ONT Re: Signs Of Pragmata -- Discussion
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SOP. Discussion Note 4
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JA = Jon Awbrey
RX = Reader X
Re: SOP 2. http://suo.ieee.org/ontology/msg05411.html
In: SOP. http://suo.ieee.org/ontology/thrd1.html#05410
JA: Consider the following two sign relations, reflecting, let us say,
the ways that Ann and Bob use the words "Ann", "Bob", "I", "you".
Table 1. Sign Relation of Interpreter A
o---------------o---------------o---------------o
| Object | Sign | Interpretant |
o---------------o---------------o---------------o
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
o---------------o---------------o---------------o
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
o---------------o---------------o---------------o
JA: Normally, the structure of the Object domain
is reconstructed on the semiotic plane S x I,
between the Sign and Interpretant columns of
a sign relational database, by partitioning
signs into equivalence classes of some sort.
RX: In the above table, I see the four relations resulting from S x I.
I am not certain what one means by denoting these 4 objects as distinct.
I also do not see how signs would be in equivalence classes, but rather,
it seems to me, that the objects have been placed into such a class,
hence the partitioning evident from the table. This ties in to
your "exception" following, but what would be a normal case?
In this extensional way of thinking, a "relation" L is a subset of
a cartesian product, which we write as L c X_1 x ... x X_k, at least,
so long as the number of factors X_j is finite.
In this particular case, the sign relation L(A) is a subset of O x S x I,
where O = {A, B}, S = {"A", "B", "i", "u"}, and I = {"A", "B", "i", "u"}.
Using the notation |X| for the cardinality of X, or the number of elements
in the set X, we have |O x S x I| = |O||S||I| = 2 * 4 * 4 = 32, out of which
we have selected the 8 triples listed in the Table to make up the relation L(A).
If we delete the object column, and don't count repeated pairs in what remains,
then we have what would be called the "projection" of L(A) on the S x I plane,
or some language to that effect, which we may write as Proj_SI (L(A)) or just
L(A)_SI, to be lazy I guess. To be even more lazy, let L = L(A)_SI for now.
L is a 2-adic relation L c S x I that has the following 8 ordered pairs:
Table 1_SI. L(A)_SI c S x I
o---------------o---------------o
| Sign | Interpretant |
o---------------o---------------o
| "A" | "A" |
| "A" | "i" |
| "i" | "A" |
| "i" | "i" |
o---------------o---------------o
| "B" | "B" |
| "B" | "u" |
| "u" | "B" |
| "u" | "u" |
o---------------o---------------o
L has the structure of an "equivalence relation".
That is, L is Reflexive, Symmetric, and Transitive.
First of all, S = I, so we can say that L c S x S.
L is reflexive, because <x, x> in L for all x in S.
L is symmetric, because <x, y> in L => <y, x> in L.
L is transitive, because <x, y> and <y, z> in L =>
<x, z> in L. When you have an equivalence relation,
the elements of the underlying set S can always be
partitioned into "equivalence classes" of elements
that are all related to each other by the relation,
while elements in different classes are not related.
In this case, the equivalence classes are {"A", "i"}
and {"B", "u"}, which codes up the fact that Agent A,
in the "No Exit" discourse that involves just A and B,
uses either "A" or "i" indifferently to denote A, and
uses either "B" or "u" indifferently to denote B. Fin.
As it happens, the equivalence classes of L(A)_SI are
in correspondence with the objects, the elements in O.
The class of signs {"A", "i"} corresponds to object A.
The class of signs {"B", "u"} corresponds to object B.
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.
And that is just what I was hinting at in the following observation:
JA: But here the partition is different for each interpreter:
Interpreter A has the signs in {"i", "A"} denoting A,
and the signs in {"u", "B"} denoting B.
Interpreter B has the signs in {"u", "A"} denoting A,
and the signs in {"i", "B"} denoting B.
Now, this is a slightly ad hoc example, but what I am
talking about here reflects a very general phenomenon.
Have to break for today ...
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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