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THE STRUCTUREOF SCIENCE

?roblems in tbe Logic ofScieutific xplanation

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Copyright @ 1979 by Ernest Nagel

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Printed in the United States of America

Second Edition Third Printing 1987

Library of Congress Catalog Card Numbcr 6G-15504

ISBN: G-9I5I44-71-l) (phk')ISBN: 0-915 144'72-7 (t'lotlr)

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The Reduction of Theories

,a / It is a-commonplace that classical mechanics is no longer/l 4 regarded as the universal and fundamental science ofI I nature, Its brilliant successes in explaining and bringingI I into systematic relations a large variety of phenomena

'r- -r- were at one time indeed unprecedented. And the belief,once widely held by physicists and philosophers, that

all the processes of nature must eventuuily fult withii trr" ,iop" or it,principles was repeatedly conffrmed by the absorption of various sectorsof physics into it. Nevertheless, the period of th-e "imperialism. of me-chanics was practically over by the Iaiter part of the nireteenth century.The difEculties which faced the extension of mechanics into still uncon-quered territory, and particularly into the domain of erectromagneticphenomena, came to be acknowleclged as insuperable.

However, new candidates for the o{fice of a-universar physical sciencclvere proposed, sometimes with the backing of a priori aiguments a,alo-gous to those once used to support the claims of -*echa,ics. To bc surc,with some few doubtful exceptions no serious student of the scicnccs to-day believes that any phystar theory can be warranted on a pri,rigrounds, or that such arguments can establish a theory in that Irigh ,f-ffce' Moreover, many outstancling physicists are fra,kly skcptic*l wrrctlrcrit is possible to realize the ideal of a cornprchensive tl,.,,i.y wlricL willintegrate all domains of natural scicncc in tcn,s of a c.,,,,,,,,1rr .s.t

T3 The Structure of Sciencere is taken to understand the expression in the sense relevant to, andluired by, the special context in which the expression has acquired ash,rse. Such alterations are particularly pfone to occur when one the-y is explained by, or reduced to, another theory; and the shifts in the:aningi of familiar expressions that often result as a consequence of theluction are not always accompanied by a clear awareness of the log-rl and experimental conditions under which the reduction has beenected. In consequence, both successful and unsuccessful attempts atluction have been occasions for comprehensive philosophical reinter'etations of the import and nature of physical science, such as those:ed in the preceding paragraph. These interpretations are in the mainghly dubious because they are commonly undertaken with little ap-6ciation for the conditions that must be fulfflled if a successful reduc'rn is to be achieved. It is therefore of some importance to state withre what these conditions are, both for the light that the discussion ofose conditions throws on the structure of scientiftc explanation andr the help which the discussion can provide toward an adequate ap'aisal of a number of widely held philosophies of science. An examina-rn of the eonditions for reduction and of their bearing on some moot;ues in the philosophy of science is the central task of the presentrapter.

The Reduction of Thermodynarnics to Statistical MechanicsReduction, in the sense in which the word is here employed, is the ex'anation of a theory or a set of experimental laws established in oneea of inquiry, by a theory usually though not invariably {o_rnulatedr somc oth"i do*uin. For the sake of brevity, we shall call the set ofeories or experimental laws that is reduced to another theory,the

*sec-

rdary scienci,o and the theory to which the reduction is efiected or.oposed the'primary science.- However, many cases Of reduction seem6e normal sieps in-the progressive expansion of a scientiftc theory andrely generate ierious perplexities or misunderstandings. It will there-," b"

"ot't"nient to distinguish, with the help of some examples, be-

ireen the two types of reduction, the ffrst of which is commonly re-rrded as being quite unproblematic and which we shall ignore in con-rquence, while the secorrd is often felt to be a source of intellectual.scomfort.

1. A theory may be formulated initially for a type of phenomenonfiibited by a somewhat restricted class of bodies, though subscqucntlyre theory may be extended to cover that phenornen()n evcn whcn mani-:sted by a more incltlsive class of things. For cxarnplc', tlto thcory ofrechanics was lirst dcvclopcd for thc lnotions tlf poirrt.rttnsscs (i.c', for

The Reduction of Theories 889the motions of bodies whose dimensions are negligibly small when com-pared with the distances between the bodies) and was eventually ex-tended to the motions of rigid as well as deformable bodies. In suchcases, if Iaws have already been established within the more inclusivedomain (perhaps on a pulely experimental basis, and before the devel-opment of the theory), these laws are then reduced to the theory. How-ever, in these cases there is a marked qualitative similarity between thephenomena occurring in the initial and the enlarged provinces of thetheory. Thus, the motions of point-masses are quite like those of rigidbodies, since the motions in both cases involve only changes in spatialposition, even though rigid bodies can exhibit a form of motion (rota-tion) that point-masses do not. Such reductions usually raise no seriousquestions as to what has been efiected by them.

Analogously, the range of application of a macroscopic theory may beextended from one domain to another homogeneous with it in the fea-tures under study, so that substantially the same concepts are employedin formulating the laws in both domains. For example, Galileo's ?rooNeut Sci,encas was a contribution to the physics of free-falling terrestrialbodies, a discipline which in his day was considered to be distinct fromthe science of celestial motions. Galileo's laws were eventually absorbedinto Newtonian mechanics and gravitational theory, which was formu-lated to cover both terrestrial and celestial motions. Although the twoclasses of motions are clearly distinct, no concepts are required for de-scribing motions in one area that are not also employed in the other.Accordingly, the reduction of the laws of terrestrial and celestial mo-tions to a single set of theoretical principles has for its outcome simplythe incorporation of two classes of qualitatively similar phenomena intoa more inclusive class whose members are likewise qualitatively homo-geneous. Because of this circumstance, the reduction again generatesno special logical puzzles, although it did in point of fact produce arevolution in men's outlook upon the world.

In reductions of the sort so far mentioned, the laws of the secondaryscience employ no descriptive terms that are not also used with approxi-mately the same meanings in the primary science. Reductions of thistype can therefore be regarded as establishing deductive relations be-tween two sets of statements that employ a homogeneous vocabulary.Since such "homogeneous" reductions are commonly accepted as phasesin the normal development of a scienee and give rise to few misconcep-tions as to what a scientiffc theory achieves, we shall pay no further at-tention to them.

2. 'l'lrc sitrrrrtirln is rrsrrirlly'tliffcrcnt in the case of a second type of1:

840 The Structure of Sciencetraits of some subject matter is assimilated to what is patently a set ofquite dissimilar traits. In such cases, the distinctive traits that are thesubject matter of the secondary science fall into the province of a theorythat may have been initially designed for handling qualitatively difier-ent materials and that does not even include some of the characteristicdescriptive terms of the secondary science in its own set of basic theo-retical distinctions. The primary science thus seems to wipe out familiardistinctions as spurious, and appears to maintain that what are primafacie indisputably difierent traits of things are really identical. The acutesense of mystiffcation that is thereby engendered is especially frequentwhen the secondary science deals with macroscopic phenomena, whilethe primary science postulates a microscopic constitution for those mac-roscopic processes. An example will show the sort oI ptrzzle that is gen-erated.

Most adults in our society know how to measure temperatures with anordinary mercury thermometer. If provided with such an instrumentthey know how to determine with reaSonable accuracy the temperatureof various bodies; and, in terms of operations that are performed withthe instrument they understand what is meant by such statements asthat the temperature of a glass of milk is 10' C. A good fraction of theseadults would doubtless be unable to explicate the meaning of the word'temperature'to the satisfaction of someone schooled in thermodynam-ics; and these same adults would probably also be unable to state ex-plicitly the tacit rules governing their use of the word. Nevertheless,most adults do know how to use the word, even if only within certainlimited contexts.

Let us now assume that some person has come to understand what ismeant by temperature' exclusively in terms of manipulating a mercurythermometer. If that individual were told that there is a substance whichmelts at a temperature of fffteen thousand degrees, he would probablybe at a loss to make sense of this statement, and he might even claimthat what has been told him is quite meaningless. In support of thisclaim he might maintain that, since a temperature can be assigned tobodies only on the basis of employing a mercuy thermometer, and sincesuch thermometers are vaporized when brought into the proximity ofbodies whose temperatures (as speciffed by a mercury thermometer)are a little above 350o C, the phrase 'temperature of fifteen thousanddegrees" has no deffned sense and is therefore meaningless. However, hispuzzlement over the information given him would be quickly removedby a little study of elementary physics. For he would then discover thatthe word temperature' is associated in physics with a more embracingset of rules of usage than the rules that controlled his own use of theword. In particular, he would Iearn that laboratory scientists employ the

The RedaAnan of Themies 841word to refer to a certain state of physical bodieq and that variatlons lnthis state aro often manifested ln other ways than by the volume expan'sion of mercury-for examplg in changes in tho electrical resistance of abody, or in the generation of electrical currents under speciffed condi-tions, Accordingly, once tho laws are explained that formulate tho rela'tions between the behaviors of instruments such as thermocouples, whicharo sometimes used to record changes in the physical state of bodiescalled their temperature,'the person understands how the word can bomeaningfully employed in situations other than those in which a mer'cury thermometer can be used. The enlargement of the word's range ofApplication then appears no more puzzling or mysterious than does theextension of the word 'lengttr,'from its primitive meaning as ffxed byusing the human foot for determining lengths, to contexts in which astandard metal bar replaces the human organism as a measuring in-strumenL

Suppose, however, that the layman for whom otemperature' thus aoquires a more generalized meaning now pursues his study of physics intothe kinetic theory of gases. Here he discovers that the temperature of agas is the mean kinetic energy of the molecules which by hypothesisconstitute the gas. This new information may then generate a fresh per-plexity, and indeed in an acute form. On the one hand, the layman hasnot forgotten his earlier lesson, according to rvhich the temperature ofa body is speciffed in terms of various overtly performed instrumentaloperations. But on the other hand, he is also assured by some authoritieshe now consults that the individual molecules of a gas cannot be saidto possess a temperature, and that the meaning of the word is identical'ty deffnition" with the meaning of 'the mean kinetic energy of molo-cules.'l Confronted by such apparently conflicting ideas, he may ther+fore ffnd a host of typically'philosophical" questions both relevant andinescapable.

If the meaning of 'temperature' is indeed the same as that of 'themean kinetic energy of molecules,'what is the plain man in the streettalking about when he says that milk has a temperature of 10o C? Most6onsumers of milk who might make such statements are surely not assert-ing anything about the energies of molecules; for even though they un-derstand and know how to use such statements, they are generally un-instructed in physics, and know nothing about the molecular composi-tion of milk. Accordingly, when the man in the street learns about mole-cules in milk, he may come to believe that he is confronted with a serl.ous issue as to what is genuine 'reality" and what is only 'appearance.'He rnay then perlraps be-persuaded by a traditional philosophical argu-

I Cf. Bomhard IJuvlnk, Tho Arutomu of Sclerrce, London, 1982, p. 09.

,42 The Structure of Sciencenent that the familiar distinctions between hot and cold (indeed; evonhe distinctions between various temperatures of bodies as speciffed tn:erms of instrumental operations), refer to matters that are 'tlubjectivo"nanifestations of an underlying but mysterious physical reality, a realityvhich cannot properly be said to possess temperatures in the common-tense meaning of the word. Or he may accept the view, supported by alifferent mode of reasoning, that it is temperature as aefrnia by pro-redures involving the use of thermometers and other such instrumentcvhich is the genuine reality, and that the molecular energies in termsrf which the kinetic theory of matter 'deffnes" temperattrie are just oictioa. Alternatively, if the layman adopts a somewhat *or" soplrirtt.ated line of thought, he may perhaps come to regard temperature as nnemergent" trait, manifested at certain "higher levels' of the organlzn-ion of nature but not at the 'lower levels' of physical reality; and honay then question whether the kinetic theory, wti"t ostensibly is con.erned only with those lower levels, does after alfiea[y explain" thoccurrence of emergent traits such as temperature.Perplexities of this sort are frequently generated by reductions of tlro

ype of which the above example is an instance. In such reductions, tlroubject matter_of the primary science appears to be qualitatively discorr.inuous with the materials studied by the secondary science. put sorrrn.rhat more precisely, in reductions of this type the secondary scicrroomploys in its formulations of laws and theories a number of distinctlvoescriptive predicates that are not included in the basic theoretical torrrrrr in the associated rules of correspondence of the primary scienco. Ilo-uctions of the ffrst or "homogeneous" type can be regarded as a sp.t:lrrlme of reductions of the second or "heterogeneouso type. But it is wlllr:ductions of the second type that we shall be

"or""rred in what folkrwr,8. To ffx our ideas, let us consider a deffnite example of a rcrlrrt.lhrrr

I this variety. The incorporation of thermodynamics *ittin mcclrurrk,rrore exactly, within statistical mechanics and the kinetic thcory ol' nrrrl:r-is a classic and generally familiar instance of such a reclucti,rr. worall therefore outline one small fragment of the argument by wlrlt,tr llrnduction is effected, on the assumption that this prit of thc argrrrrrr,rrl lrrfficiently representative of reductions of this type to scrvc as rr lrunlr)r a generalized discussion of the logic of reduction in ttrcor.r,l lr,nl:ience.

.Let us ffrst briefly recall some historical facts. Thc stucly.f llr.urrrrlhenomena goes back in moclcrn tirrcs to calilco antl lris cir

844 The Structure ol Sclencevolume is V. The gas is assumed to be composed of a large number ofperfectly elastic spherical molecules possessing equal masses and vol-umes but with dimensions that are negligible when compared with thoaverage distances between them. The molecules are further assumed tobe in constant relative motions, subject only to forces of impact betweenthemselves and the perfectly elastic walls of the containir. Thus thomolecules within their container constitute by postulation an isolated orconservative system, and the molecular motions are analyzable in termsof the principles of Newtonian mechanics. The problem now is to calcu.late the relation of other features of their motion to the pressure (orforce per unit area) exerted by the molecules on the walls of the con-tainer because of their constant bombardments.

However, since the instantaneous coordinates of state of the individualmolecules are not actually ascertainable, the usual mathematical pro.cedures of classical mechanics cannot be applied. In order to ,rakoheadway, a further assumption must be introduced-a statistical onoconcerning the positions and momenta of the molecules. This statisticalassumption takes the following formr Let the volume v of the gas bosubdivided into a very large number of smaller volumes, whose Ji*en-sions- are equal among themselves and yet large when compared withthe diameters of the molecules; and also divide the maximum range ofthe velocities that the molecules may possess into a large number of equalintervals. Now associate with each small volume all possible velocity.intervals, and call each complex obtained by associating a volume wiiha velocity-interval a *phase-cell." The statistical assumption then is thatthe probabiuty of a molecule's occupying an assigned phase-cel is thosame for all molecules and is equal to the probability of a moleculotoccupying any other phase-cell, and that (subject to certain qualiffca.tions involving among other things the total energy of the system ) thoprobability that one molecule occupies a phase-cell is independent ofthe occupation of that cell by any other molecula

If in addition to these various assumptions it is stipulated that thopressure p exerted at any instant by the molecules on the walls of tlrocontainer is the average of the instantaneous momenta transferred fromthe molecules to the walls, it is possible to deduce that the pressuro pis related in a very deffnite way to the mean kinetic

"n"rgy E of thomolecules, and that in fact p:28/3V, or pV = ZE/8. But a cr:mparisorrof this equation with the Boyle-charles' law (according to wrrich pv

-ftT, where ft is a constant for a given mass of gnr, on,l T its absorutotemperature) suggests that the law could bc clcducc

46 The Structure of Sciencea, In the highly developed science S (such as mechanics, electrody-

ramics, or thermodynamics ) there is a class T of statements consistingf the fundamental theoretical postulates of the discipline. These postu-ntes appear as premises (or partial premises) in aII deductions withini. They are not derived from other assumptions in a given codiftcationrf the science, although in an alternative exposition of S a difierent setrf logically primitive statements may be employed. Since T is adoptedn order to account for, and to direct further inquiry into, experimentalaws and observable events, there will also be a class R of coordinatingleffnitions (or rules of correspondence) for a sufficient number of theo-etical notions occurring in T or in statements formally derivable fromhose in T. Moreover, T will presumably satisfy the usual requirementsor an adequate scientiffc theory. In particular, T will be capable of ex-rlaining systematically a large class of experimental laws belonging toi; it will not contain any assumptions whose inclusion does not signiff-:antly augment the explanatory power of ? but serves merely to ac':ount for perhaps only one or two experimental laws; it will be *com-)endent" (in the sense that any pair of postulates in T will have ateast one theoretical term in common); and the postulates in T will be'simple" and not too numerous. As noted in Chapter 6, it is sometimes:onvenient to use the assumptions ? not as premises bn:t as leading prin->iples or as methodological rules of analysis. However, the issues thatuise from stressing the role of theories as guiding principles rather thans premises have already been discussed, and those issues are in anylase of no moment in the present context.

It is often possible to establish a hierarchy among the statements of Tn respect to their generality (in the sense of 'generality" examined inShapter 3). When this can be done it is then useful to distinguish thenbclass ?r, containing the most general theoretical assumptions in ?,irom the remaining subclass ?:, of more specialized ones. The most gen-lral postulates fr normally have a scope of application that is more in':lusive than the scope of the theory ? taken as a whole. Accordingly, thepostulates ?1 are comprehensive postulates of which T is but a special:ase, while the assumptions Tz are hypotheses concerning some speciallgpe of physical systems. For example, the most general theoretical as'rumptions in the kinetic theory of gases are the Newtonian axioms of mo-tion, so that they belong to ?r; and their scope is clearly more embrac-ing than is the scope of the kinetic theory. On the other hand, the postu'late that every gas is a system of perfectly elastic molecules whose di-mensions are negligible, or the postulate that all the molecules have therame probability of occupying a given phase-cell, are less general thanlhe Newtonian axioms, and belong to Ts. The assumptions Tr can thusbo regarded as variablo supplements to thoso in T1, for they can bo

The Reduction ol Thewiet U7varied without altering the content of those in T1, since the latter areapplied to difierent types of systems. For example, the Newtonian ax-ioms are supplemented by distinctive assumptions concerning the moleo'ular structures of gases, liquids, and solids, when these axioms are usedin theories about the properties of difierent states of aggregation ofmatter. Again, although the kinetic theory of gases retains the funda-mental assumptions of Newtonian mechanics when it deals with varioustypes of gases, the theory does not always postulate that gas moleculeshave negligible dimensions; moreover, the forces assumed by the theoryto be acting between the molecules depend on whether or not the gasis far removed from its point of liquefaction.

Although it may not always be possible to distinguish sharply betweenthe more general postulates Tt of a theory and the less general variablesupplements to them, some such distinction is commonly recognized.Thus, despite the fact that the primary science to which thermodynamicshas been reduced contains other postulates than those of classical me-chanics, thermodynamics is often said (even if only loosely) to be re-ducible to mechanics, presumably because the Newtonian axioms ofmotion are the most general assumptions of the kinetic theory of gases,so that they formulate the basic framework of ideas within which thespecial conclusions of the theory are embedded. Moreover, were thekinetic theory of gases able to account for some of the experimental lawsof thermodynamics only by modifying one or more of its less generalassumptions T2, it is unlikely that anyone would therefore dispute thereducibility of thermodynamics to mechanics, provided that the prin-ciples of mechanics are retained as the most general explanatory premisesof the revised theory.

b. A science S possessing a fundamental theory T will also have aclass of theorems that are logical consequences of T. Some of the theo-rems will be formally derivable exclusively from T (indeed, often fromthe most general postulates T1) without any help from the correspond-once rules R, while others can be obtained only by using R as well. Forexarirple, a familiar theorem of the ffrst kind in planetary theory is that,lf a point-mass is moving under the action of a single central force, itsorbit is a conic section; a theorem of the second kind is that, if a planetls moving under the action of the sun's gravitational fo,.ce alone, its arealvclocity is constant.

But whether or not S has a comprehensive theory, it will in generalcontain a class L of experimental laws that are conventionally regardedos falling into the special province of S. Thus, the various laws dealingwlth the reflection, refraction, and diffraction of light constitute part oftho oxpcrimental content of the science of optics. Although at any given

348 The Structure af Sciencestage of development of s the class of its experimental laws L is inprinciple unambiguously determinable, this clasiis frequently augmented(and sometimes even_diminished) with the progress ,l ir,i"*y. Nor islh"f u permanently fixed demarcation betweerith"

"rp".ir"rrtar rawst !h1t are grouped together as beronging to one brarr-ch of science sand the laws that are considered to fal into a difierent branch. Thus,it was not always understood that electrical and magnetic phenomenaare intimately related; and in older books on physics, tf,ough iot in mostof the recent ones, experimentar laws about prima facie different phe-nomena are classiffed

_as -beronging to distinct departments of exieri-mental inquly. Indeed, the limits assumed for th6 domain of a given

science, and the ratio-nale operative in classifying experimental lawsunder difierent scieni'ffc,disciprines, are often basei on^the

"*lurutoryscope of currently held theories.

-

c. Eu"ry positive science contains a Iarge class of singular statementsthat either formulate the outcome of obseiiations on thJ subyect matterregarded_as the province of the science or describe the overt prt""drr",instituted in conducting some actual inquiry within that disciprine. weshall call such singular statements "obiervation statements,,, but withthe understanding that in using this label we are not committed to anyspecial psychological or philosophical theory as to what are the *real,,data of observation. In particular, observation statements are not to beidentiffed with statements about "sense data" sometim",

"uLs"J i, b"excl.rrsive objects of 'direct experience." Thus, 'There was a toial eclipseof the sun at Sabral in North Brazir on May zg,1g1g,' and .The switchwas turned on yesterday in my office when ih" t"-p"rrture of the roomdroppgd to 50o F,' both

"o.rrri a, observation statements in the presentuse of this designation. observation statements may on occasion formu-late initial and boundary conditions for a theory ol ru*i tr,"y-*uy urrobe employed to conffrrn or refute theories and iaws.d. Many observation statements of a given science s describe the ar-

rangement and behavior of-apparatus required for conducting experi.ments in s or for testing various assumptions adopted in s. Acclordingry,lhe assertion of such observation staterients maylacitly involve the usel{ Igws c-oncerning characteristics of difierent sorts of instruments; some:f these Iaws may not falr into the generalry acknowredgeJ-frouin"" olt u1d. may not be explained by-any theory of S. For Jr*ffiL, photo-yaphic equipment attached-to terescopes is commonry

"*ptoiuiin test-,ng Newtonian gravitational theory, io that the construition of suchtpparatus, as well as the interpretation of data obtained with its help,;akes for granted theories ancr experimentar laws u"tii

"r ,pii"s and

The Reduction of Theories 349chemistry. f:lowever, the general assumptions thus taken for grantecl donot belong to the science of mechanics; and Newtonian gravitationaltheory does not pretend to explain or to warrant optical and chemicallaws. When cameras and telescopes are used in inquiries into mechanicalphenomena, distinctions and laws are therefore 'borrowed" from otherspecial disciplines. We shall refer to such laws, which are wed in a sci-ence S but are not established or explained within S itself, as *borrowedIaws" of S.

Most sciences will also contain statements that are certiffable as logi-cally true, such as those of logic and mathematics. Even if we ignorethese, we have identiffed four major classes of statements that may occurin a science S, whether or not any degree of autonomy is claimed for itrelative to other special disciplinesr (a) the theoretical postulates of S,the theorems derivable from them, and the coordinating definitions as-sociated with theoretical notions in the postulates or theorems; (b) thecxperimental laws of S; (c) the observation statements of S; and (d)the borrowed laws of S.

2. We come to the second formal point. Every statement of a scienceS can be analyzed as a linguistic structure, compounded out of moreclementary expressions in accordance with tacit or explicit rules of c.on-struction. It will be assumed that, though these elementary expressionsmay be vague in varying degrees, they are employed unambiguously inS, with meanings ffxed either by habitual usage or explicitly formulatedrules. Some of the expressions will be locutions of formal logiq arith-mctic, and other branches of mathematical analysis. We shall, however,be primarily concerned with the so-called *descriptive expressions," sig-nifying what are generally regarded as 'empirical" obiects, traits, rela-tions, or processes, rather than purely forrnal or logical entities. Althoughthcre are difficulties in developing a precise distinction between logicaland descriptive expressions, these difficulties do not impinge upon theprcsent discussion. Let us in any case consider the class D of descriptivocxpressions in S that do not occur in borrowed Iaws of S.

Many of the descriptive expressions of a science are simply taken overfrom the Ianguage of ordinary affairs, and retain their everyday mean-Ings. This is frequently true for expressions occurring in obseruationstatements, since a large fraction of the overt procedures employed evenln carefully devised Iaboratory experiments can be described in the lan-guoge of gross experience. On the other hand, other descriptive expres-sions may be speciffc to a given science; they may have a use restrictedto highly spccinlized tcclrnical contexts; and the meanings assigned tothcm in that science may eVen preclude their being employed to de-scribc mattors idcntiffablo either by direct or indirect observation. De,

50 The Struature ol Sclenaaoriptive expressions of this latter sort occur typically tn the theoreticalssumptions of a science.It is often possible to explicate the meaning of an expression in D

rith the aid of other expressions in D supplemented by logical ones.uch explications can sometimes be supplied in the form of conventionalrplicit deffnitions, though usually more complicated techniques forxing the meanings of terms are required. But whatever formal tech-iques of explication may be used, let us call the set of expressions in) which, with the help of purely logical locutions, suffice to explicatere meanings of all other expressions in D, the "primitive expressions"f S. There will always be at least one set P of primitive expressions,.nce, in the least favorable cases, when no descriptive expression cane explicated in terms of others, the set P will be identical with thelass D. On the other hand, there may be more than one such set P,)r, as is well known, expressions that are primitive in one context ofnalysis may lose their primitive status in another context; but this pos-ibility does not afiect the present discussion.However, if S has a comprehensive theory as well as observation state-

rents and expelimental laws, the explication of an expression may pro-eed in either one of two clirections that must be noted, since in generalach direction involves the use of a distinctive set of primitives.

a. Let us designate as 'observation expressions" those expressions in) that refer to things, properties, relations, and processes capable ofeing observed. The distinction between observation expressions andther desciptfue ones is admittedly vague, especially since difierentegrees of stringency may be used in difierent contexts in deciding whattatters are to count as observable ones. But despite its vagueness, theistinction is useful and is unavoidable in both scientific inquiry andveryday practice. In any event, many explications aim at specifying thereanings of descriptive expressions in terms of observable ones. Theroglam (advocated by Peirce and Bridgman among others) of ffxingre meanings of terms by giving what are currently known as *opera-onal deffnitions'for them appears to have explications of this sort for:s objective. Let us call the set P1 of observation expressions needed forxplicating in this manner the maximum number of expressions in D, therbservation primitives" of S. For example, the meaning of temperature'r frequently explained in physics in terms of the volume expansions ofquids and gases or in terms of other observable behaviors of bodies;r such cases the explication of 'temperature' is given by way of ob-rrvable primitives.

b. Let us suppose that S has a theory capable of explaining all thexperimental laws of the science; and let us designate as the "theoretical

The Reductlon ol Tlrcorles 851exprossions" of $ the descriptive expressions employed in the theoreticalpostulates (exclusive of the coordinating deffnitions) and the theoremsformally derivable from them. Many explications aim at specifying themeanings of expressions by way of theoretical ones; and we shall callthe set Pr of theoretical expressions needed for explicating in this waythe maximum number of expressions in D, the "theoretical primitives"of S. For example, the meaning of temperature' is given a theoreticalexplication in the science of heat with the help of statements describingthe Carnot cycle of heat transformations, and therefore in terms of suchtheoretical primitives as'perfect nonconductors,' inffnite heat reservoirs,'and'inffnitely slow volume expansions.'

As rve have seen in Chapter 6, the question whether theoretical expres-sions are explicitly definable in terms of observable ones has been muchdebated. If theoretical expressions were always so deftnable, they couldbe eliminated in favor of observable ones, so that the distinction wouldhave little point. However, although a negative answer to the questionhas not been demonshably established, all the available evidence sup-ports that answer. Indeed, there are good reasons for maintaining thestronger claim that theoretical expressions cannot in general be ade-quately explicated with the help of observation ones alone, even whenforms of explication other than explicit deffnitions are employed. It isnot necessary to adopt a position on these questions for the purposes ofthe present discussion. We must nevertheless not assume as a matter ofcourse that the set of observation primitives Pr is sufficient to explicateall the descriptive expressions D; and we must allow for the possibilitythat the class P of primitive expressions of S does not in general coineidewith the class P1. Accordingly, although temperaturd is explicated inthe science of heat both in terms of theoretical and of observation primi-tives, it does not follow that the word understood in the sense of theffrst explication is synonymous with temperature' construed in the senseof the second.

B. We can now turn to the third formal consideration on reduction.The primary and secondary sciences involved in a reduction generallyhave in common a large number of expressions (including statements)that are associated with the same meanings in both sciences. Statementscertiftable in formal logic and mathematics are obvious illustrations ofsuch common expressions, but there usually are many other descriptiveones as well. For example, many laws belonging to the science of me-chanics, such as Hooke's law or the laws of the lever, also appear inthe science of heat, if only as borrowed laws; and the latter science em-ploys in its own experimental laws such expressions as 'volume,' 'pres-sure,'and'work'in senses that coincide with the meanings of these wordsin mechanics. On the other hpnd, before its reduction the secondary sci-

952 The Structure ol Sctencoenc gnerally uses expressions and asserts experimental laws formulatedwjlh iheir help which clo not occur in the prinrary science, except pos.sibly in the latter's c]asses of obscrvation staternents ancl borrowccl laws.For example, the scicnce of mcchanics in its crassicar form croes notcount the Boyle-cha'les' law as one of its experimental laws; nor doesthe term'temperature' occur in the theoretical lssLrmption, oi *""rr*ni"r,though the word may sometimes be employed i" it,

"rp".imlntar in.quiries to describe the circumstances uncler which ,o-" li* of the sci-ence is being used.

It is, however, of utmost importance to note that expressions belong-ing to -a

science possess meanings trrat are ffxed by iti o*n fro""dr.",of explication. In particular, expressions distinctivl of "

;i"-"; science(such as the word 'temperaturel as employed in the scieice of heat)are intelligible in terms of the rules o, hutit, of usage or trrui branchof inquiry; and when those expressions are used in trr"i t.*"t or ,,rrdnthey rnust be understood in the senses associated with them in thatbranch, whether or not the science has been reduced to--ro.re ott

",discipline, Sometimes, to be surg the meaning of * ""pr"rrion ir, "s-cience can be explicated.wi1h the herp of the primitiies-(whether

theoretical or observational) of ro-" otlr!, science. For example, thereare ftrm grounds for the assumption that the *o.a .p."rrrr";'il ,ra"r.stood in thermodynamics is synonymous with the *.r, rr"*.r1,

", "*-plicated by way of the theoretical primitives of mechanil. It neverthe-Iess does not follow that in general every expression employed in a givenscience, in the sense specified by its own distinctive ruLs L. pro""du."r,is explicable in terms of the primitives of some other discipliie.

with these prelimina.es out of the way, we must now state the formal:equirements that must be satisffed for the reduction of one ,"i"rr"" a,mother. As has already been indicated in this chaptea u *i""nr" i,rffected when the experimental laws of the secondary science (and ift has an adequate theory, its theory as well) are shorm to be the Iogical)onsequences of the theoretical assumptions (incrusive of th"

"oorairr^ung deffnitions) of the qrimary scienc-e. It should b" ob^;J ih", *"5e n-ot stipulating that the borrowed laws of the secondary-r"i"r"" *rrtrlso be derivable from the theory of the primary science. However, ifhe laws of the secondary science contain'terms that do not occur inhe theoretical assumptions of the primary disciprine (;"J;; is theype of reduction to which *"

"gr""d "rrli". to conffne the discussion),he logical derivation of the former from the rutt", L iriri""ii"u, ,^-'ossible. The claim that the derivation is impossibre is based on thermiliar logical canon tha! save for some essentially irrelevarrt'"*""p-'ons, no term can appear in the concrusion of a forrnar demonstration

Ths Reductton ol fheortw gU0unless ths terrn also appeare ln the premlsos,, Accordtngly, when tholnwr of the secondary ocience do contain somo torm 'A'ihat Is absontfrom tho thooretical assumptlons of the prlmary scrence, thero aro twon(lcessory fornral conditions for tho reduction of the formof to the lattelr(I ) Assumptions of somo kind niust be introduced which posturate euit.-

i Posslble objections to thls loglcal canon are based for the mo6t part oa thefnct that, lu.vlew o[ eomo theorems ln modorn lormal logic, a valid diductlve ar-gurn.ent can havs a concluslon coutalulng tenns not occuriing in tho premises,

'fhere are at least two laws lo the sententlal calculus ( ot logic rf uaanalyredpruposltions) that permit tho deduction of ruch concluslons. Accordi-ng to one of them,uny statement oI the form 'If St, then S, or $r,' whero S, and Sa aro any Btatement8,ls logically true, 80 that sr or $r ls derivable from $r. But since Sr orn be chosennrbitrarily, sr or s: can be made to contaln terms not occuring ln sr, According toa second logical law, any.statement of the form'S,, lf and only {f Sr and (Si orrrot-Sr)'ls logically true; hence 'S, and (Sr or not.Sr)' ls derlvable from $r, wtththo

"same general outcome as ln tho ffrst case. Howevor, tt le clear that neitti;; type

of deductive step can yield the Boyle-charles' law from tho kinetic theory of gasis.If it could (for example, by way of substituting this law for sr in the hrst oJ thotwo logiTllaws mentioned), then, stnce S, ls eniirely arbttrary, the deduction wouldalso yield qlJe contradictory of this law; and ttrts cannot happen, unless the kinetlctlreory ltself-ls self-contradlctory. Thls.argument ls qutte geneial and appliee too.ther examples of.reduction. Accordingly,-lnsofar as ieductions make usd only ofthe logical laws of the sentential ealculus in deduclng Btatemento of tho secondaryrcielce {rom the theory of t}re prlmary sciencg It ls euficient to meet tJro objectionto-ttre_logical canon rnentloned ln tho text by emending the latter to readi In avalid deduction no term appears in the conclusion thit doee not occut in lfiopremlses, unless a term enters lnto the conclusion via loglcal laws of tls sententlalcnlculus, which permit the lntloduction of any arbttra{ term lnto the concluslon..

I-Iowever, thero aro other logtcal- laws, developed h bther parts of frirmal logtc,that also sanction conclusions with terms not ln tho premises. Substitution joivariables expressing universa[ty ls a famlliar type of such inference, For example,although the premlso "For any r,.lf.* is a planet then * shines by roflected hfht"does not contain the term 'Mars,' the state.ment "If Mars ls a pianet, then Marsshines by refected light" can be validly deduced from it, Anothei typo of such rn-ference ls illustrated by the dertvatlon from "All men are mortal' oi ihe concluston"All hungry men are hungry mortals." Nevertheless, an examlnation of tho derlvationof the Boyle-charles' Iaw reveals that the term 'temperature,' eontained tn tlrtslaw but not ln the klnetlc theory, ls not lntroducerl into tha derivation by wayof any such universally valid deductive steps; and an argument, analogous io theone presented in the preceding paragraph of this note for the caso of deductlonsIn tho sentential calculus, can be constructed to show that thls must also bo thecase ln the deduction of other laws, contalnlng distinctlve terms, of a secondaryscience that ls reducible to somo primary one, Aceordlngly, theso varlous exceptionito the logical canon of the text can be lgnored as not rJlevant to tho matters irnderdiscussion,

.

A different objectlon^ to thls^ canon ls thal formal loglc asrde, wo often do recog-nlzo arguments as valid even

-though they ostensibly vlolate the canon. Thus, '|ohnfsa cou-sin of Mary'ls said to follow from 'The uncle bf 1oh" ls tho father of Mary,' and'smith's shirt ls colored' is said to follow from 'smltli's shlrt ts red,' despito the factthat a term appears in each of the conclusions that is absent from ihe correspondingpremise, However, these examples and others like them are essentially enthyrnematiclnferences, with a tacit assumption either ln the form of an explictt dednition orsomo other kind of^a priori statement. when these suppressed assimptions aro madoexplicit, the examples no longer appear to bo exceptions to the logical canon underexamlnatlon.

954 The Structure of Sctenceable relations between whatever is signiffed by'N and traits representedby theoretical terms already present in the primary science. The natureof such assumptions remains to be examined; but without preiudging theoutcome of further discussion, it will be convenient to refer to this con-dition as the "condition of connectability," (2) With the help of theseadditional assumptions, all the laws of the secondary science, includingthose containing the term A,' must be logically derivable from the theo-retical premises and their associated coordinating deffnitions in the pri'mary discipline. Let us call this the 'condition of derivability." a

There appear to be just three possibilities as to the nature of the link'ages postulated by these additional assumptions: (1) The ftrst is thatthe links are logical connections between established meanings of ex'pressions. The assumptions then assert 'A' to be logically related (pre-sumably by synonymy or by some form of one-way analytical entail-ment) to a theoretical expression 'B' in the primary science. On thisalternative, the meaning of 'A' as ffxed by the rules or habits of usageof the secondary science must be explicable in terms of the establishedmeanings of theoretical primitives in the primary discipline. (2) Thesecond possibility is that the linkages are eonoentions, created'by delib-erate ffat. The assumptions are then coordinating deftnitions, which in.stitute a correspondence between 'A' and a certain theoretical primitive,or some construct formed out of the theoretical primitives, of the pri-mary science. On this alternative, unlike the preceding one, the mean-ing of 'A' is not being explicated or analyzed in terms of the meaningsof theoretical primitives. On the contrary, if 'A is an observation termof the secondary science, the assumptions in this case assign an experi-mental signiffcance to a certain theoretical expression of the primaryscience, consistent with other such assignments that may have beenpreviously made. (8) The third possibility is that the linkages arc factualor materinl. The assumptions then are physical hypotheses, asserting thatthe occurrence of the state of affairs signiffed by a certain theoreticalexpression ?' in the primary science is a sufficient (or necessary andsufficient) condition for the state of afiairs designated by'A! It will beevident that in this case independent evidence must in principle be

r The condition of connectability requires that theoreticaZ terms of the primary scl-ence appear in the statement of these additional assumptions. It would not suffico,for example, if these assumptions formulated an explication of 'A' by way of ob-servaUon primitives of the primary science, even if the theoretical primitives couldalso be explicated by way of the observation primitives. For it would not therebyfollow that 'A' could be explicated by way of the theoretical prlmitives' Thus, al-though 'uncle' and 'grandfather' are each dcffnable ln terms of 'malo' and 'parent,''uncle' is not deffnable in terms of 'grandfather.' In consequenco, tlto addlUoualassumption would not contributo toward tho ful8llnrent of tho condltlon tlf do-rivability.

The Red.uction of Theories 955obtainable for the occurrence of each of the two states of affairs, so thatthe expressions designating the two states must have identiffably dif.ferent meanings. On this alternative, therefore, the meaning of 'A' is notrelated analytically to the meaning of '8,' Accordingly, the additionalassumptions cannot be certiffed as true by logical analysis alone, and thehypothesis they formulate must be supported by empirical evidence.6

In the light of this discussion, let us now examine the derivation of theBoyle-Charles'law from the kinetic theory of gases. For the sake of sim-plicity let us also assume that the word'temperature" is the only term inthis law that does not occur in the postulates of that theory. However,as already noted, the deduction of the law from the theory depends onthe additional postulate that the temperature of a gas is proportionalto the mean kinetic energy of its molecules. Our problem is to decide onthe status of this postulate and to determine which if any of the threetypes of linkage we have been discussing is asserted by the postulate.

For reasons mentioned in the ftrst section of the present chapter, itIs safe to conclude that'temperature,' in the sense the word is employedln classical thermodynamics, is not synonymous with 'mean kinetic en-orgy of molecules,'nor can its meaning be extracted from the meaningof the latter expression. Certainly no standard exposition of the kineticthcory of gases pretends to establish the postulate by analyzing the rreor.lngs of the terms occurring in it. The linkage stipulated by the postulateounnot therefore be plausibly regarded as a logical one.

Ilut it is far more dificult to decide which of the remaining two types0 It follows that the condition of connectability is in general not sufficient for re-

{utrtkrn and must be supplemented by the condition of derivability. Connectabilitywould lndeed assure derivability if, as has been rightly argued by John G. KemenyErrd l'uul Oppenheim ["On Reduction," Philosoplilcal Studi.es, VoL 7 (1956), p. 10],for ovory tcrm'A'in the secondary science but not in the primary one there is a theo-rolk:nl tcrm'B'in the primary science such that A and B are linked by a biconditionallA lf nnd only if B. If the linkage has this form, 'A' can be replaced by 'B' in anylnw L of the secondary science in which'A'occurs, and so yield a warranted theo-rnlkrnl postulate L'. If L' is not itself derivable from the available theory of theprhrrlry sclcnce, the theory need only be augmented by t' to become a modifiedllrnrry, brrt nonetheless a theory of the primary science' In any event, L will berlnrhrt:llrto from a theory of the primary science with the help of the biconditionals'llrrwovor, thc linkage between A and B is not necessarily biconditional in form, andrrrnv for cxumple be only a one-way conditional: If B, then A. But in this eventuality'A' ll rot rcplnceable by '8,' and hence the secondary science will not in generalhn rlt,rlrurlble from a theory of the primary discipline. Accordingly, even if wewnlvn llrt' qucstion whcther a reduction is satisfactory when achieved by augment-hr1 llrr, lhrrory of the primarv science by a new postulate L'which is empiricallyrlrrrllrrrrr.rl lrrrt may contribute next to nothing to the explanatory power of theltltlnl tlurory,

856 The Structurc of Sclenceof linlrage is asserted !r the posturate, for there are prausibro reasonsfavoring each of these ariernati;o. i[;.g,rment in support of the craimthat the postulate is

-simpry a coordinating deffnitio, i, "rr"rrtiaily asfollows: The kinetic theor!'of gr",

"urroti" p,,t;; exieiim-entar test,unless rules of correspondlnce-ffrst arro"i"t" some of its theoretical no-tions -with experimental conc-epts. But the postulate itself cannot be sub-iected to experimentar c,ontrol. For, arthougirtrr" t"*p"*t.il'oiu g", .u,be determined bv famiriar laborato.v prol"dur"r, i#;;ir-'r;;rentry now1l of as-certaining the mean kinetic eiergy of the r,yprtr,"tiir"r*gu,

-orocules-unless, indeed,.the- temperature is stipulat"d b;, ff J;;-iJ* *"rrrr"oJ this energy' Accordingry, thi posturate ci., be r,ot*rrg o,t

"rit u., one ofthe-correspondence rures which irstitute an association between theoreticaland experimental concepts., on the other hand, the

"r"i- trrrt trre posturateis a phvsical hvpothesis is arso not an unfounded "r;;;;;;;J"".+ i, i. i,this fashion that the posturate is introduced in many technical presenta-tions of the subject The major reason

"au"n""a roiir,i,

"iri* l; that, ar_llough the postulate.cannot be tested by direct measurements on the meankinetic energy of gai molecures, the vuir" or trrir "n".gy "u, i*u".tt "t"r,be ascertained indirectry, by caiculationfrom experimental data on gasesother than data obtained by measuring temperatures. In consequence, itdoes seem possible to d-etermin"

"rp"ri**ti[y whethe, arr" ,"rip"ratureofa gas is proportional to the mean'kinetic enerw of its morecures.Despite appearances to the contrary, these arternative craims "nd srpporting reasons for them are not necessariry ir**prtiur".'ira""a, tn"alternatives illustrate what is by now a famiiiar poirri-thai the "ognitivestatus of an assumption.often depends on the rrira"

"i"p,"a"ro-, u.u*_fating a theory in a particurar con^text. The reductir" ;iH;;;olyru*ioto mechanics can undoubtedrv -be so "rpou;J;;--d;;;;ditionarpostulates about the proportionality of temperature to the mean kineticenergy of gas molecules institutes what is at ffrst the sole link betweenthe theoretical notions of the primary ."i** ,rd

"rp"rir*rr,"i "or""p,,of .the seeondary one. I., suctia

"oni* oi"rposition, the posturate can-not be subjected to experimentar test but functiorrs *-J""*ainatingleffnition' However, different -rd"; ;f ;;position are arso possibre, inrhich coordinating deffaitions "r"

ir;J;:ed for ;il';Hr" or ,n"o_:etical and experimentar c-o-ncepts. ror *r*pre, one theoreticar notion:an be made to co*esponcl t9 tle ",,n"*";,ui id"" of uir"ority, *namother can be associaLd with the "*!"ri*.","I concept of heat flown consequence, sinco the mean kinetic energy of gas ,r"r*r1". i, ,""ated' by virtue of the assu,rptions of the kinetic theory, to these other

,l?l'ur+]**ttn R' campbell, Phssics, the Elements, cambridgs England, lg2e

The Raductlon ol Theorles 857thcoretical notions, a connection may thus be indirectly established be-tween temperature and kinetic energy. Accordingly, in such a contextof exposition, it would make good sense to ask whether the temperatureof a gas is proportional to the value of the mean kinetic energy of thegas molecules, where this value is calculated in some indirect fashionfrom experimental data other than that obtained by measuring the tem-perature of the gas. In this case the postulate would have the status ofa physical hypothesis.

It is therefore not possible to decide in general whether the postulateis a coordinating deffnition or a factual assurnption, except in some givencontext in which the reduction of thermodynamics to mechanics is beingdeveloped. This circumstance does not, however, wipe out the distinc-tion between rules of correspondence and material hypotheses, nor doesit destroy the importance of the distinction. But in any event, the presentdiscussion does not require that a decision be made between these alter-native interpretations of the postulate. The essential point in this dis-cussion is that in the reduction of thermodynamics to mechanics a pos-tulate connecting temperature and mean kinetic energy of gas moleculesmust be introduced, and that this postulate cannot be warranted by sim-ply explicating the meanings of the expressions contained in it.

One objection to this central contention must be briefly considered.The redeffnition of expressions with the development of inquiry, so theobiection notes, is a recurrent feature in this history of science. Accord-ingly, though it must be admitted that in an earlier use the word tern-perature'had a meaning speciffed exclusively by the rules and proceduresof thermometry and classical thermodynamics, it is now so used thattemperature is 'identical by deffnition" with molecular energy. The de-duction of the Boyle-Charles' law does not therefore require the intro-duction of a further postulate, whether in the form of a coordinatingdeffnition or a special empirical hypothesis, but simply makes use ofthis deffnitional identity. This objection illustrates the unwitting doubletalk into which it is so easy to fall. It is certainly possible to redeffne theword 'temperaturen so that it becomes synonymous with 'mean kineticenergy of molecules.' But it is equally certain that on this redeffnedusage the word has a different meaning from the one associated with itin the classical science of heat, and therefore a meaning difierent fromthe one associated with the word in the statement of the Boyle-Charles'law. However, if 'thermodynamics is to be reduced to mechanics, it istemperature in the sense of the term in the classical science of heat whichmust be asserted to be proportional to the mean kinetic energy of gasmolecules. Accordingly, if the word temperature' is redeffned as sug-gested by the objection, the hypothesis must be invoked that the stateof bodies described as 'temperature' (in the classical thermodynamical

858 The Sttucture ol Sckncesense) ls also characterized by'temperature'in the redeffned senso ofthe term. ?his hypothesis, however, *lu then be one that does not holdas. a matter of deffnition, and will not be one for which logical necessitygan-bejgltly claimed. unless the hypothesis is adopted]it is not thlBoyle-charles'law which can be derived from the issumptions of thekinetic theory of gases. what is derivable without the hylothesis is asentence similar in syntactical structure to the standard formulation ofthat.law, but possessing a sense that is unmistakabry difierent from whatthe law asserts.

ilI. Nonformnl Conditions for Reductionwe must now turn to features of reduction that are not primarily for-

mal, though some of them have already been touched upir in passing.1. The two formal conditions for reduction discussed in the previous

section do not suffice to distinguish trivial from noteworthy icientiffcachievements. If the sole requirement for reduction were that the sec-o.ndary science is logically deducible from arbitrarily chosen premises,the requirement could be satisfied with relatively rittie difficultl,. In thehistory of signiffcant reductions, however, the premises of the'primaryscience are not ad hoc assumptions. Accordingly, although it would bL1f3r too strong condition that the premises must be knoiwn to be trugit does seem reasonable to impose is a nonformal requirement that thetheoretical_assumptions of the primary science be iupported by em-pirical evidenc_e possessin-g some degree of probativ" 6i"". The prob-lems connected with the logic of weighing

"uid"oce are dificult and atmany crucial points still unsettled. However, the issues raised by theseproblems are not exclusively relevant to the analysis of reduction; and,rycept for some brief eomments especially pertinent to the reduction oflhermodynamics to mechanics, we shall not at this place examine thenotion of adequate evidential support.

The evidence for the several assumptions of the kinetic theory of gases:omes from a variety of inquiries, only a fraction of which rail into theIomain of thermodynamics; Thus, the hypothesis of the molecular con-ititution of matter was supported by quantitative relations exhibited in:hemical interactions even before thermodynamics was reduced to me-:hanics; and it was also conffrmed by a number of laws in molar physicsrot primarily about thermal properties of bodies. The adoption ol thi,rypothesis for the new task of accounting for the thermaibehavior of,ases was therefore in line with the normal strategy of the science torxploit on a new front ideas and analogies found to be fruitful elsewhere.iimilarly, the axioms of mechanics, constituting the most general parts

Tha Rerlactton ol Theodetof the premises in the primary science to which thermodynamics is ra.duced, are supported by evidence from many ffelds quite distinct fromthe study of gases. The assumption that these axioms also hold for thehypothetical molecular components of gases thus involved the extrapo.Iation of a theory from domains in which it was already well conffrmedlnto another domain postulated to be homogeneous in important respectswith the former ones. But the point having greatest weight in this con-nection is that the combined assumptions of the primary science to whichthe science of heat was reduced have made ii possi6le to incorporatelnto a unifted system many apparently unrelated laws of the science ofheat as well as of other parts of physics. A number of gas laws hadof course been established before the reduction. However, some ofthese laws were only approximately valid for gases not satisfying certainnarrowly restrictive conditions; and most of the laws, moreover, couldbe affirmed only as so many independent facts about gases. The reduotion of thermodynamics to mechanics altered this state of afiairs in sig-nificant ways. It paved the way for a reformulation of gas Iaws so asto bring them into accord with the behaviors of gases satisfying lessrestrictive conditions; it provided leads to the discovery of new laws;and it supplied a basis for exhibiting relations of systematic dependenceamong gas laws themselves, as well as between gas laws and laws aboutbodies in other states of aggregation.

This last point deserves a brief elaboration. If the Boyle-Charles' Iawwere the sole experimental law deducible from the kinetic theory ofgases, it is unlikely that this result would be counted by most physicistsas weighty evidence for the theory. They wouid probably take the viewthat nothing of signiftcance is achieved by the deduction of only this onelaw. For prior to its deduction, so they might maintain, this law wasknown to be in good agreement with the behavior of only *ideal' gases,that is, those at temperatures far above the points at which the gasesIiquefy; and by hypothesis, nothing further follows from the theoryas to the behavior of gases at lower temperatures. Moreover, physicistswould doubtless call attention to the telling point that even the deduo.tion of this law can be effected only with the help of a special postulateconnecting temperature with the energy of gas molecules-a postulatethat, under the circumstances envisaged, has the status of an ad hocassumption, supported by no evidence other than the evidence warrant-ing the Boyle-Charles'law itself. In short, if this law were the sole ex-perimental consequence of the kinetic theory, the latter would be deadwood from which only artiffcially suspended fruit could be gathered.

In actual fact, however, the reduction of thermodynamics to the kinetictheory of gases achieves much more than the deduction of the BoyloCharles'law. There is available other evidence that counts heavily with

T'he Structure of Sctrlncert physicists as support for the theory and that removes from thocial postulate connecting temperaturei and morecurar energy evenappearance of arbitrariness. Indeed, two related sets of considera-s make the reduction a signiffcant scientiftc accomplishment. onoconsists of experimental laws, deduced from the theory, which havebeen previously established or which are in better agieement with

'ider range of facts than are laws previously accepted.-For example,Boyle-charles'law holds only for ideal gries ard is deducible fromkine-uc the_ory when some of the less general assumptions of the

;tic theory have the limiting form correqponding to a gas being anrl gas. However, these special assumptions can be replacid by olrershout modifying the fundamental ideas of the theory, ind in particularassumptions less simple than those introduced for ideal gaies. Thus,ead of-the stipulations with the aid of which the Boyre--harles'lawerivable from the theory, we can assume that the dimensions of gasecules are not negligible when compared to the mean distance be-en them, and that in addition to forces of impact there are also co.ive forces acting upon them. It is then postibre to deduce from the>ry employing these more complex special assumptions the van derals'law for gases, which formulates more adequately than does thele-charles'law the behavior of both ideal and nonideal gases. rneral, therefore, for a reduction to mark a signiffcant intelleitual ad-ce, it is not enough that previously estabushed laws of the secondarynce be represented within the theory of the primary discipline. ThLxy must also be fertile in usable suggestions for developing the sec-ary science, and must yield theorems referring to the tatter s subjectter which augment or

-correct its currently accepted body of laws.he second set of considerations in virtue of wti"t the reiuction ofmodynamics to mechanics is generally regarded as an important.evement consists of th_e intimate and frequently surprising relationslependence that can thereby be shown to obiain betweJn various:rimental laws. An obvious type of such dependence is illustratedn laws, hitherto asserted on independent evidential grounds, are de-ible from an integrated theory as a conseq.r.r"" oI the reduction.s, both the second law of thermodynamics laccording to which theopy_of

_a closed physical system never diminishes) L we[ as the

le-charles'Iaw are derivable from statisticar mechanics, although in;ical thermodynamics these laws are stated as independent priniitivemptions. In some ways a more impressive and subtler type of de-lence is illustrated when some numerical constant appeaiing in dif-rt experimental Iaws of the secondary science is exhibited as a deff-function of theoretical parameters in the primary discipline-an

ome that is particularly striking when congruous numerical values

Tha Reiluctlon ol Theorles 861can be calculated for those parameters from experimental data obtainedln independent Iines of inquiry, Thus, one of the postulates of the kinetictheory is that, under standard conditions of temperature and pressure,oqual volumes of a gas contain an equal number of molecules, irrespec-tive of the chemical nature of the gas. The number of molecules in aliter of gas under standard conditions is thus the same for all gases, andis known as Avogadro's number. Ivloreover, a certain constant appearingin several gas laws (among others, in the Boyle-Charles' law and theIaws of speciffc heats) can be shown to be a function of this numberand other theoretical parameters. On the other hand, Avogadro's num-ber can be calculated in alternative ways from experimental data gath-ered in different kinds of inquiry, e.g., from measurements in the studyof thermal phenomena, of Brownian movements, or of crystal structure;and the values obtained for the number from each of these diverse setsof data are in good agreement with one another. Accordingly, apparentlyindependent experimental laws (including thermal ones) are shown toinvolve a common invariant component, represented by a theoreticalparameter that in turn becomes ffrmly tied to several kinds of experi-mental data. In consequence, the reduction of thermodynamics to kinetictheory not only supplies a uniffed explanation for the laws of the formerdiscipline; it also integrates these laws so that directly relevant evidencefor any one of them can serye as indirect evidence for the others, andso that the available evidence for any of the laws cumulatively supportsvarious theoretical postulates of the primary science.

2. .These general comments on the considerations that determine

whether a reduction is a significant advance in the organization ofknowledge or only a formal exercise, and on the character of the evi-dence that actually supports the kinetic theory direct attention to animportant feature of sciences in active development. As has already been.suggested, different branches of science may sometimes be delimitedon the basis of the theories used as explanatory premises and leadingprinciples in their respective domains. Nevertheless, theories do not asa rule remain unaltered with the progress of inquiry; and the history ofscience provides many examples of special branches of knowledge be-coming reorganized around new types of theory. Moreover, even if adiscipline continues to retain the most general postulates of some theo-retical system, the less general ones are often modified or are augmentedby others as fresh problems arise.

Accordingly, the question whether a given science is reducible toanother cannot in the abstract be usefully raised without reference tosome particular stage of development of the two disciplines. Questionsabout reducibility can be profftably discussed only if they are made

i62 The Sbucture ol Sctenceleffnite by specifytng the established content at a given date of thociences under consideration. Thus, no practicing physicist is likely toake seriously the claim that the contemporary science of nuclear physicsr reducible to some variant of classical mechanics*even if the claimhould be accompanied by a formal deduction of tlie laws of nuclearrhysics from admittedly purely mechanical assumptions-unless thesessumptions are supported by adequate evidence available at the timehe claim is made, and also possess at that time the heuristic advantagesormally expected of the theory belonging to a proposed primary sci-nce. Again, it is one thing to say that thermodynamics is reducible torechanics when the latter counts among its recognized postulates as-umptions (including statistical ones) about molecules and their modesf action; it is quite a different thing to claim that thermodynamics is:ducible to a science of mechanics that does not countenance suchssumptions. In particular, though contemporary thermodynamics is un-oubtedly reducible to a statistical mechanics postdating 1866 (the yearr which Boltzmann succeeded in giving a statistical interpretation forre second law of thermodynamics with the help of certain statisticalypotheses), that secondary science is not reducible to the mechanicst 1700. Similarly, certain parts of nineteenth-century chemistry (anderhaps the whole of this science) is reducible to post-1g25 physics, butot to the physics of a hundred years ago.Moreover, the possibility should not be ignored that little if any new

nowledge or increased power for signiffcant research may actually berined from reducing one science to another at certain periods of theirevelopment, however great may be the potential advantages of such:duction at some later time. Thus, a discipline may be at a stage of:tive growth in which the imperative task is to survey and classify thertensive and diversiffed materials of its domain. Attempts to reduce theiscipline to another (perhaps theoretically more advanced) science,ren if successful, may then divert needed energies from what are theucial problems at this period of the discipline's expansion, without:ing compensated by effective guidance from the primary science inre conduct of further research. For example, at a time when the prime;ed of botany is to establish a systematic typology of existing plant life,re discipline may reap little advantage from adopting a physicochemicalLeory of living organisms. Again, although one science may be reducibleanother, the secondary discipline may be progressively solving its own

recial class of problems with the help of a theory expressly devised for;aling with the subject matter of that discipline. As a basis for attackingese problems, this less inclusive theory may well be more satisfactoryan the more general theory of the primary science-perhaps becausee primary science requires the use of techniques too reffned and cum-

Tlrc Reductlon of Theorles 988borsome for tho subjects under study in the secondary science, or be'oauso the initial conditions needed for applying it to these subjects arenot available, or simply because its structure does not suggest fruitfulnnalogies for handling these problems. For example, even if biology wereroducible to the physics of current quantum mechanics, at the presentrtnge of biological science the gene theory of heredity may be a moreratisfactory instrument for exploring the problems of biological inheritance than would be the quantum theory. An integrated system of ex'plnnation by some inclusive theory of a primary science may be anovcntually realizable intellectual ideal. But it does not follow that thisldeal is best achieved by reducing one science to another with an ad'mittedly comprehensive and powerful theory, if the secondary scienceat that stage of its development is not prepared to operate efiectivelywith this theory.

Much controversy over the interrelations of the special sciences, andover the limits of the explanatory power of their theories, neglects theseelementary considerations, The irreducibility of one science to anothel(for exan-rple, of biology to physics) is sometimes asserted absolutely,and without temporal qualiffcations. In any event, arguments for suchclaims often appear to forget that the sciences have a history, and thatthe reducibility (or irreducibility) of one science to another is eon-tingent upon the speciffc theory employed by the latter discipline at somestated time. On the other hand, converse claims maintaining that someparticular science is reducible to a favored discipline also do not alwaysgive sufficient heed to the fact that the sciences under considerationmust be at appropriately mature levels of development if the reductionis to be of scientiffc importance. Such claims and counterclaims are per-haps most charitably construed as debates over what is the most prom-ising direction systematic research should take at some given stage ofa science. Thus biologists who insist on the 'autonomy" of their scienceand who relect in foto so-called "mechanistic theories" of biological phe-nomena sometimes appear to adopt these positions because they believethat in the present state of physical and biological theory biology standsto gain more by carrying on its investigations in terms of distinctivelybiological categories than by abandoning them in favor of modes ofanalysis typical of modern physics. Analogously, mechanists in biologycan often be understood as recommending the reduction of biology tophysics, because in their view biological problems can now be handledmore efiectively lvithin the framework of current physical theories thanwith the help of any purely biological ones. As we shall see in the fol-lowing chapter, however, this is not the way that the issues are usuallystated by those taking sides in such debates. On the contrary, Iargelybecause of a failure to note that claims concerning the reducibility or

164 Tho Scructuro of Scloncoireducibility of a science must be temporally qualiffed, questions thatrt bottom relate to the strategy of research, or to the lof'ical relationsbetween sciences as constituted at a certain time, are commonly dis-:ussed as if they were about some ultimate and immutable structure ofhe universe

8. Throughout the present discussion stress has been praced on con-:eiving the reduction of one science to another as the deduction of oneret of empirically conffrmable statenxerrts from another such set. How-lver, the issues of reduction are frequently discussed on the suppositionhat reduction is the derivation of the praoperties of one subieci matterrom the properties of another. Thus a contemporary writer maintainshat-psychology is demonstrably an autono-orJ discipline with respecto physics and physiology, because "a headache is not an arrangementrr reaffangement of particles in one's cranium,- and "our sensation of'iolet rs not a change in the optic nerve.'Accordingly, though the minds said to be *connected mysteriously' with the physical piocesses, ..itannot be reduced to those processes, nor can ii be expriined by therws_of tlose processes." ? Another recent w-riter, in pres6nting the caseor the occurrence of *genuine novelties" in inorganlc natur{ declareshat it is an error to assume that all

.the properties of a compound can,e deduced solely from the nature of its elements." rn a similar vein, ahird contemporary author asserts that the characteristic behavior ofchemical compound, such as water, "could not, even in theory, be de-

,uced from the rnost complete knowledge of the behavior of its com-onents, taken separately or in other combinations, and of their proper-ies and arrangements in this whole." 8 we must now briefly i-ndicaterat the conception of reduction as the deduction of propirtles fromther properties is potentially misleading and generates spurious prob-)ms._The conception is misleading because it suggests that the questionf whether one science is reducible to another isto be settled by inspect-rg the *properties" or alleged 'hatures" of things rather than b| in-estigating the logical consequences of certain explicitly formuiatedrcorie_s (that is, systems of statements). For the conception ignores the:ucial point that the onatureso of things, and in particular oI th" '"1"-rentary constituents" of things, are not accessible to direct inspectionrd that we cannot read ofi by simple inspection what it is they- do oro not imply. such 'hatures' must be stated as a theory and are not thebjects of observation; and the range of the possibre 'hatures' whichtBrqrd Blanshard, "Fact,.Value^and Science,. in Science anil Man (ed. by RutI, Anshen), New York, L942, p. ZO3,8 C. D. Broad, The Mdnd, and lts pl.ace in Natureo London, Ig25, p. Sg.

fhe Rerluctlon of Tlrcorles 8SSchemlcal elements may possess ls as varled as tho difierent theories aloututomlc structures that wo can devise, Just as tho "fundamental nature"of electricity used to bo stated by Maxwell's equations, so tho funda-mental nature of molecules and atoms must bo stated as an explicitlyorticulated theory about them and their structures. Accordingly, thonupposition tha! ln order to reduce one science to another, some prop.erties must be deduced from certain other properties or 'natures' eon.verts what is eminently a logical and empirical question into a hopelesslylrresolvable speculative one. For how can we discover the "essentialnatures' of the chemical elements (or of anything else) except by con.atructing theories which postulate deffnite characteristics for these ele.ments, and then controlling the theories in the usual way by confrontingconsequences deduced from the theories with the outcome of appropriatooxperiments? And how can we know in advance that no such theory canover be constructed which will permit the various laws of chemistry tobo derived systematically from it?

Accordingly, whether a given set of 'properties' or 'behavioral traits'of macroscopic objects can be explained by, or reduced to, the 'prop-erties" or 'tehavioral traits" of atoms and molecules is a lunction ofwhatever theory is adopted for specifying the 'natures" of these ele-ments. The deduction of the "properties" studied by one science frornthe'properties" studied by another may be impossible if the latter sci-ence posfulates these properties in terms of one theory, but the reduc.tion may be quite feasible if a different set of theoretical postulates isadopted. For example, the deducUon of the laws of chemistry (e.g., of thelaw that under certain conditions hydrogen and oxygen combine to forma stable compound commonly known as water, which in turn exhibits cer-tain deffnite modes of behavior in the presence of other substances) fromthe physical theories of the atom accepted fffty years ago was rightly heldto be impossible. But what was impossible relative to one theory need notbe impossible relative to another physical theory. The reduction of vari.ous parts of chemistry to the quantum theory of atomic structure nowappears to be making steady if slow headway; and only the stupendousmathematical difficulties involved in making the relevant deductionsfrom the quantum theoretical assumptions seem to stand in the way ofcarrying the task much further along. Again, to repeat in the presentcontext a point already made in another, if the 'nafure' of moleculesis stipulated in terms of the theoretical primitives of classical statisticalmechanics, the reduction of thermodynamics is possible only if an addi-tional postulate is introduced that connects temperature and kineticenergy. However, the impossibility of the reduction without such specialhypothesis follows from purely formal considerations, and not fromsome alleged ontological hiatustetween the mechanical and the thermo-

866 The Structure of Sclencedynamical. Laplace was thus demonstrably in error when he believedthat a Divine Intelligence could foretell the future in every detail, giventhe instantaneous positions and momenta of all material particles aswell as the magnitudes and directions of the forces acting between them.At any rate, Laplace was in error if his Divine Intelligence is assumedto draw inferences in accordance with the canons of logic, and is there-fore assumed to be incapable of the blunder of asserting a statement asa conclusion of an inference when the statement contains terms not oc-curring in the premises.

However this may be, the reduction of one science to a second-e.g.,thermodynamics to statistical mechanies, or chemistry to contemporaryphysical theory-does not wipe out or transform into something insub-stantial or *merely apparent" the distinctions and types of behavior whichthe secondary discipline recognizes. Thus, if and when the detailedphysical, chemical, and physiological conditions for the occurrence ofheadaches are ascertained, headaches will not thereby be shown to beillusory. On the contrary, if in consequence of such discoveries a portionof psychology will be reduced to another science or to a combinationof other sciences, all that will have happened is that an explanation willhave been found for the occurrence of headaches. But the explanationthat will thus become available will be of essentially the same sort asthose obtainable in other areas of positive science. It urill not establisha logically necessary connection between the occurrence of headachesand the occurrence of certain events or processes specifted by physics,chemistry, and physiology. Nor will it consist in establishing the syn-onymy of the term treadache'with some expression deffned by meansof the theoretical primitives of these disciplines. It will consist in statingthe conditions, formulated by means of these primitives, under which,and as a matter of sheer contingent faet, a determinate psychologicalphenomenon takes place.

ry. The Doctrine of EmergenceThe analysis of reduction is intimately relevant to a number of cur-

rently debated theses in general philosophy, especially the doctrineknown as 'emergent evolution" or "holism." Indeed, some results ofthat analysis have already been applied in the preceding section of thischapter to some of the issues raised by the doctrine of emergence. Weshall now exarnine this doctrine more explicitly, in the Iight suppliedby the discussion of reduction.

The doctrine of emergence is sometimes formulated as a thesis aboutthe hierarchical organization of things and processes, and the conse-quent occurrence of properties at 'tigher" levels of organization which

The Recluction ol Theorles 887nro not prcclictable from properties found at "lower' levels. On the otherIrrnd, the doctrine is sometimes stated as part of an evolutiorrary cos'rnogony, according to which the simpler properties and forms of or-gunizotion already in existence make contributions to the 'creative ad-vnnce" of nature by giving birth to more complex and *irreduciblynovel" traits and structures. In one of its forms, at any rate, emergentovolu[ion is the thesis that the present variety of things in the universels the outcome of a progressive development from a primitive stage ofthe cosmos containing gnly undifferentiated and isolated elements (suchas electrons, protons, and the like), and that the future will continueto bring forth unpredictable novelties. This evolutionary version of thecmergence doctrine is not entailed by the conception of emergence asIrreducible hierarchical organization, and the two forms of the doctrinernust be distinguished. We shall ffrst consider emergence as a thesisabout the nonpredictability of certain characteristics of things, and sub.sequently examine briefly emergence as a temporal, cosmogonic process.

l. Although emergence has been invoked as an explanatory categorymost frequently in connection with social, psychological, and biologicalphenomena, the notion can be formulated in a general way so as toapply to the inorganic as well. Thus, let O be some object that is con-stituted out of certain elements a1, . . . , a, standing to each other insome complex relation R; and suppose that O possesses a deffnite classof properties P, while the elements of O possess properties belongingto the classes At . . . , An respectively. Although the elements are nu-merically distinct, they may not all be distinct in kind; moreover, theymay enter into relations with one another (or with other elements notparts of O) that are different from R, to form complex wholes difierentfrom O. However, the occurrence of the elements d1, . . . , an in therelation R is by hypothesis the necessary and sufficient condition for theoccurrence of O characterized by the properties P.

Let us next assume what proponents of the doctrine of emergence call'complete knowledge" concerning the elements of O: we know all theproperties the elements possess when they exist 'in isolation' from oneanother; and we also know all the properties exhibited by complexesother than O that are formed when some or all of these elements standto each other ( or to additional elements ) in relations other than R, aswell as all the properties of the elements in these complexes. Accordingto the doctrine of emergence, two cases must be distinguished. In theffrst case, it is possible to predict (that is, deduce) from such completeknowledge that, if the elementS 01, . , . , a* oc,cur in the relation R,then the object O will be formed and will possess the properties P. Inthe second case, there is at least one property P, in the class P such tha!

868 The Structure of Sciencedespite complete knowledge of the elements, it is impossible to predictfrom this knowledge that, if the elements stand to each other in relationR, then an object O possessing P, will be formed. In the latter case, theobject O is an 'emergent object" and P, an 'emergent property" of O,

It is this form of the emergence doctrine that underlies the passagefrom Broad cited in the preceding section of this chapter (page 864).Broad illustrates this version of emergence as follows:

Oxygen has certain properties and Hydrogen has certain other properties.They combine to form water, and the proportions in which they do this isfixed. Nothing that we know about Oxygen itself or in its combination withanything but Hydrogen could give us the least reason to suppose that itcould combine with Hydrogen at all. Nothing that we know about Hydro-gen by itself or in its combination with anything but Oxygen, could giveus the least reason to expect that it would combine with Orygen at allAnd most of the chemical and physical properties of water have no knownconnexion, either quantitative or qualitative, with those of Orygen andHydrogen, Here we have a clear instance where, so far as we can tell, theproperties of a whole composed of two constituents could not have beenpredicted from a knowledge of those properties taken separately, or fromthis combirred with a knowledge of the properties of other wholes whichcontain these constituents.e

There are several issues raised by the present version of the doctrineof emergencg though most of them have already been touched upon inthe preceding discussion of reduction and can be settled on the basis ofconsiderations which were introduced there.

a. The supposition underlying the notion of emergence is that, al-though it is possible in some cases to deduce the properties of a wholefrom the properties of its constituents, in other cases it is not possibleto do so. We have seen, however, that both the affrrmative and negativeparts of this claim rest upon incomplete and misleading formulations ofthe actual facts. It is indeed impossible to deduce the properties of water(such as viscosity or translucency) from the properties of hydrogenalone (such as that it is in a gaseous state under certain conditions ofpressure and temperature) or of oxygen alone, or of other compoundscontaining these elements as constituents (such as that hydrofluoric aciddissolves glass ). But frequent claims to the contrary notwithstanding, it isalso impossible to deduce the behavior of a elock merely from the prop-erties and organization of its constituent parts. However, the deductionis impossible for the same reasons in both cases. It is nol properfieg butstatements (or propositions) which can bo dcduced. Morcovcr, state-

o lbd., pp. 02-88.

The Reduction of Theories 369ments about properties of complex wholes can be deduced from state-ments about their constituents only if the premises contain a suitabletheory concerning these constituents-one which makes it possible to ana-lyze the behavior of such wholes as "resultants" of the assumed behav-iors of the constituents. Accordingly, all descriptive expressions occur-ring in a statement that is allegedly deducible from the theory must alsooccur among the expressions used to formulate the theory or the assump-tions adjoined to the theory when it is applied to specialized circum-stances, Thus a statement like 'Water is translucent' cannot indeed bededuced from any set of statements about hydrogen and oxygen whichdo not contain the expres