12,089
edits
Changes
Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
Revision as of 17:15, 17 August 2011
, 17:15, 17 August 2011→6.43. Reflective Extensions: + raw text
===6.43. Reflective Extensions===
===6.43. Reflective Extensions===
<pre>
This section takes up the topic of reflective extensions in a more systematic fashion, starting from the sign relations A and B once again and keeping its focus within their vicinity, but exploring the space of nearby extensions in greater detail.
Tables 77 and 78 show one way that the sign relations A and B can be extended in a reflective sense through the use of quotational devices, yielding the "first order reflective extensions", Ref1(A) and Ref1(B).
Table 77. Reflective Extension Ref1(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>
<A> <<A>> <<A>>
<B> <<B>> <<B>>
<i> <<i>> <<i>>
<u> <<u>> <<u>>
Table 78. Reflective Extension Ref1(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>
<A> <<A>> <<A>>
<B> <<B>> <<B>>
<i> <<i>> <<i>>
<u> <<u>> <<u>>
The common "world" = {objects} U {signs} of the reflective extensions Ref1 (A) and Ref1 (B) is the set of 10 elements:
W = { A, B, <A>, <B>, <i>, <u>, <<A>>, <<B>>, <<i>>, <<u>>}.
Here, I employ raised angle brackets or "supercilia" (<...>) on a par with ordinary quotation marks ("..."), using them in the context of informal discussion to configure a new sign whose object is precisely the sign they enclose.
Regarded as new sign relations in their own right, the domains of both Ref1 (A) and Ref1 (B) are constituted as follows:
O = O<1> U O<2> = { A, B } U {<A>, <B>, <i>, <u>}.
S = S<1> U S<2> = {<A>, <B>, <i>, <u>} U {<<A>>, <<B>>, <<i>>, <<u>>}.
Thus, S overlaps with O in the set of first order signs or second order objects S<1> = O<2>, exemplifying the extent to which signs have become objects in the new sign relations.
To discuss how the denotative and connotative aspects of sign relations are affected by their reflective extensions it is helpful to introduce a few abbreviations. For each sign relation R C {A, B}, define:
Den1 (R) = (Ref1 (R))SO = PrOS (Ref1 (R)),
Con1 (R) = (Ref1 (R))SI = PrSI (Ref1 (R)).
The dyadic components of sign relations can be given graph theoretic representations, namely, as "digraphs" (directed graphs), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <x, y> is called an "arc" from point x to point y, and a self loop <x, x> is called a "sling" at x.
The denotative components Den1 (A) and Den1 (B) can be viewed as digraphs on the 10 points of the world set W. The arcs of these digraphs are given as follows:
1. Den1 (A) has an arc from each point of [A]A = {<A>, <i>} to A and from each point of [B]A = {<B>, <u>} to B.
2. Den1 (B) has an arc from each point of [A]B = {<A>, <u>} to A and from each point of [B]B = {<B>, <i>} to B.
3. In the parts added by reflective extension, Den1 (A) and Den1 (B) both have arcs from <s> to s, for each s C S<1>.
Taken as transition digraphs, Den1 (A) and Den1 (B) summarize the upshots, end results, or effective steps of computation that are involved in the respective evaluations of signs in S by Ref1 (A) and Ref1 (B).
The connotative components Con1 (A) and Con1 (B) can be pictured as digraphs on the eight points of the syntactic domain S. The arcs are given as follows:
1. Con1 (A) inherits from A the structure of a SER on S<1>, having a sling on each of the points in S<1> and two way arcs on the pairs {<A>, <i>} and {<B>, <u>}. The reflective extension Ref1(A) adds a sling on each point of S<2>, creating a SER on S.
2. Con1 (B) inherits from B the structure of a SER on S<1>, having a sling on each of the points in S<1> and two way arcs on the pairs {<A>, <u>} and {<B>, <i>}. The reflective extension Ref1(B) adds a sling on each point of S<2>, creating a SER on S.
Taken as transition digraphs, Con1 (A) and Con1 (B) highlight the associations between signs in Ref1 (A) and Ref1 (B), respectively.
The SER given by Con1 (A) for interpreter A has the semantic equations:
[<A>]A = [<i>]A,
[<B>]A = [<u>]A,
and the semantic partition:
{{ <A>, <i> }, { <<A>> }, { <<i>> },
{ <B>, <u> }, { <<B>> }, { <<u>> }}.
The SER given by Con1 (B) for interpreter B has the semantic equations:
[<A>]B = [<u>]B,
[<B>]B = [<i>]B,
and the semantic partition:
{{ <A>, <u> }, { <<A>> }, { <<u>> },
{ <B>, <i> }, { <<B>> }, { <<i>> }}.
Notice that the semantic equivalences of nouns and pronouns for each interpreter do not extend to equivalences of their second order signs, exactly as demanded by the literal character of quotations. Moreover, the new sign relations for A and B coincide in their reflective parts, since exactly the same triples were added to each set.
There are many ways to extend sign relations in an effort to increase their reflective capacities. The implicit goal of a reflective project is to achieve "reflective closure", S c O, where every sign is an object.
Considered as reflective extensions, there is nothing unique about the constructions of Ref1 (A) and Ref1 (B), but their common pattern of development illustrates a typical approach toward reflective closure. In a sense it epitomizes the project of "free", "naive", or "uncritical" reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.
At first sight it seems as though the imposition of reflective closure has multiplied a finite sign relation into an infinite profusion of highly distracting and largely redundant signs, all by itself and all in one step. But this explosion of orders happens only with the complicity of another requirement, that of deterministic interpretation.
There are two types of non determinism that can affect a sign relation, denotative and connotative.
1. A sign relation R has a non deterministic denotation if its dyadic component RSO (the converse of ROS) is not a function RSO: S >O, that is, if there are signs in S with missing or multiple objects in O.
2. A sign relation R has a non deterministic connotation if its dyadic component RSI is not a function RSI: S >I, in other words, if there are signs in S with missing or multiple interpretants in I. As a rule, sign relations are rife with this variety of non determinism, but it is usually felt to be under control so long as RSI remains close to being an equivalence relation.
Thus, it is really the denotative type of indeterminacy that is felt to be a problem in this context.
The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.
As a flexible and fairly general strategy for describing reflective extensions it is convenient to take the following tack. Given a syntactic domain S, there is an independent formal language F = F(S) = S<<>>, to be called "the free quotational extension of S", that can be generated from S by embedding each of its signs to any depth of quotation marks. In F, the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every s C S, the sequence s, <s>, <<s>>, ... contains nothing but pairwise distinct elements in F no matter how far it is produced. The set F(s) = s<<>> c F that collects the elements of this sequence is called "the subset of F generated from s by quotation".
Against this background, other varieties of reflective extension can be specified by means of semantic equations (SEQ's) that are considered to be imposed on the elements of F. Taking the reflective extensions Ref1 (A) and Ref1 (B) as the first orders of a "free" project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences Refn (A) and Refn (B).
A variant pair of reflective extensions, Ref1(A|E1) and Ref1(B|E1), are presented in Tables 79 and 80, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), with the exception of those entries that are constrained by the system of semantic equations:
E1: <<A>> = <A>, <<B>> = <B>, <<i>> = <i>, <<u>> = <u>.
This has the effect of making all levels of quotation equivalent.
By calling attention to their intended status as "semantic" equations, meaning that signs are being set equal in the SEC's they inhabit or the objects they denote, I hope to emphasize that these equations are able to say something significant about objects.
??? Redo F(S) over W ??? Use WF = O U F ???
Table 79. Reflective Extension Ref1(A|E1)
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>
<A> <A> <A>
<B> <B> <B>
<i> <i> <i>
<u> <u> <u>
Table 80. Reflective Extension Ref1(B|E1)
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>
<A> <A> <A>
<B> <B> <B>
<i> <i> <i>
<u> <u> <u>
Another pair of reflective extensions, Ref1(A|E2) and Ref1(B|E2), are presented in Tables 81 and 82, respectively. These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), except for the entries constrained by the following semantic equations:
E2: <<A>> = A, <<B>> = B, <<i>> = i, <<u>> = u.
Table 81. Reflective Extension Ref1(A|E2)
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>
<A> A A
<B> B B
<i> A A
<u> B B
Table 82. Reflective Extension Ref1(B|E2)
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>
<A> A A
<B> B B
<i> B B
<u> A A
</pre>
===6.44. Reflections on Closure===
===6.44. Reflections on Closure===