Hamminga, B., (1989a), "Sneed versus Nowak: An Illustration in Economics", Erkenntnis, Vol 30, nr. 1-2, pp. 247-265

hamminga, B., (1989a), "Sneed versus Nowak: An Illustration in Economics", Erkenntnis, Vol 30, nr. 1-2, pp. 247-265

This article deals with Nowak's reconstruction of Marx' price theory in volume III of Capital. It is claimed that Marx' analysis indeed leads to a series of idealizational statements in Nowak's sense, to which an analysis is added of the logical procedure by which Marx successively arrives at each of the idealizational statements. This procedure explains the structure found by Nowak. Nowak's analysis should thus be supplemented by a "structuralist" conciousness of Marx' theory core, and structuralists can learn from Nowak's results that the claims of Marx' theory are empirically vacuous, and hence cannot form Marx's prime purpose for constructing the theory.

It is the purpose of this article to discuss Nowak's method of describing the skeleton of economic theories and compare it with the Sneed Stegm�-Balzer type of reconstruction, known as the "structuralist" approach. I claim we learn about economics by doing so. For Nowak's approach, I refer to his bookTThe Structure of Idealization?(1980). His best known reconstruction is that of the law of value in volume III of Marx's "Capital", but he also devoted attention to volume I of Capital, to some modern theories of economic growth, and to econometrics.

In the framework of the structuralist approach, we have several reconstructions of neoclassical theories (by H"ndler (1980) Balzer (1982) (1985) Haslinger (1982) (1983) and Hands (1985), that were integrated by Kuipers and Janssen in this volume, and one by hamminga and Balzer (1986) that includes production). But we also have structuralistic reconstructions of parts of Marx capital, by Diederich and Fulda (1978), Diederich (1982) and Garcia de la Sienra (1982).
Finally, Sneed (1976) has already made thoughts about a reconstruction of his own method of reconstruction, and so did Nowak. ((1980), p. 111 etc. and in this volume).
So, what I could do, is make a Sneedian reconstruction of Nowak. Or a Nowakian reconstruction of Sneed. Or, I could even try to reconstruct them both, using an even more general theory of reconstruction. (Niiniluoto (1983)).
But, since we learn much better from examples than from the abstract comparison of metatheories, and since Marx's "Capital" is more or less common ground, I will try to make my observations in connection with Marx's treatment of the law of value, in volume III of Capital, which I consider to be a very fine and not really excentric example of the average theory structure you find in economics, though it has some peculiarities. I will not mix up theTreconstruction?with anTevaluation?of that theory. It is irrelevant whether or not the reconstructor believes the theory to be true, and it is not even relevant for this article whether or not Marx believed this himself.

Marx's fundamental law of value says that for any two types of commodities x and y:

i.e. price ratio of the commodities x and y, p(x)/p(y), equals the ratio of the labour times required in their respective production w(x)/w(y) (also called the ratio of theirTvalues).? Choosing p(y) and w(y) as units of account, the law reads: for all commodities x:

p(x) = w(x)

This version of the law says that the amount of x that exchanges for one unit of y equals the time necessary for the production of one unit of x (measured in time units equal to the time necessary for the production of one unit of y). This means that p(x) is determined by the composition of w(x). For the understanding of this composition Marx appeals to the reader's power of imagination: he hopes that all imaginable production processes (only some of which correspond to really existing production processes) can be represented in such a way that w(x) consists of three parts, as in the following illustration:


Thus, w(x) is defined by w(x): = c(x) + v(x) + m(x), reflecting the following intuitions:

    It takes time to produce x. You need raw materials. It usually has taken time to produce raw materials You have to pay the workers.

And so, the value w(x) of x consists of the indirect labour time c(x) of the raw materials that go into x, and the direct labour time v(x) + m(x) that workers need to produce x. Only part of this direct labour time, is needed to pay the workers' wages. That part is v(x) (so calledTvariable capital).? What is left is m(x), theTsurplus value?that you can at this stage, identify with profit. So, we know w(x): it is the sum of these three amounts of labour time.

As Nowak pointed out, and as is very clear in Capital, Marx never states that there actually exist or have existed economies satisfying:

where E is the set of all types of goods produced in such an economy. For Marx, the expression above is an ideal law that only holds for abstract, non existing economies satisfying the following conditions: )

C : All commodities of the same type require equal labour time.
C : All types of commodities need c (raw materials) and v (labour time equivalent of wages) in the same proportion ( c(x)/v(x) is the same for all x).
C : There is no fixed capital.
C : The turnover time of every capital is one ("period of analysis").
C : There is no merchant capital.
C : The purely financial sphere (banking) receives no surplus value (there is no interest payed).
C : There is no differential ground rent.
C : There is no absolute ground rent.
Nowak's reconstruction T of Marx's fundamental idealizational statement is written:

For purposes of exposition the E-clause shall be suppressed and we adopt the following notation and terminology:

where C is a shorthand for C and C and ... and C .
These 8 conditions are called idealizing conditions. The total expression T is called an idealizational statement. It is a material implication. The consequent of this implication is called an ideal law.

With respect to the set of ideal conditions, we can observe the following:

    They are all at variance with our image of the capitalist economy (it is a common sense belief that they are false and Marx is, of course, very well aware of this). There is a definite reason for the introduction of every one of them. Nowak does not formally reconstruct these reasons, though in analysing Marx' succession of modifications of the law "of value", Nowak uses these reasons in order to explain the exact form of the modified laws. That is, these reasons cannot be found by inspecting the idealizational statement. They come from "somewhere else".

So, now I am going to give two fully specified examples of these reasons, for the first two of the conditions, and an outline of what these reasons are for the other conditions.

C is necessary under competitive conditions on the commodity markets. Proof: if commodities of same type would not require equal labour time, then they would have to sell at different prices (if the fundamental law of value is to apply). But if there is a free market for these commodities, this market would equalize prices. Therefore, the fundamental law of value can only apply under competitive conditions if commodities of same type require equal labour time. Q.E.D.

The important thing to note is that the necessity of ideal condition C is proven from competitive conditions. And this is the same for C which is the condition that gave rise to the notorious transformation problem.

The condition C that all types of commodities need c and v in the same proportion is necessary under competitive conditions on the capital market and labour market Proof: 1) if capital freely competes, then the rate of profit p'(x) in the industry x should be the same for all x (differences in such profit rates would cause capital to move and to be reinvested in the most profitable industries). And the rate if profit is defined by

2) The rate of exploitation m'(x) = m(x)/v(x) must be the same for all industries x. If not, the worker's wages would differ over industries, which would cause them to move to the industry that pays the highest wages, until wage would again be equal everywhere. 3) If, however both p'(x) and m'(x) should not differ over industries, then the "organic composition of capital" c(x)/c(x) + v(x) cannot differ over industries, because

Therefore must be the same for all x, hence also Q.E.D.
C Introduction of fixed capital logically requires strong conditions concerning the distribution of fixed capital over the industries if profits are to remain equal everywhere, and if you want the products to be sold at their value. Relaxation of these strong conditions under maintenance of profit equality everywhere, logically leads to price deviations from values, indeed, completely according to Nowak's idea of concretization.
C Turnover time differences can be proven to have the same effects as disproportionalities in fixed capital, if profit rates are to remain equal everywhere.
C Merchant capital is improductive, according to Marx, and if merchants take surplus value from the industry, and if they do not do so in very nicely suiting proportions they can be proven to cause additional price deviations if profit rates are to remain equal everywhere.
C The same holds for bankers.
C and C If there is a supply monopoly of land, the free competition between land hiring capitalist-farmers and competitive conditions on the markets for commodities produced with the help of land can be proven jointly to cause different types of land-rent, which in turn can be proven to cause different types of price deviations.
Marx' strategy in Capital, as Nowak pointed out, consists of removing successively these idealizing conditions, at the same time introducing a modification of the ideal law such that the modified form is not inconsistent with competitive conditions in situations where the ideal condition does not hold.
The analysis of ideal conditions above draws attention to the fact that behind this Nowak-series of idealizational statements T , T , T ,..... there is a coherent and uniform theory-core K that contains the main assumptions from which each of the ideal conditions occuring in T is proven logically. The axioms in this theory core are that competitive conditions rule on the capital market, the labour market and on the markets for every type of commodity.
So, I disagree with Nowaks attempt, in this volume, to assign to deduction solely the purpose of drawing consequences from idealizational statements. I think deduction is the essence of economics and that the ideal conditions occurring in the idealizational statements, the structure and sequence of which is so marvelously reconstructed by Nowak, are in fact themselves deduced. Deduced from the law of value in conjunction with theory core K that is, from the assumption of labour time determining prices and of competitive conditions ruling everywhere. You can prove logically that this conjunction is only consistent if the ideal conditions hold.
The dropping of condition 1, condition 2, etc. forces Marx logically to modify something somewhere in this conjunction and he chooses to do so by modifying the determination of prices and profits, under maintenance of competitive conditions, everywhere.
Logically, Marx could have stuck to his labour time determined prices! Then he would have been forced logically to accept profit rate differences, or differences in the rate of exploitation, or differences in prices of commodities of equal type. Marx considered it to be his job to keep the conjunction (of the law of value with the assumption of competitive conditions ruling everywhere) consistent under dropping of his special conditions. In doing so, he keeps the competitive assumptions, from the deep to the surface. But if you look for prices purely determined by labour time, you find them only in the deep. That is: under all ideal conditions. Going upwards, additional forces come in and cause "deviations". So, I think, Marx considered competitive conditions to be more essential than labour time, for the formation of profits and prices. And he says so! In the pre-capitalist economics, he writes, the whole price determination by labour time was impossible because of the absence of competitive conditions. ) Competitive conditions form Marx' basic theory of the capitalist economy. They make possible a system of exchange ratio's between commodities purely determined by labour time, though only under conditions that are never satisfied in practise. This is the logic of the problem situation in which Marx finds himself:

Read: from competitive conditions obtaining on all markets (K) and commodities being sold at their values it follows logically that C and C and ... and C . This expression is equivalent to

Read: under competitive conditions it can be proven logically that, if commodities sell at their values, then conditions C , C , ..., C must be satisfied.

Now, let us approach Nowak's arrangement of expressions more closely by introducing yet another equivalent report of this Marxian proof:

Read: under competitive conditions it can be proven logically that if at least one of the ideal conditions is not satisfied, then not all commodities will sell at their values. Let us call the proven material implication a competitive corrolary. Let us now compare this to Nowak's T (saying that if the ideal conditions hold, then the commodities will sell at their values):

As Nowak clearly pointed out, such an idealizational statement is not logically empty, but it has no testable content, since the testing conditions for the law (C to C ) are never satisfied in real capitalist societies, as everybody knows. The following truth table clarifies the difference:

Nowak, Marx, and probably the reader, too, all agree: all capitalist countries are examples of case IV of this truth table: the ideal conditions do not hold, and prices do not sell at their values. Nowak's interpretation of Marx's statements pertaining to the law of value is that Marx expresses the following statement: if the ideal conditions would actually hold somewhere, that is, if we would have a society that is on example of case I or II, then this society would be an example of case I (and not an example of case II). But the least we can say of Marx is that he actually proves, proves logically, that competitive conditions rule out case III; they imply the logical impossibility of a society where commodities sell at their values while at the same time the ideal conditions are not all satisfied. Such a case would logically falsify the competitive conditions, that is, Marx would conclude from the occurence of such a case that competitive conditions must have been violated. But not only Marx, because this is a purely logical affair. It is the affair of deduction. So, the least we can say is, that Marx deduced his ideal conditions, from competitive conditions.

And, one could say, he is not at all happy about having deduced these ideal conditions, because he had to get rid of them in order to construct a theory of value determining price under competitive conditions, that does not so obviously rule out all real capitalist economies. And this is what a large part of volume III of Capital is all about. His course of analysis can be reconstructed as consisting of the following 27 steps (to be explained below).

The whole exposition of Capital somehow starts with the idea (line 1 in the scheme above) that under the competitive conditions of a capitalist society, commodities sell at their values, but, right from the beginning, Marx concedes that this is not happening in reality. The rhetorical equipment used by Marx is the concept of value. In Capital, Marx uses the notion "law of value" ("Wertgesetz") in an ambiguous way. After having given only four pages of considerations on the first four pages of Capital, Marx concludes (Marx (1867), p. 54): "therefore the amount of socially necessary labour... determines the amount of value". This is called later on (Marx (1867), p. 202) "law of value": "according to the general law of value are, for instance, 10 Pounds of Yarn equivalent to 10 Pounds of cotton ... if the same labour time is necessary ...". This is, to my knowledge, the first time Marx uses the term "Wertgesetz". There is absolutely no mention of price, it concerns the relation between "value" and "socially necessary" labour time. This is repeated on Marx (1867), p. 337 where Marx speaks about "the law of the determination of value by labour time".

Is equating value to socially neccessary labour time meant as a law or as a definition of the concept of value? In my opinion Marx is unclear about this. Moreover I think this unclearness is deliberate, firstly because in handling many concepts in Capital, Marx reveals perfectly to understand the difference between definitions and statements. Secondly: if Marx would consider it to be a definition, then why call it a law? But if he considers it as a law, then he should have provided seperately a definition of value. Thirdly: Marx knows very well that the reader is tempted to identify "value" with "price" or "natural price" if confronted with statements like "the amount of socially necessary labour determines the amount of value". Marx ((1867), p. 557) mentions that the workers' money wage is often referred to as the value of labour ("Man spricht hier vom Wert der Arbeit"). (This, however, says Marx, is false! It is false because we now "know" that value is something different, that value is incorporated labour and that it does not make sense to say that labour has value. It is value).

Whatever Marx says about value, he carefully avoids, right from the beginning, to equate it unconditionally with price. "Amounts of value" have a "money expression" (Marx (1867) p. 116) ("Geldausdruck") which, already here, in volume I, is "more or less", dependent upon "given circumstances" (Marx, (1867), p. 117) that are not yet specified.

Yet, and obviously, the reader of Capital is (I presume: deliberately) put into the temptation to regard values of commodities as "first approximations" of their price, and, through all modifications in Volume III, the sum total of the prices of all commodities remains equal to the sum total of all values produced, and therefore the average price of all commodities remains equal to the average value of commodities. Nevertheless, individual values are not all equal to individual prices, unless C to C are satisfied, and this is what, in the second step, (L ) says. (L ) describes a mathematical state of affairs and therefore does not involve any hypothesis, be it on real or on abstract economies. The idealizational statement ) T , on the contrary, is an hypothesis, though it is one about an abstract economy, and hence cannot be tested. T cannot be deduced from L and the same holds for all transitions from (L ) to (T ). Marx deduces his ideal conditions but he does not at all deduce his idealizational statements (T ). This consistently performed "Marxian jump" from L to T has something puzzling about it, though it is a jump that many will, I think, be familiar with: if you find that one of your favourite conclusions does not hold if some conditions are violated you are inclined to maintain your conclusion, gnashing your teeth while conceding that the said conditions must be satisfied. In the truth table shown above you can see that the competitive corrollary allows case II as a logical possibility: in some capitalist economy the ideal conditions are satisfied but commodities do not sell at their values. Part 1 of the "Marxian jump" to the idealizational statement consists of excluding this case (the idealizational statement has a zero truth value for case II). Part 2 of the jump, if we stick to Nowak's reconstruction, consists of removing the zero for case III. But case III is a logical impossibility, given the competitive conclusions that forms Marx's very reason to introduce the ideal conditions. There may be reason here to consider replacing the material implication in Nowak's idealizational statement by material equivalence, which would remove part of the "Marxian jump", thereby retaining the zero that Marx logically discovered for case III (the material equivalence would yield a column [1001]). I will not consider this here and leave the "Marxian jump" as it is; a zero for case III "moves up" to case II of the truth table.

Let us get back to step 3) of the scheme above. After the jump, Nowak constructs a function f of value and other factors. This function should replace the simple expression w(x) in the law of value. The first, and main requirement for the new function is that it should not anymore involve C as a necessary condition. Secondly, f should be such that if C where satisfied, the value of f should be equal to w(x), that is, in that case the abstract law of value as expressed on the first line of the scheme above, should hold. These two requirements make the claims independent of C . Thirdly, f should be economiccally plausible. What is, and what is not plausible cannot be fully specified logically, because it depends upon the specific economic force introduced by dropping an ideal condition. In case of C , for instance, this involves the introduction of an average labour time required to produce x, which determines the value of x. Capitals producing x somewhat faster will sell x above individual value and yield surplus profit, capitals producing x slower will sell x below individual value and yield less than the average rate of profit. There is, however, one requirement of plausibility that Marx retains over all modifications: the sum of all prices q .f (w(x), ...) (where q is the amount of commodity x that is produced) should remain equal to the sum of all values q .w(x), and hence the sum of all profits should remain equal to the sum of all surplus value. ) That is: every modification f in the law of value involves a new way of redistributing the original value of commodities to arrive at prices different from these original values.

It can be seen in the scheme above that the procedure leading to f in step 4 is now iterated seven times, finally yielding the idealizational statement (T ) that does not anymore contain any of the ideal conditions C to C . The relationship between any f and f in Capital can be described generally as follows:

Requirement 1) says, in Nowak's terms, that the idealizational statement containing f should be a concretization of the idealizational statement containing f . Requirement 2) could properly be called a reducibility requirement. These two requirements guarantee logical independency of C . Requirement 3) makes sure the law of value remains true for the aggregate sum of value and price (and so for the average). In addition to these three requirements, every special modification has its own specific criteria of plausibility, for which, of course, a general logical description is not possible. This is reflected in requirement 4).

What would Nowak's achievement be if described in the language of the Sneed-Stegmuller-Balzer-type of structuralism? I wish to maintain that Nowak did not reconstruct Marx's theory in the structuralist sense of the word. He did, quite succesfully, reconstruct the sequence of claims that Marx obtains from his theory by deduction and jump. Idealizational statements should be looked upon as the claims of a theory. And so, structuralism could learn from Nowak.

    The claims of Marx's theory, Nowaks idealizational statements T , have no testable content, because they have counterfactual conditions. They are made so on purpose. There is not one claim, but a whole series of them. There are very clear logical relationships between the claims in such a series. Roughly, each next claim has less counterfacutal conditions then the previous one.

Let me illustrate the situation with the Venn diagrams below in Figure 1 and 2:


Figure 1: P (M ) The set of all sets of imaginary capitalist economics.

In figure 1, I depict P (M ), the power set of the set of all imaginary capitalist economies. For pointing at the crucial features, I do not even have to specify the sets and functions that are involved. By S(C) I denote the set for the elements of which the condition C holds. By way of exercize I analyze, in figure 1, the ideal conditions for their own sake. The smallest set, indicated by arrow 1, is the set of sets of imaginary capitalist economies for which all 8 ideal conditions hold. The set indicated by arrow 2 is the set of sets for which all ideal conditions hold except for C . The set indicated by arrow 3 is the set of sets for which all ideal conditions hold except for C and C , etc., until the set indicated by arrow 4. This is the set of sets of imaginay capitalist economics that satisfy C only.

Now, the set I of imaginary capitalist economies, the structure of which coincides with that of the real capitalist economies that occur in history, is in the peel outside even the largest set S(C ). And, everybody knows! This is not the kind of claim that Marx set out to establish.

The Venn-diagram in figure 2 analyses the sets that are defined by the idealizational statements.


Figure 2: P (M ) (Rearranged).

Now, the ideal conditions appear in the antecedent clause of a material implication. And this exactly reverses the relationships between the sets. You simply know that an idealizational statement is true for real capitalist economies because a material implication is true whenever the antecedent clause is false.

So, the fundamental, initial, idealizational statement yielding the set indicated by arrow 5 only rules out an outer peel of which everybody knows that I is not there. If you drop condition 1, you arrive at another set, indicated by arrow 6, and again, everybody in perfectly sure that I is in this set, too, because everybody knows that real economies do not only violate C , they also violate C to C , the antecedent clause therefore still is false, and the implication therefore still is true for real economies.

There is an inclusion relation between the sets indicated by arrow 5 and 6, because of the reduction relation between f and f , and similarly for every f and f . So now, all new sets are smaller then the old ones, they are included in the old ones, up to the last set, indicated by arrow 9, where all conditions C to C are removed.

Here, I must admit that these functions f can not as such be found in Capital, because they presuppose a process of sequential concretization:

Another type of concretization is isolated concretization:

This yields functions f' different from f because ideal conditions are reintroduced after it is demonstrated how they could be removed. Marx in fact practises a mixed type of concretization. His actual pathway is thus:


He does so, because in his verbal discourse it would be too cumbersome to retain all deviations caused by, say, merchant capital (C ) in the treatment of, say, differential ground rent (C ). But once the values are transformed into production prices (dropping C which leads to f ) these production prices have come to stay in the subsequent analysis.

Now what is the harvest of Marx's theory? Not that we now know that the set of real capitalist economies is in the set indicated by arrow 9 of figure 2. This is not what Marx wished to establish. If the set I would be outside the set indicated by arrow 9 then at least some ideal conditions would have to hold in reality, and we all know they are false for real economies. What does it mean to exclude that I is in the difference of the set indicated by 8 and the set indicated by 9? The final claim: would be false if there would be capitalist countries where competitive conditions rule and prices do not sell at the eighth sequential modification of values. But this also is uninteresting as a claim of the theory. Competitive conditions hold, says Marx for capitalist economies only approximately, therefore strictly,TK is false?, and we all know that: there never is completely pure competition on the capital market, the labour market, and the markets for all commodities.

So, Marx is finally prepared to throw away the ladder on which he climbed up from the deep. Also, this turns the eighth modification of the law of value into an untestable statement. Marx considers this final 8th form of the law to be established, not by testing but by the procedure of concretization itself.

A final word about the purpose of modifying the abstract law of value to such a form as is consistent with competitive conditions, that is, to the 8th form. These forms show, says Marx, following Ricardo, that value, though not the only determinant of prices, is yet a determinant, and that therefore the variations in value determine the variations in prices and profits, ceteris paribus. And it is the historical variation of value, that yields the tendency of the profit rate to fall, and allows Marx to predict the end of capitalism.

He does so with the help of trivially true claims that are attained with the help of a complex procedure of concretization. The justification for upholding the ideal law that is the consequent of the claim, is the procedure of concretization itself.

What can, concluding, be learned from studying Marx, Nowak and Sneed is that

    Nowaks series of idealizational statements should not make us forget that Marx has a coherent basic theory from which the ideal conditions are logically deduced. Though the modified forms of the law of value are not deduced from the basic theory, consistency of these laws with the basic theory is a necessary condition for a successful modification. Sneed-like claims can be constructed but are trivially true; false antecedent clauses make the law of value and all its modified forms untestable. The modified forms of the laws are considered -by Marx- to be established by the procedure of concretization itself.

In the final analysis, Marx's book, and for that matter, the book of any other theoretical economist should be looked upon as a chain of proofs. The questions considered by theoretical economists to be the crucial ones are always: what is it that the author wishes to prove, and from what conditions does he prove it, and, finally, what is the relationship between the successive proofs in the chain?


*)  The author gratefully acknowledges helpful suggestions of Wolfgang Balzer and other members of the audience of "Philosophy of Economics II".

    These conditions slightly differ from those in Nowak (1980), Ch. 1. It is not the purpose of the present paper to treat these differences. Marx (1894), p. 298. Nowak gives T to T subscript numbers in reversed order. The order in this article is chosen for convenience; this way, the "i modification" of the law of value involves function f in idealizational statement T . Exactly this metaeconomic requirement is dropped in later attempts to solve the so called transformation problem of values into prices with respect to unequal organic compositions of capital (cf. von Bortkiewicz (1906/7), (1907).


Balzer, W. (1982a) "Empirical Claims in Exchange Economics" in: Stegm�ller, W. et al. (1982) pp. 16-40.

Balzer, W. (1982b) "A Logical Reconstruction of Pure Exchange Economics" Erkenntnis 17 pp. 23-46.

Balzer, W. (1985) "The Proper Reconstruction of Exchange Economics" Erkenntnis 23 pp. 185-200.

v. Bortkiewicz, L. (1906/7) "Wertrechnung und Preisrechnung im Marxschen System", Teil I und II. Archiv f� Sozialwissenschaft und Sozialpolitik Band 23, pp. 1-50; Band 25 (1), pp. 10-51.

v. Bortkiewicz, L., (1907) "Zur berichtigung der grundlegenden theoretische Konstruction von Marx im dritten Band des 'Kapital'", Jahrb"cher f� Nationalekonomie und Statistik, Band 34, pp. 319-355.

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