big improvementsin one at the expense of the other. At theboundary it is possible to improve one mea-sure only by reducing the other. Point A isone possible choice, but it is not better thanotherchoices on the boundary with regard toboth criteria.But it seems thattheyhave a different ic-ture nmind. In Figure 1(b) we showa specialconfiguration fthe model space in which allmodels lie withinthe rectangle bounded bythe axes, the origin, and pointA. Then A isa unique best model and there s no trade-off.The burden ofproofwould be borne by anyclaimant arguing that the model spaces ofpopulationbiologyare the sort shownin Fig-ure 1(b). Orzack and Sober attemptto dothis by the device of the added variable orcoefficient iscussed below.Therefore n a formal ense, any examplesin which Orzack and Sober show modelchangesthat ncreaseall three riteria t onceare irrelevant nd would simplybe examplesofpointsB and B'.

SOME CONTEXTS OF MODELINGIn real scientific racticewe regularlymakedecisions that improveone or more ofthesedesiderataat theexpense ofanother. In eachof the contextsofmodeling below, the rela-tionships mong thedesideratachange in thecourse of an investigation. Suppose we haveto account forthe dynamics of a particularobject of study, a population of fish or themicrobial communityof a rainforest r thegeographicrangeof a bird,which has alreadybeen observed. Then we would start withwhat we know in general about such objectsand discardfrom he model what s irrelevantto theparticularcase. We would expressrela-tionships such as predation by particularequations, and estimate the appropriate pa-rametersnordertofit he model totheobser-vations.This is a modeling sequence from hegen-eral to theparticular, dapting generalknowl-edge toparticularcases by successive specifi-cations that might reach the end point ofassigningnumerical values. At each stage ofspecificationthe model is more precise andrealistic but less general.

Suppose the task is to provide a generalmodel for class ofobjects such as lakes or dryforests. his can be approached from ppositeends. If we startwith a general model ofeco-systems, the procedure is one of increasingspecification, leaving out what is relevantonly to other ecosystems n order to study heparticularproperties fthe one at hand. Fromthe other end, we might startfroma singlelake or forest, bserve its properties, nd askhow general the results are. Then we gothroughan opposite procedure of reducingspecification,relaxing assumptions, and los-ing precision to gain generality.We mightwant to examine the ecologicalor evolutionary significanceof a particularphenomenon such as mutualismorheterosis.Here we would usually startwith hephenom-enon in a pure form,withoutcomplicatingfactors.We might use several very specialcases for ease of handling in order to get afeelfor he problem. If a conclusion seems tohold for all ofthem we might tryto prove itgenerally,withfewer pecial assumptions. Orwe may opt for increased realism by usingmore complex and intractable numericalmodels that are once again precise but notgeneral. Many investigators now combinethese approaches, scanning a phenomenonwith simplifiedmodels and then verifying heresults with simulations.We mightbe trying o findways to recog-nize a given phenomenon suchas densityde-pendenceorheterosis r todistinguish mongalternative xplanations.This is similar o theprevious case except thatnow we examine theconsequences ofdifferentmodels in order tofind onclusions hatdifferentiatemong them.Suppose finally hatthe task s to showthata common senseexpectation snotnecessarilyso. Here the firstmodels are usually quitesimple. Ifwe need a life table we mightusea negative exponential. Age-specific repro-ductive values mightbe representedby a tri-angle, latency by a fixed delay. If we guessthat some resultdepends on a periodic envi-ronmentwe may chose alternating onditionsor a sinusoidal curve. Since we are trying omodel a process ratherthan a specific case,we are not concerned with additional factorsthatmightbe present n some situationsandnot others. Thus we have become inclined



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552 THE QUARTERLYREVIEW OF BIOLOGY VOLUME 68toward more general, unrealistic and precisemodels. Factorswe know to be operating areignored, circumstances such as symmetrythat are uncommon in reality re assumed inorder to reveal more clearlytheoperationofthe factorswe are interested n.These different activities are idealizedtypes. Real researchmay shift mong themin a zigzag course, now modeling the richinteractions of a system n exquisite detail,now reducingmagnification nd seeing onlythe outline structure,moving fromthe gen-eral to particular cases, finding results thatmightbe generalized, specifying he termsofa model forcomputer simulation or leavingthemqualitativefor nalyticwork. It is quitecommon now to use analytic models in con-junction with simulationmodels.As we make the model more specificwe arelosing generality n twoways: Each step thatadds new variables or new relations amongthem excludes those cases inwhichthesepar-ticular variables or relations are not opera-tive. And once we have introducednew vari-ables we may have to say somethingaboutthe mathematicalforms ftheir nteractions,the exact equations, or the magnitudes oftheirparameters.The last stage in specifica-tion is instantiation.Orzack and Sober note that n instantiatedmodel (one inwhichnumerical values are as-signed to the parameters so that numericalpredictions an be made about thevariables)is lessgeneralthanan uninstantiated ne, butthey all thisobservationtrivial.Full instanti-ation, however, is the end point of a contin-uum ofincreasing specificity,nd generalityis lost at each step.It is plausible toexpectthat s we add vari-ables, relations, or parameters to a model,we increaseprecision.Indeed this s often heintention,but not always the result. Some ofthe variables may not be measurable or onlyverypoorlymeasurable.Our understandingfphysiology ow includes emotional states hatare much more difficult o measurethan con-ventionalphysiological variables, but shouldbe included in the models. Epidemiologicalmodels may have to include the behavior ofmedicalbureaucracies in responseto thepro-gression f n epidemic. Sincemodelbuildingusually beginswithvariables we are more fa-

miliar with, the addition of variables often(not logicallyand necessarilybut historicallyand in practice) mprovesrealismbutreducesprecision.For many models increasing complexitymagnifies hesensitivity ftheoutcome of themodel to the parameters. The end point ofthisprogressioncan be chaotic trajectories.Therefore, even ifeach additional parameterismeasurable tothe same degreeofaccuracy,the outcome may be less predictable.Because of these practical difficulties,hevery arge and complexmodels used forbiomestudies, the world economy, or epidemicshave not been conspicuouslymorepreciseintheirpredictionsthan simplermodels.

UNDETERMINED COEFFICIENTSA key to theirargumentis the use of themethod ofadding undeterminedcoefficients.Each of the additional factors that may berelevant to some particular cases of a generalphenomenon can be included in the model,multiplied by a coefficient epresenting he

magnitude of its effect. f that effect an beallowed to be zero, then theexpanded modelincludes thepreviousone as a special case.Each of these factorsmay be relevant some-where, but in each particular case most ofthemwillbe zero. In thatwaya largenumberof quite different ituations can be coveredformally by the same model, but in eachmodel most of the factors have zero effect.The modelwillbecome cluttered y predatorswith predation rate zero, competitors withcompetitioncoefficients f zero, zero timelags, and so on. Mathematically,zero can bethought f as just anyold number,and whenwe fit arameters t s no big deal ifwe decidethat some are zero. Thus Orzack and Sobercompare two growth quations:

dN/dt = rNand

dN/dt = rN + oaN2.They claim thatthe second ismoregeneralthan the first since it includes the case of



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DECEMBER 1993 RESPONSE TO ORZACK AND SOBER 553ox 0.0. Mathematicallythis is true: Zero isjust anothernumber. But biologically t s nottrue. Each factor n the modelmustbe takeninto account, looked for,thought bout, per-haps excluded or enhanced experimentally.It is a serious matter o include anotherfactorin a model. And in terms of their conse-quences, nonpredation is not just a specialkind of predation, noncompetition s not aspecial kindofcompetition,nor is density n-dependence a special limiting ase ofdensitydependence.When coefficients uch as a are added, wedo not treat zero as just anotherpossible pa-rametervalue. We usually test forthe coeffi-cientbeing significantly ifferentrom ero,not differentromsay - 0.017 or any othernegativenumber. That proceduredecides be-tween density dependent and densityinde-pendent models. An alternative approachwould be simplyto accept the best estimateof a, which will almost always be differentfromzero. Then we are implicitly doptingdensitydependence. Even the smallestnega-tivea gives qualitativelydifferentesults roma = 0.0 interms fequilibrium,or ofsensitiv-ityto different ixed values of r in differentpopulations,or ofdependence on pastvaluesof r, or of correlationsbetween populationsize and environmental actors frvariesovertime, or ofthe limiting probabilitydistribu-tion ofN withrandom r.So far have been focusingthe discussionon mathematicalmodels, butmuch thesamesituationobtains in experimental design. Iftheproblem s a general one, such as theroleofcrowdingor of a fluctuating nvironment,the choice of species may be determinedbyease ofmanipulationand also representative-ness. That is, ifwe wanttobe able togeneral-ize the results we choose a species that is insome sense typical, t least inrelationtotheproblem being studied. Enclosed populationsin the laboratory can be kept at predeter-mined temperatures,whereas humidity ismore difficultomanage. Therefore, emper-ature sused as a typicalnonconsumableenvi-ronmentalfactor.Experimentalvariables canbe measured frequently, nd manipulationsperformed according to plan. The desiredphenomenon may be amplified above the

background noise of nature for asy observa-tion. Here precisionis gained at the expenseof realism. Field observations and experi-ments, on the other hand, are done undermore realistic conditions, but with less pre-cision.ROBUSTNESS

Orzack and Sober deal with robustnesswitha logical model. They distinguish hreecases: Ifwe knowthatone of a setof modelsM1, M2, . . . is true and that each M, impliesR, thenR is true. They then considercaseswherewe know that each is false, and assertthat in these cases the factthatR is impliedby all of them does not make it true. Andfinally, fwe do notknowwhether nyofthesetis true, then their oint implication s alsonotproven. They then llustrate heir nalysis.This analysis, however, is not relevanttotheproblemof robustness hatwas posed: Inorderfor model tobe useful t must be bothsimilar o and differentrom heobjectorpro-cess itmodels. Therefore tcontainspartsthatmay be true and parts that are either falseor are of much more limited validity. Theproblem is how to identify he implicationsofthetruepart either or rediction rtesting.The logical structureof the argument isquite differentromOrzack and Sober's rep-resentationof it. Let C be the common partof all the models, the core relationshipsweare either confident for wish to test. Let V,be the variable part ofthe model introducedforconvenienceor because each mightholdforsome cases. Then if C togetherwith V1(the intersection fC with V1or their ogicalproduct CV1) implies R, and if CV2 impliesRI . . ., it followsthatthe intersection f Cwiththe union ofthe V, mpliesR:

CV1 + CV2 + CV3 + ...= C(V + V2 + V3 + =>R.If thesetofV, xhaustsall the admissible alter-natives, then

V1+ V2+ V3+ = 1 and C implies R:,Vi[(E c ) >Rj = (C =>R).



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554 THE QUARTERL YREVIEW OF BIOLOGY VOLUME 68Ifthe V's do notexhaust thepossible alterna-tives, at least the more inclusive the set ofV's, themorewe can have confidencethat CimpliesR. Ifwe feelthatthe set of V,'s pansa wide enough range ofpossibilities, henwemaygeneralizetoclaim thatCusuallyimpliesR, a result hat snotvery xciting s a mathe-matical theorem but may be good biology.The point s that all systemshave morego-ing on thantheprocess we are interested n,and each systemhas some thingswe have nottaken ntoaccount. The searchforrobustthe-oremsreflectshestrategy fdetermining owmuchwe can getaway withnotknowing,andstillunderstand the system.

Orzack and Sober are worriedthat the ro-bustness strategy eems to propose a way totruth ndependentofobservation. This is notthecase. Observationentersfirstnthe choiceofthecoremodel and theselection fplausiblevariable parts, and later in thetestingofthepredictionsthatfollow fromthe core model.Multiple models sharingthatsame corehelpto findthe consequences of that core whenwe are unable to offer generalproofthat CimpliesR. Thus the search forrobustness asunderstoodhere sa valid strategy or eparat-ing conclusionsthatdepend on the commonbiological core ofa model from hesimplifica-tions,distortions nd omissions ntroduced ofacilitate heanalysis, and for rriving t theimplicationsofpartial truths.The use ofmul-tiplemodels is a commonpracticefor hisrea-son, either o strengthenheconclusionor toguide us in lookingfor a general result.CONCLUSIONS

(1) Formal analysis depends on imposinga sharp differentiationmong objects that donot have clearboundaries. It attempts o eval-uate modelsoutside of heir ontext odevelopan investigation rthehistory ftheir cience.Itextinguishes ifferencesmong quitediffer-entprocedures thatshare a formalproperty.(2) Models do not fall ntomutuallyexclu-siveclasses but ie on a multidimensional on-tinuum. Three of these axes are generality,realism and precision. Others are manage-ability nd understandability.nstantiation san end point on the axis ofprecision.(3) The location of a model on thiscontin-

uum isnotdeterminedby themodel itself utdepends on thechangingcontexts nwhich tis used. The problem to be solved and thelevel of abstractionof the investigationcanchange the generality,realismand precisionof a given model. The same formal act, forexample, adding termsto a model, may in-creaseordecrease generality, ealism,orpre-cision.(4) The same model can have differentmeanings depending not only on how it isused but also on thestageofan investigationand the state of the science. It can be takenas abstractmathematicalrelations.It can bea claim about what happens when nothingelse interferes. his may be coupled toclaimsthatnothing lsedoes interferensomepartic-ular case. It can be interpreted s identifyingone of theprocessesthatalways influencestrajectory ut only sometimes s sufficientodetermine the outcome.(5) The strategy fmodel buildingconsistsof deciding how to move along this contin-uum. It is concerned with the processes bywhichdifferent esiderata supportand inter-ferewith each other. It is notconcernedwithrankingor measuringmodels.(6) In a wide range of scientific ractices,includingmodeling a particularobject or aclass ofobjects,examiningthesignificance fa phenomenon, xploringwhether n acceptedpropositionreallyholds, deciding among al-ternativeexplanations, or determininghowto recognizea phenomenon, there s a trade-off mong generality, ealismand precision.(7) In formal nalysis,a model that spartlytrue and partlyfalse s false.Therefore,mod-els can be divided into trueand false. In sci-ence, most of the models we workwith arepartlytrue and partlyfalse.(8) We may strengthen ur confidence nthe implicationsofsome assumptionsby us-ing ensembles ofmodelsthat hare a commoncore of these assumptions but also differ swidelyas possible in assumptionsabout otheraspects.Then the more the variablepart spanstherangeofplausible assumptions,themorevalid the claim thattheconclusions sharedbyall of themdepend on the constantpart. Ifwe also have confidence hat heconstantpartis true, thenwe have strongsupportforthe



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DECEMBER 1993 RESPONSE TO ORZACK AND SOBER 555claim that the conclusion is generally true.This gives robustness to the conclusions.A formal/analyticramework s notthe ap-propriatedomain forevaluating a model ofresearchstrategy.

ACKNOWLEDGMENTI am indebted to Rosario Morales forher helpin clarifying nd expressing the concepts in thisresponse.

REFERENCESEngels, F. 1940. TheDialectics fNature. nterna-tionalPublishers,New York.Levins,R., and R. C. Lewontin. 985.The ialec-

ticalBiologist.Harvard UniversityPress, Cam-bridge.

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