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===6.9. Higher Order Sign Relations : Introduction===
===6.9. Higher Order Sign Relations : Introduction===
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When interpreters reflect on their own use of signs they require an appropriate technical language in which to pursue these reflections. For this they need signs that refer to sign relations, signs that refer to the elements and components of sign relations, and signs that refer to the properties and classes of sign relations. All of these additional signs can be placed under the description of "higher order" (HO) signs, and the extended sign relations that involve them can be referred to as "higher order" (HO) sign relations.
Whether any forms of observation and reflection can be conducted outside the medium of language is not a question I can address here. It is apparent as a practical matter, however, that stable and sharable forms of knowledge depend on the availability of an adequate language. Accordingly, there is a relationship of practical necessity that binds the conditions for reflective interpretation to the possibility of extending sign relations through higher orders. At minimum, in addition to the signs of objects originally given, there must be signs of signs and signs of their interpretants, and each of these HO signs requires a further occurrence of HO interpretants to continue and complete its meaning within a HO sign relation. In general, HO signs can arise in a number of independent fashions, but one of the most common derivations is through the specialized devices of quotation. This establishes a contingent relation between reflection and quotation.
This entire topic, involving the relationship of reflective interpreters to the realm of HO sign relations and the available operators for quotation, forms the subject of a recurring investigation that extends throughout the rest of this work. This section introduces only enough of the basic concepts, terminology, and technical machinery that is necessary to get the theory of HO signs off the ground.
By way of a first definition, a "higher order" (HO) sign relation is a sign relation, some of whose signs are "higher order" (HO) signs. If an extra degree of precision is needed, HO signs can be distinguished in a variety of different "species" or "types", to be taken up next.
In devising a nomenclature for the required species of HO signs, it is a good idea to generalize slightly, designing an analytic terminology that can be adapted to classify the HO signs of arbitrary relations, not just the HO signs of sign relations. The work of developing a more powerful vocabulary can be put to good account at a later stage of this project, when it is necessary to discuss the structural constituents of arbitrary relations and to reflect on the language that is used to discuss them. However, by way of making a gradual approach, it still helps to take up the classification of HO signs in a couple of passes, first considering the categories of HO signs as they apply to sign relations and then discussing how the same ideas are relevant to arbitrary relations.
Here are the species of HO signs that can be used to discuss the structural constituents and intensional genera of sign relations:
1. Signs that denote signs, that is, signs whose objects are signs in the same sign relation, are called "higher ascent" (HA) signs.
2. Signs that denote dyadic components of elementary sign relations, that is, signs whose objects are elemental pairs or dyadic actions having any one of the forms <o, s>, <o, i>, <s, i>, are called "higher employ" (HE) signs.
3. Signs that denote elementary sign relations, that is, signs whose objects are elemental triples or triadic transactions having the form <o, s, i>, are called "higher import" (HI) signs.
4. Signs that denote sign relations, that is, signs whose objects are themselves sign relations, are called "higher upshot" (HU) signs.
5. Signs that denote intensional genera of sign relations, that is, signs whose objects are properties or classes of sign relations, are called "higher yclept" (HY) signs.
Analogous species of HO signs can be used to discuss the structural constituents and intensional genera of arbitrary relations. In order to describe them, it is necessary to introduce a few extra notions from the theory of relations. This, in turn, occasions a recurring difficulty with the exposition that needs to be noted at this point.
The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another that makes it difficult to know where to get on and where to get off the circle of explanation. As I currently understand their relationship, it can be approached in the following order:
: Relations have types.
: Types are functions.
: Functions are relations.
In this setting, a "type" is a function from the "places" of a relation, that is, from the index set of its components, to a collection of sets that are called the "domains" of the relation.
When a relation is given an extensional representation as a collection of elements, these elements are called its "elementary relations" or its "individual transactions". The "type" of an elementary relation is a function from an index set whose elements are called the "places" of the relation to a set of sets whose elements are called the "domains" of the relation. The "arity" or "adicity" of an elementary relation is the cardinality of this index set. In general, these cardinalities can be ranked as finite, denumerably infinite, or non denumerable.
Elementary relations are also called the "effects" of a relation, more specifically, as its "maximal" or "total" effects, which are the kinds of effects that one usually intends in the absence of further qualification. More generally, a "component relation" or a "partial transaction" of a relation is a projection of one of its elementary relations on a subset of its places.
A "homogeneous relation" is a relation, all of whose elementary relations have the same type. In this case, the type and the arity are properties that are defined for the relation itself. The rest of this discussion is specialized to homogeneous relations.
When the arity of a relation is a finite number n, then the relation is called an "n place relation". In this case, the elementary relations are just the n tuples belonging to the relation. In the finite case, for example, a "non trivial properly partial transaction" is a k tuple extracted from an n tuple of the relation, where 1 < k < n. The first element of an elementary relation is called its "object" or "prelate", while the remaining elements are called its "correlates".
1. Signs that denote single correlates of an object in a relation are called "higher ascent" (HA) signs.
2. Signs that denote moderate effects in a relation, that is, signs whose objects are partial transactions or k-tuples involving more than one place but less than the full set of places in a relation, are called "higher employ" (HE) signs.
3. Signs that denote elementary relations involving all the places of a relation are called "higher import" (HI) signs.
4. Signs that denote relations are called "higher upshot" (HU) signs.
5. Signs that denote properties or classes of relations are called "higher yclept" (HY) signs.
Whenever the sense is clear, it is usually convenient to stick with the more generic terms for HO signs and HO sign relations, letting context determine the appropriate meaning. For the rest of this section, it is mainly the categories of HA signs and HI signs that come into play.
The inquiry into inquiry is not pursued for reasons of sheer narcissism, but because it is unavoidably a part of the inquiry into anything else, since critical reflection on the methods employed is implicit in the task. This means that the inquiry into inquiry must be able to formulate and critique alternative descriptions of inquiry in general, including itself. Thus, there are notions of "entelechy", of a self referent objective, a completion in self description, or an end to self actualization, that are intrinsic to the conception of inquiry, whether or not its ends in view are ever achieved. If inquiry, as a manner of thinking, is carried on in sign relations and is ever to be supported by computational means, then these reflections raise the issue of self describing sign relations and self documenting data structures.
This is where HO sign relations come in, making it possible to formalize sign relations that describe themselves and other sign relations, and thus enabling one to conceive of inquiries that inquire into themselves and other inquiries, at least in part. It is useful to approach these topics in a couple of stages, at first, by describing sign relations that describe other sign relations, and then, by describing sign relations that describe themselves. Although the implicit aim, or naive hope, is always to make these descriptions as complete as possible, it has to be recognized that partial success is all that is likely to be realized in practice. It seems to be something between rare and impossible that a non trivial sign relation could completely describe itself with respect to every facet of its being and in all the ways that it does in fact exist.
Nevertheless, "partially self describing" (PSD) sign relations and "partially self documenting" (PSD) data structures do arise in practice, and so it is incumbent on this inquiry to look into the question of how they usually develop. That is, how does a sign get itself interpreted in a sign relation in such a way that it acts as a partial self description of that selfsame sign relation? There appear to be two main ways that this can happen. Occasionally, it develops through the reflective operation or insightful turn of "retracting projections", that is, by recognizing that a feature attributed to others is also (or primarily) an aspect of oneself. More commonly, PSD sign relations are encountered already in place, as when a HO sign relation has signs that describe lower orders, partial aspects, or previous stages of itself.
A further reduction in the number of different kinds of signs to worry about can be achieved by means of a special technique — some may call it an "artful dodge" — for referring indifferently to the elements of a set without referring to the set itself. Under the designation of a "plural indefinite reference" (PIR) is included all the various ways of dealing with denominations, multiple denotations, collective references, or objective multitudes that avail themselves of this trick.
By way of definition, a sign q in a sign relation R c OxSxI is said to be, to constitute, or to make a PIR to (every element in) a set of objects, X c O, if and only if q denotes every element of X. This relationship can be expressed in a succinct formula by making use of one additional definition.
The "denotation of q in R", written as "De(q, R)", is defined as follows:
De(q, R) = Den(R).q = ROS.q = {o C O : <o, q, i> C R, for some i C I}.
Then q makes a PIR to X in R if and only if X c De(q, R). Of course, this includes the limiting case where X is a singleton, say X = {o}. When this happens then the reference is neither plural nor indefinite, properly speaking, but q denotes o uniquely.
The proper exploitation of PIR's in sign relations makes it possible to finesse the distinction between HI signs and HU signs, in other words, to provide a ready means of translating between the two kinds of signs that preserves all the relevant information, at least, for many purposes. This is accomplished by connecting the sides of the distinction in two directions. First, a HI sign that makes a PIR to many triples of the form <o, s, i> can be taken as tantamount to a HU sign that denotes the corresponding sign relation. Second, a HU sign that denotes a singleton sign relation can be taken as tantamount to a HI sign that denotes its single triple. The relation of one sign being "tantamount to" another is not exactly a full fledged semantic equivalence, but it is a reasonable approximation to it, and one that serves a number of practical purposes.
In particular, it is not absolutely necessary for a sign relation to contain a HU sign in order for it to contain a description of itself or another sign relation. As long the sign relation is "content" to maintain its reference to the object sign relation in the form of a constant name, then it suffices to use a HI sign that makes a PIR to all of its triples.
In the theory of sign relations, as in formal language theory, one tends to spend a lot of the time talking about signs as objects. Doing this requires one to have signs for denoting signs and ways of telling when a sign is being used as a sign or is just being mentioned as an object. Generally speaking, reflection on the usage of an established order of signs recruits another order of signs to denote them, and then another, and another, until a limit on one's powers of reflection is ultimately reached, and finally one is forced to conduct one's meaning in forms of interpretive practice that fail to be fully reflective in one critical respect or another. In the last resort one resigns oneself to letting the recourse of signs be guided by casually intuited inklings of their potential senses.
In this text a number of linguistic devices are used to assist the faculty of reflection, hopefully forestalling the relegation of its powers to its own natural resources for a long enough spell to observe its action. Two of the most frequently used strategies toward this purpose can be described as follows:
1. In the declaration of HO signs and the specification of their uses, one can employ the same terminology and technical distinctions that are found to be effective in describing sign relations. This turns the established terms for significant properties of world elements and the provisional terms for their relationships to each other to the ends of prescribing the relative orders of HO signs and their objects. In short, the received theory of signs, however transient it may be at any given moment of inquiry, allows one to declare the absolute types and the relative roles that all of these entities are meant to take up.
For example, if I say that x connotes y and that y denotes z, then it means I imagine myself to have an interpretation or a sign relation in mind where x and y are both signs belonging to a single order of signs and y is a sign belonging to the next higher order of signs up from z, everything being relative to that particular moment of interpretation. Of course, as far as wholly arbitrary sign relations go, there is nothing to guarantee that the interpretation I think myself to have in mind at one moment can be integrated with the interpretation I think myself to have in mind at another moment, or that a just order can be founded in the end by any manner of interpretation that "just follows orders" in this way.
2. Ordinary quotation marks ("...") function as an operator on pieces of text to create names for the signs or expressions enclosed in them. In doing this the quotation marks delay, defer, or interrupt the normal use of their subtended contents, interfering with the referential use of a sign or the evaluation of an expression in order to create a new sign. The use of this constructed sign is to mention the immediate contents of the quotation marks in a way that can serve thereafter to indicate these contents directly or allude to them indirectly.
In the informal context, however, quotation marks are used equivocally for several other purposes. In particular, they are frequently used to call attention to the immediate use of a sign, to stress it or redress it for a definitive, emphatic, or skeptical service, but without necessarily intending to interrupt or seriously alter its ongoing use. Furthermore, ordinary quotation marks are commonly taken so literally that they can inadvertently pose an obstacle to functional abstraction. For instance, if I try to refer to the effect of quotation as a mapping that takes signs to HO signs, thereby attempting to define its action by means of a lambda abstraction: x > "x", then there are modes of IL interpretation that would read this literally as a constant map, one that sends every element of the functional domain into the single code for the letter "x".
For these reasons I introduce the use of raised angle brackets (<...>), also called "arches" or "supercilia", to configure a form of quotation marker, but one that is subject to a more definite set of understandings about its interpretation. Namely, the arch marker denotes a function on signs that takes (the name of) a syntactic element located within it as (the name of) a functional argument and returns as its functional value the name of that syntactic element. The parenthetical operators in this statement reflect the optional readings that prevail in some cases, where the simple act of noticing a syntactic element as a functional argument is already tantamount to having a name for it. As a result, a quoting function that is designed to operate on the signs denoting and not on the objects denoted seems to do nothing at all, but merely uses up a moment of time to do it.
In IL contexts the arch quotes are construed together with their syntactic contents as forming a certain kind of term, one that achieves a naming function on syntactic elements by taking the enclosed text as a functional argument and giving a directly embedded indication of it. In this type of setting the name of a string of length k is a string of length k+2.
In FL contexts the arch marker denotes a function that takes the literal syntactic element bounded by it as its argument and returns the name, code, annotation, godel number, or "unique numerical identifier" (UNI) of that syntactic element. In this setting there need be no straightforward relationship between the size or complexity of the syntactic element and the magnitude of its numerical code or the form of its symbolic code.
In CL implementations the arch operation is intended to do exactly what the principal uses of ordinary quotes are supposed to do, except that it obeys restrictions that are necessary to make it work as a notation for a computable function on the identified syntactic domain.
One further remark on the uses of quotation marks is pertinent here. When using HA signs with high orders of complexity and depth, it is often convenient to revert to the use of ordinary quotes at the outer boundary of a quotational expression, in this way marking a return to the ordinary context of interpretation. For example, one observes the colloquial equivalence: <<<x>>> = "<<x>>".
In general, a good way to specify the meaning of a new notation is by means of a semantic equation, or a system of semantic equations, that expresses the function of the new signs in terms of familiar operations. If it is merely a matter of introducing new signs for old meanings, then this method is sufficient. In this vein, the intention and use of the "supercilious notation" for reflecting on signs could have its definition approximated in the following way.
Let <x> = "x" as signs for the object x, and let <<x>> = <"x"> = "<x>" as signs for the object "x", an object that incidentally happens to be sign. An alternative way of putting this is to say that the members of the set {<x>, "x"} are equivalent as signs for the object x, while the members of the set {<<x>>, <"x">, "<x>"} are equivalent as signs for the sign "x".
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===6.10. Higher Order Sign Relations : Examples===
===6.10. Higher Order Sign Relations : Examples===