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Jon Awbrey


Registered: Feb 2004
Posts: 557

Higher Order Signs

HOS. Note 1

3.4.9. Higher Order Sign Relations: Introduction

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 entire sign relations, signs
that refer to the elements and components of sign relations, and
signs that refer to the conceivable properties and classes of
sign relations. All of these additional signs can be placed
under the heading of "higher order signs" (HOS's), and the
extended sign relations that involve them can be referred
to as "higher order sign relations" (HOSR's).

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 a 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 higher order signs requires a further occurrence of
higher order interpretants to continue and complete its meaning
within a higher order sign relation. In general, higher order
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 higher order 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 (3.4.9) introduces only
enough of the basic concepts, terminology, and technical machinery that
is necessary to get the theory of higher order signs off the ground.

Jon Awbrey

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Higher Order Signs

HOS. Note 2

3.4.9. Higher Order Sign Relations: Introduction (cont.)

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 and addressed at this juncture.

Jon Awbrey

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Higher Order Signs

HOS. Note 3

3.4.9. Higher Order Sign Relations: Introduction (cont.)

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, its "maximal" or "total" effects, which are the kinds of
effects that one usually means 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 of the
    places of a relation are called "higher import" (HI) signs.

  4. Signs that denote entire relations are
    called "higher upshot" (HU) signs.

  5. Signs that denote properties of relations 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.

Jon Awbrey

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Higher Order Signs

HOS. Note 4

3.4.9. Higher Order Sign Relations: Introduction (cont.)

The inquiry into inquiry is not pursued for reasons of pure narcissism,
but because it is unavoidably a part of the inquiry into anything else,
since critical reflection on the methods used is intrinsic to the task.
This means that the inquiry into inquiry must be able to formulate and
critique competing descriptions of inquiry in general, itself included.
Thus, there are ideas of "entelechy", of a self-referent objective, of
completion in self-description, or of ends to self-actualization, that
are implied in the concept 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 fated to be enhanced with computational means,
then these reflections raise the issue of self-descript sign relations
and self-documenting data structures.

This is where higher order 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 actual practice.
It seems to be something between rare and impossible that a non-trivial
sign relation could completely describe itself with regard to every facet
of its being and in all the ways that any sign relation 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.

Jon Awbrey

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Higher Order Signs

HOS. Note 5

3.4.9. Higher Order Sign Relations: Introduction (cont.)

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 elation L c O x S x I 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, to wit:

The "denotation of q in L", written as "De(q, L)", is defined as follows:


  • De(q, L) = Den(L).q = L_OS.q =

    {o in O : <o, q, i> in L, for some i in I}.

Then q makes a PIR to X in L if and only if X c De(q, L).
Of course, this includes the limiting case where X is a
singleton set, 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 of the relevant information, at any rate doing so
for many significant 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 associated
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 constitutes 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.

Jon Awbrey

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Higher Order Signs

HOS. Note 6

3.4.9. Higher Order Sign Relations: Introduction (concl.)

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, where everything is
    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 sort 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 any number of 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 to 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 by default 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 all of 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 last 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, we have
a quoting function that is designed to operate on the signs denoting
and not on the objects denoted that seems to do nothing at all, but
merely uses up a moment of time to do it.

In IL ("informal language") 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 ("formal language") 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 type of 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 ("computational language") 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".

Jon Awbrey

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Higher Order Signs

HOS. Note 7

3.4.10. Higher Order Sign Relations: Examples

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
o-----------------------------o-----------------------------o
| ` ` ` Object` Denoted ` ` ` | ` ` `Equivalent `Signs` ` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `A` ` ` ` ` ` ` | ` ` ` <^A^> `=` `"A"` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `B` ` ` ` ` ` ` | ` ` ` <^B^> `=` `"B"` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------o-----------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "A" ` ` ` ` ` ` | <<^A^>> = <^"A"^> = "<^A^>" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "B" ` ` ` ` ` ` | <<^B^>> = <^"B"^> = "<^B^>" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "i" ` ` ` ` ` ` | <<^i^>> = <^"i"^> = "<^i^>" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` "u" ` ` ` ` ` ` | <<^u^>> = <^"u"^> = "<^u^>" |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------o-----------------------------o

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 and abuses of the old notation in
the interpretation of the new notation.

Jon Awbrey

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Higher Order Signs

HOS. Note 8

3.4.10. Higher Order Sign Relations: Examples (cont.)

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 every 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^>>, <<<^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
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` `...` ` ` ` | ` ` ` ` x ` ` ` ` | ` ` ` `...` ` ` ` |
| ` ` ` `...` ` ` ` | ` ` ` `...` ` ` ` | ` ` ` `...` ` ` ` |
| ` ` ` ` x ` ` ` ` | ` ` ` ` y ` ` ` ` | ` ` ` `...` ` ` ` |
o-------------------o-------------------o-------------------o

This is all that it takes to make x a LO sign and y 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.

Jon Awbrey

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Higher Order Signs

HOS. Note 9

3.4.10. Higher Order Sign Relations: Examples (cont.)

Despite the paucity of their materials, 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 that is suggested
in Table 39.2, 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 higher and lower order are meant
to have any type of formally invariant meaning.

Table 39.1 Sign Relation for a Succession of HO Signs (1)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` x ` ` ` ` | ` ` ` <^x^> ` ` ` | ` ` ` `...` ` ` ` |
| ` ` ` <^x^> ` ` ` | ` ` `<<^x^>>` ` ` | ` ` ` `...` ` ` ` |
| ` ` `<<^x^>>` ` ` | ` ` <<<^x^>>> ` ` | ` ` ` `...` ` ` ` |
| ` ` ` `...` ` ` ` | ` ` ` `...` ` ` ` | ` ` ` `...` ` ` ` |
o-------------------o-------------------o-------------------o

Table 39.2 Sign Relation for a Succession of HO Signs (2)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` x ` ` ` ` | ` ` ` `s_1` ` ` ` | ` ` ` `...` ` ` ` |
| ` ` ` `s_1` ` ` ` | ` ` ` `s_2` ` ` ` | ` ` ` `...` ` ` ` |
| ` ` ` `s_2` ` ` ` | ` ` ` `s_3` ` ` ` | ` ` ` `...` ` ` ` |
| ` ` ` `...` ` ` ` | ` ` ` `...` ` ` ` | ` ` ` `...` ` ` ` |
o-------------------o-------------------o-------------------o

Jon Awbrey

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Higher Order Signs

HOS. Note 10

3.4.10. Higher Order Sign Relations: Examples (cont.)

The rest of the present Section (3.4.10) 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 by way
of seeing how the process works, the sign relations L(A) and L(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 preserves
the identification of the syntactic domain as !S! = !I! and contemplates
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 an 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.

Jon Awbrey

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Higher Order Signs

HOS. Note 11

3.4.10. Higher Order Sign Relations: Examples (cont.)

The remainder of this Section (3.4.10) 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 L(A) and L(B) that have disjoint sets of objects
and signs and thus have no reflective capacity at the outset. The status
of L(A) and L(B) as the "reflective origins" of a "reflective development"
is recalled by saying that L(A) and L(B) themselves are the "zeroth order
reflective extensions" of L(A) and L(B). In symbolic terms this can be
expressed by the equations, L(A) = Ref^0 (A) and L(B) = Ref^0 (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 L(A) = Ref^0 (A)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^i^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^i^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^u^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^u^> ` ` ` |
o-------------------o-------------------o-------------------o

Table 41. Reflective Origin L(B) = Ref^0 (B)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^u^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^u^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^i^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^i^> ` ` ` |
o-------------------o-------------------o-------------------o

Tables 42 and 43 show one way that the sign relations L(A) and L(B) can
be extended in a reflective sense through the use of quotational devices,
yielding the "first order reflective extensions", Ref^1 (A) and Ref^1 (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^>.

Table 42. Higher Ascent Sign Relation Ref^1 (A)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^i^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^i^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^u^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^u^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` <^A^> ` ` ` | ` ` `<<^A^>>` ` ` | ` ` `<<^A^>>` ` ` |
| ` ` ` <^B^> ` ` ` | ` ` `<<^B^>>` ` ` | ` ` `<<^B^>>` ` ` |
| ` ` ` <^i^> ` ` ` | ` ` `<<^i^>>` ` ` | ` ` `<<^i^>>` ` ` |
| ` ` ` <^u^> ` ` ` | ` ` `<<^u^>>` ` ` | ` ` `<<^u^>>` ` ` |
o-------------------o-------------------o-------------------o

Table 43. Higher Ascent Sign Relation Ref^1 (B)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^u^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^u^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^i^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^i^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` <^A^> ` ` ` | ` ` `<<^A^>>` ` ` | ` ` `<<^A^>>` ` ` |
| ` ` ` <^B^> ` ` ` | ` ` `<<^B^>>` ` ` | ` ` `<<^B^>>` ` ` |
| ` ` ` <^i^> ` ` ` | ` ` `<<^i^>>` ` ` | ` ` `<<^i^>>` ` ` |
| ` ` ` <^u^> ` ` ` | ` ` `<<^u^>>` ` ` | ` ` `<<^u^>>` ` ` |
o-------------------o-------------------o-------------------o

Notice that the semantic equivalences of nouns and pronouns referring to each
of the interpreters A and B 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 L(A) and L(B)
coincide in their reflective parts, since exactly the same triples are
being added to each set.

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 set-theoretic 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, a sign relation driven to grow by a reflective process
is forced to keep expanding. In particular, the process that is "freely"
suggested by the formation of Ref^1 (A) and Ref^1 (B) cannot reach closure
if it continues as indicated, without further constraints.

Jon Awbrey

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Higher Order Signs

HOS. Note 12

3.4.10. Higher Order Sign Relations: Examples (cont.)

Tables 44 and 45 present "HI extensions" of L(A) and L(B), respectively.
These are HO sign relations that are constructed by adding selections of
HI signs and their denoted objects to the underlying set of triples from
L(A) and L(B). One way of understanding these extensions is as follows.
The interpretive agents 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
a noun is used as a canonical form that refers indifferently to all of the
<o, s, i> triples in the sign relation so named, thus affording a pragmatic
way that a sign relation can refer to itself and to other sign relations.

Table 44. Higher Import Sign Relation HI^1 (A)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^i^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^i^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^u^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^u^> ` ` ` |
o-------------------o-------------------o-------------------o
| <A, <^A^>, <^A^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <A, <^A^>, <^i^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <A, <^i^>, <^A^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <A, <^i^>, <^i^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
o-------------------o-------------------o-------------------o
| <B, <^B^>, <^B^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <B, <^B^>, <^u^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <B, <^u^>, <^B^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <B, <^u^>, <^u^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
o-------------------o-------------------o-------------------o
| <A, <^A^>, <^A^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <A, <^A^>, <^u^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <A, <^u^>, <^A^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <A, <^u^>, <^u^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
o-------------------o-------------------o-------------------o
| <B, <^B^>, <^B^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <B, <^B^>, <^i^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <B, <^i^>, <^B^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <B, <^i^>, <^i^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
o-------------------o-------------------o-------------------o

Table 45. Higher Import Sign Relation HI^1 (B)
o-------------------o-------------------o-------------------o
| ` ` `Object ` ` ` | ` ` ` Sign` ` ` ` | ` Interpretant` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^A^> ` ` ` | ` ` ` <^u^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^A^> ` ` ` |
| ` ` ` ` A ` ` ` ` | ` ` ` <^u^> ` ` ` | ` ` ` <^u^> ` ` ` |
o-------------------o-------------------o-------------------o
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^B^> ` ` ` | ` ` ` <^i^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^B^> ` ` ` |
| ` ` ` ` B ` ` ` ` | ` ` ` <^i^> ` ` ` | ` ` ` <^i^> ` ` ` |
o-------------------o-------------------o-------------------o
| <A, <^A^>, <^A^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <A, <^A^>, <^i^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <A, <^i^>, <^A^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <A, <^i^>, <^i^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
o-------------------o-------------------o-------------------o
| <B, <^B^>, <^B^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <B, <^B^>, <^u^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <B, <^u^>, <^B^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
| <B, <^u^>, <^u^>> | ` ` ` <^A^> ` ` ` | ` ` ` <^A^> ` ` ` |
o-------------------o-------------------o-------------------o
| <A, <^A^>, <^A^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <A, <^A^>, <^u^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <A, <^u^>, <^A^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <A, <^u^>, <^u^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
o-------------------o-------------------o-------------------o
| <B, <^B^>, <^B^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <B, <^B^>, <^i^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <B, <^i^>, <^B^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
| <B, <^i^>, <^i^>> | ` ` ` <^B^> ` ` ` | ` ` ` <^B^> ` ` ` |
o-------------------o-------------------o-------------------o

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 HI^1 (A) and HI^1 (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 interpretive agents
A and B, insofar as their sign relations L(A) and L(B) are extended by
the new transactions in HI^1 (A) and HI^1 (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 HI^1 (A) and HI^1 (B) are not very discriminating
in their descriptions of the sign relations L(A) and L(B), since
they refer 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 interpretive agent little good to have the HI signs
for referring to triples of objects, signs, and interpretants if it does not
also possess the HA signs for referring to each triple's syntactic portions.
Consequently, the HO sign relations that one is most likely to observe in
practice are typically a mixed affair, 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 for the various components.

Jon Awbrey

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Higher Order Signs

HOS. Note 13

3.4.10. Higher Order Sign Relations: Examples (cont.)

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 the task of creating a CRE is to imagine
oneself beginning with a sign relation L c !O! x !S! x !I!, and to
consider its denotative component Den_L = L_OS c !O! x !S!. As a
general rule, one typically has a "naming function", call it "Nom",
that maps objects into signs:


    Nom c Den_L c !O! x !S!, 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 relatively natural
and "easy", perhaps on account of the "nominal" nature of signs
themselves.

Jon Awbrey

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Higher Order Signs

HOS. Note 14

3.4.10. Higher Order Sign Relations: Examples (cont.)

The effect of reflection on the original sign relation L c !O! x !S! x !I! 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:


    L' c !O!' x !S!' x !I!',

    where !O!' = !O! |_| !O!_1

    and !S!' = !S! |_| !S!_1.

In this construction !O!_1 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 !O!_1 c !O!'. The sign domain !S! is extended to
a new sign domain !S!' by the addition of these HO signs, namely, the
set !S!_1. Using the arch quotes (<^...^>), the mapping from !O!_1
to !S!_1 can be defined as follows:

    Nom_1 : !O!_1 -> !S!_1 such that Nom_1 : x ~> <^x^>.

Finally, the reflectively extended naming function
Nom' : !O!' -> !S!' is defined as Nom' = Nom |_| Nom_1.

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 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 Nom_1, 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.

Jon Awbrey

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Jon Awbrey


Registered: Feb 2004
Posts: 557

Higher Order Signs

HOS. Note 15

3.4.10. Higher Order Sign Relations: Examples (concl.)

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 allied 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 required 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 be assessed only 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 to
a quoted character or when a variable is set equal to a quoted character,
as in the following sequence of Pascal expressions:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` const ` ` comma = ',' ; ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` var x;` ` x := comma` ; ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o

In this passage, the sign <^comma^> is made to denote whatever it is that
the sign <^','^> denotes, and the variable x is then set equal to this value.

Jon Awbrey

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