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Flagpoles, Shadows and Deductive Explanation Author(s): Michael E. Levin and Margarita Rosa Levin Reviewed work(s): Source: Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 32, No. 3 (Oct., 1977), pp. 293-299 Published by: Springer Stable URL: http://www.jstor.org/stable/4319174 . Accessed: 13/02/2012 22:31 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

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MICHAEL E. LEVIN AND MARGARITA

ROSA LEVIN

FLAGPOLES, SHADOWS AND DEDUCTIVE EXPLANATION

(Received 21 October, 1976)

The Deductive-Nomologicalaccount of explanationholdsthat a necessaryand sufficient condition for an event e to be explained is that e's occurrencebe deduced from laws and initial conditions.Thereare in the literaturea number of counterexamplesto the sufficient condition component of this account (henceforth the sufficiency thesis), that deduction from laws and initial conditions is sufficient for explanation. These counterexamplesare taken to be cases in which the deduction-from-laws requirementis met but, intuitively, we do not have an explanation.We intend to show that severalpopularsuch counterexamplesare unsuccessful,and that this suggeststhat a certainbroad class of counterexamplesall fail uniforrnly. We begin with two of the most interestingcases that have been deployed in the recentliteratureagainstthe sufficiencythesis: 1)

2)

A mouse is 3 feet from the base of a flagpole 4 feet tall, at the top of which is an owl. These initial conditions, plus the PythagoreanTheorem, entail that the owl is 5 feet from the mouse. We have the right sort of deduction;but, intuitively, we do not yet know why the owl is 5 feet from the mouse. (Adapted by K. Lehrerfrom S. Bromberger'.) A flagpole casts a shadow 3 feet long whose endpoint is 5 feet from the top of a flagpole. These initial conditions, plus the PythagoreanTheorem,entail that the flagpole is 4 feet tall; but, intuitively,we do not know why the flagpoleis 4 feet tall.2

(For convenience,let us referto "theinitialconditionsplusthe Pythagorean Theorem"in both cases as "I + P.") Hempel and others3 have tried with variousdegreesof success to rescue the sufficiency thesis by explainingaway these intuitions. A more convincing way to rescue the sufficiency thesis is to acknowledgethese intuitions and show them compatible with it. This can be done by (a) showing that the Philosophical Studies 32 (1977) 293-299. All Rights Reserved Copyright ? 1977 by D. Reidel Publishing Company, Dordrecht-Holland


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explanandumof the explanationwe intuitively think is missingis not entailed by I + P; and (b) finding an explanandumwhich is entailed by I + P and which, intuitively, is explainedby I + P. This strategyrequiresdemonstrating that the explanandain the foregoingcasesareambiguous,or at least susceptible to two interpretations,and this claim can be made good. Consider, first, the question 'Whyis the owl 5 feet from the mouse?'An explanationin case 1) should answer this question. But in fact this question has two possible interpretations.The first is, simply 1a)

Given their spatial positions, why is the owl 5 feet from the mouse?

However,this is not how Lehrerintends the question, for he says "the matter requires some explanation - owls eat mice!" (op. cit., p. 167). So there is clearlya furtherconstructionof the question to be made: 1b)

Given the predatoryhabits of owls, why is the owl inhibitedfrom attackingthe mouse?

Similarly,the explanation-seekingwhy-questionattached to case 2) has both a purely geometrical and a non-geometricalinterpretation.One ignorantof geometrymight literallybe asking 2a)

Given that the flagpole has a height, and spaceis Euclidean,why is (the measureof) the height 4 feet?

Or else, he could be asking 2b)

Whywas this flagpolemade4 feet tall?

Thereis, moreover,an independenttest whichdetermineswhichof these two sorts of interpretationthe why-questionsareintendedto have.If you areasking the geometricalquestion 1a), which is basicallythe question why a line connecting the points at which the owl and mouse are situatedis 5 feet long, you won't care if the explanandumin case 1) is replacedby anotherexplanandum in which two other objects are where the owl and the mouse are. Giventhat a rock and a radishare where the owl and the mouse were, you will accept 'Whyis the rock 5 feet from the radish?'as essentially the same question as 'Whyis the owl 5 feet from the mouse?' and you will be as satisfiedwith an answerto the first as an answerto the second. (Similarly,if an explanationseeker in case 2) is asking2a) he will allow the flagpole to be replacedby a


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broomstick.) If, on the other hand, he will accept no substitutesfor the owl and the mouse, he is asking about them specifically; he is asking lb). (Similarly, if he will not accept replacingthe flagpole by a broomstick,he wants to know why this flagpole is 4 feet high; he is asking2b:)) This test is seen to work in other contexts. Suppose someone asks 'Whyare applesred?' This could either be a request to know why apples are red ratherthan some other color; or, it could be a requestto know why things are red - a request for a reductiveexplanationof color. If the former is meant, the interlocutor will not allow 'Why are fire engines red?' as an acceptablesubstitute for his question. If the latter is meant, he will be just as happy with a reductive explanation of why some other objects, like fire engines, have the property of redness. It is clear enough that la) and 2a) are explanandathat are entailed by I + P. Moreover, intuition accepts I + P as genuinely explaining them. As a truth of physical geometry, the PythagoreanTheorem really does explain why certain points of physical space have the metrical relationsthey do.4 If the explanation seems thin, so is the explanandum.However,I + P cannot entail the b) cases, since physicalgeometryis mum about the constructionof flagpoles and the mechanismsthat inhibit predatorybehavior.Moreover,it seems evident that one will find I + P not an explantion of why the owl is 5 feet from the mouse just in case one will not exchange this explanandum for the geometrically congruent rock-radishexplanandum. For note that Lehrer says that the geometricalexplantion won't work because owls eat mice - a point he couldn't make about the rock and the radish. By the foregoing test, this means that I + P fails to be explanatoryjust when the explanandum of the case is given the b) interpretation.(Of course, some other laws and initial conditions will explain the b) explanandum;see p. 297 below.) Putting these steps together, we find that those cases where we intuitively feel I + P fails to explain arejust those cases whereit also does not entail the alleged explanandum.Therefore, neither 1) nor 2) nor cases like them qualify as convincingcounterexamplesto sufficiency thesis. Whatare 'caseslike' 1) and2)? Thecrucialfeatureof the law P whichgenerated cases 1) and 2) is a feature shared by the laws involved in a wide class of counterexamplesto the sufficiency thesis: P is a law of co-existence.5 This suggeststhat all counterexamplesof this type can be handledby the strategy lately deployed against1) and 2). Wewill try to substantiatethis by using our


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strategy againsta typical such counterexample,superficiallyratherdifferent from the cases examinedso far. We need to define some terms. A law of co-existencespecifiesthe value of some variablex at time t to be a functionf of the value of some other variable y at time t, its generalform being (x, t) = f(y, t). Lawsof coexistence present a difficulty for the sufficiency thesis because, very often, a deductionof the value of x at t from such a law and informationabout the value of y at t is not, intuitively, explantory. (A furthercomplicationis that the inverseof the function f is sometimes both functionaland explantory.)Takethe law of the pendulumH, which connects the period p of a pendulumwith its length1 by the formula p = 21rv'/Ig, g being the gravitationalconstant. We can decude from H that the length I of a pendulumwith period 21r/Y\gsec. is 1 foot, but such a deductionhardlyseems to answerthe question 3)

Why does this pendulumhave a length of 1 foot?6

As usual,we denote the conjunctionof H andthe initialconditionp = 2rr/</g by 'I + H.' 3) turns out to be ambiguous;underone interpretationit is both entailed and explained by I + H; under the other interpretationit does not follow from I + H, and the substitution test indicates that this is the interpretation under which I + H leaves 3) intuitively unexplained. To see this, we can follow the precedents of examples 1) and 2) and decompose 3) into two questions: 3a) Given that this pendulumhas a length and a period of 2ir//g, why is (the measureof) its length 1 foot? 3b) How did this pendulumget to be 1 foot long? You are asking 3a), which amounts to the question why the measure of length of a pendulumwith a period of 21r/V/gsec. is 1 foot, just in case you won't mind replacingthis explanandumby anotherwhich featuresa different pendulumwith the same period. If you are as satisfiedwith an answerto the same question about this other pendulum as you are with an answerto 3) proper, you are asking3a). If, however, you will accept no substitute questions about anotherpendulumwith the same period, then you are asking3b); you want to know how this pendulumcame to be 1 foot long, how it got a 1 foot shaft. It is clear that (the explanandumassociatedwith) 3a) is entailedby I + H.


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Moreover,intuition accepts I + H as an explanation of 3a); for H is why the measureof the length of a pendulumwith period 2rriVg is 1 foot, just as P is why the measure of the line connecting where the owl is to where the mouse is is 5 feet. (We offer below an account of the sort of explanationthis is.) I + H does not entail (the explanandumassociatedwith) 3b), since H is mum about the construction of pendula. Finally, the substitution test indicates that one who finds I + H not answerto 3) must be intending3) as 3b). For what I + H does not tell him is how this pendulumgot to be 1 foot long, a point he could not make about anotherpedulum. Why do our explanatory intuitions (as well as critics of the sufficiency thesis) take explananda that laws of co-existence do not even purport to explain to be consequencesof laws of coexistence?Whyis (x, t) = f(y, t) and y = Yo (& t = to) taken to be an (intuitively unsatisfying) answer to the question 'How did x get the valuexo = f(yo, to)?' The reason,we conjecture, is this. When(x, t) = f(y, t) holds, there is almost always some other law or cluster of lawlike statements which connect the value of x at t with the value of some other variableat a time earlierthan t. Thereis almost alwaysavailable some law of succession of the form (x, t) = g(z, t - At), and it is this law of succession which both entails and explainshow x got to be xo at t. Examples of such 'laws' are the laws of inhibition which explain why the owl is still on the flagpole and the principles governingthe manufactureof flagpoles and pendula. Whena law of co-existenceentailsan answerto an a)-questionthere is usually a known answerto a b)-question, and since both questions can be formulatedthe same (ambiguous)way, it appearsthat the law of co-existence entails an answerto a b)-question. Once this slip has been made, we have an appearentcounterexampleto the sufficiencythesis. This conjecture can be tested, for it leads us to expect that, were we to have a law of co-existence (x, t) =f(y, t) unaccompaniedby any law of succession to yield answersto b)-questionsabout valuesof x, intuition would accept the law of co-existence as supplyingadequateanswersto a)-questions, ratherthan reject it as supplyinginadequateanswersto b)-questions.Such a test case is supplied by the system of field equationsof GeneralRelativity. Roughly speaking, these equations correlate the value of the stress-energy tensor T at a point s of space-timewith the radiusof curvatureR of spacetime at s. Understoodas differentialequations,they correlatethe distribution of energy throughoutspace-timewith the geometricalstructureof space-time itself. This system of equationsis a law of co-existence, the most fundamantal


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such law in science. In particular,given the value To of T at point s of spacetime, one can deduce from the field equations that the value of R at s is, say, Ro. (The time at which R(s) = Ro must be equal to that at which T(s) = To because s already contains a time co-ordinate.)Whatkind of explanation,if any, is this of the fact that R (s) = Ro ? It is, to be sure, difficult to speak with confidence of explanatoryintuitions in a context this abstruse. Nonetheless, it seems to us that this deduction - quite parallelto the deduction of 1 = 1 foot from I + H and the deduction of the height of the flagpole fromI + P - is an answerto the question of why s has the radiusof curvature Ro, and scientists evidently accept it as such.7 And the one significantdifference between this case and the earlier ones is just that no-one has the faintest idea of even how to addressthe b)-interpretationof this why-question. There is at present nothing faintly like a law of successionwhich correlates the value of R at a point s with the value of anothervariableat a time earlierthan (the time co-ordinateof) s. Nobody has any idea of how spacetime gets a geometry, and consequentlythe b)-questiongoes begging.Nor do scientistsregardthe deduction of Ro as a b)-type answerto the why-question; rather,they look at the field equationsas imposinga "consistencyconstraint"8 on the geometrodynamicstructure of the world. These equations say precisely that the world is such that an orderedpair (To, Ro) can be true of s only if they satisfy the field equations. Giventhat T(s) = To, this is why the value of R at s must be Ro. And this is to say that the field equationssupply answers to the a)-interpretationsof why-questionsabout the geometry of space. In their more modest ways,P andH amountto consistencyconstraints on the world: that is how they answer why-questionsabout the values of parameters. We have gained some ground for the sufficiency thesis and the DeductiveNomological model by showing them to be immune to counter-examples based on laws of co-existence, given the recurrenceof this sort of counterexample in the literature. But a sceptic might ask how this immunizesthe sufficiency thesisagainstother counterexamplesfrom quite differentquarters. In the nature of the case this is something that cannot be proven. To prove that there will never be a counterexample to the DN thesis is, after all, tantamountto provingthe DN thesis itself, and no-one reallyexpects the DN thesis to be derivablefrom essentiallymore generaland independentlymore plausiblepremises.In this it is like most other philosophicaltheses;Church's


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andso on. The only way of defendthe DNthesis,apart Thesis,Nominalism, usage,is to deal its consiliencewith relevantpreanalytic fromdemonstrating in job lots. Successin this showsthe DN thesisto be with counterexamples simpleand systematic;and simplicityand systematicityare hallmarksof truth. City College,City Universityof New York and Universityof Minnesota NOTES 1

Lehrer,K., Knowledge,Oxford, UniversityPress,Oxford, 1974, pp. 166-7; see also Bromberger,'Why-Questions',in R. Colodny (ed.), Mindand Cosmos,PittsburghUniversityPress,Pittsburgh,1966. 2 One recent use of this example to underminethe sufficiency thesis is by Friedman, M., 'Explanationand Scientific Understanding',Journalof Philosophy LXXI (1974), 5-19. 3 See Hempel, C., Aspects of Scientific Explanation, Free Press, New York, 1965; especiallythe title eassy.See also Gasking,D., 'Causationand Recipes',Mind1955. ' We ignore the tangle of questions involvingpure vs. interpretedgeometry, the ontologicalstatusof space,etc. The most convincingBrombergerexamplesarebasedon laws of co-existence. 6 This exampletroublesHempel;see op. cit., p. 352. For example,Mgller,in Theoryof Relativity,OxfordUniversityPress,Oxford,1962, considers the mathematicalconsequencesof assumingthat nothing moves, and says "hence the geometry is only approximatelyEuclidean"(Chapter7, Section 120). He also considersthe importantcase of the gravitationalfield inside a rotatingsphereand says the result "explainsthe negativeresult of the experimentactuallyperformedby Friedlander"(ibid., p. 320). 8 This is L. Sklar's apt phrase; see his Space, Time, and Space-Time,Universityof California,1974, pp. 75, 215-216.

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