Alan writes:
I begin with an apparently minor point: as Gil points out (4B) =
reflexivity can
be deduced from symmetry and transitivity. (proof: suppose aRb, then bRa =
by
symmetry, hence aRa by transitivity). Steve makes the same point.
Only one conclusion follows from the above result, namely, we can reduce =
the
axiom set by one axiom.
<end quote>
Reflexivity doesn't actually follow from symmetry and transitivity in =
general, because it demands that for all x, xRx, whereas symmetry and =
transitivity together imply only that if there exists y different from =
x, such that xRy, then xRx. But in the case of exchangeability via =
money, a reflexivity axiom is not needed anyway, because there is a =
universal equivalent.
Here's an axiom set for exchangeability via money.=20
First we start with the set of alienable products of labour. We assume =
that there is an equivalence relation defined on them that groups them =
into commodities of particular definite kinds. This relation is required =
before exchange can take place. When we speak of e.g. an apple having a =
definite price, this price applies to each the many actual physical =
apples by virtue of their inclusion in an equivalence class. When we =
speak of "10 apples" being exchangeable for something, we mean that any =
set of 10 of the apples which are members of the "apple" equivalence =
class can be exchanged. Apples are equivalent to each other with respect =
to this relation and unequal to oranges.
0. Definition: commodity bundle kx is the set containing k alienable =
useful products of human labour, each of which is a member of the =
equivalence class denoted commodity x.
Now we have exchangeable things, we need two relations, direct =
exchangeability D, and exchangeability via money R. D is defined on the =
set of commodity bundles; R is defined on the same set excluding money. =
These axioms I've more or less taken from Chapter 1, except that I've =
used the more general D-composition instead of Marx's linearity (because =
it implies linearity and allows us to deal with bundles of different =
commodities). Axioms 0 to 3 come from the elementary form of value, 4 =
and 5 come from the general form.
1. D-symmetry: xDy =3D> yDx=20
2. D-composition: xDy, zDa =3D> (xUz)D(yUa)
3. D-definiteness: xDy, (xUa)Dy =3D> a =3D {}
4. Universal equivalent: for any non-money commodity x, there exists a =
positive integer k such that xDkm where m is the unit of money
5. Definition of R: xRy <=3D> xDkm, kmDy
These axioms allow us to deduce R-symmetry, R-transitivity and =
R-reflexivity. So R is an equivalence.
R-reflexivity: for all x, xDkm (axiom 4) =3D> kmDx (axiom 1), =3D> xRx =
(axiom 5)
R-symmetry: xRy =3D> xDkm, kmDy (axiom 5) =3D> yDkm, kmDx (axiom 1) =
=3D> yRx (axiom 5)
R-transitivity: xRy, yRz =3D> xDkm, kmDy, yDjm, jmDz (axiom 5) =3D> jmDy =
(axiom 1) =3D> j=3Dk (axiom 3) =3D> kmDz =3D> xRz (axiom 5)
Other consequences include R-composition: xRy, zRa =3D> (xUz)R(yUa) and =
hence R-linearity: xRy =3D> kxRky where k is any positive integer. =
R-composition: xRy, zRa =3D> xDkm, kmDy, zDjm, jmDa (axiom 5), =3D> =
(xUz)D(k+j)m, (k+j)mD(yUa) (axiom 2), =3D> (xUz)R(yUa) (axiom 5).
This composition property plus equivalence amounts to proof that R =
defines a positive integer quantity associated with each commodity.=20
This reproduces Marx's contention that his argument in Chapter 1 showed =
how values are "computed" in the exchange relationship, how the number =
which is fetishistically assigned to the commodity as an attribute like =
its weight or colour, is actually socially calculated.
But the analysis of commodity exchange he uses, and the axiom set above, =
make no reference to the actual quantitive prices of particular =
commodities at all, and hence they would be consistent with any price =
structure at all. Since we know that not just any price structure could =
arise in practice, it is shown that the exchange relationship computes =
(measures) a quantity that is not formed in exchange and hence must have =
been formed in production.=20
Brendan
P.S. Another consequence is universality, that any non-money commodity =
can be exchanged for any other. (For all x, y there exist positive =
integers j, k such that jxRky.) Axioms 3 and 4 together amount to the =
law of one price.
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