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, 18:04, 19 August 2011→7.2.3. Propositional Reasoning About Relations: + text
====7.2.3. Propositional Reasoning About Relations====
====7.2.3. Propositional Reasoning About Relations====
Suppose A and B, besides just referring to themselves in various ways, want to say something significant about the nature of their interpretive practices. That is, they begin to inquire into the structure of their own systems of interpretation. It can be imagined that the course of this inquiry leads A and B to generate a series of models and theories, each of which attempts to empirically summarize or to rationally describe the observable varieties of their own interpretive usage.
Tables ** and ** illustrate a couple of the many possible ways that the SOI's associated with A and B might develop under the impetus of such an inquiry. These Tables present the empirical models of interpretation that A and B could conceivably form at a subsequent stage of inquiry, in this case taking the form of the "higher order" (HO) sign relations A' and B', respectively.
A sign relation is a "higher order" (HO) sign relation if some of its objects, signs, or interpretants are themselves sign relations. Thus, A' and B' are classified as HO sign relations by dint of the fact that some of their objects are elementary sign relations, namely, triples of the form <o, s, i> coming from the sign relations A and B, collectively. Note that this definition allows an arbitrary HO sign relation R to have subsets, "sections", or "subrelations" Q c R that are not in themselves HO sign relations, but purely "lower order" (LO) sign relations. Thus, it is often convenient to indicate different subsets of HO sign relations as being their "properly" HO sections or else as being their LO sections, depending on the evident complexity in each case.
In the Tables of A' and B', I have allowed the separate inquiries of A and B to develop in an asymmetric and fragmentary fashion, solely in order to demonstrate the different kinds of structure that can occur. Perhaps the point of these examples is best understood in this way: They serve more as pegs to hang conceptual tools on when these tools are not in use than they function as the objects of actual application. In accord with this organizing role, that gradually hews their features toward complementing rather than imitating the intended characteristics of any real objectives, it is best if these classical and classifying examples are not turned out in too fine a form, and it should not be expected to see in them the same kind of realism that one finds in the prospective subjects or gives to the worked material under discussion. With this purpose in mind, I can now elaborate a number of important structural concepts embodied in the HO sign relations A' and B'.
<pre>
Table **. Higher Order Sign Relation A'
Object Sign Interpretant
A "A" "i"
A "i" "i"
<A, "A", "A"> "A" "A"
<A, "A", "i"> "A" "A"
<A, "i", "A"> "A" "A"
<A, "i", "i"> "A" "i"
<B, "B", "B"> "A" "A"
<B, "B", "u"> "A" "A"
<B, "u", "B"> "A" "A"
<B, "u", "u"> "A" "A"
B "B" "u"
B "u" "u"
<A, "A", "A"> "B" "B"
<A, "A", "u"> "B" "B"
<A, "u", "A"> "B" "B"
<A, "u", "u"> "B" "B"
<B, "B", "B"> "B" "B"
<B, "B", "i"> "B" "B"
<B, "i", "B"> "B" "B"
<B, "i", "i"> "B" "B"
Table **. Higher Order Sign Relation B'
Object Sign Interpretant
A "u" "A"
A "A" "A"
<A, "A", "A"> "A" "A"
<A, "A", "i"> "A" "A"
<A, "i", "A"> "A" "A"
<A, "i", "i"> "A" "A"
<B, "B", "B"> "A" "A"
<B, "B", "u"> "A" "A"
<B, "u", "B"> "A" "A"
<B, "u", "u"> "A" "A"
B "i" "B"
B "B" "B"
<A, "A", "A"> "B" "B"
<A, "A", "u"> "B" "B"
<A, "u", "A"> "B" "B"
<A, "u", "u"> "B" "B"
<B, "B", "B"> "B" "B"
<B, "B", "i"> "B" "B"
<B, "i", "B"> "B" "B"
<B, "i", "i"> "B" "B"
</pre>
A syntactic element is "stable" with respect to a process iff ...
A syntactic element is "select" with respect to a purpose iff ...
A syntactic element is "standard" with respect to a <process, purpose> if and only if it is stable and select with respect to the process and the purpose, respectively.
With respect to the connotative component of a sign relation, a syntactic element is said to be "stable" if ...
In the LO sections of A' and B' the SER's that existed in the connotative components of A and B have devolved into less symmetric, more directed dyadic relations.
<pre>
Con A': "A" connotes only "i" and "i" connotes only "i",
"B" connotes only "u" and "u" connotes only "u".
Con B': "u" connotes only "A" and "A" connotes only "A",
"i" connotes only "B" and "B" connotes only "B".
</pre>
Interpreter A is apparently biased toward using personal pronouns, while interpreter B has a habit of using proper names to refer to everybody in the third person, including himself.
Unless A and B develop suitable fragments of the language of set theory within their SOI's they cannot avail themselves of syntactic elements that can denote the whole sign relations A and B under singular terms. Until then, they are forced to resort faute de mieux to the device of "plural indefinite reference" or "partially informed reference" (PIR), interpreting the same signs in systematically ambiguous ways to denote extended pluralities of objects indifferently. This way of letting an interpreter represent a set in action, by "being the set", as it were, is actually the preferred strategy according to some tastes, since it does not multiply abstract entities beyond necessity. Combined with a proper treatment of "normal forms" for syntactic elements, it leads to SOI's with sufficient power to deal with almost all of their practical set theoretic needs.
====7.2.4. Dynamic and Evaluative Frameworks====
====7.2.4. Dynamic and Evaluative Frameworks====
