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# Inquiry Driven Systems • Part 15

Author: Jon Awbrey

## Reflective Interpretive Frameworks (concl.)

### Prospective Indices : Pointers to Future Work

In the effort to unify dynamical, connectionist, and symbolic approaches to intelligent systems, indices supply important stepping stones between the sorts of signs that remain bound to circumscribed theaters of action and the kinds of signs that can function globally as generic symbols. Current technology presents an array of largely accidental discoveries that have been brought into being for implementing indexical systems. Bringing systematic study to bear on this variety of accessory devices and trying to discern within the wealth of incidental features their essential principles and effective ingredients could help to improve the traction this form of bridge affords.

In the points where this project addresses work on the indexical front, a primary task is to show how the actual connections promised by the definition of indexical signs can be translated into system-theoretic terms and implemented by means of the class of dynamic connections that can persist in realistic systems.

An offshoot of this investigation would be to explore how indices like pointer variables could be realized within “connectionist” systems. There is no reason in principle why this cannot be done, but I think that pragmatic reasons and practical success will force the contemplation of higher orders of connectivity than those currently fashioned in two-dimensional arrays of connections. To be specific, further advances will require the generative power of genuinely triadic relations to be exploited to the fullest possible degree.

To avert one potential misunderstanding of what this entails, computing with triadic relations is not really a live option unless the algebraic tools and logical calculi needed to do so are developed to greater levels of facility than they are at present. Merely officiating over the storage of “dead letters” in higher dimensional arrays will not do the trick. Turning static sign relations into the orders of dynamic sign processes that can support live inquiries will demand new means of representation and new methods of computation.

To fulfill their intended roles, a formal calculus for sign relations and the associated implementation must be able to address and restore the full dimensionalities of the existential and social matrices in which inquiry takes place. Informational constraints that define objective situations of interest need to be freed from the locally linear confines of the “dia-matrix” and reposted within the realm of the “tri-matrix”, that is, reconstituted in a manner that allows critical reflection on their form and content.

The descriptive and conceptual architectures needed to frame this task must allow space for interlacing forms of “open work”, projects that anticipate the desirability of higher order relations and build in the capability for higher order reflections at the very beginning, and do not merely hope against hope to arrange these capacities as afterthoughts.

### Dynamic and Evaluative Frameworks

The sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ are lacking in several dimensions of realistic properties that would ordinarily be more fully developed in the kinds of sign relations that are found to be involved in inquiry. This section initiates a discussion of two such dimensions, the dynamic and the evaluative aspects of sign relations, and it treats the materials that are organized along these lines at two broad levels, either within or between particular examples of sign relations.

1. The dynamic dimension deals with change. Thus, it details the forms of diversity that sign relations distribute in a temporal process. It is concerned with the transitions that take place from element to element within a sign relation and also with the changes that take place from one whole sign relation to another, thereby generating various types and levels of sign process.
2. The evaluative dimension deals with goals. Thus, it details the forms of diversity that sign relations contribute to a definite purpose. It is concerned with the comparisons that can be made on a scale of values between the elements within a sign relation and also between whole sign relations themselves, with a view toward deciding which is better for a designated purpose.

At the primary level of analysis, one is concerned with the application of these two dimensions within particular sign relations. At every subsequent level of analysis, one deals with the dynamic transitions and evaluative comparisons that can be contemplated between particular sign relations. In order to cover all these dimensions, types, and levels of diversity in a unified way, there is need for a substantive term that can allow one to indicate any of the above objects of discussion and thought — including elements of sign relations, particular sign relations, and states of systems — and to regard it as an “object, sign, or state in a certain stage of construction”. I will use the word station for this purpose.

In order to organize the discussion of these two dimensions, both within and between particular sign relations, and to coordinate their ordinary relation to each other in practical situations, it pays to develop a combined form of dynamic evaluative framework (DEF), similar in design and utility to the objective frameworks set up earlier.

A dynamic evaluative framework (DEF) encompasses two dimensions of comparison between stations:

1. A dynamic dimension, as swept out by a process of changing stations, permits comparison between stations in terms of before and after on a scale of temporal order.

A terminal station on a dynamic dimension is called a stable station.

2. An evaluative dimension permits comparison between stations on a scale of values.

A terminal station on an evaluative dimension is called a canonical station or a standard station.

A station that is both stable and standard is called a normal station.

Consider the following analogies or correspondences that exist between different orders of sign relational structure:

1. Just as a sign represents its object and becomes associated with more or less equivalent signs in the minds of interpretive agents, the corpus of signs that embodies a SOI represents in a collective way its own proper object, intended objective, or try at objectivity (TAO).
2. Just as the relationship of a sign to its semantic objects and interpretive associates can be formalized within a single sign relation, the relation of a dynamically changing SOI to its reference environment, developmental goals, and desired characteristics of interpretive performance can be formalized by means of a higher order sign relation, one that further establishes a grounds of comparison for relating the growing SOI, not only to its former and future selves, but to a diverse company of other SOIs.

From an outside perspective the distinction between a sign and its object is usually regarded as obvious, though agents operating in the thick of a SOI often act as though they cannot see the difference. Nevertheless, as a rule in practice, a sign is not a good thing to be confused with its object. Even in the rare and usually controversial cases where an identity of substance is contemplated, usually only for the sake of argument, there is still a distinction of roles to be maintained between the sign and its object. Just so, …

Although there are aspects of inquiry processes that operate within the single sign relation, the characteristic features of inquiry do not come into full bloom until one considers the whole diversity of dynamically developing sign relations. Because it will be some time before this discussion acquires the formal power it needs to deal with higher order sign relations, these issues will need to be treated on an informal basis as they arise, and often in cursory and ad hoc manner.

### Elective and Motive Forces

The ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ example, in the fragmentary aspects of its sign relations presented so far, is unrealistic in its simplification of semantic issues, lacking a full development of many kinds of attributes that almost always become significant in situations of practical interest. Just to mention two related features of importance to inquiry that are missing from this example, there is no sense of directional process and no dimension of differential value defined either within or between the semantic equivalence classes.

When there is a clear sense of dynamic tendency or purposeful direction driving the passage from signs to interpretants in the connotative project of a sign relation, then the study moves from sign relations, statically viewed, to genuine sign processes. In the pragmatic theory of signs, such processes are usually dignified with the name semiosis and their systematic investigation is called semiotics.

Further, when this dynamism or purpose is consistent and confluent with a differential value system defined on the syntactic domain, then the sign process in question becomes a candidate for the kind of clarity-gaining, canon-seeking process, capable of supporting learning and reasoning, that I classify as an inquiry driven system.

There is a mathematical turn of thought that I will often take in discussing these kinds of issues. Instead of saying that a system has no attribute of a particular type, I will say that it has the attribute, but in a degenerate or trivial sense. This is merely a strategy of classification that allows one to include null cases in a taxonomy and to make use of continuity arguments in passing from case to case in a class of examples. Viewed in this way, each of the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ can be taken to exhibit a trivial dynamic process and a trivial standard of value defined on the syntactic domain.

### Sign Processes : A Start

To articulate the dynamic aspects of a sign relation, one can interpret it as determining a discrete or finite state transition system. In the usual ways of doing this, the states of the system are given by the elements of the syntactic domain, while the elements of the object domain correspond to input data or control parameters that affect transitions from signs to interpretant signs in the syntactic state space.

Working from these principles alone, there are numerous ways that a plausible dynamics can be invented for a given sign relation. I will concentrate on two principal forms of dynamic realization, or two ways of interpreting and augmenting sign relations as sign processes.

One form of realization lets each element of the object domain ${\displaystyle O}$ correspond to the observed presence of an object in the environment of the systematic agent. In this interpretation, the object ${\displaystyle x}$ acts as an input datum that causes the system ${\displaystyle Y}$ to shift from whatever sign state it happens to occupy at a given moment to a random sign state in ${\displaystyle [x]_{Y}.}$ Expressed in a cognitive vein, ${\displaystyle {}^{\backprime \backprime }Y~\mathrm {notes} ~x{}^{\prime \prime }.}$

Another form of realization lets each element of the object domain ${\displaystyle O}$ correspond to the autonomous intention of the systematic agent to denote an object, achieve an objective, or broadly speaking to accomplish any other purpose with respect to an object in its domain. In this interpretation, the object ${\displaystyle x}$ is a control parameter that brings the system ${\displaystyle Y}$ into line with realizing a target set ${\displaystyle [x]_{Y}.}$

Tables 78 and 79 show how the sign relations for ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ can be filled out as finite state processes in conformity with the interpretive principles just described. Rather than letting the actions go undefined for some combinations of inputs in ${\displaystyle O}$ and states in ${\displaystyle S,}$ transitions have been added that take the interpreters from whatever else they might have been thinking about to the semantic equivalence classes of their objects. In either modality of realization, cognitive-oriented or control-oriented, the abstract structure of the resulting sign process is exactly the same.

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\end{matrix}}}$

Treated in accord with these interpretations, the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ constitute partially degenerate cases of dynamic processes, in which the transitions are totally non-deterministic up to semantic equivalence classes but still manage to preserve those classes. Whether construed as present observation or projective speculation, the most significant feature to note about a sign process is how the contemplation of an object or objective leads the system from a less determined to a more determined condition.

On reflection, one observes that these processes are not completely trivial since they preserve the structure of their semantic partitions. In fact, each sign process preserves the entire topology — the family of sets closed under finite intersections and arbitrary unions — that is generated by its semantic equivalence classes. These topologies, ${\displaystyle \mathrm {Top} ({\text{A}})}$ and ${\displaystyle \mathrm {Top} ({\text{B}}),}$ can be viewed as partially ordered sets, ${\displaystyle \mathrm {Poset} ({\text{A}})}$ and ${\displaystyle \mathrm {Poset} ({\text{B}}),}$ by taking the inclusion ordering ${\displaystyle (\subseteq )}$ as ${\displaystyle (\leq ).}$ For each of the interpreters ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}},}$ as things stand in their respective orderings ${\displaystyle \mathrm {Poset} ({\text{A}})}$ and ${\displaystyle \mathrm {Poset} ({\text{B}}),}$ the semantic equivalence classes of ${\displaystyle {}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}$ and ${\displaystyle {}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}$ are situated as intermediate elements that are incomparable to each other.

 ${\displaystyle {\begin{array}{lllll}\mathrm {Top} ({\text{A}})&=&\mathrm {Poset} ({\text{A}})&=&\{\varnothing ,\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\},\{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\},S\}.\\[6pt]\mathrm {Top} ({\text{B}})&=&\mathrm {Poset} ({\text{B}})&=&\{\varnothing ,\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\},\{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\},S\}.\end{array}}}$

In anticipation of things to come, these orderings are germinal versions of the kinds of semantic hierarchies that will be used in this project to define the ontologies, perspectives, or world views corresponding to individual interpreters.

When it comes to discussing the stability properties of dynamic systems, the sets that remain invariant under iterated applications of a process are called its attractors or basins of attraction.

Note. More care needed here. Strongly and weakly connected components of digraphs?

The dynamic realizations of the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ augment their semantic equivalence relations in an “attractive” way. To describe this additional structure, I introduce a set of graph-theoretical concepts and notations.

The attractor of ${\displaystyle x}$ in ${\displaystyle Y.}$

 ${\displaystyle Y~{\text{at}}~x~=~\mathrm {At} [x]_{Y}~=~[x]_{Y}\cup \{{\text{arcs into}}~[x]_{Y}\}.}$

In effect, this discussion of dynamic realizations of sign relations has advanced from considering semiotic partitions as partitioning the set of points in ${\displaystyle S}$ to considering attractors as partitioning the set of arcs in ${\displaystyle S\times I=S\times S.}$

### Reflective Extensions

This section takes up the topic of reflective extensions in a more systematic fashion, starting from the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ once again and keeping its focus within their vicinity, but exploring the space of nearby extensions in greater detail.

Tables 80 and 81 show one way that the sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ can be extended in a reflective sense through the use of quotational devices, yielding the first order reflective extensions, ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}}).}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }\\{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\end{matrix}}}$

The common world ${\displaystyle W}$ of the reflective extensions ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}})}$ is the totality of objects and signs they contain, namely, the following set of 10 elements.

 ${\displaystyle W=\{{\text{A}},{\text{B}},{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle },{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\}.}$

Raised angle brackets or supercilia ${\displaystyle ({}^{\langle }\ldots {}^{\rangle })}$ are here being used on a par with ordinary quotation marks ${\displaystyle ({}^{\backprime \backprime }\ldots {}^{\prime \prime })}$ to construct a new sign whose object is precisely the sign they enclose.

Regarded as sign relations in their own right, ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}})}$ are formed on the following relational domains.

 ${\displaystyle {\begin{array}{ccccl}O&=&O^{(1)}\cup O^{(2)}&=&\{{\text{A}},{\text{B}}\}~\cup ~\{{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle }\}\\[8pt]S&=&S^{(1)}\cup S^{(2)}&=&\{{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle }\}~\cup ~\{{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\}\\[8pt]I&=&I^{(1)}\cup I^{(2)}&=&\{{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle }\}~\cup ~\{{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle },{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\}\end{array}}}$

It may be observed that ${\displaystyle S}$ overlaps with ${\displaystyle O}$ in the set of first-order signs or second-order objects, ${\displaystyle S^{(1)}=O^{(2)},}$ exemplifying the extent to which signs have become objects in the new sign relations.

To discuss how the denotative and connotative aspects of a sign related are affected by its reflective extension it is useful to introduce a few abbreviations. For each sign relation ${\displaystyle L}$ in ${\displaystyle \{L_{\text{A}},L_{\text{B}}\}}$ the following operations may be defined.

 ${\displaystyle {\begin{array}{lllll}\mathrm {Den} ^{1}(L)&=&(\mathrm {Ref} ^{1}(L))_{SO}&=&\mathrm {proj} _{OS}(\mathrm {Ref} ^{1}(L))\\[6pt]\mathrm {Con} ^{1}(L)&=&(\mathrm {Ref} ^{1}(L))_{SI}&=&\mathrm {proj} _{SI}(\mathrm {Ref} ^{1}(L))\end{array}}}$

The dyadic components of sign relations can be given graph-theoretic representations, namely, as digraphs (directed graphs), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge ${\displaystyle (x,y)}$ is called an arc from point ${\displaystyle x}$ to point ${\displaystyle y,}$ and a self-loop ${\displaystyle (x,x)}$ is called a sling at ${\displaystyle x.}$

The denotative components ${\displaystyle \mathrm {Den} ^{1}(L_{\text{A}})}$ and ${\displaystyle \mathrm {Den} ^{1}(L_{\text{B}})}$ can be viewed as digraphs on the 10 points of the world set ${\displaystyle W.}$ The arcs of these digraphs are given as follows.

1. ${\displaystyle \mathrm {Den} ^{1}(L_{\text{A}})}$ has an arc from each point of ${\displaystyle [{\text{A}}]_{\text{A}}=\{{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle }\}}$ to ${\displaystyle {\text{A}}}$ and from each point of ${\displaystyle [{\text{B}}]_{\text{A}}=\{{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle }\}}$ to ${\displaystyle {\text{B}}.}$
2. ${\displaystyle \mathrm {Den} ^{1}(L_{\text{B}})}$ has an arc from each point of ${\displaystyle [{\text{A}}]_{\text{B}}=\{{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle }\}}$ to ${\displaystyle {\text{A}}}$ and from each point of ${\displaystyle [{\text{B}}]_{\text{B}}=\{{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle }\}}$ to ${\displaystyle {\text{B}}.}$
3. In the parts added by reflective extension ${\displaystyle \mathrm {Den} ^{1}(L_{\text{A}})}$ and ${\displaystyle \mathrm {Den} ^{1}(L_{\text{B}})}$ both have arcs from ${\displaystyle {}^{\langle }s{}^{\rangle }}$ to ${\displaystyle s,}$ for each ${\displaystyle s\in S^{(1)}.}$

Taken as transition digraphs, ${\displaystyle \mathrm {Den} ^{1}(L_{\text{A}})}$ and ${\displaystyle \mathrm {Den} ^{1}(L_{\text{B}})}$ summarize the upshots, end results, or effective steps of computation that are involved in the respective evaluations of signs in ${\displaystyle S}$ by ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}}).}$

The connotative components ${\displaystyle \mathrm {Con} ^{1}(L_{\text{A}})}$ and ${\displaystyle \mathrm {Con} ^{1}(L_{\text{B}})}$ can be viewed as digraphs on the eight points of the syntactic domain ${\displaystyle S.}$ The arcs of these digraphs are given as follows.

1. ${\displaystyle \mathrm {Con} ^{1}(L_{\text{A}})}$ inherits from ${\displaystyle L_{\text{A}}}$ the structure of a semiotic equivalence relation on ${\displaystyle S^{(1)},}$ having a sling on each point of ${\displaystyle S^{(1)},}$ arcs in both directions between ${\displaystyle {}^{\langle }{\text{A}}{}^{\rangle }}$ and ${\displaystyle {}^{\langle }{\text{i}}{}^{\rangle },}$ and arcs in both directions between ${\displaystyle {}^{\langle }{\text{B}}{}^{\rangle }}$ and ${\displaystyle {}^{\langle }{\text{u}}{}^{\rangle }.}$ The reflective extension ${\displaystyle \mathrm {Ref} ^{1}(L_{\text{A}})}$ adds a sling on each point of ${\displaystyle S^{(2)},}$ creating a semiotic equivalence relation on ${\displaystyle S.}$
2. ${\displaystyle \mathrm {Con} ^{1}(L_{\text{B}})}$ inherits from ${\displaystyle L_{\text{B}}}$ the structure of a semiotic equivalence relation on ${\displaystyle S^{(1)},}$ having a sling on each point of ${\displaystyle S^{(1)},}$ arcs in both directions between ${\displaystyle {}^{\langle }{\text{A}}{}^{\rangle }}$ and ${\displaystyle {}^{\langle }{\text{u}}{}^{\rangle },}$ and arcs in both directions between ${\displaystyle {}^{\langle }{\text{B}}{}^{\rangle }}$ and ${\displaystyle {}^{\langle }{\text{i}}{}^{\rangle }.}$ The reflective extension ${\displaystyle \mathrm {Ref} ^{1}(L_{\text{B}})}$ adds a sling on each point of ${\displaystyle S^{(2)},}$ creating a semiotic equivalence relation on ${\displaystyle S.}$

Taken as transition digraphs, ${\displaystyle \mathrm {Con} ^{1}(L_{\text{A}})}$ and ${\displaystyle \mathrm {Con} ^{1}(L_{\text{B}})}$ highlight the associations between signs in ${\displaystyle \mathrm {Ref} ^{1}(L_{\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}(L_{\text{B}}),}$ respectively.

The semiotic equivalence relation given by ${\displaystyle \mathrm {Con} ^{1}(L_{\text{A}})}$ for interpreter ${\displaystyle {\text{A}}}$ has the following semiotic equations.

 ${\displaystyle [{}^{\langle }{\text{A}}{}^{\rangle }]_{\text{A}}}$ ${\displaystyle =}$ ${\displaystyle [{}^{\langle }{\text{i}}{}^{\rangle }]_{\text{A}}}$ ${\displaystyle [{}^{\langle }{\text{B}}{}^{\rangle }]_{\text{A}}}$ ${\displaystyle =}$ ${\displaystyle [{}^{\langle }{\text{u}}{}^{\rangle }]_{\text{A}}}$ or ${\displaystyle {}^{\langle }{\text{A}}{}^{\rangle }}$ ${\displaystyle =_{\text{A}}}$ ${\displaystyle {}^{\langle }{\text{i}}{}^{\rangle }}$ ${\displaystyle {}^{\langle }{\text{B}}{}^{\rangle }}$ ${\displaystyle =_{\text{A}}}$ ${\displaystyle {}^{\langle }{\text{u}}{}^{\rangle }}$

These equations induce the following semiotic partition.

 ${\displaystyle \{\{{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle }\},\{{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle }\},\{{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }\},\{{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }\},\{{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }\},\{{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\}\}.\!}$

The semiotic equivalence relation given by ${\displaystyle \mathrm {Con} ^{1}(L_{\text{B}})}$ for interpreter ${\displaystyle {\text{B}}}$ has the following semiotic equations.

 ${\displaystyle [{}^{\langle }{\text{A}}{}^{\rangle }]_{\text{B}}}$ ${\displaystyle =}$ ${\displaystyle [{}^{\langle }{\text{u}}{}^{\rangle }]_{\text{B}}}$ ${\displaystyle [{}^{\langle }{\text{B}}{}^{\rangle }]_{\text{B}}}$ ${\displaystyle =}$ ${\displaystyle [{}^{\langle }{\text{i}}{}^{\rangle }]_{\text{B}}}$ or ${\displaystyle {}^{\langle }{\text{A}}{}^{\rangle }}$ ${\displaystyle =_{\text{B}}}$ ${\displaystyle {}^{\langle }{\text{u}}{}^{\rangle }}$ ${\displaystyle {}^{\langle }{\text{B}}{}^{\rangle }}$ ${\displaystyle =_{\text{B}}}$ ${\displaystyle {}^{\langle }{\text{i}}{}^{\rangle }}$

These equations induce the following semiotic partition.

 ${\displaystyle \{\{{}^{\langle }{\text{A}}{}^{\rangle },{}^{\langle }{\text{u}}{}^{\rangle }\},\{{}^{\langle }{\text{B}}{}^{\rangle },{}^{\langle }{\text{i}}{}^{\rangle }\},\{{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }\},\{{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }\},\{{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }\},\{{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }\}\}.\!}$

Notice that the semiotic equivalences of nouns and pronouns for each interpreter do not extend to equivalences of their second-order signs, exactly as demanded by the literal character of quotations. Moreover, the new sign relations for interpreters ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ coincide in their reflective parts, since exactly the same triples are added to each set.

There are many ways to extend sign relations in an effort to increase their reflective capacities. The implicit goal of a reflective project is to achieve reflective closure, ${\displaystyle S\subseteq O,}$ where every sign is an object.

Considered as reflective extensions, there is nothing unique about the constructions of ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}})}$ but their common pattern of development illustrates a typical approach toward reflective closure. In a sense it epitomizes the project of free, naive, or uncritical reflection, since continuing this mode of production to its closure would generate an infinite sign relation, passing through infinitely many higher orders of signs, but without examining critically to what purpose the effort is directed or evaluating alternative constraints that might be imposed on the initial generators toward this end.

At first sight it seems as though the imposition of reflective closure has multiplied a finite sign relation into an infinite profusion of highly distracting and largely redundant signs, all by itself and all in one step. But this explosion of orders happens only with the complicity of another requirement, that of deterministic interpretation.

There are two types of non-determinism, denotative and connotative, that can affect a sign relation.

1. A sign relation ${\displaystyle L}$ has a non-deterministic denotation if its dyadic component ${\displaystyle {L_{SO}}}$ is not a function ${\displaystyle L_{SO}:S\to O,}$ in other words, if there are signs in ${\displaystyle S}$ with missing or multiple objects in ${\displaystyle O.}$
2. A sign relation ${\displaystyle L}$ has a non-deterministic connotation if its dyadic component ${\displaystyle L_{SI}}$ is not a function ${\displaystyle L_{SI}:S\to I,}$ in other words, if there are signs in ${\displaystyle S}$ with missing or multiple interpretants in ${\displaystyle I.}$ As a rule, sign relations are rife with this variety of non-determinism, but it is usually felt to be under control so long as ${\displaystyle L_{SI}}$ remains close to being an equivalence relation.

Thus, it is really the denotative type of indeterminacy that is felt to be a problem in this context.

The next two pairs of reflective extensions demonstrate that there are ways of achieving reflective closure that do not generate infinite sign relations.

As a flexible and fairly general strategy for describing reflective extensions, it is convenient to take the following tack. Given a syntactic domain ${\displaystyle S,}$ there is an independent formal language ${\displaystyle F=F(S)=S\langle {}^{\langle \rangle }\rangle ,}$ called the free quotational extension of ${\displaystyle S,}$ that can be generated from ${\displaystyle S}$ by embedding each of its signs to any depth of quotation marks. Within ${\displaystyle F,}$ the quoting operation can be regarded as a syntactic generator that is inherently free of constraining relations. In other words, for every ${\displaystyle s\in S,}$ the sequence ${\displaystyle s,{}^{\langle }s{}^{\rangle },{}^{\langle \langle }s{}^{\rangle \rangle },\ldots }$ contains nothing but pairwise distinct elements in ${\displaystyle F}$ no matter how far it is produced. The set ${\displaystyle F(s)=s\langle {}^{\langle \rangle }\rangle \subseteq F}$ that collects the elements of this sequence is called the subset of ${\displaystyle F}$ generated from ${\displaystyle s}$ by quotation.

Against this background, other varieties of reflective extension can be specified by means of semantic equations that are considered to be imposed on the elements of ${\displaystyle F.}$ Taking the reflective extensions ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}})}$ as the first orders of a “free” project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences ${\displaystyle \mathrm {Ref} ^{n}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{n}({\text{B}}).}$

A variant pair of reflective extensions, ${\displaystyle \mathrm {Ref} ^{1}({\text{A}}|E_{1})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}}|E_{1}),}$ is presented in Tables 82 and 83, respectively. These are identical to the corresponding free variants, ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}}),}$ with the exception of those entries that are constrained by the following system of semantic equations.

 ${\displaystyle {\begin{matrix}E_{1}:&{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }={}^{\langle }{\text{A}}{}^{\rangle },&{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }={}^{\langle }{\text{B}}{}^{\rangle },&{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }={}^{\langle }{\text{i}}{}^{\rangle },&{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }={}^{\langle }{\text{u}}{}^{\rangle }.\end{matrix}}}$

This has the effect of making all levels of quotation equivalent.

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$

Another pair of reflective extensions, ${\displaystyle \mathrm {Ref} ^{1}({\text{A}}|E_{2})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}}|E_{2}),}$ is presented in Tables 84 and 85, respectively. These are identical to the corresponding free variants, ${\displaystyle \mathrm {Ref} ^{1}({\text{A}})}$ and ${\displaystyle \mathrm {Ref} ^{1}({\text{B}}),}$ except for the entries constrained by the following semantic equations.

 ${\displaystyle {\begin{matrix}E_{2}:&{}^{\langle \langle }{\text{A}}{}^{\rangle \rangle }={\text{A}},&{}^{\langle \langle }{\text{B}}{}^{\rangle \rangle }={\text{B}},&{}^{\langle \langle }{\text{i}}{}^{\rangle \rangle }={\text{i}},&{}^{\langle \langle }{\text{u}}{}^{\rangle \rangle }={\text{u}}.\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{B}}\\{\text{A}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{B}}\\{\text{A}}\\{\text{B}}\end{matrix}}}$

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\\{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\langle }{\text{A}}{}^{\rangle }\\{}^{\langle }{\text{B}}{}^{\rangle }\\{}^{\langle }{\text{i}}{}^{\rangle }\\{}^{\langle }{\text{u}}{}^{\rangle }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{B}}\\{\text{B}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{B}}\\{\text{B}}\\{\text{A}}\end{matrix}}}$

By calling attention to their intended status as semantic equations, meaning that signs are being set equal in the semantic equivalence classes they inhabit or the objects they denote, I hope to emphasize that these equations are able to say something significant about objects.

Question. Redo ${\displaystyle F(S)}$ over ${\displaystyle W}$? Use ${\displaystyle W_{F}=O\cup F}$?

### Reflections on Closure

The previous section dealt with a formal operation that was dubbed reflection and found that it was closely associated with the device of quotation that makes it possible to treat signs as objects by making or finding other signs that refer to them. Clearly, an ability to take signs as objects is one component of a cognitive capacity for reflection. But a genuine and less superficial species of reflection can do more than grasp just the isolated signs and the separate interpretants of the thinking process as objects — it can pause the fleeting procession of signs upon signs and seize their generic patterns of transition as valid objects of discussion. This involves the conception and composition of not just higher order signs but also higher type signs, orders of signs that aspire to catch whole sign relations up in one breath.

### Intelligence ⇒ Critical Reflection

It is just at this point that the discussion of sign relations is forced to contemplate the prospects of intelligent interpretation. For starters, I consider an intelligent interpreter to be one that can pursue alternative interpretations of a sign or text and pick one that makes sense. If an interpreter can find all of the most sensible interpretations and order them according to a scale of meaningfulness, but without losing the time required to act on their import, then so much the better.

Intelligent interpreters are a centrally important species of intelligent agents in general, since hardly any intelligent action at all can be taken without the ability to interpret signs and texts, even if read only in the sense of “the text of nature”. In other words, making sense of dubious signs is a central component of all sensible action.

Thus, I regard the determining trait of intelligent agency to be its response to non-deterministic situations. Agents that find themselves at junctures of unavoidable uncertainty are required by objective features of the situation to gather together the available options and select among the multitude of possibilities a few choices that further their active purposes.

Reflection enables an interpreter to stand back from signs and view them as objects, that is, as objective possibilities for choice to be followed up in a critical and experimental fashion rather than pursued as automatic reactions whose habitual connections cannot be questioned.

The mark of an intelligent interpreter that is relevant in this context is the ability to face (encounter, countenance) a non-deterministic juncture of choices in a sign relation and to respond to it as such with actions appropriate to the uncertain nature of the situation.

[Variants]

An intelligent interpreter is one that can follow up several different interpretations at once, experimenting with the denotations and connotations that are available in a non-deterministic sign relation, …

An intelligent interpreter is one that can face a situation of non deterministic choice and choose an interpretation (denotation or connotation) that fits the objective and syntactic context.

An intelligent interpreter is one that can deal with non-deterministic situations, that is, one that can follow up several lines of possible meaning for signs and read between the lines to pick out meanings that are sensitive to both the objective situation and the syntactic context of interpretation.

An intelligent interpreter is one that can reflect critically on the process of interpretation. This involves a capacity for standing back from signs and interpretants and viewing them as objects, seeing their connections as objective possibilities for choice, to be compared with each other and tested against the objective and syntactic contexts, rather than taking the usual paths responding in a reflexive manner with the …

To do this it is necessary to interrupt the customary connections and favored associations of signs and interpretants in a sign relation and to consider a plurality of interpretations, not merely to pursue many lines of meaning in a parallel or experimental fashion, but to question seriously whether anything at all is meant by a sign.

… follow up alternatives in an experimental fashion, evaluate choices with a sensitivity to both the objective and syntactic contexts.

The mark of intelligence that is relevant to this context is the ability to comprehend a non deterministic situation of choice precisely as it is, …

If a species of determinism is nevertheless expected, then the extra measure of determination must be attributed to a worldly context of objects and signs extending beyond those taken into account by the sign relation in question, or else to powers of choice as yet unformalized in the character of interpreters.

This means that the recursions involved in the process of interpretation, besides having recourse to the inner resources of interpreters, will also recur to interfaces with objective situations and syntactic contexts. Interpretation, to be intelligent, must have the capacity to address the full scope of objects and signs and must be given the room to operate interactively with everything up to and including the undetermined horizons of the external world.

On the whole throughout this project, the “meta” issue that has been raised here will be treated at three different levels of sophistication.

1. The way I have chosen to deal with this issue in the present case is not by injecting more features of the informal discussion into the dialogue of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}},}$ but by trying to imagine how agents like ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ might be enabled to reflect on these aspects of their own discussion.
2. In the series of examples that I will use to develop further aspects of the ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ dialogue, several different ways of extending the sign relations for ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ will be explored. The most pressing task is to capture facts of the following sort.

 ${\displaystyle {\text{A}}}$ knows that ${\displaystyle {\text{B}}}$ uses ${\displaystyle {}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}$ to denote ${\displaystyle {\text{B}}}$ and ${\displaystyle {}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}$ to denote ${\displaystyle {\text{A}}.}$ ${\displaystyle {\text{B}}}$ knows that ${\displaystyle {\text{A}}}$ uses ${\displaystyle {}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}$ to denote ${\displaystyle {\text{A}}}$ and ${\displaystyle {}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}$ to denote ${\displaystyle {\text{B}}.}$

Toward this aim, I will present a variety of constructions for motivating extended, indexed, or situated sign relations, all designed to meet the following requirements.

1. To incorporate higher components of “meta-knowledge” about language use as it works in a community of interpreters, in reality the most basic ingredients of pragmatic competence.
2. To amalgamate the fragmentary sign relations of individual interpreters into “broader-minded” sign relations, in the use and understanding of which a plurality of agents can share.

Work at this level of concrete investigation will proceed in an incremental fashion, augmenting the discussion of A and B with features of increasing interest and relevance to inquiry. The plan for this series of developments is as follows.

1. I start by gathering materials and staking out intermediate goals for investigation. This involves making a tentative foray into ways that dimensions of directed change and motivated value can be added to the sign relations initially given for ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}.}$
2. With this preparation, I return to the dialogue of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ and pursue ways of integrating their independent selections of information into a unified system of interpretation.
1. First, I employ the sign relations ${\displaystyle L_{\text{A}}}$ and ${\displaystyle L_{\text{B}}}$ to illustrate two basic kinds of set theoretic merges, the ordinary or simple union and the indexed or situated union of extensional relations. On review, both forms of combination are observed to fall short of what is needed to constitute the desired characteristics of a shared sign relation.
2. Next, I present two other ways of extending the sign relations ${\displaystyle L_{\text{A}}}$ and ${\displaystyle L_{\text{B}}}$ into a common system of interpretation. These extensions succeed in capturing further aspects of what interpreters know about their shared language use. Although motivated on different grounds, the alternative constructions that develop coincide in exactly the same abstract structure.
3. As this project begins to take on sign relations that are complex enough to convey the impression of genuine inquiry processes, a fuller explication of this issue will become mandatory. Eventually, this will demand a concept of higher-order sign relations, whose objects, signs, and interpretants can all be complete sign relations in their own rights.

In principle, the successive grades of complexity enumerated above could be ascended in a straightforward way, if only the steps did not go straight up the cliffs of abstraction. As always, the kinds of intentional objects that are the toughest to face are those whose realization is so distant that even the gear needed to approach their construction is not yet in existence.

### Mutually Intelligible Codes

Before this complex of relationships can be formalized in much detail, I must introduce linguistic devices for generating higher order signs, used to indicate other signs, and situated signs, indexed by the names of their users, their contexts of use, and other types of information incidental to their usage in general. This leads to the consideration of systems of interpretation (SOIs) that maintain recursive mechanisms for naming everything within their purview. This “nominal generosity” gives them a new order of generative capacity, producing a sufficient number of distinctive signs to name all the objects and then name the names that are needed in a given discussion.

Symbolic systems for quoting inscriptions and ascribing quotations are associated in metamathematics with gödel numberings of formal objects, enumerative functions that provide systematic but ostensibly arbitrary reference numbers for the signs and expressions in a formal language. Assuming these signs and expressions denote anything at all, their formal enumerations become the codes of formal objects, just as programs taken literally are code names for certain mathematical objects known as computable functions. Partial forms of specification notwithstanding, these codes are the only complete modes of representation that formal objects can have in the medium of mechanical activity.

In the dialogue of ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ there happens to be an exact coincidence between signs and states. That is, the states of the interpretive systems ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ are not distinguished from the signs in ${\displaystyle S}$ that are imagined to be mediating, moment by moment, the attentions of the interpretive agents ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ toward their respective objects in ${\displaystyle O.}$ So the question arises: Is this identity bound to be a general property of all useful sign relations, or is it only a degenerate feature occurring by chance or unconscious design in the immediate example?

To move toward a resolution of this question I reason as follows. In one direction, it seems obvious that a sign in use (SIU) by a particular interpreter constitutes a component of that agent's state. In other words, the very notion of an identifiable SIU refers to numerous instances of a particular interpreter's state that share in the abstract property of being such instances, whether or not anyone can give a more concise or illuminating characterization of the concept under which these momentary states are gathered. Conversely, it is at least conceivable that the whole state of a system, constituting its transitory response to the entirety of its environment, history, and goals, can be interpreted as a sign of something to someone. In sum, there remains an outside chance of signs and states being precisely the same things, since nothing precludes the existence of an interpretive framework (IF) that could make it so.

Still, if the question about the distinction or coincidence between signs and states is restricted to the domains where existential realizations are conceivable, no matter whether in biological or computational media, then the prerequisites of the task become more severe, due to the narrower scope of materials that are admitted to answer them. In focusing on this arena the problem is threefold:

1. The crucial point is not just whether it is possible to imagine an ideal SOI, an external perspective or an independent POV, for which all states are signs, but whether this is so for the prospective SOI of the very agent that passes through these states.
2. To what extent can the transient states and persistent conduct of each agent in a community of interpretation take on a moderately public and objective aspect in relation to the other participants?
3. How far in this respect, in the common regard for this species of outward demeanor, can each agent's behavior act as a sign of genuine objects in the eyes of other interpreters?

The special task of a nuanced hermeneutic approach to computational interpretation is to realize the relativity of all formal codes to their formal coders, and to seek ways of facilitating mutual intelligibility among interpreters whose internal codes can be thoroughly private, synchronistically keyed to external events, and even a bit idiosyncratic.

Ultimately, working through this maze of “meta” questions, as posed on the tentative grounds of the present project, leads to a question about the logical reference frames or metamathematical coordinate systems that are supposed to distinguish “objective” from “symbolic” entities and are imagined to discriminate a range of gradations along their lines. The question is: Whether any gauge of objectivity or scale of virtuality has invariant properties discoverable by all independent interpreters, or whether all is vanity and inane relativism, and everything concerning a subjective point of view is sheer caprice?

Thus, the problem of mutual intelligibility turns on the question of common significance: How can there be signs that are truly public, when the most natural signs that distinct agents can know, their own internal states, have no guarantee and very little likelihood of being related in systematically fathomable ways? As a partial answer to this, I am willing to contemplate certain forms of pre-established harmony, like the common evolution of a biological species or the shared culture of an interpretive community, but my experience has been that harmony, once established, quickly corrupts unless active means are available to maintain it. So there still remains the task of identifying these means. With or without the benefit of a prior consensus, or the assumption of an initial but possibly fragile equilibrium, an explanation of robust harmony must detail the modes of maintaining communication that enable coordinated action to persist in the meanest of times.

The formal character of these questions, in the potential complexities that can be forced on contemplation in the pursuit of their answers, is independent of the species of interpreters that are chosen for the termini of comparison, whether person to person, person to computer, or computer to computer. As always, the truth of this kind of thesis is formal, all too formal. What it brings is a new refrain of an old motif: Are there meaningful, if necessarily formal series of analogies that can be strung from the patterns of whizzing electrons and humming protons, whose controlled modes of collective excitation form and inform the conducts of computers, all the way to the rather different patterns of wizened electrons and humbled protons, whose deliberate energies of communal striving substantiate the forms of life known to be intelligible?

A full consideration of the geometries available for the spaces in which these levels of reflective abstraction are commonly imagined to reside leads to the conclusion that familiar distinctions of “top down” versus “bottom up” are being taken for granted in an arena that has not even been established to be orientable. Thus, it needs to be recognized that the distinction between objects and signs is relative to a definite system of interpretation. The pragmatic theory of signs is designed, in part, precisely to deal with the circumstance that thoroughly objective states of systems can be signs of each other, undermining any pretended distinction between objects and signs that one might propose to draw on essential grounds.

From now on, I will reuse the ancient term gnomon in a technical sense to refer to the gödel numbers or code names of formal objects. In other words, a gnomon is a gödel numbering or enumeration function that maps a domain of objects into a domain of signs, ${\displaystyle \mathrm {Gno} :O\to S.}$ When the syntactic domain ${\displaystyle S}$ is contained within the object domain ${\displaystyle O,}$ then the part of the gnomon that maps ${\displaystyle S}$ into ${\displaystyle S,}$ providing names for signs and expressions, is usually regarded as a quoting function.

In the pluralistic contexts that go with pragmatic theories of signs, it is no longer entirely appropriate to refer to the gnomon of any object. At any moment of discussion, I can only have so-and-so's gnomon or code word for each thing under the sun. Thus, apparent references to a uniquely determined gnomon only make sense if taken as enthymemic invocations of the ordinary context and all that is comprehended to be implied in it, promising to convert tacit common sense into definite articulations of what is understood. Actually achieving this requires each elliptic reference to the gnomon to be explicitly grounded in the context of informal discussion, interpreted with respect to the conventional basis of understanding assumed in it, and relayed to the indexing function taken for granted by all parties to it.

In computational terms, this brand of pluralism means that neither the gnomon nor the quoting function that forms a part of it can be viewed as well-defined unless it is indexed, explicitly or implicitly, by the name of a particular interpreter. I will use either one of the equivalent notations ${\displaystyle {}^{\backprime \backprime }\mathrm {Gno} _{i}(x){}^{\prime \prime }}$ or ${\displaystyle {}^{\backprime \backprime \langle }x,i{}^{\rangle \prime \prime }}$ to indicate the gnomon of the object ${\displaystyle x}$ with respect to the interpreter ${\displaystyle i.}$ The value ${\displaystyle \mathrm {Gno} _{i}(x)={}^{\langle }x,i{}^{\rangle }\in S}$ is the nominal sign in use or the name in use (NIU) of the object ${\displaystyle x}$ with respect to the interpreter ${\displaystyle i,}$ and thus it constitutes a component of ${\displaystyle i}$'s state.

In the special case where ${\displaystyle x}$ is a sign or expression in the syntactic domain, then ${\displaystyle \mathrm {Gno} _{i}(x)={}^{\langle }x,i{}^{\rangle }}$ is tantamount to the quotation of ${\displaystyle x}$ by and for the use of the interpreter ${\displaystyle i,}$ in short, the nominal sign to ${\displaystyle i}$ that makes ${\displaystyle x}$ an object for ${\displaystyle i.}$ For signs and expressions, it is usually only the quoting function that makes them objects. But nothing is an object in any sense for an interpreter unless it is an object of a sign relation for that interpreter. Therefore, …

If it is now asked what measure of invariant understanding can be enjoyed by diverse parties of interpretive agents, then the discussion has come upon an issue with a familiar echo in mathematical analysis. The organization of many local coordinate frames into systems capable of supporting communicative references to relatively “objective” objects is usually handled by means of the concept of a manifold. Therefore, the analogous task that is suggested for this project is to arrive at a workable definition of sign relational manifolds.

The discrete nature of the ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ dialogue renders moot the larger share of issues of interest in continuous and differentiable manifolds. However, it is still possible to get things moving in this direction by looking at simple structural analogies that connect the pragmatic theory of sign relations with the basic notions of analysis on manifolds.

### Discourse Analysis : Ways and Means

Before the discussion of the ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ dialogue can proceed to richer veins of semantic structure it will be necessary to extract the relevant traces of embedded sign relations from their environments of informally interpreted syntax.

On the substantive front, sign relations serving as raw materials of discourse need to be refined and their content assayed, but first their identifying signatures must be sounded out, carved out, and lifted from their embroiling inclusions in the dense strata of obscure intuitions that sediment ordinary discussion. On the instrumental front, sign relations serving as primitive tools of discourse analysis need to be identified and improved by a deliberate examination of their designs and purposes.

So far, the models and methods made available to formal treatment were borrowed outright, with little hesitation and less recognition, from the context of casual discussion. Thus, these materials and mechanisms have come to the threshold of critical reflection already in play, devoid of concern for the presuppositions and consequences associated with their use, and only belatedly turned to the effortful work and tedious formalities of self-conscious exposition.

To reflect on the properties of complex and higher order sign relations with any degree of clarity it is necessary to arrange a clearer field of investigation and a less cluttered staging area for analytic work than is commonly provided. Habitual processes of interpretation that typically operate as automatic routines and uncritical defaults in the informal context of discussion have to be selectively inhibited, slowed down, and critically examined as objective possibilities, instead of being taken for granted as absolute necessities.

In other words, an apparatus for critical reflection does not merely add more mirrors to the kaleidoscopic fun-house of interpretive discourse, but it provides transient moments of equanimity, or balanced neutrality, and a moderately detached perspective on alternative points of view. A scope so limited does not by any means grant a god's eye view, but permits a sufficient quantity of light to consider how the original array of sights and reflections might have been created otherwise.

Ordinarily, the extra degree of attention to syntax that is needed for critical reflection on interpretive processes is called into play by means of syntactic operators and diacritical devices acting at the level of individual signs and elementary expressions. For example, quotation marks are used to force one type of “semantic ascent”, causing signs to be treated as objects and marking points of interpretive shift as they occur in the syntactic medium. But these operators and devices must be symbolized, and these symbols must be interpreted. Consequently, there is no way to avoid the invocation of a cohering interpretive framework, one that needs to be specialized for analytic purposes.

The best way to achieve the desired type of reflective capacity is by attaching a parameter to the interpretive framework used as an instrument of formal study, specifying certain choices or interpretive presumptions that affect the entire context of discussion. The aesthetic distance needed to arrive at a formal perspective on sign relations is maintained, not by jury-rigging ordinary discussion with locally effective syntactic devices, but by asking the reader to consider certain dimensions of parametric variation in the global interpretive frameworks used to comprehend the sign relations under study.

The interpretive parameter of paramount importance to this work is one that is critical to reflection. It can be presented as a choice between two alternative conventions, affecting the way one reflexively regards each sign in a text: (1) as a sign provoking interest only in passing, exchanged for the sake of a meaningful object it is always taken for granted to have, or (2) as a sign comprising an interest in and of itself, a state of a system or a modification of a medium that can signify an external value but does not necessarily denote anything else at all. I will name these options for responding to signs according to the aspects of character that are most appreciated in their net effects, whether signs for the sake of objects, or signs for their own sake, respectively.

The first option I call the object convention, recognizing it as the natural default of informal language use. In the ordinary language context it is the automatic assumption that signs and expressions are intended to denote something external to themselves, and even though it is quite obvious to all interpreters that the medium is filled with the appearances of signs and not with the objects themselves, this fact passes for little more than transitory interest in the rush to cash out tokens for their indicated values.

The object convention, as appropriate to an introduction that needs to begin in the context of ordinary discussion, is the parametric choice that was left in force throughout the treatment of the A and B example. Doing things this way is like trying to roller skate in a buffalo herd, that is, it attempts to formalize a fragment of discussion on a patchwork of local scales without interrupting the automatic routines and default assumptions that prevail on a global basis in the informal context. Ultimately, one cannot avoid stumbling over the hoofprints ${\displaystyle ({}^{\backprime \backprime }\,{}^{\prime \prime })}$ of overly cited and opaquely enthymematic textual deposits.

The second option I call the sign convention, observing it to be the treatment of choice in programming and formal language studies. In the formal language context it is necessary to consider the possibility that not all signs and expressions are assured to denote or even connote much of anything at all. This danger is amplified in computational frameworks where it resonates with a related theme, that not all programs are guaranteed to terminate normally with a definite result. In order to deal with these eventualities, a more cautious approach to sign relations is demanded to cover the risk of generating nonsense, in other words, to guard against degenerate forms of sign relations that fail to serve any significant purpose in communication or inquiry.

Whenever a greater degree of care is required, it becomes necessary to replace the object convention with the sign convention, which presumes to take for granted only what can be obvious to all observers, namely, the phenomenal appearances and temporal occurrences of objectified states of systems. To be sure, these modulations of media are still presented as signs, but only potentially as signs of other things. It goes with the territory of the formal language context to constantly check the inveterate impulses of the literate mind, to reflect on its automatic reflex toward meaning, to inhibit its uncontrolled operation, and to pause long enough in the rush to judgment to question whether its constant presumption of a motive is itself innocent.

In order to deal with these issues of discourse analysis in an explicit way, it is necessary to have in place a technical notation for marking the very kinds of interpretive assumptions that normally go unmarked. Thus, I will describe a set of devices for annotating certain kinds of interpretive contingencies, namely, the discourse analysis frames or the global interpretive frames that may be operative at any given moment in a particular context of discussion.

To mark a context of discussion where a particular set ${\displaystyle J}$ of interpretive conventions is being maintained, I use labeled brackets of the following two forms: “unitary”, as ${\displaystyle \{J|\ldots |J\},}$ or “divided”, as ${\displaystyle \{J|\ldots |\ldots |J\}.}$ The unitary form encloses a context of discussion by delimiting a range of text whose reading is subject to the interpretive constraints ${\displaystyle J.}$ The divided form specifies the objects, signs, and interpretive information in accord with which a species of discussion is generated. Labeled brackets enclosing contexts can be nested in their scopes, with interpretive data on each outer envelope applying to every inclusion. Labeled brackets arranging the conversation pieces or the generators and relations of a topic can lead to discussions that spill outside their frames, and thus are permitted to constitute overlapping contexts.

For the present, I will consider two types of interpretive parameters to be used as indices of labeled brackets.

1. Names of interpreters or other references to context can be used to indicate the provenance of the objects and signs that make up the assorted contents of brackets. On occasion, I will use the first person singular pronoun to signify the immediate context of informal discussion, as in ${\displaystyle \{I|\ldots |I\},}$ but more often than not this context goes unmarked.
2. Two other modifiers can be used to toggle between the options of the object convention, more common in casual or ordinary contexts, and the sign convention, more useful in formal or sign theoretic contexts.
1. The brackets ${\displaystyle \{o|\ldots |o\}}$ mark a context of informal language use or ordinary discussion, where the object convention applies. To specify the elements of a sign relation under these conditions, I use a form of presentation like the following:

 ${\displaystyle \{o|~{\text{A}},{\text{B}}~|||~{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }~|o\}.}$

Here, the names of objects are placed on the left side and the names of signs on the right side of the central divide, and the outer brackets stipulate that the object convention is in force throughout the discussion of a sign relation that is generated on these elements.

2. The brackets ${\displaystyle \{s|\ldots |s\}}$ mark a context of formal language use or controlled discussion, where the sign convention applies. To specify the elements of a sign relation in this case, I use a form like:

 ${\displaystyle \{s|~[{\text{A}}],[{\text{B}}]~|||~{\text{A}},{\text{B}},{\text{i}},{\text{u}}~|s\}.}$

Again, expressions for objects are placed on the left and expressions of signs on the right, but formal language conventions are now invoked to let the alphabet letters and the lexical items of a formal vocabulary stand for themselves, and denotation brackets ${\displaystyle {}^{\backprime \backprime }[\dots ]{}^{\prime \prime }}$ are placed around signs to indicate the corresponding objects, when they exist.

When the information carried by labeled brackets becomes more involved and more extensive, a set of convenient abbreviations and suggestions for “pretty printing” can be followed. When the bracket labels become too long to bother repeating, I will leave the last label blank or use ditto marks, as with ${\displaystyle \{a,b,c~|~\ldots ~|{}^{\prime \prime }\}.}$ When it is necessary to break labeled brackets over several lines, multiple dividers and dittos can be used to fill out corresponding columns, as in the following text:

 ${\displaystyle {\begin{array}{*{12}{c}}\{&I&,&o&|&{\text{A}}&,&{\text{B}}&&&&\\|&|&|&|&|&{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }&,&{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\|&{}^{\prime \prime }&{}^{\prime \prime }&{}^{\prime \prime }&\}&&&&&&&\end{array}}}$

A notation for discourse analysis ought to find a crucial test of its usefulness in whether it can help to disclose structural properties of interpretive frameworks that would otherwise escape the attention due. If the dimensions of interpretive choice that are represented by these devices are to serve a useful function, then …

Although these devices for discourse analysis are bound to seem a bit ad hoc at this point, they have been designed with a sign relational bootstrap in mind, that is, with a view to being formalized and recognized as a species within the domain of sign relations itself, where this is the very domain that is laid out as their field of application.

One note of caution may help to prevent a common misunderstanding. It is futile to imagine that any system of interpretive markers for discourse can become totally self sufficient, like the Worm Uroboros, determining all aspects of interpretation and eliminating all ambiguity. The ultimate appeal of signs, and signs upon signs, is always to an intelligent interpreter, a reader who knows there are more interpretive choices to make than could ever be surrendered to signs, and whose free responsibility to appropriate interpretations cannot be abdicated to any text or abridged by any gloss on it, no matter how fit or finished.

In a sense, at least at first, nothing is being created that could not have been noticed without signs. It is merely that actions are being articulated that were not articulated before, and hopefully in ways that make transient insights easier to remember and reuse on new occasions. Instead, the requirement here is to devise a language, the marks of which can reflect the ambient light of observation on its own process. It is not unusual to succeed at this in artificial environments crafted especially for the purpose, but to achieve the critical angle in vivo, in the living context of a natural language, takes more art.

### Combinations of Sign Relations

At a point like this in the development of a formal subject matter, it is customary to introduce elements of a logical calculus that can be used to describe relevant aspects of the formal structures involved and to expedite reasoning about their manifold combinations and decompositions. I will hold off from doing this for sign relations in any formal way at present. Instead, I consider the informal requirements and the foreseeable ends that a suitable calculus for sign relations might be expected to meet, and I present as tentative alternatives a few different ways of proceeding to formalize these intentions.

The first order of business in the “comparative anatomy” and “developmental biology” of sign relations is to undertake a pair of closely related tasks: (1) to examine the structural articulation of highly complex sign relations in terms of the primitive constituents that are found available, and (2) to explain the functional genesis of formal (that is, reflectively considered and critically regarded) sign relations as they naturally arise within the informal context of representational and communicational activities.

Converting to a political metaphor, how does the “republic” constituted by a sign relation — the representational community of agents invested with a congeries of legislative, executive, and interpretive powers, employing a consensual body of conventional languages, encompassing a commonwealth of comprehensible meanings, diversely but flexibly manifested in the practical administration of abiding and shared representations — how does all of this first come into being?

… and their development from primitive/ rudimentary to highly structured …

The grasp of the discussion between ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ that is represented in their separate sign relations can best be described as fragmentary. It fails to capture what everyone knows ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ would know about each other's language use.

How can the fragmentary system of interpretation (SOI) constituted by the juxtaposition of individual sign relations ${\displaystyle L({\text{A}})}$ and ${\displaystyle L({\text{B}})}$ be combined or developed into a new SOI that represents what agents like ${\displaystyle {\text{A}}}$ and ${\displaystyle {\text{B}}}$ are sure to know about each other's language use? In order to make it clear that this is a non-trivial question, and in the process to illustrate different ways of combining sign relations, I begin by considering a couple of obvious suggestions for their integration that immediate reflection will show to miss the mark.

The first thing to try is the set-theoretic union of the sign relations. This leads to a “confused” or “confounded” combination of the component sign relations. For example, the sign relation defined as ${\displaystyle L_{\text{C}}=L_{\text{A}}\cup L_{\text{B}}}$ is shown in Table 86. Interpreted as a transition digraph on the four points of the syntactic domain ${\displaystyle S=\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\},}$ the sign relation ${\displaystyle L_{\text{C}}}$ specifies the following behavior for the conduct of its interpreter:

1. ${\displaystyle {\text{A}}\!\cdot \!L_{\text{C}}}$ has a sling at each point of ${\displaystyle \{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\}}$ and two-way arcs on the pairs ${\displaystyle \{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\}}$ and ${\displaystyle \{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\}.}$
2. ${\displaystyle {\text{B}}\!\cdot \!L_{\text{C}}}$ has a sling at each point of ${\displaystyle \{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\}}$ and two-way arcs on the pairs ${\displaystyle \{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\}}$ and ${\displaystyle \{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\}.}$

These sub-relations do not form equivalence relations on the relevant sets of signs. If closed up under transitive compositions, then ${\displaystyle \{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\}}$ are all equivalent in the presence of object ${\displaystyle {\text{A}},}$ but ${\displaystyle \{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime },{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\}}$ are all equivalent in the presence of object ${\displaystyle {\text{B}}.}$ This may accurately represent certain types of political thinking, but it does not constitute the kind of sign relation that is wanted here.

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\\{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\\{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$ ${\displaystyle {\begin{matrix}{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }\\{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }\end{matrix}}}$

Reflecting on this disappointing experience with using simple unions to combine sign relations, it appears that some type of indexed union or categorical co-product might be demanded. Table 87 presents the results of taking the disjoint union ${\displaystyle \textstyle L_{\text{D}}=L_{\text{A}}\coprod L_{\text{B}}}$ to constitute a new sign relation.

 ${\displaystyle {\text{Object}}}$ ${\displaystyle {\text{Sign}}}$ ${\displaystyle {\text{Interpretant}}}$ ${\displaystyle {\begin{matrix}{\text{A}}_{\text{A}}\\{\text{A}}_{\text{A}}\\{\text{A}}_{\text{A}}\\{\text{A}}_{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{A}}_{\text{B}}\\{\text{A}}_{\text{B}}\\{\text{A}}_{\text{B}}\\{\text{A}}_{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{A}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}_{\text{A}}\\{\text{B}}_{\text{A}}\\{\text{B}}_{\text{A}}\\{\text{B}}_{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{A}}\\{{}^{\backprime \backprime }{\text{u}}{}^{\prime \prime }}_{\text{A}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{B}}_{\text{B}}\\{\text{B}}_{\text{B}}\\{\text{B}}_{\text{B}}\\{\text{B}}_{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{B}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{B}}{}^{\prime \prime }}_{\text{B}}\\{{}^{\backprime \backprime }{\text{i}}{}^{\prime \prime }}_{\text{B}}\end{matrix}}}$