The Reduction of Phenomenological to Kinetic Thermostatics University of Groningen The Reduction of Phenomenological to Kinetic Thermostatics Kuipers, Theo AF Published in: Philosophy of Science DOI: 10.1086/289037 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1982 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Kuipers, T. AF. (1982). The Reduction of Phenomenological to Kinetic Thermostatics. Philosophy of Science, 49(1), 107-119. https://doi.org/10.1086/289037 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). 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Kuipers Source: Philosophy of Science, Vol. 49, No. 1 (Mar., 1982), pp. 107-119 Published by: The University of Chicago Press on behalf of the Philosophy of Science Association Stable URL: https://www.jstor.org/stable/186883 Accessed: 30-10-2018 07:14 UTC 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. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms Philosophy of Science Association, The University of Chicago Press are collaborating with JSTOR to digitize, preserve and extend access to Philosophy of Science This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THE REDUCTION OF PHENOMENOLOGICAL TO KINETIC THERMOSTATICS* THEO A. F. KUIPERSt Department of Philosophy University of Groningen Standard accounts of the micro-reduction of phenomenological to kinetic thermostatics, based on the postulate relating empirical absolute temperature to mean kinetic energy (I = (3/2)kT), face two problems. The standard postulate also allows 'reduction' in the other direction and it can be criticized from the point of view that reduction postulates need to be ontological identities. This paper presents a detailed account of the reduction, based on the postulate that thermal equilibrium is ontologically identical to having equal mean kinetic energy. In particular, it is shown that this postulate enables reduction only in the appropriate direction, but leaves room for 'evidence transport' in the other. Moreover, it makes possible the derivation (explanation) of the standard pos- tulate, using the existential kinetic hypothesis and phenomenological laws with which it turns out to be laden. 1. Introduction. In this paper a detailed account will be given of the micro-reduction relation between basic phenomenological laws and rel- evant kinetic hypotheses concerning equilibrium states of gases. Leaving out all reference to asymptotic behavior the reduction goes, according to textbook expositions, roughly as follows. The phenomenological ideal gas law states that any mole of pure gas satisfies PV = RT (P:pressure, V:volume, R: the universal gas constant, T:empirical absolute tempera- ture). Mechanical considerations lead to the kinetic hypothesis PV = (2/3)Nu (N:Avogadro's number of molecules in one mole, u:mean kinetic translatory energy). By introducing the reduction postulate, here called the kinetic temperature relation KTR,t = (3/2) (R/N)T =df (3/2)kT (k:Boltzmann's constant), we are able to derive the ideal gas law from the kinetic hypothesis. Nagel's well-known account in The Structure of Science (1961, Ch. 11) differs from the above only in that he does not use the concept of mole. But this is of no consequence in the following two problems inherent to all standard accounts. A moment's reflection shows that we can reverse the standard argu- ment: from the ideal gas law and KTR we can derive the kinetic hypoth- esis. Hence the reduction seems to work in both directions. Intuitively, *Received October 1980; revised May 1981. tThe author would like to thank Theo Pots and Henk Zandvoort for their criticism. Philosophy of Science, 49 (1982) pp. 107-119. Copyright ? 1982 by the Philosophy of Science Association. 107 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THEO A. F. KUIPERS however, in case of micro-reduction we are inclined to expect, almost by definition, asymmetry: although the ideal gas law provides evidence for the kinetic hypothesis, the latter may not be derivable from the former and the proper reduction postulate. If this intuition is correct the standard bridge principle KTR must be too strong qua reduction postulate, al- though it may be valid as such. The second problem of the standard account needs some more intro- duction. In his analysis Nagel pays very much attention to the cognitive status of KTR and concludes that it may be conceived either as an em- pirical hypothesis or as a so-called correspondence principle, depending on our willingness to accept claims to independent evidence for the re- ducing theory, i.e., the kinetic theory. However, in the general literature on reduction in the last two decades the ontological status of reduction postulates has come to the fore. In particular, it has been argued fre- quently, and in our opinion correctly, that the core of any reduction is that the reduction postulates do not represent causal connections but on- tological identities. Robert L. Causey has elaborated this point of view in logical detail in a number of articles, culminating in his Unity of Sci- ence (1977). Although we will pay little attention to logical questions, the present study is in the spirit of Causey's work. A main problem of the identity view is of course how to distinguish causal connections from ontological identities. There seems to be little more than what appears at first sight to be a question-begging criterion: whereas causal connections ask for further causal explanation to bridge the gap, ontological identities do not because there is supposed to be no gap. Of course, this does not mean that a reduction postulate, if it indeed states an ontological identity, cannot be empirically supported. On the contrary, if there is independent evidence for the reducing theory as well as for the theory or laws to'be reduced, the total evidence supports the relevant reduction postulates, in which case they are (supported) empir- ical hypotheses. However, in cases of micro-reduction, independent evi- dence may not (yet) be available, in which case Nagel calls the reduction postulates correspondence principles. But even in this case they are not simply correspondence principles: if the reducing theory is interpreted realistically they state ontological identities and not causal connections. Hence, the crucial question is whether or not KTR can be considered as (representing) an ontological identity. The criterion mentioned above leads to the question "does KTR ask for further causal explanation?". Such a question, however, can only be answered in an acceptable way when we have found a satisfactory causal explanation, in which case we would say that KTR did indeed ask for further causal explanation and hence that KTR represents a causal connection. We hasten to add that we did not look for a causal explanation for KTR, 108 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THE REDUCTION OF THERMOSTATICS let alone find one, for its being a causal connection is too counterintuitive. Nonetheless, we had a great deal of intuitive resistance, for two reasons, to accepting that KTR would be an ontological identity. As was shown before (the first problem) KTR leads directly to the 'reverse reduction' and, hence, it must be laden with (be part of) the kinetic hypothesis. Moreover, KTR uses the notions of absolute temperature T and the gas constant R, and it is well-known that these notions are based, in some way or other, on some phenomenological laws, i.e., KTR is also laden with them. Now the obvious question arises whether 'ontological identity' is the only alternative to 'causal connection'. Or, to put it differently, a bridge principle might ask for further explanation, though no causal explanation. If this makes sense, we may call such a bridge principle a theoretical identity, i.e., an identity which can (only) be explained by appeal to one or more ontological identities and some laws from one or both sides. Hence, the particular question is whether KTR is a theoretical identity, i.e., whether it can be explained further in the described non-causal way. In this article, it will be shown that KTR can indeed be explained on the basis of 1) the existential hypothesis that there is at least one gas satisfying the kinetic hypothesis, 2) the phenomenological laws providing the appropriate existence and uniqueness condition for the introduction of absolute temperature and the gas constant, and 3) the reduction pos- tulate stating that thermal equilibrium between two states of gases is iden- tical to having equal mean kinetic energy. Moreover, it will be shown that this reduction postulate is sufficient for the reduction in the appro- priate direction, but insufficient for the other direction, although it trans- ports evidence in that direction. Finally, it will become clear from the exposition that the new reduction postulate is neither laden with the re- ducing hypotheses nor with the reduced laws. This suggests the claim that it indeed represents an ontological identity, for the quest for a further explanation seems absurd. Before we start the presentation some technical introductory remarks need to be made. The analysis will be given in terms of asymptotic be- havior of gases because this provides a more realistic and yet sufficient foundation for the present reduction. As Causey stresses rightly in his book (1977, p. 87) the case of re- duction under study also requires a reduction postulate with respect to pressure: the identification of phenomenological and kinetic pressure. Its status as ontological identity, and hence as not further explainable, will not be disputed. Usually it remains entirely implicit in textbook exposi- tions. In what follows we will restrict verbal comments, supposing familiarity with elementary thermostatics. Although there is no sharp distinction be- 109 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THEO A. F. KUIPERS tween kinetic theory and statistical mechanics the present analysis cer- tainly belongs to kinetic theory. Hence we will not enter into the statis- tical foundations and presuppositions underlying the primitive concepts of kinetic theory. In the final evaluation we will make some remarks about the relevance of the present analysis for the reduction of thermo- dynamics to statistical mechanics. Finally, predicates will be used to in- dicate the corresponding sets and vice versa. 2. Common Background. As common background we presuppose some macroscopic and molecular, but non-thermal and non-kinetic, no- tions: G :the set of all pure gases, Z(g) :the set of equilibrium states of an isolated mole of g in G, Z :the union of Z(g) for all g in G, V:Z-->IR :V(z) being the (measurable) volume 'of state z' The notion of equilibrium state is here considered as a primitive one, leaving aside the problem of how to determine that a gas is in an equi- librium state. Of course, moles of the same (pure) gas are assumed to behave in the same way, which justifies our speaking of g as if it were a (representative) mole. The concept of mole does not presuppose any thermal theory. To be precise, the concepts of mole and pure gas are based on some macro- scopic laws concerning masses of gasmixtures, some molecular hy- potheses concerning types of molecules and their relative molecular- masses and the trivial, but crucial, reduction postulates that a sample of gas is a sample of molecules and that the mass of an amount of gas is equal to the sum-total of the absolute masses of the molecules. In prin- ciple it is now possible to determine whether an amount of gas is pure, i.e., is of one type of molecule. A mole amount of pure gas is then defined as an amount with mass in grams equal to the relative mass of the molecules of this gas. The molecular hypotheses, if true, assure that a mole of any pure gas contains the same number of molecules: Avo- gadro's number N. Of course, determination of the value of N is another story, but we will not assume to know this value. 3. Phenomenological Thermostatics. The basic notion of any thermal theory is the empirical relation of thermal equilibrium between states. This relation happens to be an equivalence relation (zeroth law of ther- modynamics). The generated equivalence classes constitute: S :the set of thermal states. 110 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THE REDUCTION OF THERMOSTATICS Further there are supposed to be unique functions: P:Z--IR+ :P(z) being the (measurable) pressure of z, t:Z->S :t(z) being the (measurable, i.e., identifiable) thermal state of z. The following simplicity assumption will be helpful: for all g :Range (t [Z(g)) = S. Moreover, in the context of a particular g, z is supposed to range over Z(g) and the functions V, P, t, and those to be introduced later, are sup- posed to be restricted to Z(g). The following two notions remain implicit in textbook expositions, but they will turn out to be of fundamental importance. Def. 1.1 g in G is an Asymptotic Boyle Gas (ABG) iff there is a one- one function Bg:S -> IR+ such that P(z)V(z) - Bg(s) if V(z) -> oo and t(z) = s, 1.2 g and g' in ABG are Comparable iff Bg = Bg,. Now we are able to formulate the three empirical laws which appear to be, in conjunction, equivalent to the (asymptotic) ideal gas law. The first and the second provide the required existence and uniqueness con- dition for the introduction of the notions of (empirical) absolute temper- ature and the (universal) gas constant. Addition of the third law completes the final claim. E-FL some G are ABG (Existential-F-Law) C-FL all ABG are Comparable (Comparability-F-Law) U-FL all G are ABG (Universal-F-Law). By EC-, EU-, UC- and EUC-FL we indicate in an obvious way con- junctions of F-Laws. Note that the addition of 'E' to 'U' gives a universal law existential import. Def. 2 For arbitrary g in ABG 2.1 the (empirical) absolute temperature is the function T:S -- IR+ defined by T(s) = (Bg (s)/Bg (s3)). T(s3) with s3 and T(s3) conventional; current convention: S3 is the (thermal state of the) triplepoint of water and T(s3) is 273.16 ?Kel- vin, 2.2 the (universal) gas constant R is the quotient Bg (s3)/T(s3) (and hence RT (s) = Bg (s)). 1ll This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THEO A. F. KUIPERS 2.3 By natural extension we define T:Z -> IR+ by T(z) = T(t(z)) (implying T(z) = T(z') iff t(z) = t(z')). It is evident that E-FL provides the required existence condition for the definition of T and R and it is easy to check that C-FL provides in addition the uniqueness condition. To be precise, T is already unique on the basis of the law that all ABG are quasi-comparable, i.e., Bg and Bg, have con- stant proportion. However, this law, together with E-FL, implies C-FL if R is also required to be unique. In conclusion, T and R presuppose E- and C-FL, i.e., EC-FL. Supposing EC-FL we may now define Def. 3 g in ABG is an Asymptotic Ideal Gas (AIG) iff P(z)V(z) -- RT(z) if V(z) -- oo. Some reflection shows that EC-FL is now equivalent to the law: (i) some G are ABG and all ABG are AIG. If we use 'all G are ABG', i.e., U-FL, we can derive from (i) the law (ii) some G are AIG and all G are AIG. Using the fact that 'all AIG are ABG' and 'all ABG are G' are true by definition we see that (ii) implies not only (i), and hence EC-FL, but also U-FL. Therefore (ii) is equivalent to EUC-FL. But (ii) amounts precisely to the ideal gas law if that law is considered as a universal one, restricted to asymptotic behavior and with existential import. Hence, for the re- duction of the ideal gas law it suffices to reduce EUC-FL or, equivalently, its three component laws. At the end we will show that the solution of this reduction problem leads to the reduction of the ideal gas law if that law is extended (ideal- ized) to non-asymptotic behavior. Note, however, given the previous analysis, that this extension is in no way required for the notions of ab- solute temperature and the gas constant. Moreover, it will turn out that the reduction postulates to be introduced for the asymptotic case are suf- ficient for the extended, non-asymptotic, case. 4. Kinetic Thermostatics. The kinetic theory considers an equilibrium state of an isolated amount of gas as a statistically stable state, i.e., as a continuous sequence of momentary states for which the following local quantities are constant in time: the local (number-) density of molecules, and the local mean kinetic (translatory) energy in all directions. It is to be noted that the definition of these local quantities needs to be related to appropriate volume-elements in order to be able to be constant in time. Now it is possible to show that such a stable state is only possible, in 112 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THE REDUCTION OF THERMOSTATICS view of the mutual interactions and the interactions with the wall, if these interactions are elastic and if, in addition, the following invariance con- ditions are satisfied: the local mean kinetic energy is the same in all di- rections (isotropy), and the local total mean kinetic energy is everywhere the same (homogeneity). On the basis of these assumptions it now follows that there are unique functions for the states of moles of pure gas: u:Z -> IR+ :t(z) being the (total) mean kinetic energy, p:Z IR+ :p(z) being the kinetic pressure, i.e., the resultant force per surface-unit exerted by the molecules on the wall. Again a simplicity assumption will be helpful: for all g,g': Range (u I Z (g)) = Range (ti f Z(g')). By a well-known argument it can now be shown that, if the effects of mutual interactions are neglected, the kinetic pressure p(z) is equal to (2/3)(N/V)tu(z), where N is Avogadro's number. Of course, the more dilute the gas the more realistic is the neglect of mutual interactions. This suggests the following definition: Def. 4 g in G is an Asymptotic Perfect Gas (APG) iff p(z)V(z) -> (2/3)Nt(z) if V(z) -> oo. Although the crucial argument about asymptotic perfect behavior seems applicable to any gas if applicable to some, it turns out, in order to dis- cover the asymmetry of the reduction, to be of crucial importance to dis- tinguish the existential from the universal kinetic hypothesis about perfect behavior. E-KH Some G are APG (Existential K-Hypothesis) U-KH All G are APG (Universal-K-Hypothesis). By EU-KH we indicate the conjunction, i.e., the universal kinetic hy- pothesis with existential import. The kinetic theory leads also of course to other specific kinetic hy- potheses, e.g., the Maxwell velocity distribution. But our subject is the reduction of phenomenological laws and hence we restrict our attention to reducing kinetic hypotheses, i.e., hypotheses which play a role in that reduction. 5. The Core of the Reduction. In the introduction we announced the following reduction postulates: RP1 P(z) = p(z): pressure is identified with kinetic pressure, i.e., 113 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THEO A. F. KUIPERS the pressure we measure is, ontologically speaking, the kinetic pressure, RP2 t(z) = t(z') iff t(z) = u(z'): thermal equilibrium is identified with having equal mean kinetic energy, i.e., that is what is the case, ontologically speaking, at thermal equilibrium. By RP we will indicate the conjunction of RP1 and RP2. Note that RP1 implies its structural analogue to RP2: P(z) = P(z') iff p(z) = p(z'), and hence that both also relate states of different gases. Note also that RP2 suggests directly the definition of the unique one-one function of thermal energy of a thermal state: u:S -- IR+ :u(s) = a(z) for arbitrary z with t(z) = s. The basic theorem for the reduction is: Th. 1 RP implies: all APG are Comparable ABG. Proof: Suppose g is an APG; with RP1 we get that P(z)V(z) -> (2/3)Na(z) if V(z) -- oo. The condition t(z) = s in the definition of ABG corresponds, according to RP2, with a(z) is constant, viz. u(s). Hence Bg(s) =df (2/3)Nu(s) provides the one-one function guaranteeing that g is an ABG. Because this Bg does not depend on g, all APG are even mutually comparable ABG, q.e.d. It is important to note, and easy to check from this proof, that, assuming either RP1 or RP2, the other is not only sufficient to derive the basic claim, but also necessary. The following reduction theorems are direct consequences of Th. 1. Th.2.1 E-KH RP-implies E-FL, Th.2.2 U-KH RP-implies UC-FL. They state that the F-laws are derivable (reducible) from (to) the K-hy- potheses, using RP. The central reduction claim, viz., the reduction of the asymptotic ideal gas law, which was shown to be equivalent to EUC- FL, is obtained by conjunction: Th.2 EU-KH RP-implies EUC-FL. Similar to the proof of Th. 1 is the proof of the evidence transport theo- rem: Th.3 E-KH and UC-FL RP-imply U-KH (and then trivially EU-KH and EUC-FL) It states that UC-FL, i.e., all G are Comparable ABG, is sufficient evi- dence, assuming RP, for the claim that all G are APG (i.e., U-KH), provided there is at least one APG (i.e., E-KH). However, it is clearly 114 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THE REDUCTION OF THERMOSTATICS not possible to derive E-KH from the F-laws and RP. Hence, in com- bination with Th.2 we see that RP leaves an essential asymmetry between the F-laws and the K-hypotheses: the former can be reduced to the latter, but not vice versa. We conclude this section by indicating some points that could have been included in the analysis. To begin with, given the symbolic for- mulation of RP2, which uses t, it is trivial to claim that RP2 implies that thermal equilibrium is an equivalence relation (the zeroth law). From the verbal formulation, however, it is clear that we could have avoided t, in which case the zeroth law would have been a non-trivial consequence, enabling the introduction of S and t. The second point is that the relation on S, defined by u(s) > u(s'), corresponds to the empirical linear order relation 'higher thermal state than' on S. The latter is based on the empirical relation 'warmer than' on Z, the former on the relation 'higher mean kinetic energy than' on Z. Again we claim that these relations are ontologically identical. If we had included this identity as a reduction postulate it would have been possible to derive the additional F-law that all ABG are monotone, i.e., Bg(s) > Bg(s') iff s is in a higher thermal state than s'. 6. The Kinetic Temperature Relation (KTR). One way of formulating KTR, the bridge principle used in standard expositions, is the following: KTR RT(s) = (2/3)Nu(s) or, equivalently, u(s) = (3/2)kT(s), with k =df R/N, i.e., Boltzmann's constant. Note that KTR in this formulation not only presupposes EC-FL (because T and R occur in it) but also RP2, because the thermal energy function u (with argument s) occurs in it. A second formulation is RT(t(z)) = RT(z) = (2/3)Nu(z) which also presupposes EC-FL, but not RP2. Nonetheless, it implies RP2 because T(z) = T(z') iff t(z) = t(z'). This, however, is not the case for the first formulation. For this reason and because it may even be a more adequate interpretation of the standard expositions we will assume the first formulation from now on. It is not difficult to check that Th. 1, and hence the reduction theorem Th.2, can be proved more easily with KTR, although it is still necessary to use RP2. That is, if S(tandard) RP indicates the conjunction of RP1 and RP2 and KTR, we have Th.4 EU-KH SRP-implies EUC-FL. But, with SRP we can also prove the 'reverse reduction': 115 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THEO A. F. KUIPERS Th.5 EUC-FL SRP-implies EU-KH (where E-FLS RP-implies E- KH is crucial). For its proof KTR is not only useful but even essential, just as RP2 (and RP1). Hence we may conclude that KTR is responsible for this reverse reduction and that it is, in view of Th.2, not necessary for the reduction in the appropriate direction. From the foregoing, it may not be concluded that we dispute the va- lidity of KTR. On the contrary, we have Th.6 E-KH and EC-FL RP-imply KTR. The proof is again elementary. From that proof it is clear that the premises E-KH and C-FL are essential (note that E-FL is, according to Th.2.1, an RP-consequence of E-KH). Hence, our conclusion is straightforward: KTR can and must be explained by appeal to E-KH, in addition to EC- FL and RP. Therefore, KTR does not represent an ontological identity, and hence it is not adequate as a reduction postulate, but follows as an explainable bridge principle, to be called a theoretical identity, i.e., an identity laden with a reducing hypothesis (E-KH) and also with reduced laws (EC-FL). This leads to the following general picture of micro-reduction of macro- laws. The proper reduction postulates, being ontological identities, lead to asymmetric reduction. That is, they enable only the derivation of the macro-laws from the relevant part of the micro-theory, the so-called re- ducing micro-hypotheses, notwithstanding evidence transport in the other direction. However, it may be that the reduction postulates, combined with a reducing micro-hypothesis (and some macro-laws, but this may not be essential) generate additional (or stronger) bridge principles, called theoretical identities, which conceal the asymmetry. The conjunction of ontological and theoretical identities may enable the derivation of the re- ducing micro-hypotheses from the macro-laws, i.e., the 'reverse reduc- tion'. In this description 'the derivation ... from' may be replaced by 'the reduction ... to' and it will be clear how to amend it if there is a macro-theory on the macro-side. 7. The (Extended) Ideal Gas Law. We remarked already that Th.2 states in fact the reduction of the asymptotically interpreted ideal gas law. In view of the definition of an Asymptotic Ideal Gas (AIG, Def. 3) this theorem may be restated as: Th.7 EU-KH RP-implies 0 - G = ABG = AIG = APG, that is, if the kinetic theory is true, at least EU-KH, and if RP is correct, then, as a matter of theory, the notion AIG has the same extension as the 116 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THE REDUCTION OF THERMOSTATICS notion APG, viz.G, i.e., all gases. Hence, Th.7 provides the legitimation of the practice of equating asymptotic ideal and perfect behavior. In philosophical expositions the ideal gas law is usually stated without the restriction to asymptotic behavior, i.e., as a strong idealization in- deed. It will be shown, in an indirect way, that our reduction postulates are sufficient for the reduction of this extended ideal gas law. The following notions, laws and hypotheses are all straightforward generalizations of the asymptotic ones. Def. 5.1 g in G is a Boyle Gas (BG) iff there is a one-one function Bg:S -> IR+ such that P(z)V(z) = Bg(s) if t(z) = s, 2 g in G is an Ideal Gas (IG) iff P(z)V(z) = RT(z), 3 g in G is a Perfect Gas (PG) iff p(z)V(z) = (2/3) Nu(z). E-FL some G are BG, C-FL all BG are Comparable, U-FL all G are BG, E-KH some G are PG, U-KH all G are PG. Again it is easy to see that the law: some G areIG and all G are IG is not only a straightforward interpretation of the extended ideal gas law, with existential import, but also that it is equivalent to EUC-FL. Once we have agreed upon the localization of the asymmetry in the reduction and the presuppositions of KTR (according to Th.6:E-KH, EC- FL and RP) we may of course use SRP (i.e., RP and KTR) to legitimate extended 'reduction' claims. Analogous to Th.1,2 (or 4),3,5 and 7 we get: Th.8.1 SRP implies: all PG are Comparable BG, 2 EU-KH SRP-implies EUC-FL, i.e., the extended ideal gas law, 3 E-KH and UC-FL SRP-imply U-KH, 4 EUC-FL SRP-implies EU-KH, 5 EU-KH SRP-implies 0 + G = BG = IG = PG. It is to be noted, however, that the role of KTR in the respective proofs may be quite different. In Th.8.1 KTR is helpful, but not essential. It is 117 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THEO A. F. KUIPERS only essential in Th.8.4; to be precise, in view of the premise, it is neces- sary to use E-KH. In Th.8.2,8.3 and 8.5 KTR is already RP-implied by the premises. In particular, from the fact that EU-KH obviously implies EU-KH and Th.2 and 6 we may conclude that Th.8.2 may be strength- ened to Th.9 EU-KH RP-implies EUC-FL, and, hence, that RP is also sufficient for the reduction of the extended ideal gas law. 8. Concluding Remarks. The foregoing analysis may not be surprising for physicists. In particular, we guess, on the basis of the verbal com- ments in some textbooks, that many physicists already share the convic- tion that the identification of thermal equilibrium with equal mean kinetic energy is crucial for the reduction, notwithstanding the introduction of KTR as a postulate in the technical part of textbook presentations. Our analysis legitimates this conviction. But apart from the value of a detailed account of the present case of reduction for its own sake, it provides a non-trivial illustration of, and hence support for, the ontological-identity-view on (micro-) reduction, including some further articulations of that view. To summarize once more; specifically for micro-reduction: a. the reduction postulates are (or represent) ontological identities, and hence are not further explainable, b. they enable only the reduction in the appropriate direction, c. notwithstanding transport of evidence in the other direction, d. and the possibility of theoretical identities, laden with the micro- theory, enabling the 'reverse reduction'. Still outstanding is the problem of the reduction of (equilibrium) ther- modynamics to classical and quantum (equilibrium) statistical mechanics. In fact, the present research was started as a small preparatory study. We do not claim that our analysis is directly applicable to the big reduction problem. Nonetheless, a detailed analysis, in the spirit of the present one, may well show a similar structure. For, after all, the textbook bridge principles also allow here the reverse reduction, and it is difficult to ac- cept them as ontological identities. If we find a similar structure it will also become clear what the precise relation is with the present case. We may expect at least compatibility. At the moment it seems to make little sense to write down our further speculations on all these points. For, from the present paper, it will be clear that the process of trying to prove such speculations leads to many corrections, until the puzzle is solved. More- 118 This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms THE REDUCTION OF THERMOSTATICS 119 over, we do claim that the present analysis does not depend on the so- lution of the big reduction problem. REFERENCES Causey, Robert L. (1977), Unity of Science. Dordrecht: Reidel Publishing Company. Nagel, Ernest (1961), Structure of Science. New York: Harcourt Brace Jovanovich. This content downloaded from 129.125.148.19 on Tue, 30 Oct 2018 07:14:09 UTC All use subject to https://about.jstor.org/terms Contents image 1 image 2 image 3 image 4 image 5 image 6 image 7 image 8 image 9 image 10 image 11 image 12 image 13 Issue Table of Contents Philosophy of Science, Vol. 49, No. 1, Mar., 1982 Front Matter Reason, Reference, and the Quest for Knowledge [pp. 1 - 23] From the Descriptive to the Normative in Psychology and Logic [pp. 24 - 42] Counterfactual Definiteness and Local Causation [pp. 43 - 50] Probability and Determinism [pp. 51 - 66] Natural Kinds and Freaks of Nature [pp. 67 - 90] Split Brains and Atomic Persons [pp. 91 - 106] The Reduction of Phenomenological to Kinetic Thermostatics [pp. 107 - 119] Discussion The Failure to Be Rational [pp. 120 - 124] How Not to Reduce a Functional Psychology [pp. 125 - 137] Book Reviews untitled [pp. 138 - 140] untitled [pp. 140 - 142] untitled [pp. 142 - 144] untitled [pp. 144 - 146] untitled [pp. 146 - 147] untitled [pp. 147 - 149] untitled [pp. 149 - 150] Back Matter [pp. 151 - 157]