Volume One: Contributed Papers || The Apparent Inconsistency of Moulines' Treatment of Equilibrium Thermodynamics

The Apparent Inconsistency of Moulines' Treatment of Equilibrium ThermodynamicsAuthor(s): John H. HarrisSource: PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association,Vol. 1982, Volume One: Contributed Papers (1982), pp. 304-311Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/192675 .

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The Apparent Inconsistency of Moulines' Treatment of Eqcuilibrium Thermodynamics

John H. Harris

University of Otago

1. Introduction

One of the main advantages that has been claimed for the Sneedian "structuralist" approach is that with it one can actually represent real scientific theories. (I consider a "structuralist" approach to be one in which the objects of study are structures in the model- theoretic sense, with at least some such structures thought of as representing physical situations or processes.) Using just Suppes' structuralist approach, one can already give a reasonable representation of many features of scientific theories. Examples of such are the representations of classical particle mechanics (CPM) in McKinsey, Sugar & Suppes (1953), relativistic particle mechanics in Rubin & Suppes (1954), and rigid body mechanics in Adams (1959). Then with Sneed's (1971) modifications of Suppes' approach, in particular with Sneed's introduction of the notion of a constraint between structures, the representational strength of a structuralist approach seems to increase dramatically. Kuhn in his (1976, p.182) describes the pur- pose of Sneedian constraints as insuring that "the values assumed by theoretical functions in one application must, for example, be compatible with those assumed in others." He goes on to say the following. "Together with the correlated notion of applications, that of constraints constitutes what I take to be the central conceptual innova- tion of Sneed's formalism." (Kuhn 1976, p.182).

Though Sneed includes constraints with what he calls the mathematical formalism of a scientific theory, he never shows that they play a key role in the presentation or theoretical development of such formalism (as versus playing a key role in how the formalism is related to applications of the theory.) It is in the work of Moulines (1975), and so far mainly in this paper, that one finds convincing evidence that Sneedian constraints play an essential role even in the purely theoretical development of the mathematical formalism of at least some scientific theories, in particular, for simple equilibrium thermodynamics (S.E.T.). In ?2 I will attempt to show that Moulines' treat-

PSA 1982, Volume 1, pp. 304-311 Copyright Q 1982 by the Philosophy of Science Association



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ment of S.E.T., call it S.E.T.M, is either inconsistent or unintelli-

gible, with the problem being due to the way Moulines represents constraints. In ?3 I will ouline an alternative approach to representing constraints which is both formally intelligible and consistent.

2. The Apparent Inconsistency of S.E.T.m

S.E.T.m is rather technical, as it rightly should be. But the

problem with S.E.T.m is a very basic one; it is even an obvious one

once you see it. So fortunately we will not need to review and use very much of the complicated Sneedian formalism.

A Sneedian (representation of a) scientific theory is an ordered pair <K,I> , where K is a class called the core of the theory and supposedly represents the mathematical formalism of the theory, while I is roughly speaking the set of (representations of) successful applications of the formalism. But Moulines never works with I and just focuses on the core part K.

A Sneedian core K is a 5-tuple (M ,M ,r,M,C> where the nature

of the various components are described in any good treatment of the Sneedian approach. (For two of the latest cf. Stegmrtller (1979) or Niiniluoto (1979).) Moulines (1975, p. 122) -- from now on all page references will be to that paper - presents the Sneedian core

T ~ ~ T for S.E.T. as K = <ET,SS,r T,SET, C > (where the IT' is, I presume,

to be suggestive of 'thermodynamics'). The components SS, ET and SET are well-defined (apparently proper) classes, being the extensions of three earlier defined set-theoretic predicates: SS(x); ET(x) - x is an equilibrium thermodynamics system; SET(x) - x is a simple ET system (cf., Dl-D3, pp.111-112). The third component, rT, is quite superfluous,

T being definable in terms of ET and SS. The last component, e , the "Sneedian constraint class", is loosely defined on p.122 in terms of eight constraint conditions, represented by Cl-C8, which Moulines introduced earlier. (pp. 119-122)

Though the problem with S.E.T.m has to do with constraints, it T T has nothing to do with c for Moulines never makes use of e

In fact I don't see how anyone could and I wonder whether Sneed even thinks that anyone could make use of a constraint class (as versus specific constraints) when using (as versus just trying to abstractly characterize) the formalism of a specific scientific theory.

The problem with S.E.T.m also has nothing to do with Moulines'

choice of constraints. The ones he lists seem quite reasonable when he describes them verbally. The problem with S.E.T.m is caused by the

combination of the way Moulines formally represents constraints and the way he uses these representations.



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His representations of constraints are the rather formal set- theoretic formulas C1-C8 mentioned earlier. (We will soon list and consider two of these.) And he really does make use of these formulas. In particular he uses C1-C8 in giving rather formal looking proofs of two very significant results that a physicist or physical chemist would expect to hold in any good formulation of S.E.T., namely the positivity of thermodynamic temperature (Th.4, p.123) and Clausius' principle (Th.5, p.124). For example in his proof of Th.4 he says that step (6) is justified "by [steps] (2),(3),(5),C8"; that step (7) is justified "by (2),(5),c4', etc. Now in any proof the various steps are justified by axioms, previous steps and/or rules of inference. Since constraints don't seem to be rules of inference, they or at least their representations Cl-C8 must be treated as axioms, you would think.

There is a problem with Sneedians using C1-C8 as axioms: it seems to lead to S.E.T.m being inconsistent. And in such cases we don't even

need C1-C8 collectively to get an inconsistency. Any one of these formulas, if conjoined as an axiom with set theory seems to lead to an inconsistent theory.

The easiest way to get the idea of what seems to be wrong with adding any of the formulas Cl-C8 as axioms is to give an example not in

S.E.T.m but in a simpler mathematical theory where everyone is familiar

with the notions. For instance, rather than considering ET(x), let us instead consider the predicate 'x is a group'. In particular, say we define x is a group iff x = (G,O> for some non-empty set G and some binary operation c:G XG -> G satisfying the usual group axioms. Let . be the class of all groups. And instead of considering one of Moulines' constraints on ET-structures, let us consider a constraint on

9-structures, i.e., on groups. In particular, consider the constraint that group operations associated with groups of the type we want to consider should agree where their domains overlap. In the spirit of Moulines, such a constraint would be expressed by say

CG (A x,y,G,GO, )((G,-> E9 A <G,o') E9. A x E GnG /A y C GnG

e xoy = xc)'y

But the formula CG is incompatible with the formalism of group theory that we have introduced because we can find, within set theory, examples of groups not satisfying CG, hence in set theory we can prove

CG (Vx,y,G,G',Q') (<G,O> GG AG/,&'>eqA x G GnG' ye GAG

A xfy / xo'y

which contradicts CG. Thus conjoining the constraint CG with the theory of groups gives a formally inconsistent system.

The above way of generating an inconsistent theory is a rather glaring one. Let us now look at Moulines' work and see if he has indeed made a similar mistake. To set the stage we first need to introduce some of Moulines' special notation. He defines ET to be the class of all 8-tuples x = <Z,V,I,N,P,E,U,S> satisfying certain



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conditions such as 0 / E C Z, U:Z -*ER and S:Z -e R. And he defines SET-structures to be ET-structures satisfying certain additional conditions, what you might call the fundamental laws of S.E.T. (In the ET- and SET-structures we are interested in, the first (Z), sixth (E), seventh (U) and eighth (S) components will represent a continuous process, the set of equilibrium states of that process, the energy function, and the entropy function on the states that make up that process, respectively.) He (p. 118) also introduces a suggestive indexing notation such that, for example, Sz denotes the entropy func-

tion and Ez, the set of equilibrium states associated with process Z.

We now have sufficient notation to present two of Moulines' constraints exactly as he did. They are:

Cl A z,z,z'( z E AzA E Z' -e z E EV.)

c5 A z,z,z' ( z E z nz' - u Z(z) = uZ, (z) A SZ(Z) = S,(z)).

Before we consider how to interpret these formulas, let us note that Moulines' subscripting notation is actually unacceptable. After all, given two ET-structures with the same first components, there is no guarantee that they will have the same second through eighth components. But this is what his notation 'Ez ','Uz ','Sz', etc., presupposes.

In fact what would legitimize such notation are the very constraints Cl, c5 and some others which he hasn't given. To avoid confusion, we will use a better notation. Define eight functions Z,V,I,N,P,E,U,S on ET as follows. For any x E ET, let Z be the first component of x,..*

and S be the eighth component of x (itself a function from Z into

the reals). In this new notation Moulines' Cl and C5 become

Cil A z,x,y (z C E A Z C Z -*z C E x y y C5' A z, x,y(zezfnZ - UX(Z) = U(Z) A S(Z) = S(Z) ).

Now to discuss the intuitive constraints which the formulas Cl and C5 (or Cl' and C5') are supposed to represent. As Moulines puts it (p. 119) Cl is to represent the idea that "the property of being an equilibrium state is independent of the system to which the state belongs." And according to what he says (p. 120) we should think of c5 (or our c5') as representing - he says "being" - "...the identity constraint for energy and entropy analogous to the identity constraint for mass in mechanics stated by Sneed." So in words, what c5 (or C5') is meant to say is that the energy and entropy functions associated with two ET-structures of the type we want to consider should agree where their domains overlap. But his notation doesn't seem to cope with the "of the type we want to consider" aspect, which is the very aspect constraints are supposed to cope with.

Let us look at the formulas Cl' and Cs' (or Cl and c5). Now the very first thing a logician or even a mathematician is going to ask when he looks at the 'A z,x,y' part (or the 'A x,Z,Z' part) of the formulas is "What are the ranges of those quantified variables?" The only



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two classes of x's already defined by Moulines for which the Z , ,x ,

Ex, U and S appearing inside Cl' and C5' make sense are ET and SET.

So as a first guess one would assume that Moulines means x and y to range over one of these classes, take your pick. But it is then not

difficult to construct examples of ET-, even SET-structures disobeying

Cl' and C5', hence it is easy to prove within "SET-theory"

Cl V z,x,y ( z E E / z C Z ^ z ' E ) and X y y VZXy( Z (E Znz A (UX(Z) # Uy(Z)Sx(Z)S(z)))

where x and y are interpreted as ranging over ET or SET, whichever one you chose. Clearly Cl' and C5' are incompatible with Cl' and C5'

respectively, being the negations of Cl' and C5'; hence SET-theory conjoined with Cl' or C5' is inconsistent, hence S.E.T.m seems to be inconsistent, as I claimed.

One of the weak points in our argument was the need to guess at what Moulines would have meant the range of x and y to be in Cl' and C5' when we quantified over them in the form '(A z,x,y)'. We guessed it to

be ET or SET. Maybe he meant some other class. Let us rewrite Cl' and C5' to allow for this possibility. For any X C ET, consider

Cl' (Az)(A x,y C X)(z C Ex A Z C Z o z E Ey) ;

C5' (A z) (Ax,y C X) (z C zxnzy- UX (z) = UY (z) A SX (z) = s y(z)). Now for any class X that we might define, it is clear that using just set theory we will probably be able to prove Cl' , in which case there

is no point in stating it as a constraint, or prove the negation of Cli, in which case adding the constraint Cl' as an axiom would lead to x an inconsistent formalism; and similarly for C5'. (I say 'probably'

because there would be those rare examples of a defined X where Cl' is undecidable within any present standard version of set-theory.) X

So generalizing our previous argument about the inconsistency of

S.E.T.m to the case when Cl' or C5' are replaced by Cl' or C5' for any

definable X you want to choose doesn't improve the situation: if we gain anything by conjoining such constraint axioms, we gain an inconsistent system.

The other weak point in our argument, the one I imagine Sneedians would jump on considering their extreme anti-linguistic bent, is our assumption that Cl and C5 are axioms or could even be represented by axioms. But since Cl-C8 are used to justify various steps of some of Moulines' proofs, if they aren't axioms, then they must be (functioning as) rules of inference, at least according to the present day canons of what constitutes a proof. But if they are rules of inference, then they don't seem to be of any type studied so far by logicians, and apparently they wouldn't be finitary rules of inference. To put Moulines' treatment of S.E.T. on a firm foundation, do we really need



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to go to such extremes as leaving the realm of finitary proof theory? And Sneedians can't pull out that over-worked escape clause that they are using acceptable informal set theory, so everything is alright. Informal it may be, but it sure doesn't seem to be acceptable.

In summary, by the canons of what mathematicians and especially logicians accept these days, S.E.T.m seems to be either inconsistent or unintelligible.

Maybe Sneedians have a simple solution to the problems raised above. I don't know. But in the next section I will present a very simple non-Sneedian solution to these problems.

3. A Consistent Approach to Representing Constraints

Whether one prefers a "statement" or a "non-statement" view of scientific theories, one surely must admit that scientists do use the mathematical formalism of a scientific theory to carry out deductive style arguments, sloppy and full of hidden premises though they may be. It is very clear on Suppes' "non-statement structuralist" view how one should represent and think of such derivations. one defines certain set-theoretical predicates such as SET(x) and maybe the corresponding

classes of set-theoretic structures such as SET. One can then derive within set-theory (augmented only by various definitional axioms) various statements about such structures. Since such set-theoretical statements are analytically true, following as they do from just set theory, everything is still in keeping with a "non(-contingent)- statement" view.

When it comes to representing an actual scientific theory on a Sneedian "non(-contingent)-statement" view, one has a problem as we have seen: How do we represent the constraints-part of the mathematical formalism in a way that could actually be used in derivations carried out by scientists and philosophers of science interested in foundations? Moulines' approach to this problem in the case of S.E.T. seems to fail as we have shown in ?2. Another approach, following the method of representing scientific theories used in Harris (1978) or (1983), seems to solve this representational problem in a particularly simple way, as we will now see.

All those who favour a "non-statement" view, either of the type proposed by Suppes or Sneed, labour under what seems to me to be the needless constraint of requiring the total separation of the mathematical formalism of a scientific theory from any indication of its interpretation. Let us see how easy things become if one drops this constraint. In the case of representing S.E.T. via statements about -111-1 T SET-structures, why not think of there being a class, I , of "intended"

T SET-structures which S.E.T. is meant to be about. We can't define I just as we can't define Sneed's class I in a Sneedian theory <K,I>.

T In fact we might not even have a good heuristic idea of what is in I



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But that can't stop us from introducing 'I ' as a class symbol in the T

language of set-theory to represent I , as vague as it is, and pro- T T

posing some axioms about I - i.e., about the use of 'I ' - which will T

narrow down the classes which 'I ' could represent. One such axiom r T _1 - T would be I . SET (Thus I wouldn't be the same as Sneed's class I

since in the case of S.E.T. Sneed would require I C SS, or in the latest version I know of, I CP(SS) ,scf.,Stegmilller (1979, p.90); and SS-structures are 5-tuples whereas SET-structures are 8-tuples.) Other

axioms would be to the effect that structures in I are related by Moulines' constraints. For example, we could use Cl' and C5' with

T ~ ~ ~~~~~~~x x T

X = I , call them Cl" and C55'. It would be trivial to go through Moulines' paper and convert C2 to a corresponding C2", etc. And note: the Sneedian constraints are now being fruitfully represented by axioms. With just such a simple change as suggested above it seems that we could salvage all of Moulines' work.

Note how our method of showing S.E.T. inconsistent is no longer

available. We can still find ET-, even SET-structures x,y such that

(*) (z E BE A z E Z A Z B E ) for some z.

But we obviously can't show such structures are in I because we

haven't pinned I down that much. In fact Cl" claims that I doesn't contain any ET-structures for which (*) holds.

All of the above is compatible with a "non-statement" view if one T r T --- '

assumes that axioms about I such as I C SET , Cl" and C5" are T T

meaning postulates on I (or on the use of 'I '), hence are analytically true. In this case anything derived from just these axioms would also be analytically true, hence would represent a non- contingent statement. My suggestion also seems to be compatible with Niiniluoto's idea - see his (1979, p.16) - of modifying a Sneedian theory to a triple < K,I,J> where in the case of S.E.T. the set J

T of "theoretical applications of the core K" would be I . But my suggested approach to modifying S.E.T. is also compatible with a

statement view when some of the axioms on I are considered to be proper (contingent) axioms. In either case we definitely get a contingent theory when we also add some axiom to the effect that a particular Sneedian non-theoretical type structure - some particular SS-structure in the case of S.E.T. - can be extended to an T I -structure.



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References

Adams, E.W. (1959). "The Foundations of Rigid Body Mechanics and the Derivation of its Laws from Those of Particle Mechanics.n In The Axiomatic Method. Edited by L. Henkin, et al. Amsterdam: North Holland. Pages 250-265.

Harris, J.H. ( 1978) . nA Semantical Alternative to the Sneed- Stegmuller-Kuhn Conception of Scientific Theories.n In The Logic and EDistemologv of Scientific ChanRze (Acta Philosophica Fernica 30.) Edited by I. Niiniluoto and R. Tuomela. Amsterdam: North Holland. Pages 184-204.

?-.------. (1983). "Existential Claims in Physical Theories.n Forthcoming in Boston Studies in the Philoso2hy of Science.

Kuhn, T.S. (1976). "Theory-Change as Structure Change: Comments on the Sneed Formalism." Erkenntnis 10: 179-199.

McKinsey, J.C.C., Sugar, A.C., and Suppes, P. (1953). "Axiomatic Foundations of Classical Particle Mechanics.n Journal of Rational Mechanics and Analysis 2: 253-272.

Moulines, C.U. (1975). "A Logical Reconstruction of Simple Equilibrium Thermodynamics.n Erkenntnis 9: 101-130.

Niiniluoto, I. (1979). "The Growth of Theories: Comments on the Structuralist Approach.n In Proceedings of the Second International Congress for History and Philosophy of Science. Edited by E. Agassi and J. Hintikka. Dordrecht: Reidel. Pages 3-46.

Rubin, H. and Suppes, P. (1954). "Transformations of Systems of Relativistic Particle Mechanics." Pacific Journal of Mathematics 4: 563-601.

Sneed, J. (1971). The Lozical Structure of Mathematical Physics. Dordrecht: Reidel.

Stegmiiller, W. (1979). The Structuralist View of Theories. Berlin: Springer-Verlag.



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Volume One: Contributed Papers || The Apparent Inconsistency of Moulines' Treatment of Equilibrium Thermodynamics