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===6.35. Reducibility of Sign Relations===
===6.35. Reducibility of Sign Relations===
<pre>
This section introduces a topic of fundamental importance to the whole theory of sign relations, namely, the question of whether triadic relations are "determined by", "reducible to", or "reconstructible from" their dyadic projections.
Suppose R c XxYxZ is an arbitrary triadic relation and consider the information about R that is provided by collecting its dyadic projections. To formalize this information define the "projective triple" of R as follows:
Proj (R) = Pr2(R) = <Pr12(R), Pr13(R), Pr23(R)>.
If R is visualized as a solid body in the 3 dimensional space XxYxZ, then Proj (R) can be visualized as the arrangement or ordered collection of shadows it throws on the XY, XZ, and YZ planes, respectively.
There are a couple of set theoretic constructions that are useful here, in particular for describing the source and target domains of Proj.
1. The set of subsets of a set S is called the "power set" of S. This object is denoted by either of the forms "Pow (S)" or "2S" and defined as follows:
Pow (S) = 2S = {T : T c S}.
The power set notation can be used to provide an alternative description of relations. In the case where S is a cartesian product, say S = X1x...xXn, then each n place relation described as a subset of S, say as R c X1x...xXn, is equally well described as an element of Pow (S), in other words, as R C Pow (X1x...xXn).
2. The set of triples of dyadic relations, with pairwise cartesian products chosen in a pre arranged order from a triple of three sets <X, Y, Z>, is called the "dyadic explosion" of XxYxZ. This object is denoted by "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by 2's", or more simply as "X, Y, Z, choose 2", and defined as follows:
Explo (X, Y, Z; 2) = Pow (XxY) x Pow (XxZ) x Pow (YxZ).
This domain is defined well enough to serve the immediate purposes of this section, but later it will become necessary to examine its construction more closely.
By means of these constructions the operation that forms Proj (R) for each triadic relation R c XxYxZ can be expressed as a function:
Proj : Pow (XxYxZ) > Explo (X, Y, Z; 2).
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 Proj is injective. The mapping Proj : Pow (XxYxZ) > Explo (X, Y, Z; 2) is said to "preserve information" about the triadic relations R C Pow (XxYxZ) 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 R C Pow (XxYxZ), it can be said that R is "determined by" ("reducible to" or "reconstructible from") its dyadic projections if and only if Proj 1(Proj (R)) is the singleton set {R}. Otherwise, there exists an R' such that Proj (R) = Proj (R'), and in this case R 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 Proj.
The next series of Tables illustrates the operation of Proj by means of its actions on the sign relations A and B. For ease of reference, Tables 69.1 and 70.1 repeat the contents of Tables 1 and 2, respectively, while the dyadic relations comprising Proj (A) and Proj (B) are shown in Tables 69.2 to 69.4 and Tables 70.2 to 70.4, respectively.
Table 69.1 Sign Relation of Interpreter A
Object Sign Interpretant
A "A" "A"
A "A" "i"
A "i" "A"
A "i" "i"
B "B" "B"
B "B" "u"
B "u" "B"
B "u" "u"
Table 69.2 Dyadic Projection AOS
Object Sign
A "A"
A "i"
B "B"
B "u"
Table 69.3 Dyadic Projection AOI
Object Interpretant
A "A"
A "i"
B "B"
B "u"
Table 69.4 Dyadic Projection ASI
Sign Interpretant
"A" "A"
"A" "i"
"i" "A"
"i" "i"
"B" "B"
"B" "u"
"u" "B"
"u" "u"
Table 70.1 Sign Relation of Interpreter B
Object Sign Interpretant
A "A" "A"
A "A" "u"
A "u" "A"
A "u" "u"
B "B" "B"
B "B" "i"
B "i" "B"
B "i" "i"
Table 70.2 Dyadic Projection BOS
Object Sign
A "A"
A "u"
B "B"
B "i"
Table 70.3 Dyadic Projection BOI
Object Interpretant
A "A"
A "u"
B "B"
B "i"
Table 70.4 Dyadic Projection BSI
Sign Interpretant
"A" "A"
"A" "u"
"u" "A"
"u" "u"
"B" "B"
"B" "i"
"i" "B"
"i" "i"
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).
As it happens, each of the relations R C {A, B} is uniquely determined by its projective triple Proj (R). This can be seen as follows. Consider any coordinate position <s, i> in the plane SxI. If <s, i> is not in RSI then there can be no element <o, s, i> in R, therefore we may restrict our attention to positions <s, i> in RSI, knowing that there exist at least |RSI| = 8 elements in R, and seeking only to determine what objects o exist such that <o, s, i> is an element in the objective "fiber" of <s, i>. In other words, for what o C O is <o, s, i> C PrSI 1(<s, i>)? The fact that ROS has exactly one element <o, s> for each coordinate s C S and that ROI has exactly one element <o, i> for each coordinate i C I, plus the "coincidence" of it being the same o at any one choice for <s, i>, tells us that R has just the one element <o, s, i> over each point of SxI. This proves that both A and B are reducible in an informational sense to triples of dyadic relations, that is, they are "dyadically reducible".
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===6.36. Irreducibly Triadic Relations===
===6.36. Irreducibly Triadic Relations===
