Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1
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1.3.4.3. Semiotic Equivalence Relations
If one examines the sign relations \(L_\text{A}\) and \(L_\text{B}\) that are associated with the interpreters \(\text{A}\) and \(\text{B}\), 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 \(L_\text{A}\) and \(L_\text{B}\) is that their connotative components \(\text{A}_{SI}\) and \(\text{B}_{SI}\) 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 (SERs) since they equate signs that mean the same thing to somebody. Each of the SERs, \(\text{A}_{SI}, \text{B}_{SI} \subseteq S \times I = S \times S\), partitions the whole collection of signs into semiotic equivalence classes (SECs). This makes for a 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 of their semiotic partitions (SEPs) of the syntactic domain.
The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for \(\text{A}\) and \(\text{B}\) 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.
Information about the different forms of semiotic equivalence induced by the interpreters \(\text{A}\) and \(\text{B}\) is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant by saying that the SEPs for \(\text{A}\) and \(\text{B}\) are orthogonal to each other.
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To discuss these types of situations further, I introduce the square bracket notation \([x]_E\) for 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 \(S\), 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 semiotic equivalence relation \(L_{SI}\) 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 one 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 \(\text{A}\) yields the semiotic equations:
| \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{A}\) | \(=\) | \([{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{A}\) | \([{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{A}\) | \(=\) | \([{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{A}\) | ||
| or | \({}^{\backprime\backprime} \text{A} {}^{\prime\prime}\) | \(=_\text{A}\) | \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\) | \({}^{\backprime\backprime} \text{B} {}^{\prime\prime}\) | \(=_\text{A}\) | \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\) |
and the semiotic partition\[\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \}\].
In contrast, the SER for interpreter \(\text{B}\) yields the semiotic equations:
| \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{B}\) | \(=\) | \([{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{B}\) | \([{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{B}\) | \(=\) | \([{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{B}\) | ||
| or | \({}^{\backprime\backprime} \text{A} {}^{\prime\prime}\) | \(=_\text{B}\) | \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\) | \({}^{\backprime\backprime} \text{B} {}^{\prime\prime}\) | \(=_\text{B}\) | \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\) |
and the semiotic partition\[\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} \}\].
