ONT Re: Sign Relations
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Sign Relation SIG,
I continue with the discussion of sign relations in the medium
of concrete examples, as illustrated by the "Story of A and B".
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| Document History:
|
| Subject: Inquiry Driven Systems: An Inquiry Into Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 8.61
| Created: 23-Jun-1996
| Revised: 04-Sep-2001
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Subsection 1.3.4.3 (Semiotic Equivalence Relations)
1.3.4.3 Semiotic Equivalence Relations
If one examines the sign relations L(A) and L(B) that are associated with
the interpreters A and B, respectively, one observes that they have many
contingent properties that are not possessed by sign relations in general.
One nice property possessed by the sign relations L(A) and L(B) is that
their connotative components A_SI and B_SI constitute a pair of equivalence
relations on their common syntactic domain S = I. It is convenient to refer to
such structures as "semiotic equivalence relations" (SER's) since they equate signs
that mean the same thing to somebody. Each of the SER's, A_SI, B_SI c SxI = SxS,
partitions the whole collection of signs into "semiotic equivalence classes" (SEC's).
This makes for an especially strong form of representation in that the structure of
the participants' common object domain is reflected or reconstructed, part for part,
in the structure of each one of their "semiotic partitions" (SEP's) of the shared
syntactic domain.
The main trouble with this notion of semantics in the present situation
is that the two semiotic partitions for A and B are not the same, indeed,
they are orthogonal to each other. This makes it difficult to interpret
either one of the partitions or equivalence relations on the syntactic
domain as corresponding to any sort of objective structure or invariant
reality, independent of the individual interpreter's point of view (POV).
Information about the different forms of semiotic equivalence that are
induced by the interpreters A and B is summarized in Tables 3 and 4.
The form of these Tables should suffice to explain what is meant by
saying that the SEP's for A and B are orthogonal to each other.
Table 3. Semiotic Partition of Interpreter A
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| "A" "i" |
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| "u" "B" |
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Table 4. Semiotic Partition of Interpreter B
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| "A" | "i" |
| | |
| "u" | "B" |
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To discuss this situation further, I introduce the square bracket notation "[x]_E"
to denote "the equivalence class of the element x under the equivalence relation E".
A statement that the elements x and y are equivalent under E is called an "equation",
and can be written in either one of two ways, as "[x]_E = [y]_E" or as "x =_E y".
In the application to sign relations I extend this notation in the following ways.
When L is a sign relation whose "syntactic projection" or connotative component L_SI
is an equivalence relation on S, then I write "[s]_L" for "the equivalence class of s
under L_SI". A statement that the signs x and y are synonymous under a SER L_SI is
called a "semiotic equation" (SEQ), and can be written in either of the forms:
"[x]_L = [y]_L" or "x =_L y".
In many situations there is a further adaptation of the square bracket notation that
can be useful. Namely, when there is known to exist a particular triple <o, s, i>
in L, it is permissible to use "[o]_L" to mean the same thing as "[s]_L". These
modifications are designed to make the notation for semiotic equivalence classes
harmonize as well as possible with the frequent use of similar devices for the
denotations of signs and expressions.
In these terms, the SER for interpreter A yields the semiotic equations:
| ["A"]_A = ["i"]_A
|
| ["B"]_A = ["u"]_A
or
| "A" =_A "i"
|
| "B" =_A "u"
and the semiotic partition: {{"A", "i"}, {"B", "u"}}.
In contrast, the SER for interpreter B yields the semiotic equations:
| ["A"]_B = ["u"]_B
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| ["B"]_B = ["i"]_B
or
| "A" =_B "u"
|
| "B" =_B "i"
and the semiotic partition: {{"A", "u"}, {"B", "i"}}.
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