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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 69.1 and 70.1 repeat the contents of Tables 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 69.2 to 69.4 and Tables 70.2 to 70.4, respectively.
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Table 69.1 Sign Relation of Interpreter A
Table 69.1 Sign Relation of Interpreter A
Object Sign Interpretant
Object Sign Interpretant
B "u" "B"
B "u" "B"
B "u" "u"
B "u" "u"
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Table 69.2 Dyadic Projection AOS
Table 69.2 Dyadic Projection AOS
Object Sign
Object Sign
B "B"
B "B"
B "u"
B "u"
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Table 69.3 Dyadic Projection AOI
Table 69.3 Dyadic Projection AOI
Object Interpretant
Object Interpretant
B "B"
B "B"
B "u"
B "u"
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Table 69.4 Dyadic Projection ASI
Table 69.4 Dyadic Projection ASI
Sign Interpretant
Sign Interpretant
"u" "B"
"u" "B"
"u" "u"
"u" "u"
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Table 70.1 Sign Relation of Interpreter B
Table 70.1 Sign Relation of Interpreter B
Object Sign Interpretant
Object Sign Interpretant
B "i" "B"
B "i" "B"
B "i" "i"
B "i" "i"
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Table 70.2 Dyadic Projection BOS
Table 70.2 Dyadic Projection BOS
Object Sign
Object Sign
B "B"
B "B"
B "i"
B "i"
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Table 70.3 Dyadic Projection BOI
Table 70.3 Dyadic Projection BOI
Object Interpretant
Object Interpretant
B "B"
B "B"
B "i"
B "i"
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Table 70.4 Dyadic Projection BSI
Table 70.4 Dyadic Projection BSI
Sign Interpretant
Sign Interpretant
"i" "B"
"i" "B"
"i" "i"
"i" "i"
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A comparison of the corresponding projections in Proj (A) and Proj (B) shows that the distinction between the triadic relations A and B is preserved by Proj, and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections. However, to say that a triadic relation R C Pow (OxSxI) is reducible in this sense it is necessary to show that no distinct R' C Pow (OxSxI) exists such that Proj (R) = Proj (R'), and this can take a rather more exhaustive or comprehensive investigation of the space Pow (OxSxI).
A comparison of the corresponding projections in Proj (A) and Proj (B) shows that the distinction between the triadic relations A and B is preserved by Proj, and this circumstance allows one to say that this much information, at least, can be derived from the dyadic projections. However, to say that a triadic relation R C Pow (OxSxI) is reducible in this sense it is necessary to show that no distinct R' C Pow (OxSxI) exists such that Proj (R) = Proj (R'), and this can take a rather more exhaustive or comprehensive investigation of the space Pow (OxSxI).
