On the equivalence vs. equality question, Paul writes:
> Paul
> ----
> I am not sure that this gets us much further. Given a set S, an
> arbitary binary equivalence operator which we will write ~,and
> an element e in S, then the triple {S, ~, e} defines the set of
> elements in S equivalent to e under ~.
>
> Let us define S to be the set of commodity bundles, such that
> each element of the set is a vector of commodities. (This is a
> generalisation of Marx's argument where he deals only with
> quantities of individual commodities. Marx's quantities of
> individual commodities constitute the basis vectors of the
> commodity bundle space.) Let us further define ~ to be the
> operator meaning, exchanges for. Given a point in commodity
> bundle space, [ 1000kg flour, 800kg rice, 4000litre paraffin],
> the ~ defines all bundles of commodities that will exchange with
> this bundle.
>
> Since ~ is an equivalence relation, it induces a partitioning of
> the commodity bundle space.
>
> Gil says that two entities are equal if they are equivalent in
> all dimensions or characteristics alowed by the relevant axiom
> set.
>
> The question becomes what is the relevant axiom set. In
> discussing equivalences between commodities, properties such as
> weight are clearly irrelevant. The only thing that is relevant
> is what they will exchange for, in these terms the elements of
> the set induced by ~e are equal to one another. So the
> equivalence relation is an equality relation.
Paul's conclusion is at best problematic if "equality relation" is
understood in the same sense Marx uses it in Volume I, Chapter 1. To
see this, note that nothing in the logical structure of his
argument requires that the elements of set S be limited to
commodities, i.e., exchangeables which are products of labor; they
need only satisfy the relation ~, "exchanges for". So let's include
in Paul's example 1 acre of uncultivated land, on the realistic
ground that the latter can be exchanged. Then, following his
subsequent argument exactly, the relevant vector including
uncultivated land satisfies an "equality relation" with other vectors
of commodities, in Marx's sense of the term, and thus uncultivated
land, following his argument, must be a product of labor,
contradicting both reality and the starting assumption.
I emphasize that whether or not Marx intended to exclude
non-commodities from his argument is irrelevant to the foregoing.
Otherwise, the structure of his Chapter 1 argument must be understood
to begin with the postulate:
Assume that logically relevant counterexamples to a deductive
argument can be disposed of by fiat.
Paul continues:
> But Marx's argument goes beyond this, he says that the elements
> of the equivalence sets are equal because each element of the
> equivalence set belongs to it by virtue of a relationship they
> share to another space, a scalar dimension ( quantity of common
> substance) which he identifies with labour time.
No, he does not say this. Just the reverse: he says that commodities
enjoy this "relationship to another space, a scalar dimension
(quantity of common substance)..." *because* they are equated, which
is the relationship at issue. I quote:
" Let us now take two commodities, for example corn and iron.
Whatever their exchange relation may be, *it can always be
represented by an equation in which a given quantity of corn is
**equated** to some quantity of iron...What does this **equation
signify**? **It signifies** that a **common element of identical
magnitude** exists in two different things....of which they represent a
greater or a lesser quantity.* [I, p. 127 (Penguin); emphasis added.]
Thus Paul has Marx's argument exactly backwards.
He continues:
>
> There are two questions here:
>
> 1) Is it valid to deduce that the partitioning of commodity
> bundle space is associated with a distinct scalar dimension?
>
> 2) Is it empirically reasonable to conclude that this scalar
> dimension is that of labour time.
>
> I will argue that the answer to both questions is yes.
I will address these questions later. They are necessarily separate:
one could affirm both claims and still logically deny that
equivalence relationships established by exchange "express something equal",
in the sense used by Marx.
In solidarity, Gil Skillman