[Inquiry] Re: Logic Of The Sciences -- Discussion

[Jon Awbrey]

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LOTS.  Discussion Note 14

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BM = Bernard Morand
JA = Jon Awbrey

Re: LOTS-DIS 13.  http://stderr.org/pipermail/inquiry/2005-March/002437.html
In: LOTS-DIS.     http://stderr.org/pipermail/inquiry/2005-March/thread.html#2416

JA: Let us try to separate the substantive matters from the terminological issues.
    I think that most of the difficulties here are purely terminological, but if
    one reifies the nomenclature one will tend to confuse terms with things.
    To keep matters simple I will use the language that I understand best.
    This relies on a distinction between a formal or mathematical object
    that is called a "relation" and a logical or linguistic expression
    that is called a "relative term".

BM: So, you agree that there is an interplay
    between the two disciplines on this subject?

I can't imagine how you'd imagine I'd imagine otherwise,
as I'd hardly call them different disciplines except for
their listing as such in the average college catalog, but
for the sake of this discursive locale I've sifted out the
object-oriented focus of mathematics from the sign-fixated
concern of logic, as partial as those two axes are in their
own orthogonal orthodoxies.

BM: But, you seem to account for such a subject precisely in terms of the
    terminological issues you are trying to escape from.  Why do you relate
    what is called "relation" with mathematics while associating "relative term"
    to logics or (unfortunately) to linguistics?  At least in the quote above CSP
    suggests a character of "relation" as a "relationship considered as something
    that may be said to be true of one of the objects, the others being separated
    from the relationship yet kept in view".  The predicative aspect attributed to
    relations makes them logical entities I think.  If there is some mathematical
    concept, it is the "relationship" one.  In fact I used to make an analogy for
    "fundamentum relationis" with the concept of numerical base in arithmetic.
    I don't know if the analogy is really accurate.

I'm not saying anything about the way it ought to be -- I am merely
making a sociological observation based on my contingent experience
with typical self-labelled -- I would say "soi-disant" but for fear
of antagonizing somebody with an alien language -- practitioners of
the various and sundry dark arts of logic and math.  If you've seen
the opposite tendencies, then, well, that's what "contingent" means.

I do not have the intensional fortitude to be a language reformer --
if I dreamed there was any hope of getting people to use the term
"fundamentum relationis" instead of "relation", I might just try,
but I'm afraid some compromises are inevitable in this situation.

JA: The relation can be given in extension as a subset of a cartesian product,
    taking the form L c X_1 x ... x X_k, where "c" marks the subset relation
    and "x" marks the cartesian product operation.

BM: This is the Jon Awbrey's interpretation of the whole thing.
    It has won its spurs in set theory that is presupposed here
    by way of the concepts of subset and cartesian product on
    domains.  But I remain uncertain whether CSP had something
    like that in mind with is Logic of Relatives.  Note that 
    independently of the answer to the previous question I
    accept that one can take it as his own starting point.

Well, I can hardly take credit for inventing this -- it is simply one
of the working tools in the everyday toolbox of math and computer sci --
but it is not really in doubt that Peirce had the same concept, under
one name or another, from very early on in the game, as we find ample
evidence of it in his publications from 1870 on.  It is also found in
the work that he did with his father on linear algebra, in as much as
a special case of this general form of extensional representation was
already stock in trade for the mathematics of the time.

JA: We may think of the relation as being denoted by many different
    styles of relative terms, and it will go toward saving us from
    the triune evil of nominalism, logicism, syntacticism if we
    try to treat the relation as the primary thing and the
    relative term as relatively secondary in importance.

BM: Agreed

JA: The "canonical" form for a k-adic relative term is actually one
    with k-1 blanks, since the whole expression, when filled in, is
    regarded as denoting the relate.  Here we may think of examples
    like:  "father of __", "giver of __ to __", "sign of __ to __".

BM: Yes, you are right I think.  The k-adic expression with k-1 blanks is 
    precisely what CSP calls a relation and the number k is the number of
    objects upon which the relationship is grounded.  Thus the consequence
    that with a relationship you get as many relations as there are blanks
    (note that set theory generally contents itself with binary relations).

Strickly speaking, what Peirce calls a "relation" in 1897 -- since his terms
vary quite a bit over time even if most of the basic concepts remain the same --
and that we might approximate with the word "role", would be called a "flag".
It is what you get if you mark or pin down one domain of a k-place relation,
leaving everything else intact.  But let's save this for when we have more
basic materials pinned down.

JA: When we transform to forms like "__ is a sign of __ to __", with
    k blanks for a k-adic relation, then the items filling the slots
    are successive elements in one k-tuple of the relation itself.

BM: This is where something hurts me.  The law of succession is
    not at random in the sign relation:  each slot is filled with
    an element the weight of which depends on the place of the slot
    in the whole form.  In the sign relation case, "standing for" is
    not interchangeable with "standing to".  I know anything like this
    with the operation of cartesian product.  Formally, each variable
    can take place anywhere.  Observe that with the application of
    this mathematical model in relational databases, the order of
    the attributes (columns) matters:  that's why they need to
    be named.

I'm not sure I understand the problem here.  The names and numbers of the slots --
sign as 1, object as 2, interpretant as 3 -- are arbitrary and do not make for
a correspondence between sign relational roles and categories.  The names get
their meaning from a suitable definition of the relation, and not the other
way around.  For each convened on ordering of the roles there are numerous
alternative orderings and alternative expressions as relative terms and
predicates that convey the same information, that might as well have
been convened on instead, and so we have to consider how to recover
the invariant information itself from the diverse expressions of it.

JA: Looking back over the history of logic, a lot of confusion has been
    engendered by what may have seemed like convenient abbreviations at
    the time, for example, using "general", "individual", "relative" for
    general term, individual term, relative term, respectively, or indeed,
    using "interpretant" for interpretant sign.  So this is something that
    we have to watch out for.

JA: The way that Peirce uses the term "relation" in the quotation above
    is already better served by the term "role" or "relational role",
    so I will stick with that.

BM: Yes, agreed.  This gives: a sign is the role played by something in
    standing for something to something.  Yet CSP called it a "relation"
    and it looks to be far from the relation we learn within set theory.

Okay, all the same furniture of the modern decorator is there in Peirce's
house, merely under different names in different parts of the world today:
couch, divan, sofa, chesterfield, davenport, ...

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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