Changes

In this setting the issue of whether triadic relations are ''reducible to'' or ''reconstructible from'' their dyadic projections, both in general and in specific cases, can be identified with the question of whether <math>\operatorname{Proj}^{(2)}\!</math> is injective.  The mapping <math>\operatorname{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \operatorname{Pow}(X \times Y \times Z)\!</math> if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections.  Given a specific instance of a triadic relation <math>L \in \operatorname{Pow}(X \times Y \times Z),\!</math> it can be said that <math>L\!</math> is ''determined by'' (''reducible to'' or ''reconstructible from'') its dyadic projections if and only if <math>(\operatorname{Proj}^{(2)})^{-1}(\operatorname{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math>  Otherwise, there exists an <math>L'\!</math> such that <math>\operatorname{Proj}^{(2)}L = \operatorname{Proj}^{(2)}L',\!</math> and in this case <math>L\!</math> is said to be ''irreducibly triadic'' or ''genuinely triadic''.  Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under <math>\operatorname{Proj}^{(2)}.\!</math>

In this setting the issue of whether triadic relations are ''reducible to'' or ''reconstructible from'' their dyadic projections, both in general and in specific cases, can be identified with the question of whether <math>\operatorname{Proj}^{(2)}\!</math> is injective.  The mapping <math>\operatorname{Proj}^{(2)}\!</math> is said to ''preserve information'' about the triadic relations <math>L \in \operatorname{Pow}(X \times Y \times Z)\!</math> if and only if it is injective, otherwise one says that some loss of information has occurred in taking the projections.  Given a specific instance of a triadic relation <math>L \in \operatorname{Pow}(X \times Y \times Z),\!</math> it can be said that <math>L\!</math> is ''determined by'' (''reducible to'' or ''reconstructible from'') its dyadic projections if and only if <math>(\operatorname{Proj}^{(2)})^{-1}(\operatorname{Proj}^{(2)}L)\!</math> is the singleton set <math>\{ L \}.\!</math>  Otherwise, there exists an <math>L'\!</math> such that <math>\operatorname{Proj}^{(2)}L = \operatorname{Proj}^{(2)}L',\!</math> and in this case <math>L\!</math> is said to be ''irreducibly triadic'' or ''genuinely triadic''.  Notice that irreducible or genuine triadic relations, when they exist, naturally occur in sets of two or more, the whole collection of them being equated or confounded with one another under <math>\operatorname{Proj}^{(2)}.\!</math>

The next series of Tables illustrates the operation of <math>\operatorname{Proj}^{(2)}\!</math> by means of its actions on the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}.\!</math>  For ease of reference, Tables&nbsp;69.1 and 70.1 repeat the contents of Tables&nbsp;1 and 2, respectively, while the dyadic relations comprising <math>\operatorname{Proj}^{(2)}L_\text{A}\!</math> and <math>\operatorname{Proj}^{(2)}L_\text{B}\!</math> are shown in Tables&nbsp;69.2 to 69.4 and Tables&nbsp;70.2 to 70.4, respectively.

The next series of Tables illustrates the operation of <math>\operatorname{Proj}^{(2)}\!</math> by means of its actions on the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}.\!</math>  For ease of reference, Tables&nbsp;72.1 and 73.1 repeat the contents of Tables&nbsp;1 and 2, respectively, while the dyadic relations comprising <math>\operatorname{Proj}^{(2)}L_\text{A}\!</math> and <math>\operatorname{Proj}^{(2)}L_\text{B}\!</math> are shown in Tables&nbsp;72.2 to 72.4 and Tables&nbsp;73.2 to 73.4, respectively.

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 69.1} ~~ \text{Sign Relation of Interpreter A}\!</math>

|+ style="height:30px" | <math>\text{Table 72.1} ~~ \text{Sign Relation of Interpreter A}\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| width="33%" | <math>\text{Object}\!</math>

| width="33%" | <math>\text{Object}\!</math>

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<pre>

Table 69.2  Dyadic Projection AOS

Table 72.2  Dyadic Projection AOS

Object Sign

Object Sign

A "A"

A "A"

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Table 69.3  Dyadic Projection AOI

Table 72.3  Dyadic Projection AOI

Object Interpretant

Object Interpretant

A "A"

A "A"

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<pre>

Table 69.4  Dyadic Projection ASI

Table 72.4  Dyadic Projection ASI

Sign Interpretant

Sign Interpretant

"A" "A"

"A" "A"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 70.1} ~~ \text{Sign Relation of Interpreter B}\!</math>

|+ style="height:30px" | <math>\text{Table 73.1} ~~ \text{Sign Relation of Interpreter B}\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| width="33%" | <math>\text{Object}\!</math>

| width="33%" | <math>\text{Object}\!</math>

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<pre>

Table 70.2  Dyadic Projection BOS

Table 73.2  Dyadic Projection BOS

Object Sign

Object Sign

A "A"

A "A"

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<pre>

Table 70.3  Dyadic Projection BOI

Table 73.3  Dyadic Projection BOI

Object Interpretant

Object Interpretant

A "A"

A "A"

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<pre>

Table 70.4  Dyadic Projection BSI

Table 73.4  Dyadic Projection BSI

Sign Interpretant

Sign Interpretant

"A" "A"

"A" "A"