[OPE-L:21] [OPE-L:243] Re: Re: Chapter 1

Gil Skillman (gskillman@wesleyan.edu)
Wed, 28 Oct 1998 23:03:53 -0500

Brendan, Paul, Fred, and now Chris have all written posts asserting, in
various ways, that equivalence relations satisfying (something like)
reflexivity, symmetry, and transitivity constitute a sort of equality
relationship, thus seeming to refute Steve's and my claim that there is a
qualitative distinction between equivalence and equality that invalidates
Marx's argument in Ch.1 (or any similar argument).

Here, I think, is the crux of the issue. It is clearly the case that a
pre-ordering satisfying reflexivity, symmetry, and transitivity can be
interpreted as implying equality among elements in a given set, **but only
with respect to the dimension explicitly referred to in the definition of
the equivalence relation.** No necessary inference can be made about the
connection among elements in the set as measured along *any other*
dimension besides the one explicitly given.

Marx's argument, in contrast, or any argument seeking to establish a
common "substance" undergirding exchange values, asserts exactly what must
be *proven*, that equality with respect to a given dimension has
implications for equality in any other dimension, let alone the *specific*
dimension of socially necessary labor time.


1) Following Fred, a relationship of preference indifference can be
construed to mean that bundles in the indifference set yield equal marginal
utility. True, but this is simply a restatement of the definition of
indifference (on the assumption that a utility function can be invoked to
represent preferences, which requires the additional condition of
completeness). In *no other* sense need bundles along an indifference
curve be considered "equal". They may not include the same goods, the
included goods may not all be commodities, and indeed the included goods
may not all even be tangible.

2) Following Paul, alternative bundles Ax, By, and Cz may be deemed equal
in terms of the metric of exchange value, measured perhaps in some common
monetary unit. But the equality is *only* defined with respect to that
metric, and no implication about equality with respect to measures in
*other* dimensions (e.g., utility or socially necessary labor time) follows.

3) Following Marx, in Chapter 1, areas of different geometric shapes can
ofen be expressed in terms of basic underlying formulas. But this example
doesn't serve his point, because all of these formulas are expressible in a
common unit of measure--say, square meters--while he wants to assert that
equality of exchange values implies a connection among elements in a set
within some other distinct dimension--for instance, inferring claims about
the decibel level of two compact disks from a comparison of their surface

The passages from Russell and Tarski cited by Chris don't alter this
The "common property" they refer to is simply the one dictated by the
statement of the RST pre-ordering: e.g., indifference implies equal
marginal utilities; equal exchange value implies equal prices in monetary
terms. But neither implies equality along a dimension not referred to in
the original statement of the pre-ordering.


>Several posters deny that if one assumed exchange was ideally a net of
>relations characterised by relexivity, symmetry and transitivity anything
>interesting would follow. Russell and Tarski both seemed to think that
>equality followed. If anyone has a reference to more up to date literature
>on the logic of relations I would be interested to hear of it. There follow
>some key quotations.
>Russell *Principles of Methematics*
>p.166 "The principle of abstraction asserts that ,whenever a relation, of
>which there are instances, has the two properties of being symmetrial and
>transitive, then the relation in question is not primitive, but is
>analysable into sameness of relation to some other term.... Such relations
>... are always constituted by possession of a common property.... [a] third
>term to which both have one and the same relation."
>p.219 "Relations which are both symmetrical and transitive are formally of
>the nature of equality."
>p.220 repeats in other words the above on the principle of abstraction and
>adds: "Symmetrical transitive relations always spring from identity of
>Tarski *An Introduction to Modern Logic*
>sec 30 "Every relation which is at the same time relexivie symmetrical, and
>transitive is thought of as some kind of equality."
>{the above propositions seem strikingly similar to those employed by marx
>in section 1.}
> Chris Arthur