[Inquiry] Re: Sign Relations -- Discussion

[Jon Awbrey]

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SR.  Discussion Note 6

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JA = Jon Awbrey
KM = Kirsti Määttänen

Re: SR-COM 12.  http://stderr.org/pipermail/inquiry/2005-January/002259.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-January/thread.html#2242

Amended here:

JA: In cases of sign relations like the ones we are considering,
    the denotative component and the connotative component exist
    in a coherent relationship to one another.  If we examine the
    situation with the sign relations L(A) and L(B) we can see that
    the denotation relations L(A)_SO and L(B)_SO map the equivalence
    classes of the connotation relations L(A)_SI and L(B)_SI onto the
    objects of !O! in such a way that all of the signs in a distinct
    equivalence class are mapped onto the same distinct object of !O!.
    For the interpreter A, the class of signs {"A", "i"} maps to the
    object A and the class of signs {"B", "u"} maps to the object B.
    For the interpreter B, the class of signs {"A", "u"} maps to the
    object A and the class of signs {"B", "i"} maps to the object B.

JA: Now this is very pretty, and some people get so enamored of it that
    they would even say you can now do away with the objects themselves,
    having "explained them away" or "reconstructed" them as equivalence
    classes of syntactic entities.  Some folks read Frege this way, for
    instance.  But there are several good reasons for stopping short of
    that extreme.  One reason is the non-uniqueness of the construction,
    in other words, the partition into equivalence classes is different
    for each interpreter.  This is a very general phenomenon, betraying
    a certain "point of view relativity" in the way that the structures
    of an objective world are represented in the structures of language.

KM: Here I have a feeling of understanding -- or rather:
    a feeling of mutual agreement.  But I'm not sure.
    "Partition into equivalence classes"?

An equivalence relation E on a set S is a 2-adic relation E c S x S that is
reflexive, symmetric, and transitive.  A partition of a non-empty set S is
a set of mutually disjoint non-empty subsets of S whose union is all of S.
Any equivalence relation E on a set S induces a partition of S into subsets
that are called the "equivalence classes" under E.  The members of a given
equivalence class are all pairwis equivalent under E, while elements of S
from different equivalence classes are inequivalent under E.  Equivalence
relations on S can be depicted as directed graphs or "digraphs" on the
point set S, drawing an arc from x to y in S if and only if the <x, y>
is an ordered pair in E.  Since E is reflexive there is a loop at each
point.  Since E is symmetric, each arc x->y is paired with an arc y->x.

In the case of the sign relations L(A) and L(B), their projections
on the SI plane, L(A)_SI and L(B)_SI, respectively, yield equivalence
relations  on the set of signs !S! = {"A", "B", "i", "u"} that I call
"semiotic equivalence relations" (SER's).  The corresponding partitions
I call "semiotic partitions" (SEP's).

The SEP for interpreter A consists of the sets {"A", "i"} and {"B", "u"}.
The SEP for interpreter B consists of the sets {"A", "u"} and {"B", "i"}.

Jon Awbrey

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