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===6.10. Higher Order Sign Relations : Examples===
===6.10. Higher Order Sign Relations : Examples===
<pre>
In considering the HO sign relations that stem from the examples A and B, it appears that annexing the first level of HA signs is tantamount to adjoining or instituting an auxiliary interpretive framework, one that has the semantic equations shown in Table 36.
Table 36. Semantics for Higher Order Signs
Object Denoted Equivalent Signs
A <A> = "A"
B <B> = "B"
"A" <<A>> = <"A"> = "<A>"
"B" <<B>> = <"B"> = "<B>"
"i" <<i>> = <"i"> = "<i>"
"u" <<u>> = <"u"> = "<u>"
However, there is an obvious problem with this method of defining new notations. It merely provides alternate signs for the same old uses. But if the original signs are ambiguous, then equating new signs to them cannot remedy the problem. Thus, it is necessary to find ways of selectively reforming the uses of the old notation in the interpretation of the new notation.
Table 37.1 Sign Relational Schema C
Object Sign Interpretant
x "x" "x"
"x" "x" "x"
Table 37.2 Sign Relational Schema D
Object Sign Interpretant
x "x" "x"
Table 37.3 Sign Relational Schema E
Object Sign Interpretant
"x" "x" "x"
Table 37.4 Sign Relational Schema D'
Object Sign Interpretant
x "x" "x"
x "x" <x>
x <x> "x"
x <x> <x>
The invocation of "higher order" (HO) signs raises an important point, having to do with the typical ways that signs can become the objects of further signs, and the relationship that this type of semantic ascent bears to the interpretive agent's capacity for so called "reflection". This is a topic that will recur again as the discussion develops, but a speculative foreshadowing of its character will have to serve for now.
Any object of an interpreter's experience and reasoning, no matter how vaguely and casually it initially appears, up to and including the merest appearance of a sign, is already, by virtue of these very circumstances, on its way to becoming the object of a formalized sign, so long as the signs are made available to denote it. The reason for this is rooted in each agent's capacity for reflection on its own experience and reasoning, and the critical question is only whether these transient reflections can come to constitute signs of a more permanent use.
The immediate purpose of the "arch" operation is to equip the text with a syntactic mechanism for constructing "higher order" (HO) signs, that is, signs denoting signs. But the step of reflection that the arch device marks corresponds to a definite change on the part of the interpreter, affecting the "pragmatic stance" or the "intentional attitude" that the interpreter takes up with respect to the affected signs. Accordingly, because of its connection to the interpreter's capacity for critical reflection, the arch operation, whether signified by arches or quotes, opens up a topic of wide importance to the larger question of inquiry. Unfortunately, there is much to do before this issue can be taken up in detail, and immediate concerns make it necessary to break off further discussion for now.
A general understanding of HO signs would not depend on the special devices that are used to construct them, but would define them as any signs that behave in certain ways under interpretation, that is, as any signs that are interpreted in a particular manner, yet to be specified. A proper definition of HO signs, including a generic description of the operations that construct them, cannot be achieved at the present stage of discussion. Doing this correctly depends on carrying out further developments in the theories of formal languages and sign relations. Until this discussion reaches that point, much of what it says about HO signs will have to be regarded as a provisional compromise.
The development of reflection on interpretation leads to the generation of "higher order" (HO) signs that denote "lower order" (LO) signs as their objects. This process is illustrated by the following "eponymy" of progressively HO signs, all of which stem from a plain precursor and ultimately refer back to their "eponymous ancestor":
x, <x>, <<x>>, <<<x>>>, ...
The intent of this succession, as interpreted in FL environments, is that <<x>> denotes or refers to <x>, which denotes or refers to x. Moreover, its computational realization, as implemented in CL environments, is that <<x>> addresses or evaluates to <x>, which addresses or evaluates to x.
The designations "LO" and "HO" are attributed to signs in a casual, local, and transitory way. At this point they signify nothing beyond the occurrence in a sign relation of a pair of triples having the form shown in Table 38.
Table 38. Sign Relation Containing a Higher Order Sign
Object Sign Interpretant
... s ...
... ... ...
s t ...
This is all it takes to make s a LO sign and t a HO sign in relation to each other at the moments in question. Whether a global ordering of a more generally justifiable sort can be constructed from an arbitrary series of such purely local impressions is another matter altogether.
Nevertheless, the preceding observations do show a way to give a definition of HO signs that does not depend on the peculiarities of quotational devices. For example, consider the previously described "eponymy" of x, that is, the "higher archy" of increasingly HO signs stemming from the object x. Table 39.1 shows how this succession can be transcribed into the form of a sign relation. But this is formally no different from the sign relation suggested in Table 39.2, one whose individual signs are not constructed in any special way. Both of these representations of sign relations, if continued in a consistent manner, would have the same abstract structure. If one of them is HO then so is the other, at least, if the attributes of order are meant to have any formally invariant meaning.
Table 39.1 Sign Relation for a Succession of HO Signs (1)
Object Sign Interpretant
x <x> ...
<x> <<x>> ...
<<x>> <<<x>>> ...
... ... ...
Table 39.2 Sign Relation for a Succession of HO Signs (2)
Object Sign Interpretant
x s1 ...
s1 s2 ...
s2 s3 ...
... ... ...
The rest of this section discusses the relationship between HO signs and a concept called the "reflective extension" of a sign relation. Reflective extensions will be subjected to a more detailed study in a later part of this work. For now, just to see how the process works, the sign relations A and B are taken as starting points to illustrate the more common forms of reflective development.
In the most typical scenario, HO sign relations come into being as the "reflective extensions" of simpler, possibly unreflective sign relations. Conversely, the incorporation of HO signs within a sign relation leads to a larger sign relation that constitutes one of its "reflective extensions". In general, there are many different ways that a reflective extension can get started and many different structures that can result.
In the initial slice of semantics presented for the dialogue of A and B, the sign domain S is identical to the interpretant domain I, and this set is disjoint from the object domain O. In order for this discussion to develop more interesting examples of sign relations these constraints will need to be generalized. As a start in this direction, one can preserve the identification of the syntactic domain as S = I and contemplate ways of varying the pattern of intersection between S and O.
One direction of generalization is motivated by the desire to give interpreters a measure of "reflective capacity". This is a property of sign relations that can be associated with the overlap of O and S and gauged by the extent to which S is contained in O. In intuitive terms, interpreters are said to have a "reflective capacity" to the extent that they can refer to their own signs independently of their denotations. An interpretive system with a sufficient amount of reflective capacity can support the maintainence and manipulation of textual objects like expressions and programs without necessarily having to evaluate the expressions or execute the programs.
In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text. In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values. A quoting function, given a sign or expression as its single argument, needs to accomplish two things: first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.
The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects. In each case, these extensions are assumed to begin with sign relations like A and B that have disjoint sets of objects and signs and thus have no reflective capacity at the outset. The status of A and B as the "reflective origins" of a "reflective development" is recalled by saying that A and B themselves are the "zeroth order reflective extensions" of A and B, in symbols, A = Ref0(A) and B = Ref0(B).
The following set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions. For ease of reference, Tables 40 and 41 repeat the contents of Tables 1 and 2, respectively, merely replacing ordinary quotes with arch quotes.
Table 40. Reflective Origin Ref0(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 41. Reflective Origin Ref0(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>
Tables 42 and 43 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). These extensions add one layer of HA signs and their objects to the sign relations for A and B, respectively. The new triples specify that, for each <x> in the set {<A>, <B>, <i>, <u>}, the HA sign of the form <<x>> connotes itself while denoting <x>.
Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their HO signs, exactly as demanded by the literal character of quotations. Also notice that the reflective extensions of the sign relations A and B coincide in their reflective parts, since exactly the same triples were added to each set.
Table 42. Higher Ascent Sign Relation 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 43. Higher Ascent Sign Relation 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>>
There are many ways to extend sign relations in an effort to develop their reflective capacities. The implicit goal of a reflective project is to reach a condition of "reflective closure", a configuration satisfying the inclusion S c O, where every sign is an object. It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation. This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds. As long as there are HO signs that remain distinct from all LO signs, the sign relation driven by a reflective process is forced to keep expanding. In particular, the process that is "freely" suggested by the formation of Ref1(A) and Ref1(B) cannot reach closure if it continues as indicated, without further constraints.
Table 44. Higher Import Sign Relation HI1(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>> <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>
<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>
Tables 44 and 45 present "HI extensions" of A and B, respectively. These are just HO sign relations that add selections of HI signs and their objects to the underlying set of triples in A and B. One way to understand these extensions is as follows. The interpreters A and B each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named. In this form of development, using a noun as a canonical form that refers indifferently to all the <o, s, i> triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.
Table 45. Higher Import Sign Relation HI1(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>> <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>
<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>
Several important facts about the class of HO sign relations in general are illustrated by these examples. First, the notations appearing in the object columns of HI1(A) and HI1(B) are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them. The agents A and B, as extended by the transactions of HI1(A) and HI1(B), respectively, are still restricted to their original syntactic domain {"A", "B", "i", "u"}. This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as HO. Indeed, the sign relations HI1(A) and HI1(B) are not very discriminating in their descriptions of the sign relations A and B, referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.
In practice, it does an interpreter little good to have the HI signs for referring to triples of objects, signs, and interpretants if it does not also have the HA signs for referring to each triple's syntactic portions. Consequently, the HO sign relations that one is likely to observe in practice are typically a mixed bag, having both HA and HI sections. Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.
These reflections raise the issue of how articulate a HO sign relation is in its depiction of its object signs and object sign relations. For now, I can do little more than note the dimension of articulation as a feature of interest, contributing to the scale of aesthetic utility that makes some sign relations better than others for a given purpose, and serving as a drive that motivates their continuing development.
The technique illustrated here represents a general strategy, one that can be exploited to derive certain benefits of set theory without having to pay the overhead that is needed to maintain sets as abstract objects. Using an identified type of a sign as a canonical form that can refer indifferently to all the members of a set is a pragmatic way of making plural reference to the members of a set without invoking the set itself as an abstract object. Of course, it is not that one can get something for nothing by these means. One is merely banking on one's recurring investment in the setting of a certain sign relation, a particular set of elementary transactions that is taken for granted as already funded.
As a rule, it is desirable for the grammatical system that one uses to construct and interpret HO signs, that is, signs for referring to signs as objects, to mesh in a comfortable fashion with the overall pragmatic system that one uses to assign syntactic codes to objects in general. For future reference, I call this requirement the problem of creating a "conformally reflective extension" (CRE) for a given sign relation. A good way to think about this task is to imagine oneself beginning with a sign relation R c OxSxI, and to consider its denotative component DenR = ROS c OxS. Typically one has a "naming function", call it "Nom", that maps objects into signs:
Nom c DenR c OxS, such that Nom : O -> S.
Part of the task of making a sign relation more reflective is to extend it in ways that turn more of its signs into objects. This is the reason for creating HO signs, which are just signs for making objects out of signs. One effect of progressive reflection is to extend the initial Nom through a succession of new naming functions Nom', Nom'', and so on, assigning unique names to larger allotments of the original and subsequent signs. With respect to the difficulties of construction, the "hard" core or the "adamant" part of creating an extended naming function is in the initial portion Nom that maps objects of the "external world" into signs in the "internal world". The subsequent task of assigning conventional names to signs is supposed to be comparatively natural and "easy", perhaps on account of the "nominal" nature of signs themselves.
The effect of reflection on the original sign relation R c OxSxI can be analyzed as follows. Suppose that a step of reflection creates HO signs for a subset of S. Then this step involves the construction of a newly extended sign relation:
R' c O'xS'xI', where O' = O U O1 and S' = S U S1.
In this construction O1 c S is that portion of the original signs S for which HO signs are created in the initial step of reflection, thereby being converted into O1 c O'. The sign domain S is extended to a new sign domain S' by the addition of these HO signs, namely, the set S1. Using the arch quotes (<...>), the mapping from O1 to S1 can be defined as follows:
Nom1 : O1 -> S1 such that Nom1 : x -> <x>.
Finally, the reflectively extended naming function Nom' : O' > S' is defined as Nom' = Nom U Nom'.
A few remarks are necessary to see how this way of defining a CRE can be regarded as legitimate.
In the present context an application of the arch notation "<x>" is read on analogy with the use of any other functional notation "f(x)", where "f" is the name of a function f, "f( )" is the context of its application, "x" is the name of an argument x, and where the functional abstraction "x > f(x)" is just another name for the function f.
It is clear that some form of functional abstraction is being invoked in the definition of Nom1, above. Otherwise, the expression "x > <x>" would indicate a constant function, one that maps every x in its domain to the same sign or code for the letter "x". But if this is allowed, then it seems either to invoke a more powerful concept, lambda abstraction, than the one being defined or else to attempt an improper definition of the naming function in terms of itself.
Although it appears that this form of functional abstraction is being used to define the CRE in terms of itself, trying to extend the definition of the naming function in terms of a definition that is already assumed to be available, in actuality this only uses a finite function, a finite table look up, to define the naming function for an unlimited number of HO signs.
In CL contexts, especially in the Lisp tradition, the quotation operator is recognized as an "evaluation inhibitor" and implemented as a function that maps each syntactic element into its unique numerical identifier or "godel number". Perhaps one should pause to marvel at the fact that a form of delay, deference, and interruption akin to an inhibition should be associated with the creation of signs that refer in meaningful ways.
On reflection, though, the connection between attribution and inhibition, or acknowledgment and deference, begins to appear less remarkable, and in time it can even be understood as natural and necessary. For one thing, psychoanalytic and psychodynamic theories of mental functioning have long recognized that symbol formation and symptom formation are closely akin, being the twin founders of civilization and many of its discontents. For another thing, the following etymology can be rather instructive: The English word "memory" derives from the Latin "memor" for "mindful", which is akin to the Latin "mora" for "delay", the Greek "mermera" for "care", and the Sanskrit "smarati" for "he remembers". To explore the verbal complex a bit further, it merits remembering that the ideas of "merit" and "membership", besides being connected with the due proportions, earned shares, and just deserts that are parceled out on parchment, are also tied up with the particular kind of care that is needed to take account of things part for part. (The Latin "merere" for "earn" or "deserve", along with "membrana" for "skin" or "parchment" and "memor" for "mindful", are all akin to the Greek "merizein" for "divide" and "meros" for "part".) Although the voices of psychology and etymology are seldom heard at this depth in the wilderness of formal abstraction, I think it is worth heeding them on this point.
In CL environments of the Pascal variety there are several different ways that HO signs can be created. In these settings HO signs, or signs for referring to signs as objects, can be implemented as the "codes" that serve as numerical identifiers of characters or the "pointers" that serve as accessory indices of symbolic expressions.
But not all the signs that are needed for referring to other signs can be constructed by means of quotation. Other forms of HO signs have to be generated "de novo", that is, constructed independently of previous successions and introduced directly into their appropriate orders. Among other things, this obviates the "obvious" strategy for telling the order of a sign by counting its quota of quotation marks. Failing the chances of exploiting such a naive measure in absolute terms, and in the absence of a natural order for the construction of signs, the relative orders of signs can only be assessed by examining the complex network of denotative and connotative relationships that connect them, or the gaps that arise when they fail to do so.
In a CL context this often occurs when a constant is declared equal or a variable is set equal to a quoted character, as in the following sequence of Pascal expressions:
const comma = ',' ;
var x; x := comma ;
In this passage, the sign <comma> is made to denote whatever it is that sign <','> denotes, and the variable x is then set equal to this value.
</pre>
===6.11. Higher Order Sign Relations : Application===
===6.11. Higher Order Sign Relations : Application===