[Inquiry] Re: Sign Relations -- Commentary

[Jon Awbrey]

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SR.  Commentary Note 11

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If we ignore the object column in a sign relational table, and
focus only on the 2-adic relation between signs and interpretants,
we end up with what, for the momentary lack of better name, can be
called the "connotation relation" that is derived or projected from
the sign relation in question.  The interpretant sign is often said
to be an equivalent or implied sign in relation to the sign that it
interprets.  Thus we may expect sign relations, if they are fully
filled out, to yield connotation relations that are equivalence
relations, that is, reflexive, symmetric, transitive relations.
Indeed, this is just what happens in the case of L(A) and L(B).

Tables 7 and 8 show the results of deleting the object columns from
Tables 1 and 2, respectively, ignoring repeated pairs in what remains.
This gives us what is called the "projection" of L(A) and L(B) on the
SI plane, which may be notated as Proj_SI (L(A)) and Proj_SI (L(B))
or more simply as L(A)_SI and L(B)_SI, respectively.

Table 7.  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

Table 8.  L(B)_SI  c  !S! x !I!
o---------------o---------------o
| Sign          | Interpretant  |
o---------------o---------------o
| "A"           | "A"           |
| "A"           | "u"           |
| "u"           | "A"           |
| "u"           | "u"           |
o---------------o---------------o
| "B"           | "B"           |
| "B"           | "i"           |
| "i"           | "B"           |
| "i"           | "i"           |
o---------------o---------------o

Each connotation relation takes on the structure of an "equivalence relation".
In other words, L(A)_SI and L(B)_SI are reflexive, symmetric, and transitive.
First, !S! = !I!, so we can say that both relations are subsets of !S! x !S!.
Each relation L is reflexive, since the pair <x, x> is in L for all x in !S!.
Each relation L is symmetric, because <x, y> in L implies that <y, x> is too.
Each relation L is transitive, because <x, y> and <y, z> in L => <x, z> in L.

When you have an equivalence relation in !S! x !S!, 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 so related.  In this case, the
equivalence classes are {"A", "i"} and {"B", "u"} for interpreter A,
while they are {"A", "u"} and {"B", "i"} for interpreter B.

Jon Awbrey

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