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

[Jon Awbrey]

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LOTS.  Discussion Note 13

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BM = Bernard Morand

Re: LOTS-DIS 11.  http://stderr.org/pipermail/inquiry/2005-March/002435.html
In: LOTS-DIS.     http://stderr.org/pipermail/inquiry/2005-March/thread.html#2416

BM: Apologies for not having enough time  to respond to all the
    very interesting points you are making the one and the other,
    particularly about the concept of ground.  However this message
    from Jon struck me enough to react briefly.  See at the end please.

BM: I don't think so but may be I misunderstood you, Jon.
    The sign is the first, the object the second and the
    interpretant is the third.  In the canonical blank
    form we get:

    ___ (representamen = 1) stands for ___ (its object = 2) to ___ (its interpretant = 3)

BM: Thus the names have meanings that are determined by the order
    furnished by the prescission frame.  This doesn't prevent to
    consider them under other auspices that is to say another
    ordering, making for example the object a first.

BM: I think that the shift from the blank form (with its relate and correlates) to
    <s,o,i> is parallel to the shift from mathematics to logics.  In the blank form
    the correlates are "anything you please", may be what mathematicians would call
    "variables" if this designation was not disputed itself.  In the sign definition,
    the filling of the blanks with names makes it relevant to logics.  Contrary to
    the usage in set theory that can only consider relations between sets AFTER
    having defined set membership, Peirce (qua logician?) FIRST defines the
    concept of relation and THEN fills the correlates with their inhabitants.

BM: This is how I understand the following quote (note particularly
    the distinction between fundamentum relationis and relation).

| A relative, then, may be defined as the equivalent of a word or phrase which,
| either as it is (when I term it a complete relative), or else when the verb "is"
| is attached to it (and if it wants such attachment, I term it a nominal relative),
| becomes a sentence with some number of proper names left blank.  A relationship,
| or fundamentum relationis, is a fact relative to a number of objects, considered
| apart from those objects, as if, after the statement of the fact, the designations
| of those objects had been erased.  A relation is 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.  Thus, for each relationship
| there are as many relations as there are blanks.  For example, corresponding
| to the relationship which consists in one thing loving another there are two
| relations, that of loving and that of being loved by.  There is a nominal
| relative for each of these relations, as "lover of --," and "loved by --."
| These nominal relatives belonging to one relationship, are in their relation
| to one another termed correlatives. In the case of a dyad, the two correlatives,
| and the corresponding relations are said, each to be the converse of the other.
| The objects whose designations fill the blanks of a complete relative are called
| the correlates.  The correlate to which a nominal relative is attributed is called
| the relate.  In the statement of a relationship, the designations of the correlates
| ought to be considered as so many logical subjects and the relative itself as the
| predicate.  The entire set of logical subjects may also be considered as a collective
| subject, of which the statement of the relationship is predicate.
|
| C.S. Peirce, 'Collected Papers', CP 3.466-467 (1897)

BM: Please Jon and others, tell me where I am wrong!

Bernard,

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".  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.  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.

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 __".

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.

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.

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.

Jon Awbrey

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