I am convinced by Gil Skillman's contention that if the formal properties of the exchange relation imply the conservation of a substance (and I think they do), then what is conserved in the actual practice of exchange is not social labour, since non-commodities like unimproved land have a price but no labour content.
He sees this as a refutation of Marx's approach, because in his view Marx's Chapter 1 restriction of exchange to commodities i.e. products of labour, is illegitimate, a tautology.
I don't think this argument follows however, since it seems to me that the elements of Marx's analysis are not the concrete acts of exchange that take place in capitalist society, but the social relations of exchange, "exchangeability", that is, exchange as a production relation.
The chain of Marx's argument seems to me to run in the opposite direction from the argument Gil wants to impute to him. Gil wants an analysis of concrete exchanges to lead via abstraction from troublesome details like arbitrage to a formal model of exchange, which leads in turn to a conserved substance, which in turn is shown to be abstract labour (of course Gil himself denies the feasibility of such an argument). But in Marx's argument social labour is primary and concrete exchanges are derived. Is this valid?
Let's look at this as a problem in historical materialism, a problem in the natural history of the human race, a population of apes inhabiting this very planet. Production is the life-activity of this population, and undergoes a historical evolution.
But in the social production of their life, humans enter into definite relations that are independent of their wills, relations of production which correspond to a definite state of development of their material productive forces. Production of commodities is one such relation.
Commodity production developed within pre-existing human societies and, becoming the dominant relation of production, enabled an unparalleled development of the division of labour. But in any form of society, the dominant relations of production face a performance requirement: they have to allocate the total labour resources approximately in proportion to social needs. This constitutes fitness in the selection process of the evolution of human social forms.
The historical fact that exchange has rapidly expanded the division of labour shows its relative success at this task, relative that is to other production relations. Commodity production packages up of the products of social labour in privately owned use-values that are then alienated, being exchanged for others. Unless those commodities are exchanged in proportions that correspond approximately to their labour content, exchange would not serve to distribute social labour among different branches of social production in proportion to social needs. Unless this approximation to labour content was a good one, commodity exchange would not have shown the unprecedented success it has shown, at developing the division of labour and productivity.
This is the background that saves Marx from having to "prove" the labour theory of value in chapter 1. His project is a different one, to analyse the core determinations of exchange production relations to show how they generate, "compute", this approximation to social labour contents of commodities in the prices that are formed in exchange. Then by adding more determinations to construct a more concrete analysis of exchange as a social relationship, he can show how the divergence of prices from values arises.
Marx's abstraction from e.g. arbitrage or the positive price of unimproved land in Chapter 1 is not a selection of a desired set of exchanges from among all real exchanges. It's a selection among some of the properties of the social relationship of exchange. Properties that determine exchange's ability to generate prices embodying abstract labour are expounded first in order, and the properties that modify this identity to an approximation are brought in later.
But if exchange considered as a production relation is not the set of all concrete exchanges, then what is it? Exchange the relation of production, exchange as a social relationship, exchangeability, is a nexus of socially constructed roles, at base the roles of exchanger (and later, with money), buyer and seller.
Let's look at the "forms of value" in Chapter 1. They amount to analyses of definite stages (or moments) of the roles of buyer and seller. In the first "Elementary or accidental" form of value, Marx introduces the qualitive fact of alienation of use-values. The exchanger is one who has unwanted use-values that can be alienated in return for wanted use-values owned by the other. This weak (and early) form of the social role of commodity exchanger already defines some relational properties of exchange, without recourse to generalisation from concrete exchanges. For example, the fact of exchange directly implies symmetry, and the fact of exchange ratios (arbitrary though they may be in this form) imposes linearity. "It therefore follows that the elementary value-form is also the primitive form under which a product of labour appears historically as a commodity, and that the gradual transformation of such products into commodities, proceeds pari passu with the development of the v!
alue-form".(p67).
The elementary form of value has "relative" and "equivalent" subforms that correspond to the two distinct roles in exchange, the one who alienates the unwanted use-value and the one who provides the wanted use-value.
The second, "Total or expanded" form of value corresponds to the generalisation of the first form. The exchange relationship broadens to the point that the exchanger sees their commodity as exchangeable with "numberless" other commodities in definite proportions. Here exchangeability of a particular use-value extends to a wide range (potentially all) of other use-values, and hence the given use-value appears as the embodiment of social labour. The value of their commodity (embodied form not in a scalar but in the vector of exchange ratios with other commodities) appears now to the exchanger as distinct from its use-value.
The "General" form of value corresponds to the stage of equating the value of all commodities through the ratios with which they exchange with a particular commodity, a universal equivalent. It's this stage where the social relation of exchange becomes able to enforce transitivity (more or less successfully), and hence calculate a consistent vector of exchange ratios. Here value begins to obey the all the 4 axioms of of RST and composition, and thereby becomes a definite positive scalar associated with each commodity.
The Money form corresponds to the specialisation of a particular commodity, money, in the role of universal equivalent. The universal exchangeability of money now becomes its primary use-value, and the social role of exchanger diverges into buyer and seller (the "relative" and "equivalent" subforms of the money form).
I won't go further with this interpretation here, it's pretty clear how it works, and how it extends through further specification to the roles of worker, capitalist and landowner.
How does this reading of Marx's theory relate to our project of formalising and developing conceptual models? To me it suggests that our formalisation should start from a formal specification of the social roles, as algorithms or repertoires. Our theory should start from the simulation (and analysis) of the results of the interaction of multiple interacting processes implementing definite (and simple) roles.
Simulations of populations of commodity exchangers with appropriate programs should be able to generate actual exchanges that obey e.g. the equivalence relation laws; in particular they could evolve definite prices. These "evolved" laws would not take the form of binding rules, disprovable by a single counter-example. They would be attractors in the space of possible sets of concrete exchanges computed by the simulacra.
For example, suppose that our simulated exchangers were simple commodity producers. The vector of values of their products would be an attractor within the space of price vectors that they actually did compute during simulation.
Since Marx never had a computer, the option of simulation was not available as a way of demonstrating the implications of the social roles he described via people's concrete behaviour to systemic properties. He had to use analysis. We can use both techniques and use one to inform the other.
To wind up, I can now return to Gil Skillman's accusation of tautology. Marx's argument runs from a description of a subset of the properties of the social relation of commodity exchange to the existence of commodity fetishism, the appearance of the social labour content of products as an "objective character stamped upon the product of that labour". That's the kind of "tautology" that deserves the name "theorem".
As for the relationship between actual price ratios and embedded labour, that quite validly appears at a later stage of the analysis, after the social relations (surrounding the commodity "labour-power") which govern the divergence between these vectors have been added to his incremental analysis of capitalist production relations.
Brendan