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Revision as of 19:54, 14 September 2010
, 19:54, 14 September 2010→1.3.4.2. Sign Relations : A Primer
In the present example, <math>S = I = \text{Syntactic Domain}</math>.
In the present example, <math>S = I = \text{Syntactic Domain}</math>.
The sign relation associated with a given interpreter ''J'' is denoted ''L''<sub>''J'' </sub> or ''L''(''J''). Tables 1 and 2 give the sign relations associated with the interpreters ''A'' and ''B'', respectively, putting them in the form of ''[[relational database]]s''. Thus, the rows of each Table list the ordered triples of the form ‹''o'', ''s'', ''i''› that make up the corresponding sign relations, ''L''<sub>''A'' </sub>, ''L''<sub>''B'' </sub> ⊆ ''O'' × ''S'' × ''I''. It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.
The sign relation associated with a given interpreter <math>J\!</math> is denoted <math>L_J</math> or <math>L(J)</math>. Tables 1 and 2 give the sign relations associated with the interpreters <math>\text{A}</math> and <math>\text{B}</math>, respectively, putting them in the form of ''[[relational database]]s''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)</math> that make up the corresponding sign relations, <math>L_\text{A}, L_\text{B} \subseteq</math><math>O \times S \times I</math>. It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
<br>
|+ Table 1. Sign Relation of Interpreter ''A''
|- style="background:paleturquoise"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Sign Relation of Interpreter A}</math>
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}</math>
| <math>\text{Sign}</math>
| <math>\text{Interpretant}</math>
|-
|-
| ''B'' || "u" || "B"
|
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|-
|-
| ''B'' || "u" || "u"
|
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|}
|}
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
|+ Table 2. Sign Relation of Interpreter ''B''
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}</math>
|- style="background:paleturquoise"
|- style="height:40px; background:#f0f0ff"
| <math>\text{Object}</math>
| <math>\text{Sign}</math>
| <math>\text{Interpretant}</math>
|-
|-
| ''B'' || "i" || "B"
|
<math>\begin{matrix}
\text{A}
\\
\text{A}
\\
\text{A}
\\
\text{A}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{A} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{u} {}^{\prime\prime}
\end{matrix}</math>
|-
|-
| ''B'' || "i" || "i"
|
<math>\begin{matrix}
\text{B}
\\
\text{B}
\\
\text{B}
\\
\text{B}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|
<math>\begin{matrix}
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{B} {}^{\prime\prime}
\\
{}^{\backprime\backprime} \text{i} {}^{\prime\prime}
\end{matrix}</math>
|}
|}
<br>
<br>
