[OPE-L:7150] [OPE-L:653] Re: Re: Re: Equality and Equivalence: Response

Gil Skillman (gskillman@mail.wesleyan.edu)
Thu, 11 Mar 1999 13:54:48 -0500

In response to this passage from me,

>>(They go on to say (p. 28) that equality implies RST and it implies a
>>substitution property Z (such as Alan's Axiom 4). But they never assert
>>the converse, that RST plus Z imply equality, contrary to Alan's

Chris writes:

>My logic books do make this last inference. In Tarski it is called 'the
>principle of abstraction' ; what I cannot quite figure out is if this means
>we have here simply an abbreviated way of talking, or is it an inference to
>some substanital property.
>Chris Arthur

Mine don't make this inference, and to my reading Tarski's treatment of the
distinction between "equivalence" and "equality" is much more nuanced than
Chris suggests. Before reproducing and commenting on the passage that
Chris refers to, let me quote Tarski on the connection between 'identity'
and 'equality':

"Among the logical concepts not belonging to sentential calculus, the
concept of IDENTITY or EQUALITY is probably the one which has the greatest
importance. It occurs in phrases such as: 'x is identical with y,' 'x is
the same as y,' 'x equals y.' To all three of these expressions the same
meaning is ascribed; for the sake of brevity, they will be replaced by the
symbolic expression: 'x =y.'
[...] The general laws involving the above expressions constitute a part of
logic which may be called the THEORY OF IDENTITY.

"Among the logical laws concerning the concept of identity the most
fundamental is the following:

I. 'x = y if, and only if, x has every property which y has, and y has
every property which x has.'

We could also say more simply:

'x = y if, and only if, x and y have **every property** in common.'
[emphasis added]

"Law I was first stated by LEIBNIZ (although in somewhat different terms)
and hence may be called LEIBNIZ'S LAW." [Tarski, 1965, pp 54-55]

Gil's comments: I note how closely this parallel's Birkhoff and MacLane's
treatment, and why the interpretation of "equality" as "identity" in the
sense stated above is so important to this discussion. Simply put, it is
the only interpretation of "equality" which is demonstrably sufficient to
support Marx's inference that "[This equation] 1 quarter of corn = x cwt of
iron...signifies that a common element of identical magnitude exists in two
different things...", which gets him to the conclusion that abstract labor
is in some sense the basis of exchange value.

Tarski goes on to note that Leibniz's Law *implies* reflexivity, symmetry,
and transitivity [pp 56-7] (not vice-versa, note) and then comments on the
difficulties some people have with the notion of equality, for instance
taking exception to the claim that '3 = 2+1' because the symbol "3" is not
identical to the symbols "2+1"[pp 58-59]. He suggests using quotation
marks around the relevant symbols to get at the latter point, so that
although ' "3" does not equal "2+1",
still it's true that 3 = 2+1.

He then goes on to note the distinction between "equality" as used in
algebra and "congruence" as used in geometry, concluding "...in order to
avoid any confusion, **it would be recommendable consistently to avoid the
term 'equality' in all those cases where it is not a question of logical
identity [pp 62-3], and to speak of geometrically equal figures rather as
of congruent figures**" [p 63]

This is what I've been arguing. RST plus Alan's Axiom 4 establish a
relation of congruence, not equality in the above sense, and Marx's
argument requires something more than congruence.

Getting nearer to the passage that Chris refers to, Tarski focuses on
relations which satisfy the conditions of reflexivity, symmetry, and
transitivity (RST), writing:

"Every relation which is at the same time reflexive, symmetrical and
transitive is thought of as some kind of equality. Instead of saying,
therefore, that such a relation holds between two things, one can, in this
sense, also say that these things are equal **in such and such a respect**,
or--in a more precise mode of speech--that *certain properties** of things
are identical. Thus, instead of stating that two segments are congruent,
or two people equally old, or two words synonymous, it may just as well be
stated that the segments are equal in respect of their lenght, that the
people have the same age, or that the meanings of the words are identical."
[p 95]

Gil comments: Notice the point, which I've stated before: an equivalence
relation (i.e. one satisfying RST) can be thought of as establishing an
equality in **a particular dimension**, while Marx requires the inference
that equality or commonality is established **along a separate dimension**.

In any case, it is with respect to the preceding, **contingent** sense of
equality that Tarski invokes the principle of abstraction that Chris refers

"It can be shown with little difficulty that the same procedure is
applicable to any reflexive, symmetrical and transitive relation. There is
even a logical law, called the PRINCIPLE OF ABSTRACTION, that supplies a
general theoretical foundation for the procedure we have been
considering...."[p. 96]

Contrary to Chris's reading, this passage does not support the inference
that RST implies "equality." Indeed, Tarski goes on to reaffirm the
distinction I've been making all along:

" There is, so far, no universally accepted term denoting the totality of
relations which are at the same time reflexive, symmetrical, and
transitive. Sometimes they have generally been called EQUALITIES or
EQUIVALENCES. But the term 'equality' is also sometimes reserved for
particular relations of the category under consideration, and two things
are then called equal if such a relation holds between them. For instance,
in geometry...congruent segments are often referred to as equal segments.
**We will emphasize here once more that is preferable to avoid such
expressions altogether; their use merely leads to ambiguities, and it
violates a convention in accordance with which we consider the terms '
equality' and 'identity' as synonymous.**" [p. 96]

Thus I do not see Tarski as endorsing the claim that RST (plus some
substitution property) establish a relation of equality. To the contrary,
he makes the same point Steve and I have been making: these properties
establish an equivalence relation, which can be understood to establish an
equality along a particular dimension, but not across multiple dimensions,
as Marx's Chapter 1 argument requires.