|
\(\begin{matrix}
[ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}}
& = &
[ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}}
\'"`UNIQ-MathJax20-QINU`"'
: Rubric of Universal Equality'"`UNIQ-MathJax21-QINU`"'
Working under either of these assumptions, \(G\!\) can be provided with a simplified form of presentation:
| \({G = \{ G_j \} = \{ G_j : j \in J \} ~\text{with}~ G_j \subseteq X \times X ~ (\forall j \in J)}.\!\)
|
However, it serves a purpose of this project to preserve the individual indexing of relational domains for while longer, or at least to keep this usage available as an alternative formulation. Generally speaking, it is always possible in principle to form the union required by the universal inclusion convention, or without loss of generality to assume the equality imposed by the universal equality convention. The problem is that the unions and equalities invoked by these rubrics may not be effectively definable or testable in a computational context. Further, even when these sets or tests can be constructed or certified by some computational agent or another, the pertinent question at any interpretive moment is whether each collection or constraint is actively being apprehended or warranted by the particular interpreter charged with responsibility for it by the indicated assignment of domains.
But an overall purpose of this formalism is to represent the objects and constituencies known to specific interpreters at definite moments of their interpretive proceedings, in other words, to depict the information about objective existence and constituent structure that is possessed, recognized, responded to, acted on, and followed up by concrete agents as they move through their immediate contexts of activity. Accordingly, keeping individual tabs on the relational domains \({P_j}\!\) and \({Q_j},\!\) though it does not solve this array of problems, does serve to mark the concern with particularity and to keep before the mind the issues of individual attention and responsibility that are appropriate to interpretive agents. In short, whether or not domains appear with explicit subscripts, one should always be ready to answer Who subscribes to these domains?
It is important to emphasize that the index set \(J\!\) and the particular attachments of indices to dyadic relations are part and parcel to \(G\!\), befitting the concrete character intended for the concept of an objective genre, which is expected to realistically embody in the character of each \(G_j\!\) both a local habitation and a name. For this reason, among others, the \(G_j\!\) can safely be referred to as individual dyadic relations. Since the classical notion of an individual as a perfectly determinate entity has no application in finite information contexts, it is safe to recycle this term to distinguish the terminally informative particulars that a concrete index \(j\!\) adds to its thematic object \(G_j\!\).
Depending on the prevailing direction of interest in the genre \(G\!\), \({\lessdot}\!\) or \({\gtrdot},\!\) the same symbol is used equivocally for all the relations \({G_j}.\!\) The \(G_j\!\) can be regarded as formalizing the objective motives that make up the genre \(G\!\), provided it is understood that the information corresponding to the parameter \(j\!\) constitutes an integral part of the motive or motif of \({G_j}.\!\)
In this formulation, \(G\!\) constitutes ontological hierarchy of a plenary type, one that determines the complete array of objects and relationships that are conceivable and describable within a given discussion. Operating with reference to the global field of possibilities presented by \(G\!\), each \(G_j\!\) corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.
Another way to formalize the defining structure of an objective genre can be posed in terms of a relative membership relation or a notion of relative elementhood. The constitutional structure of a particular genre can be set up in a flexible manner by taking it in two stages, starting from the level of finer detail and working up to the big picture:
- Each OM is constituted by what it means to be an object within it. What constitutes an object in a given OM can be fixed as follows:
- In absolute terms, by specifying the domain of objects that fall under its purview. For the present, I assume that each OM inherits the same object domain \(X\!\) from its governing OG.
- In relative terms, by specifying a converse pair of dyadic relations that (redundantly) determine two sets of facts:
- What is an instance, example, member, or element of what, relative to the OM in question.
- What is a property, quality, class, or set of what, relative to the OM in question.
- The various OMs of a particular OG can be unified under its aegis by means of a single triadic relation, one that names an OM and a pair of objects and that holds when one object belongs to the other in the sense identified by the relevant OM. If it becomes absolutely essential to emphasize the relativity of elements, one may resort to calling them relements, in this way jostling the mind to ask: Relement to what?
The last and perhaps the best way to form an objective genre \(G\!\) is to present it as a triadic relation:
| \(G = \{ (j, p, q) \} \subseteq J \times P \times Q,\!\)
|
or:
| \(G = \{ (j, x, y) \} \subseteq J \times X \times X.\!\)
|
Given an objective genre \(G\!\) whose motives are indexed by a set \(J\!\) and whose objects form a set \(X\!\), there is a triadic relation among a motive and a pair of objects that exists when the first object belongs to the second object according to that motive. This is called the standing relation of the genre, and it can be taken as one way of defining and establishing the genre. In the way that triadic relations usually give rise to dyadic operations, the associated standing operation of the genre can be thought of as a brand of assignment operation that makes one object belong to another in a certain sense, namely, in the sense indicated by the designated motive.
There is a partial converse of the standing relation that transposes the order in which the two object domains are mentioned. This is called the propping relation of the genre, and it can be taken as an alternate way of defining the genre.
| \(G\!\uparrow ~=~ \{ (j, q, p) \in J \times Q \times P : (j, p, q) \in G \},\!\)
|
or:
| \(G\!\uparrow ~=~ \{ (j, y, x) \in J \times X \times X : (j, x, y) \in G \}.\!\)
|
The following conventions are useful for discussing the set-theoretic extensions of the staging relations and staging operations of an objective genre:
The standing relation of a genre is denoted by the symbol \(:\!\lessdot\), pronounced set-in, with either of the following two type-markings:
| \(:\!\lessdot ~\subseteq~ J \times P \times Q,\!\)
|
| \(:\!\lessdot ~\subseteq~ J \times X \times X.\!\)
|
The propping relation of a genre is denoted by the symbol \(:\!\gtrdot\), pronounced set-on, with either of the following two type-markings:
| \(:\!\gtrdot ~\subseteq~ J \times Q \times P,\!\)
|
| \(:\!\gtrdot ~\subseteq~ J \times X \times X.\!\)
|
Often one's level of interest in a genre is purely generic. When the relevant genre is regarded as an indexed family of dyadic relations, \(G = \{ G_j \},\!\) then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.
| \(\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j ~ (\exists j \in J) \}.\!\)
|
When the relevant genre is contemplated as a triadic relation, \({G \subseteq J \times X \times X},\!\) then one is dealing with the projection of \(G\!\) on the object dyad \(X \times X.\!\)
| \(G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G ~ (\exists j \in J) \}.\!\)
|
On these occasions, the assertion that \((x, y)\!\) is in \({\cup_J G = G_{XX}}\!\) can be indicated by any one of the following equivalent expressions:
| \({G : x \lessdot y},\!\)
|
\({x \lessdot_G y},\!\)
|
\({x \lessdot y : G},\!\)
|
| \({G : y \gtrdot x},\!\)
|
\({y \gtrdot_G x},\!\)
|
\({y \gtrdot x : G}.\!\)
|
At other times explicit mention needs to be made of the interpretive perspective or individual dyadic relation that links two objects. To indicate that a triple consisting of a motive \(j\!\) and two objects \(x\!\) and \(y\!\) belongs to the standing relation of the genre, in symbols, \({(j, x, y) \in ~ :\!\lessdot},\) or equally, to indicate that a triple consisting of a motive \(j\!\) and two objects \(y\!\) and \(x\!\) belongs to the propping relation of the genre, in symbols, \({(j, y, x) \in ~ :\!\gtrdot},\) all of the following notations are equivalent:
| \({j : x \lessdot y},\!\)
|
\({x \lessdot_j y},\!\)
|
\({x \lessdot y : j},\!\)
|
| \({j : y \gtrdot x},\!\)
|
\({y \gtrdot_j x},\!\)
|
\({y \gtrdot x : j}.\!\)
|
Assertions of these relations can be read in various ways, for example:
| \(j : x \lessdot y\!\)
|
\(j : y \gtrdot x\!\)
|
| \(x \lessdot_j y\!\)
|
\(y \gtrdot_j x\!\)
|
| \(x \lessdot y : j\!\)
|
\(y \gtrdot x : j\!\)
|
|
| \(j ~\text{sets}~ x ~\text{in}~ y.\!\)
|
\(j ~\text{sets}~ y ~\text{on}~ x.\!\)
|
| \(j ~\text{makes}~ x ~\text{an instance of}~ y.\!\)
|
\(j ~\text{makes}~ y ~\text{a property of}~ x.\!\)
|
| \(j ~\text{thinks}~ x ~\text{an instance of}~ y.\!\)
|
\(j ~\text{thinks}~ y ~\text{a property of}~ x.\!\)
|
| \(j ~\text{attests}~ x ~\text{an instance of}~ y.~\!\)
|
\(j ~\text{attests}~ y ~\text{a property of}~ x.~\!\)
|
| \(j ~\text{appoints}~ x ~\text{an instance of}~ y.\!\)
|
\(j ~\text{appoints}~ y ~\text{a property of}~ x.\!\)
|
| \(j ~\text{witnesses}~ x ~\text{an instance of}~ y.\!\)
|
\(j ~\text{witnesses}~ y ~\text{a property of}~ x.\!\)
|
| \(j ~\text{interprets}~ x ~\text{an instance of}~ y.\!\)
|
\(j ~\text{interprets}~ y ~\text{a property of}~ x.\!\)
|
| \(j ~\text{contributes}~ x ~\text{to}~ y.\!\)
|
\(j ~\text{attributes}~ y ~\text{to}~ x.\!\)
|
| \(j ~\text{determines}~ x ~\text{an example of}~ y.\!\)
|
\(j ~\text{determines}~ y ~\text{a quality of}~ x.\!\)
|
| \(j ~\text{evaluates}~ x ~\text{an example of}~ y.\!\)
|
\(j ~\text{evaluates}~ y ~\text{a quality of}~ x.\!\)
|
| \(j ~\text{proposes}~ x ~\text{an example of}~ y.\!\)
|
\(j ~\text{proposes}~ y ~\text{a quality of}~ x.\!\)
|
| \(j ~\text{musters}~ x ~\text{under}~ y.\!\)
|
\(j ~\text{marshals}~ y ~\text{over}~ x.\!\)
|
| \(j ~\text{indites}~ x ~\text{among}~ y.~\!\)
|
\(j ~\text{ascribes}~ y ~\text{about}~ x.~\!\)
|
| \(j ~\text{imputes}~ x ~\text{among}~ y.\!\)
|
\(j ~\text{imputes}~ y ~\text{about}~ x.\!\)
|
| \(j ~\text{judges}~ x ~\text{beneath}~ y.\!\)
|
\(j ~\text{judges}~ y ~\text{beyond}~ x.\!\)
|
| \(j ~\text{finds}~ x ~\text{preceding}~ y.\!\)
|
\(j ~\text{finds}~ y ~\text{succeeding}~ x.\!\)
|
| \(j ~\text{poses}~ x ~\text{before}~ y.\!\)
|
\(j ~\text{poses}~ y ~\text{after}~ x.\!\)
|
| \(j ~\text{forms}~ x ~\text{below}~ y.\!\)
|
\(j ~\text{forms}~ y ~\text{above}~ x.\!\)
|
|
In making these free interpretations of genres and motifs, one needs to read them in a logical rather than a cognitive sense. A statement like “\(j\!\) thinks \(x\!\) an instance of \(y\!\)” should be understood as saying that “\(j\!\) is a thought with the logical import that \(x\!\) is an instance of \(y\!\)”, and a statement like “\(j\!\) proposes \(y\!\) a property of \(x\!\)” should be taken to mean that “\(j\!\) is a proposition to the effect that \(y\!\) is a property of \(x\!\)”.
These cautions are necessary to forestall the problems of intentional attitudes and contexts, something I intend to clarify later on in this project. At present, I regard the well-known opacities of this subject as arising from the circumstance that cognitive glosses tend to impute an unspecified order of extra reflection to each construal of the basic predicates. The way I plan to approach this issue is through a detailed analysis of the cognitive capacity for reflective thought, to be developed to the extent possible in formal terms by using sign relational models.
By way of anticipating the nature of the problem, consider the following examples to illustrate the contrast between logical and cognitive senses:
- In a cognitive context, if \(j\!\) is a considered opinion that \(S\!\) is true, and \(j\!\) is a considered opinion that \(T\!\) is true, then it does not have to automatically follow that \(j\!\) is a considered opinion that the conjunction \(S ~\operatorname{and}~ T\!\) is true, since an extra measure of consideration might conceivably be involved in cognizing the conjunction of \(S\!\) and \(T.\!\)
- In a logical context, if \(j\!\) is a piece of evidence that \(S\!\) is true, and \(j\!\) is a piece of evidence that \(T\!\) is true, then it follows by these very facts alone that \(j\!\) is a piece of evidence that the conjunction \(S ~\operatorname{and}~ T\!\) is true. This is analogous to a situation where, if a person \(j\!\) draws a set of three lines, \(AB,\!\) \(BC,\!\) and \(AC,\!\) then \(j\!\) has drawn a triangle \(ABC,\!\) whether \(j\!\) recognizes the fact on reflection and further consideration or not.
Some readings of the staging relations are tantamount to statements of (a possibly higher order) model theory. For example, consider the predicate \(P : J \to \mathbb{B}\!\) defined by the following equivalence:
| \(P(j) \quad \Leftrightarrow \quad j\ \text{proposes}\ x\ \text{an instance of}\ y.\!\)
|
Then \(P\!\) is a proposition that applies to a domain of propositions, or elements with the evidentiary import of propositions, and its models are therefore conceived to be certain propositional entities in \(J.\!\) And yet all of these expressions are just elaborate ways of stating the underlying assertion which says that there exists a triple \((j, x, y)\!\) in the genre \(G (:\!\lessdot).\)
Application of OF : Generic Level
Given an ontological framework that can provide multiple perspectives and moving platforms for dealing with object structure, in other words, that can organize diverse hierarchies and developing orders of objects, attention can now return to the discussion of sign relations as models of intellectual processes.
A principal aim of using sign relations as formal models is to be capable of analyzing complex activities that arise in nature and human domains. Proceeding by the opportunistic mode of analysis by synthesis, one generates likely constructions from a stock of favored, familiar, and well-understood sign relations, the supply of which hopefully grows with time, constantly matching their formal properties against the structures encountered in the “wilds” of natural phenomena and human conduct. When salient traits of both the freely generated products and the widely gathered phenomena coincide in enough points, then the details of the constructs one has built for oneself can help to articulate a plausible hypothesis as to how the observable appearances might be explained.
A principal difficulty of using sign relations for this purpose arises from the very power of productivity they bring to bear in the process, the capacity of triadic relations to generate a welter of what are bound to be mostly arbitrary structures, with only a scattered few hoping to show any promise, but the massive profusion of which exceeds from the outset any reason's ability to sort them out and test them in practice. And yet, as the phenomena of interest become more complex, the chances grow slimmer that adequate explanations will be found in any of the thinner haystacks. In this respect, sign relations inherit the basic proclivities of set theory, which can be so successful and succinct in presenting and clarifying the properties of already found materials and hard won formal insights, and yet so overwhelming to use as a tool of random exploration and discovery.
The sign relations of \(\text{A}\!\) and \(\text{B},\!\) though natural in themselves as far as they go, were nevertheless introduced in an artificial fashion and presented by means of arbitrary stipulations. Sign relations that arise in more natural settings usually have a rationale, a reason for being as they are, and therefore become amenable to classification on the basis of the distinctive characters that make them what they are.
Consequently, naturally occurring sign relations can be expected to fall into species or natural kinds, and to have special properties that make them keep on occurring in nature. Moreover, cultivated varieties of sign relations, the kinds that have been converted to social purposes and found to be viable in actual practice, will have identifiable and especially effective properties by virtue of which their signs are rendered significant.
In the pragmatic theory of sign relations, three natural kinds of signs are recognized, under the names of icons, indices, and symbols. Examples of indexical or accessional signs figured significantly in the discussion of \(\text{A}\!\) and \(\text{B},\!\) as illustrated by the pronouns \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!\) and \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!\) in \(S.\!\) Examples of iconic or analogical signs were also present, though keeping to the background, in the very form of the sign relation Tables that were used to schematize the whole activity of each interpreter. Examples of symbolic or conventional signs, of course, abide even more deeply in the background, pervading the whole context and making up the very fabric of this discussion.
In order to deal with the array of issues presented so far in this subsection, all of which have to do with controlling the generative power of sign relations to serve the specific purposes of understanding, I apply the previously introduced concept of an objective genre (OG). This is intended to be a determinate purpose or a deliberate pattern of analysis and synthesis that one can identify as being active at given moments in a discussion and that affects what one regards as the relevant structural properties of its objects.
In the remainder of this subsection the concept of an OG is used informally, and only to the extent needed for a pressing application, namely, to rationalize the natural kinds that are claimed for signs and to clarify an important contrast that exists between icons and indices.
The OG I apply here is called the genre of properties and instances. One moves through its space, higher and lower in a particular ontology, by means of two dyadic relations, upward by taking a property of and downward by taking an instance of whatever object initially enters one's focus of attention. Each object of this OG is reckoned to be the unique common property of the set of objects that lie one step below it, objects that are in turn reckoned to be instances of the given object.
Pretty much the same relational structures could be found in the genre or paradigm of qualities and examples, but the use of examples here is polymorphous enough to include experiential, exegetic, and executable examples. This points the way to a series of related genres, for example, the OGs of principles and illustrations, laws and existents, precedents and exercises, and on to lessons and experiences. All in all, in their turn, these modulations of the basic OG show a way to shift the foundations of ontological hierarchies toward bases in individual and systematic experience, and thus to put existentially dynamic rollers under the blocks of what seem to be essentially invariant pyramids.
Any object of these OGs can be contemplated in the light of two potential relationships, namely, with respect to its chances of being an object quality or an object example of something else. In future references, abbreviated notations like \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\!\) or \(\operatorname{OG} = (\operatorname{Prop}, \operatorname{Inst})\!\) will be used to specify particular genres, giving the intended interpretations of their generating relations \(\{ \lessdot,\gtrdot \}.\!\)
With respect to this OG, I can now characterize icons and indices. Icons are signs by virtue of being instances of properties of objects. Indices are signs by virtue of being properties of instances of objects.
Because the initial discussion seems to flow more smoothly if I apply dyadic relations on the left, I formulate these definitions as follows:
|
\(\begin{array}{llll}
\text{For Icons:} &
\operatorname{Sign} (\operatorname{Obj}) & = &
\operatorname{Inst} (\operatorname{Prop} (\operatorname{Obj})). \\
\text{For Indices:} &
\operatorname{Sign} (\operatorname{Obj}) & = &
\operatorname{Prop} (\operatorname{Inst} (\operatorname{Obj})). \\
\end{array}\)
|
Imagine starting from the sign and retracing steps to reach the object, in this way finding the converses of these relations to be as follows:
|
\(\begin{array}{llll}
\text{For Icons:} &
\operatorname{Obj} (\operatorname{Sign}) & = &
\operatorname{Inst} (\operatorname{Prop} (\operatorname{Sign})). \\
\text{For Indices:} &
\operatorname{Obj} (\operatorname{Sign}) & = &
\operatorname{Prop} (\operatorname{Inst} (\operatorname{Sign})). \\
\end{array}\)
|
In spite of the apparent duality between these patterns of composition, there is a significant asymmetry to be observed in the way that the insistent theme of realism interrupts the underlying genre. In order to understand this, it is necessary to note that the strain of pragmatic thinking I am using here takes its definition of reality from the word's original Scholastic sources, where the adjective real means having properties. Taken in this sense, reality is necessary but not sufficient to actuality, where actual means “existing in act and not merely potentially” (Webster's). To reiterate, actuality is sufficient but not necessary to reality. The distinction between the ideas is further pointed up by the fact that a potential can be real, and that its reality can be independent of any particular moment in which the power acts.
These abstract considerations would probably remain distant from the present concern, were it not for two points of connection:
- Relative to the present genre, the distinction of reality, that can be granted to certain objects of thought and not to others, fulfills an analogous role to the distinction that singles out sets among classes in modern versions of set theory. Taking the membership relation \(\in\!\) as a predecessor relation in a pre-designated hierarchy of classes, a class attains the status of a set, and by dint of this becomes an object of more determinate discussion, simply if it has successors. Pragmatic reality is distinguished from both the medieval and the modern versions, however, by the fact that its reality is always a reality to somebody. This is due to the circumstance that it takes both an abstract property and a concrete interpreter to establish the practical reality of an object.
- This project seeks articulations and implementations of intelligent activity within dynamically realistic systems. The individual stresses placed on articulation, implementation, actuality, dynamics, and reality collectively reinforce the importance of several issues:
- Systems theory, consistently pursued, eventually demands for its rationalization a distinct ontology, in which states of being and modes of action form the principal objects of thought, out of which the ordinary sorts of stably extended objects must be constructed. In the “grammar” of process philosophy, verbs and pronouns are more basic than nouns. In its influence on the course of this discussion, the emphasis on systematic action is tantamount to an objective genre that makes dynamic systems, their momentary states and their passing actions, become the ultimate objects of synthesis and analysis. Consequently, the drift of this inquiry will be turned toward conceiving actions, as traced out in the trajectories of systems, to be the primitive elements of construction, more fundamental in this objective genre than stationary objects extended in space. As a corollary, it expects to find that physical objects of the static variety have a derivative status in relation to the activities that orient agents, both organisms and organizations, toward purposeful objectives.
- At root, the notion of dynamics is concerned with power in the sense of potential. The brand of pragmatic thinking that I use in this work permits potential entities to be analyzed as real objects and conceptual objects to be constituted by the conception of their actual effects in practical instances. In the attempt to unify symbolic and dynamic approaches to intelligent systems (Upper and Lower Kingdoms?), there remains an insistent need to build conceptual bridges. A facility for relating objects to their actualizing instances and their instantiating actions lends many useful tools to an effort of this nature, in which the search for understanding cannot rest until each object and phenomenon has been reconstructed in terms of active occurrences and ways of being.
- In prospect of form, it does not matter whether one takes this project as a task of analyzing and articulating the actualizations of intelligence that already exist in nature, or whether one views it as a goal of synthesizing and artificing the potentials for intelligence that have yet to be conceived in practice. From a formal perspective, the analysis and the synthesis are just reciprocal ways of tracing or retracing the same generic patterns of potential structure that determine actual form.
Returning to the examination of icons and indices, and keeping the criterion of reality in mind, notice the radical difference that comes into play in recursive settings between the two types of contemplated moves that are needed to trace the respective signs back to their objects, that is, to discover their denotations:
- Icon → Object. Taking the iconic sign as an initial instance, try to go up to a property and then down to a different or perhaps the same instance. This form of ascent does not require a distinct object, since reality of the sign is sufficient to itself. In other words, if the sign has any properties at all, then it is an icon of a real object, even if that object is only itself.
- Index → Object. Taking the indexical sign as an initial property, try to go down to an instance and then up to a different or perhaps the same property. This form of descent requires a real instance to substantiate it, but not necessarily a distinct object. Consequently, the index always has a real connection to its object, even if that object is only itself.
In sum: For icons a separate reality is optional, for indices a separate reality is obligatory. As often happens with a form of analysis, each term under the indicated sum appears to verge on indefinite expansion:
- For icons, the existence of a separate reality is optional. This means that the question of reality in the sign relation can depend on nothing more than the reality of each sign itself, on whether it has any property with respect to the OG in question. In effect, icons can rely on their own reality to faithfully provide a real object.
- For indices, the existence of a separate reality is obligatory. And yet this reality need not affect the object of the sign. In essence, indices are satisfied with a basis in reality that need only reside in an actual object instance, one that establishes a real connection between the object and its index with regard to the OG in question.
Finally, suppose that \(M\!\) and \(N\!\) are hypothetical sign relations intended to capture all the iconic and indexical relationships, respectively, that a typical object \(x\!\) enjoys within its genre \(G.\!\) A sign relation in which every sign has the same kind of relation to its object under an assumed form of analysis is appropriately called a homogeneous sign relation. In particular, if \(H\!\) is a homogeneous sign relation in which every sign has either an iconic or an indexical relation to its object, then it is convenient to apply the corresponding adjective to the whole of \(H.\!\)
Typical sign relations of the iconic or indexical kind generate especially simple and remarkably stable sorts of interpretive processes. In arity, they could almost be classified as approximately dyadic, since most of their interesting structure is wrapped up in their denotative aspects, while their connotative functions are relegated to the tangential role of preserving the directions of their denotative axes. In a metaphorical but true sense, iconic and indexical sign relations equip objective frameworks with “gyroscopes”, helping them maintain their interpretive perspectives in a persistent orientation toward their objective world.
Of course, every form of sign relation still depends on the agency of a proper interpreter to bring it to life, and every species of sign process stays forever relative to the interpreters that actually bring it to term. But it is a rather special circumstance by means of which the actions of icons and indices are able to turn on the existence of independently meaningful properties and instances, as recognized within an objective framework, and this means that the interpretive associations of these signs are not always as idiosyncratic as they might otherwise be.
The dispensation of consensual bonds in a common medium leaves room for many individual interpreters to inhabit a shared frame of reference, and for a diversity of transient interpretive moments to take up and consolidate a continuing perspective on a world of mutual interests. This further increases the likelihood that differing and developing interpreters will become able to participate in compatible views and coherent values in relation to the aggregate of things, to collate information from a variety of sources, and to bring concerted action to bear on an appreciable distribution of extended realities and intended objectives. Instead of the disparities due to parallax leading to disorder and paralysis, accounting for the distinctive points of view behind the discrepancies can give rise to stereoscopic perspectives. In a community of interpretation and inquiry that has all these virtues, each individual try at objectivity is a venture that all the interpreters are nonetheless able to call their own.
Is this prospect a utopian vision? Perhaps it is exactly that. But it is the hope that inquiry discovers resting first and last within itself, quietly guiding every other aim and motive of inquiry.
Turning to the language of objective concerns, what can now be said about the compositional structures of the iconic sign relation \(M\!\) and the indexical sign relation \(N\!\)? In preparation for this topic, a few additional steps must be taken to continue formalizing the concept of an objective genre and to begin developing a calculus for composing objective motifs.
I recall the objective genre of properties and instances and re-introduce the symbols \({\lessdot}\!\) and \({\gtrdot}\!\) for the converse pair of dyadic relations that generate it. Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms “property of \(x\!\)” and “instance of \(x\!\)” by means of a case inflection on \(x,\!\) that is, as “\(x\!\)’s property” and “\(x\!\)’s instance”, respectively. Described in this way, \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}) = \langle \lessdot, \gtrdot \rangle,\!\) where:
|
\(\begin{array}{lllllll}
x \lessdot & = &
x \operatorname{'s~Property} & = &
\operatorname{Property~of}\ x & = &
\operatorname{Object~above}\ x. \\
x \gtrdot & = &
x \operatorname{'s~Instance} & = &
\operatorname{Instance~of}\ x & = &
\operatorname{Object~below}\ x. \\
\end{array}\)
|
A symbol like \(^{\backprime\backprime} x \lessdot ^{\prime\prime}\!\) or \(^{\backprime\backprime} x \gtrdot ^{\prime\prime}\!\) is called a catenation, where \(^{\backprime\backprime} x ^{\prime\prime}\!\) is the catenand and \(^{\backprime\backprime} \lessdot ^{\prime\prime}\!\) or \(^{\backprime\backprime} \gtrdot ^{\prime\prime}\!\) is the catenator. Due to the fact that \(^{\backprime\backprime} \lessdot ^{\prime\prime}\!\) and \(^{\backprime\backprime} \gtrdot ^{\prime\prime}\!\) indicate dyadic relations, the significance of these so-called unsaturated catenations can be rationalized as follows:
|
\(\begin{array}{lllll}
x \lessdot & = &
x\ \operatorname{is~the~Instance~of~what?} & = &
x \operatorname{'s~Property}. \\
x \gtrdot & = &
x\ \operatorname{is~the~Property~of~what?} & = &
x \operatorname{'s~Instance}. \\
\end{array}\)
|
In this fashion, the definitions of icons and indices can be reformulated:
|
\(\begin{array}{lllll}
x \operatorname{'s~Icon} & = &
x \operatorname{'s~Property's~Instance} & = &
x \lessdot \gtrdot \\
x \operatorname{'s~Index} & = &
x \operatorname{'s~Instance's~Property} & = &
x \gtrdot \lessdot \\
\end{array}\)
|
According to the definitions of the homogeneous sign relations \(M\!\) and \(N,\!\) we have:
|
\(\begin{array}{lllll}
x \operatorname{'s~Icon} & = &
x \cdot M_{OS} \\
x \operatorname{'s~Index} & = &
x \cdot N_{OS} \\
\end{array}\)
|
Equating the results of these equations yields the analysis of \(M\!\) and \(N\!\) as forms of composition within the genre of properties and instances:
|
\(\begin{array}{lllll}
x \operatorname{'s~Icon} & = &
x \cdot M_{OS} & = &
x \lessdot \gtrdot \\
x \operatorname{'s~Index} & = &
x \cdot N_{OS} & = &
x \gtrdot \lessdot \\
\end{array}\)
|
On the assumption (to be examined more closely later) that any object \(x\!\) can be taken as a sign, the converse relations appear to be manifestly identical to the originals:
|
\(\begin{array}{llllll}
\text{For Icons:} &
x \operatorname{'s~Object} & = &
x \cdot M_{SO} & = &
x \lessdot \gtrdot \\
\text{For Indices:} &
x \operatorname{'s~Object} & = &
x \cdot N_{SO} & = &
x \gtrdot \lessdot \\
\end{array}\)
|
Abstracting from the applications to an otiose \(x\!\) delivers the results:
|
\(\begin{array}{llllll}
\text{For Icons:} & M_{OS} & = & M_{SO} & = & \lessdot \gtrdot \\
\text{For Indices:} & N_{OS} & = & N_{SO} & = & \gtrdot \lessdot \\
\end{array}\)
|
This appears to suggest that icons and their objects are icons of each other, and that indices and their objects are indices of each other. Are the results of these symbolic manipulations really to be trusted? Given that there is no mention of the interpretive agent to whom these sign relations are supposed to appear, one might well suspect that these results can only amount to approximate truths or potential verities.
Application of OF : Motive Level
Now that an adequate variety of formal tools have been set in order and the workspace afforded by an objective framework has been rendered reasonably clear, the structural theory of sign relations can be pursued with greater precision. In support of this aim, the concept of an objective genre and the particular example provided by \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\!\) have served to rough out the basic shapes of the more refined analytic instruments to be developed in this subsection.
The notion of an objective motive or objective motif (OM) is intended to specialize or personalize the application of objective genres to take particular interpreters into account. For example, pursuing the pattern of \(\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\!\), a prospective OM of this genre does not merely tell about the properties and instances that objects can have in general, it recognizes a particular arrangement of objects and supplies them with its own ontology, giving “a local habitation and a name” to the bunch. What matters to an OM is a particular collection of objects (of thought) and a personal selection of links that go from each object (of thought) to higher and lower objects (of thought), all things being relative to a subjective ontology or a live hierarchy of thought, one that is currently known to and actively pursued by a designated interpreter of those thoughts.
The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind. As always, it is best to regard OGs and OMs as filters and reticles, as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.
With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to factor the facets or decompose the components of sign relations in a more systematic fashion. Given a homogeneous sign relation \(H\!\) of iconic or indexical type, the dyadic projections \(H_{OS}\!\) and \(H_{OI}\!\) can be analyzed as compound relations over the basis supplied by the \(G_j\!\) in \(G\!\). As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.
To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type \(\langle \lessdot, \gtrdot \rangle\!\) and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.
Let \(X\!\) collect the objects of thought that fall within a particular OM, and let \(X\!\) include the whole world of a sign relation plus everything needed to support and contain it. That is, \(X\!\) collects all the types of things that go into a sign relation, \(O \cup S \cup I = W \subseteq X\!\), plus whatever else in the way of distinct object qualities and object exemplars is discovered or established to be generated out of this basis by the relations of the OM.
In order to keep this \(X\!\) simple enough to contemplate on a single pass but still make it deep enough to cover the issues of interest at present, I limit \(X\!\) to having just three disjoint layers of things to worry about:
| The top layer is the relevant class of object qualities:
|
| \(Q = X_0 \lessdot = W \lessdot\!\)
|
| The middle layer is the initial collection of objects and signs:
|
| \(X_0 = W\!\)
|
| The bottom layer is a suitable set of object exemplars:
|
| \(E = X_0 \gtrdot = W \gtrdot\!\)
|
Recall the reading of the staging relations:
| \(h : x \lessdot m\!\)
|
\(\Leftrightarrow\!\)
|
\(h ~\operatorname{regards}~ x ~\operatorname{as~an~instance~of}~ m.\!\)
|
| \(h : m \gtrdot y\!\)
|
\(\Leftrightarrow\!\)
|
\(h ~\operatorname{regards}~ m ~\operatorname{as~a~property~of}~ y.\!\)
|
| \(h : x \gtrdot n\!\)
|
\(\Leftrightarrow\!\)
|
\(h ~\operatorname{regards}~ x ~\operatorname{as~a~property~of}~ n.\!\)
|
| \(h : n \lessdot y\!\)
|
\(\Leftrightarrow\!\)
|
\(h ~\operatorname{regards}~ n ~\operatorname{as~an~instance~of}~ y.\!\)
|
Express the analysis of icons and indices as follows:
| \(\text{For Icons:}\!\)
|
|
\(M_{OS}\!\)
|
\(\colon\!\)
|
\(x \lessdot \gtrdot x \operatorname{'s~Sign}.\!\)
|
| \(\text{For Indices:}\!\)
|
|
\(N_{OS}\!\)
|
\(\colon\!\)
|
\(x \gtrdot \lessdot x \operatorname{'s~Sign}.\!\)
|
|
Let \(j\!\) and \(k\!\) be hypothetical interpreters that do the jobs of \(M\!\) and \(N,\!\) respectively:
|
\(\begin{array}{llllll}
\text{For Icons:} &
x \operatorname{'s~Sign} & = &
x \cdot M_{OS} & = &
x \lessdot_j \gtrdot_j \\
\text{For Indices:} &
x \operatorname{'s~Sign} & = &
x \cdot N_{OS} & = &
x \gtrdot_k \lessdot_k \\
\end{array}\)
|
Factor out the names of the interpreters \(j\!\) and \(k\!\) to serve as identifiers of objective motifs:
| \(\text{For Icons:}\!\)
|
|
\(j\!\)
|
\(\colon\!\)
|
\(x \lessdot \gtrdot x \operatorname{'s~Sign}.\!\)
|
| \(\text{For Indices:}\!\)
|
|
\(k\!\)
|
\(\colon\!\)
|
\(x \gtrdot \lessdot x \operatorname{'s~Sign}.\!\)
|
|
Finally, the constant motif names \(j\!\) and \(k\!\) can be collected to one side of a composition or distributed to its individual links:
|
\(\begin{array}{llllll}
j : x \lessdot \gtrdot y &
\Leftrightarrow &
j : x \lessdot m &
\operatorname{and} &
j : m \gtrdot y &
(\exists m \in Q). \\
k : x \gtrdot \lessdot y &
\Leftrightarrow &
k : x \gtrdot n &
\operatorname{and} &
k : n \lessdot y &
(\exists n \in E). \\
\end{array}\)
|
These statements can be read to say:
- \(j\!\) thinks \(x\!\) an icon of \(y\!\) if and only if there is an \(m\!\) such that \(j\!\) thinks \(x\!\) an instance of \(m\!\) and \(j\!\) thinks \(m\!\) a property of \(y\!\).
- \(k\!\) thinks \(x\!\) an index of \(y\!\) if and only if there is an \(n\!\) such that \(k\!\) thinks \(x\!\) a property of \(n\!\) and \(k\!\) thinks \(n\!\) an instance of \(y\!\).
Readers who object to the anthropomorphism or the approximation of these statements can replace every occurrence of the verb thinks with the phrase interprets … as, or even the circumlocution acts in every formally relevant way as if, changing what must be changed elsewhere. For the moment, I am not concerned with the exact order of reflective sensitivity that goes into these interpretive linkages, but only with a rough outline of the pragmatic equivalence classes that are afforded by the potential conduct of their agents.
In the discussion of the dialogue between \(\text{A}\!\) and \(\text{B}\!\) it was allowed that the same signs \({}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!\) and \({}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!\) could reference the different categories of things they name with a deliberate duality and a systematic ambiguity. Used informally as a part of the peripheral discussion, they indicate the entirety of the sign relations themselves. Used formally within the focal dialogue, they denote the objects of two particular sign relations. In just this way, or an elaboration of it, the signs \({}^{\backprime\backprime} j {}^{\prime\prime}\!\) and \({}^{\backprime\backprime} k {}^{\prime\prime}\!\) can have their meanings extended to encompass both the objective motifs that inform and regulate experience and the object experiences that fill out and substantiate their forms.
The Integration of Frameworks
A large number of the problems arising in this work have to do with the integration of different interpretive frameworks over a common objective basis, or the prospective bases provided by shared objectives. The main concern of this project continues to be the integration of dynamic and symbolic frameworks for understanding intelligent systems, concentrating on the kinds of interpretive agents that are capable of being involved in inquiry.
Integrating divergent IFs and reconciling their objectifications is, generally speaking, a very difficult maneuver to carry out successfully. Two factors that contribute to the near intractability of this task can be described and addressed as follows.
- The trouble is partly due to the ossified taxonomies and obligatory tactics that come through time and training to inhabit the conceptual landscapes of agents, especially if they have spent the majority of their time operating according to a single IF. The IF informs their activity in ways they no longer have to think about, and thus rarely find a reason to modify. But it also inhibits their interpretive and practical conduct to the customary ways of seeing and doing things that are granted by that framework, and it restricts them to the forms of intuition that are suggested and sanctioned by the operative IF. Without critical reflection, or a mechanism to make amendments to its own constitution, an IF tends to operate behind the scenes of observation in such a way as to obliterate any inkling of flexibility in thought or practice and to obstruct every hint or threat (so perceived) of conceptual revision.
- Apparently it is so much easier to devise techniques for taking things apart than it is to find ways of putting them back together that there seem to be only a few heuristic strategies of general application that are available to guide the work of integration. A few of the tools and materials needed for these constructions have been illustrated in concrete form throughout the presentation of examples in this section. An overall survey of their principles can be summed up as follows.
- One integration heuristic is the lattice metaphor, also called the partial order or common denominator paradigm. When IFs can be objectified as OFs that are organized according to the principles of suitable orderings, then it is often possible to lift or extend these order properties to the space of frameworks themselves, and thereby to enable construction of the desired kinds of integrative frameworks as upper and lower bounds in the appropriate ordering.
- Another integration heuristic is the mosaic metaphor, also called the stereoscopic or inverse projection paradigm. This technique has been illustrated especially well by the methods used throughout this section to analyze the three-dimensional structures of sign relations. In fact, the picture of any sign relation offers a paradigm in microcosm for the macroscopic work of integration, showing how reductive aspects of structure can be projected from a shared but irreducible reality. The extent to which the full-bodied structure of a triadic sign relation can be reconstructed from its dyadic projections, although a limited extent in general, presents a near perfect epitome of the larger task in this situation, namely, to find an integrated framework that embodies the diverse facets of reality severally observed from inside the individual frameworks. Acting as gnomonic recipes for the higher order processes they limn and delimit, sign relations keep before the mind the ways in which a higher dimensional structure determines its fragmentary aspects but is not in general determined by them.
To express the nature of this integration task in logical terms, it combines elements of both proof theory and model theory, interweaving: (1) A phase that develops theories about the symbolic competence or knowledge of intelligent agents, using abstract formal systems to represent the theories and phenomenological data to constrain them; (2) A phase that seeks concrete models of these theories, looking to the kinds of mathematical structure that have a dynamic or system-theoretic interpretation, and compiling the constraints that a recursive conceptual analysis imposes on the ultimate elements of their construction.
The set of sign relations \(\{ L_\text{A}, L_\text{B} \}\!\) is an example of an extremely simple formal system, encapsulating aspects of the symbolic competence and the pragmatic performance that might be exhibited by potentially intelligent interpretive agents, however abstractly and partially given at this stage of description. The symbols of a formal system like \(\{ L_\text{A}, L_\text{B} \}\!\) can be held subject to abstract constraints, having their meanings in relation to each other determined by definitions and axioms (for example, the laws defining an equivalence relation), making it possible to manipulate the resulting information by means of the inference rules in a proof system. This illustrates the proof-theoretic aspect of a symbol system.
Suppose that a formal system like \(\{ L_\text{A}, L_\text{B} \}\!\) is initially approached from a theoretical direction, in other words, by listing the abstract properties one thinks it ought to have. Then the existence of an extensional model that satisfies these constraints, as exhibited by the sign relation tables, demonstrates that one's theoretical description is logically consistent, even if the models that first come to mind are still a bit too abstractly symbolic and do not have all the dynamic concreteness that is demanded of system-theoretic interpretations. This amounts to the other side of the ledger, the model-theoretic aspect of a symbol system, at least insofar as the present account has dealt with it.
More is required of the modeler, however, in order to find the desired kinds of system-theoretic models (for example, state transition systems), and this brings the search for realizations of formal systems down to the toughest part of the exercise. Some of the problems that emerge were highlighted in the example of \(\text{A}\!\) and \(\text{B}\!\). Although it is ordinarily possible to construct state transition systems in which the states of interpreters correspond relatively directly to the acceptations of the primitive signs given, the conflict of interpretations that develops between different interpreters from these prima facie implementations is a sign that there is something superficial about this approach.
The integration of model-theoretic and proof-theoretic aspects of physical symbol systems, besides being closely analogous to the integration of denotative and connotative aspects of sign relations, is also relevant to the job of integrating dynamic and symbolic frameworks for intelligent systems. This is so because the search for dynamic realizations of symbol systems is only a more pointed exercise in model theory, where the mathematical materials made available for modeling are further constrained by system-theoretic principles, like being able to say what the states are and how the transitions are determined.
Recapitulation : A Brush with Symbols
A common goal of work in artificial intelligence and cognitive simulation is to understand how is it possible for intelligent life to evolve from elements available in the primordial sea. Simply put, the question is: “What's in the brine that ink may character?”
Pursuant to this particular way of setting out on the long-term quest, a more immediate goal of the current project is to understand the action of full-fledged symbols, insofar as they conduct themselves through the media of minds and quasi-minds. At this very point the quest is joined by the pragmatic investigations of signs and inquiry, which share this interest in chasing down symbols to their precursive lairs.
In the pragmatic theory of signs a symbol is a strangely insistent yet curiously indirect type of sign, one whose accordance with its object depends sheerly on the real possibility that it will be so interpreted. Taking on the nature of a bet, a symbol's prospective value trades on nothing more than the chance of acquiring the desired interpretant, and thus it can capitalize on the simple fact that what it proposes is not impossible. In this way it is possible to see that a formal principle is involved in the success of symbols. The elementary conceivability of a particular sign relation, the pure circumstance that renders it logically or mathematically possible, means that the formal constraint it places on its domains is always really and potentially there, awaiting its discovery and exploitation for the purposes of representation and communication.
In this question about the symbol's capacity for meaning, then, is found another contact between the theory of signs and the logic of inquiry. As C.S. Peirce expressed it:
|
Now, I ask, how is it that anything can be done with a symbol, without reflecting upon the conception, much less imagining the object that belongs to it? It is simply because the symbol has acquired a nature, which may be described thus, that when it is brought before the mind certain principles of its use — whether reflected on or not — by association immediately regulate the action of the mind; and these may be regarded as laws of the symbol itself which it cannot as a symbol transgress.
|
| (Peirce, CE 1, 173).
|
|
Inference in general obviously supposes symbolization; and all symbolization is inference. For every symbol as we have seen contains information. And … all kinds of information involve inference. Inference, then, is symbolization. They are the same notions. Now we have already analyzed the notion of a symbol, and we have found that it depends upon the possibility of representations acquiring a nature, that is to say an immediate representative power. This principle is therefore the ground of inference in general.
|
| (Peirce, CE 1, 280).
|
|
A symbol which has connotation and denotation contains information. Whatever symbol contains information contains more connotation than is necessary to limit its possible denotation to those things which it may denote. That is, every symbol contains more than is sufficient for a principle of selection.
|
| (Peirce, CE 1, 282).
|
|
The information of a term is the measure of its superfluous comprehension. That is to say that the proper office of the comprehension is to determine the extension of the term. …
Every addition to the comprehension of a term, lessens its extension up to a certain point, after that further additions increase the information instead. …
And therefore as every term must have information, every term has superfluous comprehension. And, hence, whenever we make a symbol to express any thing or any attribute we cannot make it so empty that it shall have no superfluous comprehension.
I am going, next, to show that inference is symbolization and that the puzzle of the validity of scientific inference lies merely in this superfluous comprehension and is therefore entirely removed by a consideration of the laws of information.
|
| (Peirce, CE 1, 467).
|
A full explanation of these statements, linking scientific inference, symbolization, and information together in such an integral fashion, would require an excursion into the pragmatic theory of information that Peirce was already presenting in lectures at Harvard as early as 1865. For now, let it suffice to say that this anticipation of the information concept, fully recognizing the reality of its dimension, would not sound too remote from the varieties of law abiding constraint exploitation that have become increasingly familiar since the dawn of cybernetics.
But more than this, Peirce's notion of information supplies an array of missing links that joins together in one scheme the logical roles of terms, propositions, and arguments, the semantic functions of denotation and connotation, and the practical methodology needed to address and measure the quantitative dimensions of information. This is precisely the kind of linkage that I need in this project to integrate the dynamic and symbolic aspects of inquiry.
Not by sheer coincidence, the task of understanding symbolic action, working up through icons and indices to the point of tackling symbols, is also one of the ultimate aims that the interpretive and objective frameworks being proposed here are intended to subserve.
An OF is a convenient stage for those works that have progressed far enough to make use of it, but in times of flux it must be remembered that an OF is only a hypostatic projection, that is, the virtual image, reified concept, or phantom limb of the IF that tentatively extends it.
When the IF and the OF sketched here have been developed far enough, I hope to tell wherein and whereof a sign is able, by its very character, to address itself to a purpose, one determined by its objective nature and determining, in a measure, that of its intended interpreter, to the extent that it makes the other wiser than the other would otherwise be.
C'est Moi
From the emblem unfurled on a tapestry to tease out the working of its loom and spindle, a charge to bind these frameworks together is drawn by necessity from a single request: To whom is the sign addressed? The easy, all too easy answer comes To whom it may concern, but this works more to put off the question than it acts as a genuine response. To say that a sign relation is intended for the use of its interpreter, unless one has ready an independent account of that agent's conduct, only rephrases the initial question about the end of interpretation.
The interpreter is an agency depicted over and above the sign relation, but in a very real sense it is simply identical with the whole of it. And so one is led to examine the relationship between the interpreter and the interpretant, the element falling within the sign relation to which the sign in actuality tends. The catch is that the whole of the intended sign relation is seldom known from the beginning of inquiry, and so the aimed for interpretant is often just as unknown as the rest.
These eventualities call for the elaboration of interpretive and objective frameworks in which not just the specious but the speculative purpose of a sign can be contemplated, permitting extensions of the initial data, through error and retrial, to satisfy emergent and recurring questions.
At last, even with the needed frameworks only partly shored up, I can finally ravel up and tighten one thread of this rambling investigation. All this time, steadily rising to answer the challenge about the identity of the interpreter, Who's there?, and the role of the interpretant, Stand and unfold yourself, has been the ready and abiding state of a certain system of interpretation, developing its character and gradually evolving its meaning through a series of imputations and extensions. Namely, the MOI (the SOI experienced as an object) can answer for the interpreter, to whatever extent that conduct can be formalized, and the IM (the SOI experienced in action, in statu nascendi) can serve as a proxy for the momentary thrust of interpretive dynamics, to whatever degree that process can be explicated.
To put a finer point on this result I can do no better at this stage of discussion than to recount the “metaphorical argument” that Peirce often used to illustrate the same conclusion.
|
I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing, every word, concept, symbol has an equivalent term — or one which has become identified with it, — in short, has an interpretant.
Consider, what a word or symbol is; it is a sort of representation. Now a representation is something which stands for something. … A thing cannot stand for something without standing to something for that something. Now, what is this that a word stands to? Is it a person?
We usually say that the word homme stands to a Frenchman for man. It would be a little more precise to say that it stands to the Frenchman's mind — to his memory. It is still more accurate to say that it addresses a particular remembrance or image in that memory. And what image, what remembrance? Plainly, the one which is the mental equivalent of the word homme — in short, its interpretant. Whatever a word addresses then or stands to, is its interpretant or identified symbol. …
The interpretant of a term, then, and that which it stands to are identical. Hence, since it is of the very essence of a symbol that it should stand to something, every symbol — every word and every conception — must have an interpretant — or what is the same thing, must have information or implication.
|
| (Peirce, CE 1, 466–467).
|
It will take a while to develop the wealth of information that a suitably perspicacious and persistent IF would find implicit in this unassuming homily. The main innovations that this project can hope to add to the story are as follows:
- To prescribe a context of effective systems theory (C'EST), one that can provide for the computational formalization of each intuitively given interpreter as a determinate model of interpretation (MOI). An appropriate set of concepts and methods would deal with the generic constitutions of interpreters, converting paraphrastic and periphrastic descriptions of their interpretive practice into relatively complete and concrete specifications of sign relations.
- To prepare a fully dynamic basis for actualizing interpretants. This means that an interpretant addressed by the interpretation of a sign would not be left in the form of a detached token or abstract memory image to be processed by a hypothetical but largely nondescript interpreter, but realized as a definite type of state configuration in a qualitative dynamic system. To fathom what should be the symbolic analogue of a state with momentum has presented this project with difficulties both conceptual and terminological. So far in this project, I have attempted to approach the character of an active sign-theoretic state in terms of an interpretive moment (IM), information state (IS), attended token (AT), situation of use (SOU), or instance of use (IOU). A successful concept would capture the transient dispositions that drive interpreters to engage in specific forms of inquiry, defining their ongoing state of uncertainty with regard to objects and questions of immediate concern.
Entr'acte
Have I pointed at this problem from enough different directions to convey an idea of its location and extent? Here is one more variation on the theme. I believe that our theoretical empire is bare in spots. There does not exist yet in the field a suitably comprehensive concept of a dynamic system moving through a variable state of information. This conceptual gap apparently forces investigators to focus on one aspect or the other, on the dynamic bearing or the information borne, but leaves their studies unable to integrate the several perspectives into a full-dimensioned picture of the evolving knowledge system.
It is always possible that the dual aspects of transformation and information are conceptually complementary and even non-orientable. That is, there may be no way to arrange our mental apparatus to grasp both sides at the same time, and the whole appearance that there are two sides may be an illusion of overly local and myopic perspectives. However, none of this should be taken for granted without proof.
Whatever the case, to constantly focus on the restricted aspects of dynamics adequately covered by currently available concepts leads one to ignore the growing body of symbolic knowledge that the states of systems potentially carry. Conversely, to leap from the relatively secure grounds of physically based dynamics into the briar patch of formally defined symbol systems often marks the last time that one has sufficient footing on the dynamic landscape to contemplate any form of overarching law, or any rule to prospectively govern the evolution of reflective knowledge. This is one of the reasons I continue to strive after the key ideas here. If straw is all that one has in reach, then ships and shelters will have to be built from straw.
Discussion of Formalization : Specific Objects
|
“Knowledge” is a referring back: in its essence a regressus in infinitum. That which comes to a standstill (at a supposed causa prima, at something unconditioned, etc.) is laziness, weariness —
|
| — Nietzsche, The Will to Power, [Nie, S575, 309]
|
With this preamble, I return to develop my own account of formalization, with special attention to the kind of step that leads from the inchoate chaos of casual discourse to a well-founded discussion of formal models. A formalization step, of the incipient kind being considered here, has the peculiar property that one can say with some definiteness where it ends, since it leads precisely to a well-defined formal model, but not with any definiteness where it begins. Any attempt to trace the steps of formalization backward toward their ultimate beginnings can lead to an interminable multiplicity of open-ended explorations. In view of these circumstances, I will limit my attention to the frame of the present inquiry and try to sum up what brings me to this point.
It begins like this: I ask whether it is possible to reason about inquiry in a way that leads to a productive end. I pose my question as an inquiry into inquiry, and I use the formula \({}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!\) to express the relationship between the present inquiry, \(y_0,\!\) and a generic inquiry, \(y.\!\) Then I propose a couple of components of inquiry, discussion and formalization, that appear to be worth investigating, expressing this proposal in the form \(y >\!\!= \{ d, f \}.\!\) Applying these components to each other, as must be done in the present inquiry, I am led to the current discussion of formalization, \(y_0 = y \cdot y >\!\!= f \cdot d.\!\)
There is already much to question here. At least, so many repetitions of the same mysterious formula are bound to lead the reader to question its meaning. Some of the more obvious issues that arise are these:
- The term generic inquiry is ambiguous. Its meaning in practice depends on whether the description of an inquiry as being generic is interpreted literally or merely as a figure of speech. In the literal case, the name \({}^{\backprime\backprime} y {}^{\prime\prime}\!\) denotes a particular inquiry, \(y \in Y,\!\) one that is assumed to be prototypical in yet to be specified ways. In the figurative case, the name \({}^{\backprime\backprime} y {}^{\prime\prime}\!\) is simply a variable that ranges over a collection \(Y\!\) of nominally conceivable inquiries.
- First encountered, the recipe \(y_0 = y \cdot y\!\) appears to specify that the present inquiry is constituted by taking everything denoted by the most general concept of inquiry that the present inquirer can imagine and inquiring into it by means of the most general capacity for inquiry that this same inquirer can muster.
- Contemplating the formula \({}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!\) in the context of the subordination \(y >\!\!= \{ d, f \}\!\) and the successive containments \(F \subseteq M \subseteq D,\!\) the \(y\!\) that inquires into \(y\!\) is not restricted to examining \(y \operatorname{'s}\!\) immediate subordinates, \(d\!\) and \(f,\!\) but it can investigate any feature of \(y \operatorname{'s}\!\) overall context, whether objective, syntactic, interpretive, and whether definitive or incidental, and finally it can question any supporting claim of the discussion. Moreover, the question \(y\!\) is not limited to the particular claims that are being made here, but applies to the abstract relations and the general concepts that are invoked in making them. Among the many kinds of inquiry that suggest themselves, there are the following possibilities:
- Inquiry into propositions about application and equality. One may well begin with the forms of application and equality that are invoked in the formula \({}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!\) itself.
- Inquiry into application \((\cdot),\!\) for example, the way that the term \({}^{\backprime\backprime} y \cdot y {}^{\prime\prime}\!\) indicates the application of \(y\!\) to \(y\!\) in the formula \({}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!\)
- Inquiry into equality \((=),\!\) for example, the meaning of the equal sign in the formula \({}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}.\!\)
- Inquiry into indices, for example, the significance of \({}^{\backprime\backprime} 0 {}^{\prime\prime}\!\) in \({}^{\backprime\backprime} y_0 {}^{\prime\prime}.\!\)
- Inquiry into terms, specifically, constants and variables. What are the functions of \({}^{\backprime\backprime} y {}^{\prime\prime}\!\) and \({}^{\backprime\backprime} y_0 {}^{\prime\prime}\!\) in this respect?
- Inquiry into decomposition or subordination, for example, as invoked by the sign \({}^{\backprime\backprime} >\!\!= {}^{\prime\prime}\!\) in the formula \({}^{\backprime\backprime} y >\!\!= \{ d, f \} {}^{\prime\prime}.\!\)
- Inquiry into containment or inclusion. In particular, examine the assumption that formalization \(F\!\), mediation \(M\!\), and discussion \(D\!\) are ordered as \(F \subseteq M \subseteq D\!\), a claim that determines the chances that a formalization has an object, the degree to which a formalization can be carried out by means of a discussion, and the extent to which an object of formalization can be conveyed by a form of discussion.
If inquiry begins in doubt, then inquiry into inquiry begins in doubt about doubt. All things considered, the formula \({}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!\) has to be taken as the first attempt at a description of the problem, a hypothesis about the nature of inquiry, or an image that is tossed out by way of getting an initial fix on the object in question. Everything in this account so far, and everything else that I am likely to add, can only be reckoned as hypothesis, whose accuracy, pertinence, and usefulness can be tested, judged, and redeemed only after the fact of proposing it and after the facts to which it refers have themselves been gathered up.
A number of problems present themselves due to the context in which the present inquiry is aimed to present itself. The hypothesis that suggests itself to one person, as worth exploring at a particular time, does not always present itself to another person as worth exploring at the same time, or even necessarily to the same person at another time. In a community of inquiry that extends beyond an isolated person and in a process of inquiry that extends beyond a singular moment in time, it is therefore necessary to consider the nature of the communication process that the discussion of inquiry in general and the discussion of formalization in particular need to invoke for their ultimate utility.
Solipsism is no solution to the problems of community, since even an isolated individual, if ever there were such a thing, would have to maintain the lines of communication that it takes to integrate past, present, and prospective selves — or translating everything into the present, the parts of one's actually present self that involve actual experiences and present observations, present expectations as reflective of actual memories, and present intentions as reflective of actual hopes. So the dialogue that one holds with oneself is every bit as problematic as the dialogue that one holds with others. Others but surprise us in other ways than we ordinarily surprise ourselves.
I recognize inquiry as beginning with a surprising phenomenon or a problematic situation, more briefly described as a surprise or a problem, respectively. These are the kinds of moments that try our souls, the instances of events that instigate inquiry as an effort to achieve their own resolution. Surprises and problems are experienced as afflicted with an irritating uncertainty or a compelling difficulty, one that calls for a response on the part of the agent in question:
- A surprise calls for an explanation to resolve the uncertainty that is present in it. This uncertainty is associated with a difference between observations and expectations.
- A problem calls for a plan of action to resolve the difficulty that is present in it. This difficulty is associated with a difference between observations and intentions.
To express this diversity in a unified formula, both types of inquiry begin with a delta \((\Delta),\!\) a compact symbol that admits a spectrum of expansions: debt, difference, difficulty, discrepancy, dispersion, distribution, doubt, duplicity, or duty.
Expressed another way, inquiry begins with a doubt about one's object, whether this means what is true of a case, an object, or a world, what to do about reaching a goal, or whether the hoped-for goal is really good for oneself — with all that these questions lead to in essence, in action, or in fact.
Perhaps there is an inexhaustible reality that issues in these apparent mysteries and recurrent crises, but, by the time I say this much, I am already indulging in a finite image, a hypothesis about what is going on. If nothing else, then, one finds again the familiar pattern, where the formative relation between the informal and the formal merely serves to remind one anew of the relation between the infinite and the finite.
The Will to Form
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The power of form, the will to give form to oneself. “Happiness” admitted as a goal. Much strength and energy behind the emphasis on forms. The delight in looking at a life that seems so easy. — To the French, the Greeks looked like children.
|
| — Nietzsche, The Will to Power, [Nie, S94, 58]
|
Let me see if I can summarize as quickly as possible the problem that I see before me. On each occasion that I try to express my experience, to lend it a form that others can recognize, to put it in a shape that I myself can later recall, or to store it in a state that allows me the chance of its re-experience, I generate an image of the way things are, or at least a description of how things seem to me. I call this process reflection, since it fabricates an image in a medium of signs that reflects an aspect of experience. Very often this experience is said to be of — what? — something that exists or persists at least partly outside the immediate experience, some action, event, or object that is imagined to inform the present experience, or perhaps some conduct of one's own doing that obtrudes for a moment into the world of others and meets with a reaction there. In all of these cases, where the experience is everted to refer to an object and thus becomes the attribute of something with an external aspect, something that is thus supposed to be a prior cause of the experience, the reflection on experience doubles as a reflection on that conduct, performance, or transaction that the experience is an experience of. In short, if the experience has an eversion that makes it an experience of an object, then its reflection is again a reflection that is once again of this object.
Just on the point of becoming lost in the morass of words for describing experience and the nuances of their interpretation, one can adopt a formal perspective, and realize that the relation among objects, experiences, and reflective images is formally analogous to the relation among objects, signs, and interpretant signs that is covered by the pragmatic theory of signs. The problem remains: How are the expressions of experience everted to form the exterior faces of extended objects and exploited to embed them in their external circumstances, and no matter whether this object with an outer face is oneself or another? Here, one needs to understand that expressions of experience include the original experiences themselves, at least, to the extent that they permit themselves to be recognized and reflected in ongoing experience. But now, from the formal point of view, How means only: To describe the formal conditions of a formal possibility.
The Forms of Reasoning
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The most valuable insights are arrived at last; but the most valuable insights are methods.
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| — Nietzsche, The Will to Power, [Nie, S469, 261]
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A certain arbitrariness has to be faced in the terms that one uses to talk about reasoning, to split it up into different parts and to sort it out into different types. It is like the arbitrary choice that one makes in assigning the midpoint of an interval to the subintervals on its sides. In setting out the forms of a nomenclature, in fitting the schemes of my terminology to the territory that it disturbs in the process of mapping, I cannot avoid making arbitrary choices, but I can aim for a strategy that is flexible enough to recognize its own alternatives and to accommodate the other options that lie within their scope.
If I make the mark of deduction the fact that it reduces the number of terms, as it moves from the grounds to the end of an argument, then I am due to devise a name for the process that augments the number of terms, and thus prepares the grounds for any account of experience.
- What name hints at the many ways that signs arise in regard to things?
- What name covers the manifest ways that a map takes over its territory?
- What name fits this naming of names, these proceedings that inaugurate
a sign in the first place, that duly install it on the office of a term?
- What name suits all the actions of addition, annexation, incursion, and
invention that instigate the initial bearing of signs on an object domain?
In the interests of a maximal analytic precision (MAP), it is fitting that I should try to sharpen this notion to the point where it applies purely to a simple act, that of entering a new term on the lists, in effect, of enlisting a new term to the ongoing account of experience. Thus, let me style this process as adduction or production, in spite of the fact that the aim of precision is partially blunted by the circumstance that these words have well-worn uses in other contexts. In this way, I can isolate to some degree the singular step of adding a term, leaving it to a later point to distinguish the role that it plays in an argument.
As it stands, the words adduction and production could apply to the arbitrary addition of terms to a discussion, whether or not these terms participate in valid forms of argument or contribute to their mediation. Although there are a number of auxiliary terms, like factorization, mediation, or resolution, that can help to pin down these meanings, it is also useful to have a word that can convey the exact sense meant. Therefore, I coin the term obduction to suggest the type of reasoning process that is opposite or converse to deduction and that introduces a middle term in the way as it passes from a subject to a predicate.
Consider the adjunction to one's vocabulary that is comprised of these three words: adduction, production, obduction. In particular, how do they appear in the light of their mutual applications to each other and especially with respect to their own reflexivities? Notice that the terms adduction and production apply to the ways that all three terms enter this general discussion, but that obduction applies only to their introduction only in specific contexts of argument.
Another dimension of variation that needs to be noted among these different types of processes is their status with regard to determinism. Given the ordinary case of a well-formed syllogism, deduction is a fully deterministic process, since the middle term to be eliminated is clearly marked by its appearance in a pair of premisses. But if one is given nothing but the fact that forms this conclusion, or starts with a fact that is barely suspected to be the conclusion of a possible deduction, then there are many other middle terms and many other premisses that might be construed to result in this fact. Therefore, adduction and production, for all their uncontrolled generality, but even obduction, in spite of its specificity, cannot be treated as deterministic processes. Only in degenerate cases, where the number of terms in a discussion is extremely limited, or where the availability of middle terms is otherwise restricted, can it happen that these processes become deterministic.
A Fork in the Road
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On “logical semblance” — The concepts “individual” and “species” equally false and merely apparent. “Species” expresses only the fact that an abundance of similar creatures appear at the same time and that the tempo of their further growth and change is for a long time slowed down, so actual small continuations and increases are not very much noticed (— a phase of evolution in which the evolution is not visible, so an equilibrium seems to have been attained, making possible the false notion that a goal has been attained — and that evolution has a goal —).
|
| — Nietzsche, The Will to Power, [Nie, S521, 282]
|
It is worth trying to discover, as I currently am, how many properties of inquiry can be derived from the simple fact that it needs to be able to apply to itself. I find three main ways to approach the problem of inquiry's self-application, or the question of inquiry's reflexivity:
- One way attempts to continue the derivation in the manner of a necessary deduction, perhaps by reasoning in the following vein: If self-application is a property of inquiry, then it is sensible to inquire into the concept of application that could make this conceivable, and not just conceivable, but potentially fruitful.
- Another way breaks off the attempt at a deductive development and puts forth a full-scale model of inquiry, one that has enough plausibility to be probated in the court of experience and enough specificity to be tested in the context of self-application.
- The last way is a bit ambivalent in its indications, seeking as it does both the original unity and the ultimate synthesis at one and the same time. Perhaps it goes toward reversing the steps that lead up to this juncture, marking it down as an impasse, chalking it up as a learning experience, or admitting the failure of the imagined distinction to make a difference in reality. Whether this form of egress is read as a backtracking correction or as a leaping forward to the next level of integration, it serves to erase the distinction between demonstration and exploration.
Without a clear sense of how many properties of inquiry are necessary consequences of its self-application and how many are merely accessory to it, or even whether some contradiction still lies lurking within the notion of reflexivity, I have no choice but to follow all three lines of inquiry wherever they lead, keeping an eye out for the synchronicities, the constructive collusions and the destructive collisions that may happen to occur among them.
The fictions that one devises to shore up a shaky account of experience can often be discharged at a later stage of development, gradually coming to be replaced with primitive elements of less and less dubious characters. Hypostases and hypotheses, the creative terms and the inventive propositions that one coins to account for otherwise ineffable experiences, are tokens that are subject to a later account. Under recurring examination, many such tokens are found to be ciphers, marks that no one will miss if they are canceled out altogether. The symbolic currencies that tend to survive lend themselves to being exchanged for stronger and more settled constructions, in other words, for concrete definitions and explicit demonstrations, gradually leading to primitive elements of more and more durable utilities.
A Forged Bond
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The form counts as something enduring and therefore more valuable; but the form has merely been invented by us; and however often “the same form is attained”, it does not mean that it is the same form — what appears is always something new, and it is only we, who are always comparing, who include the new, to the extent that it is similar to the old, in the unity of the “form”. As if a type should be attained and, as it were, was intended by and inherent in the process of formation.
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| — Nietzsche, The Will to Power, [Nie, S521, 282]
|
A unity can be forged among the methods by noticing the following connections among them. All the while that one proceeds deductively, the primitive elements, the definitions and the axioms, must still be introduced hypothetically, notwithstanding the support they get from common sense and widespread assent. And the whole symbolic system that is constructed through hypothesis and deduction must still be tested in experience to see if it serves any purpose to maintain it.
A Formal Account
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Form, species, law, idea, purpose — in all these cases the same error is made of giving a false reality to a fiction, as if events were in some way obedient to something — an artificial distinction is made in respect of events between that which acts and that toward which the act is directed (but this “which” and this “toward” are only posited in obedience to our metaphysical-logical dogmatism: they are not “facts”).
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| — Nietzsche, The Will to Power, [Nie, S521, 282]
|
In this Section (1.3.5), I am considering the step of formalization that takes discussion from a large scale informal inquiry to a well-defined formal inquiry, establishing a relation between the implicit context and the explicit text.
In this project as a whole, formalization is used to produce formal models that represent relevant features of a phenomenon or process of interest. Thus, the formal model is what constitutes the image of formalization.
The role of formalization splits into two different cases depending on the intended use of the formal model. When the phenomenon of interest is external to the agent that is carrying out the formalization, then the model of that phenomenon can be developed without doing any great amount of significant reflection on the formalization process itself. This is usually a more straightforward operation, since it can avail itself of automatic competencies that are not themselves in question. But when the phenomenon of interest is entangled with the conduct of the agent in question, then the formal modeling of that conduct will generally involve a more or less difficult component of reflection.
In a recursive context, a principal benefit of the formalization step is to find constituents of inquiry with reduced complexities, drawing attention from the context of informal inquiry, whose stock of questions may not be grasped well enough ever to be fruitful and the scope of whose questions may not be focused well enough ever to see an answer, and concentrating effort in an arena of formalized inquiry, where the questions are posed well enough to have some hope of bearing productive answers in a finite time.
Analogs, Icons, Models, Surrogates
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One should not understand this compulsion to construct concepts, species, forms, purposes, laws (“a world of identical cases”) as if they enabled us to fix the real world; but as a compulsion to arrange a world for ourselves in which our existence is made possible: — we thereby create a world which is calculable, simplified, comprehensible, etc., for us.
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| — Nietzsche, The Will to Power. [Nie, S521, 282]
|
This project makes pivotal use of certain formal models to represent the conceived structure in a phenomenon of interest (POI). For my purposes, the phenomenon of interest is typically a process of interpretation or a process of inquiry, two nominal species of process that will turn out to evolve from different points of view on the very same form of conduct.
Commonly, a process of interest presents itself as the trajectory that an agent describes through an extended space of configurations. The work of conceptualization and formalization is to represent this process as a conceptual object in terms of a formal model. Depending on the point of view that is taken from moment to moment in this work, the model of interest (MOI) may be cast either as a model of interpretation or as a model of inquiry. As might be anticipated, it will turn out that both descriptions refer essentially to the same subject, but this will take some development to become clear.
In this work, the basic structure of each MOI is adopted from the pragmatic theory of signs and the general account of its operation is derived from the pragmatic theory of inquiry. The indispensable utility of these formal models hinges on the circumstance that each MOI, whether playing its part in interpretation or in inquiry, is always a model in two important senses of the word. First, it is a model in the logical sense that its structure satisfies a formal theory or an abstract specification. Second, it is a model in the analogical sense that it represents an aspect of the structure that is present in another object or domain.
Steps and Tests of Formalization
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This same compulsion exists in the sense activities that support reason — by simplification, coarsening, emphasizing, and elaborating, upon which all “recognition”, all ability to make oneself intelligible rests. Our needs have made our senses so precise that the “same apparent world” always reappears and has thus acquired the semblance of reality.
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| — Nietzsche, The Will to Power, [Nie, S521, 282]
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A step of formalization moves the active focus of discussion from the presentational object or the source domain that constitutes the phenomenon of interest to the representational object or the target domain that makes up the relevant model of interest. If the structure in the source context is already formalized then the step of formalization can itself be formalized in an especially elegant and satisfying way as a structure-preserving map, a homomorphism, or an arrow in the sense of mathematical category theory.
The test of a formalization being complete is that a computer program could in principle carry out the steps of the process being formalized exactly as represented in the formal model or image. It needs to be appreciated that this test is a criterion of sufficiency to formal understanding and not of necessity directed toward a material re-creation or a concrete simulation of the formalized process. The ordinary agents of informal discussion who address the task of formalization do not disappear in the process of completing it, since it is precisely for their understanding that the step is undertaken. Only if the phenomenon or process at issue were by its very nature solely a matter of form could its formal analogue constitute an authentic reproduction. However, this potential consideration is far from the ordinary case that I need to discuss at present.
In ordinary discussion, agents of inquiry and interpretation depend on the likely interpretations of others to give their common notions and their shared notations a meaning in practice. This means that a high level of implicit understanding is relied on to ground each informal inquiry in practice. The entire framework of logical assumptions and interpretive activities that is needed to shore up this platform will itself resist analysis, since it is precisely to save the effort of repeating routine analyses that the whole infrastructure is built.
A Puckish Referee
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Our subjective compulsion to believe in logic only reveals that, long before logic itself entered our consciousness, we did nothing but introduce its postulates into events: now we discover them in events — we can no longer do otherwise — and imagine that this compulsion guarantees something connected with “truth”.
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| — Nietzsche, The Will to Power, [Nie, S521, 282–283]
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In a formal inquiry of the sort projected here, the less the discussants need to depend on the compliance of understanding interpreters the more they will necessarily understand at the end of the formalization step.
It might then be thought that the ultimate zero of understanding expected on the part of the interpreter would correspond to the ultimate height of understanding demanded on the part of the formalizer, but this assumption neglects the negative potential of misunderstanding, the sheer perversity of interpretation that our human creativity can bring to bear on any text.
But computers are initially just as incapable of misunderstanding as they are of understanding. Therefore, it actually forms a moderate compromise to address the task of interpretation to a computational system, a thing that is known to begin from a moderately neutral initial condition.
Partial Formalizations
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It is we who created the “thing”, the “identical thing”, subject, attribute, activity, object, substance, form, after we had long pursued the process of making identical, coarse and simple. The world seems logical to us because we have made it logical.
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| — Nietzsche, The Will to Power, [Nie, S521, 283]
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In many discussions the source context remains unformalized in itself, taking form only according to the image it receives in one or another individual MOI. In cases like these, the step of formalization does not amount to a total function but is limited to a partial mapping from the source to the target. Such a partial representation is analogous to a sampling operation. It is not defined on every point of the source domain but assigns values only to a proper selection of source elements. Thus, a partial formalization can be regarded as achieving its form of simplification in a loose way, ignoring elements of the source domain and collapsing material distinctions in irregular fashions.
A Formal Utility
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Ultimate solution. — We believe in reason: this, however, is the philosophy of gray concepts. Language depends on the most naive prejudices.
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| — Nietzsche, The Will to Power, [Nie, S522, 283]
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The usefulness of the MOI as the upshot of the formalization arrow is that it provides discussion with a compact image of the source domain. In formalization one strives to extract a simpler image of the larger inquiry, a context of participatory action that one is too embroiled in carrying out step by step to see as a whole. Seen in this light, the purpose of formalization is to identify a simpler version of the problematic phenomenon or to fashion a simpler image of the difficult inquiry, one that is well-defined enough and simple enough to assure its termination in a finite interval of space and time. As a result, one of the main benefits of adopting the objective of formalization is that it equips discussion with a pre-set termination criterion, or a stopping rule.
In the context of the recursive inquiry that I have outlined, the step of formalization is intended to bring discussion appreciably closer to a solid base for the operational definition of inquiry.
A Formal Aesthetic
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Now we read disharmonies and problems into things because we think only in the form of language — and thus believe in the “eternal truth” of “reason” (e.g., subject, attribute, etc.)
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| — Nietzsche, The Will to Power, [Nie, S522, 283]
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Recognizing that the Latin word forma means not just form but also beauty supplies a clue that not all formal models are equally valuable for a purpose of interest. There is a certain quality of formal elegance, or select character, that is essential to the practical utility of the model.
The virtue of a good formal model is to provide discussion with a fitting image of the whole phenomenon of interest. The aim of formalization is to extract from an informal discussion or locate within a broader inquiry a clearer and simpler image of the whole activity. If the formalized image or precis is unusually apt then it might be prized as a gnomon or a recapitulation and be said to capture the essence, the gist, or the nub of the whole affair.
A pragmatic qualification of this virtue requires that the image be formed quickly enough to take decisive action on it. So the quality of being a result often takes precedence over the quality of the result. A definite result, however partial, is frequently reckoned as better than having to wait for a definitive picture that may never develop.
But an overly narrow or premature formalization, where the nature of the phenomenon of interest is too much denatured in the formal image, may result in destroying all interest in the result that does result.
A Formal Apology
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We cease to think when we refuse to do so under the constraint of language; we barely reach the doubt that sees this limitation as a limitation.
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| — Nietzsche, The Will to Power, [Nie, S522, 283]
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Seizing the advantage of this formal flexibility makes it possible to take abstract leaps over a multitude of material obstacles, to reason about many properties of objects and processes from a knowledge of their form alone, without having to know everything about their material content down to the depths that matter can go.
A Formal Suspicion
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Rational thought is interpretation according to a scheme that we cannot throw off.
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| — Nietzsche, The Will to Power, [Nie, S522, 283]
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I hope that the reader has arrived by now at an independent suspicion that the process of formalization is a microcosm nearly as complex as the whole subject of inquiry itself. Indeed, the initial formulation of a problem is tantamount to a mode of representational inquiry. In many ways this very first effort, that stirs from the torpor of ineffable unease to seek out any sort of unity in the manifold of fragmented impressions, is the most difficult, subtle, and crucial kind of inquiry. It begins in doubt about even so much as a fair way to represent the problematic situation, but its result can predestine whether subsequent inquiry has any hope of success. There is very little in this brand of formal engagement and participatory representation that resembles the simple and disinterested act of holding a mirror, flat and featureless, up to nature.
If formalization really is a form of inquiry in itself, then its formulations have deductive consequences that can be tested. In other words, formal models have logical effects that reflect on their fitness to qualify as representations, and these effects can cause them to be rejected merely on the grounds of being a defective picture or a misleading conception of the source phenomenon. Therefore, it should be appreciated that software tailored to this task will probably need to spend more time in the alterations of backtracking than it will have occasion to trot out parades of ready-to-wear models.
Impelled by the mass of assembled clues from restarts and refits to the gathering form of a coherent direction, the inkling may have gradually accumulated in the reader that something of the same description has been treated in the pragmatic theory of inquiry under the heading of abductive reasoning. This is distinguished from inductive reasoning, that goes from the particular to the general, in that abductive reasoning must work from a mixed collection of generals and particulars toward a middle term, a formal intermediary that is more specific than the vague allusions gathered about its subject and more generic than the elusive instances fashioned to illustrate its prospective predicates.
In a recursive context, the function of formalization is to relate a difficult problem to a simpler problem, breaking the original inquiry into two parts, the step of formalization and the rest of the inquiry, both of which branches it is hoped will be nearer to solid ground and easier to grasp than the original question.
The Double Aspect of Concepts
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Nothing is more erroneous than to make of psychical and physical phenomena the two faces, the two revelations of one and the same substance. Nothing is explained thereby: the concept “substance” is perfectly useless as an explanation. Consciousness in a subsidiary role, almost indifferent, superfluous, perhaps destined to vanish and give way to a perfect automatism —
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| — Nietzsche, The Will to Power, [Nie, S523, 283]
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This project is a particular inquiry into the nature of inquiry in general. As a consequence, every concept that appears in it takes on a double aspect.
To illustrate, let us take the concept of a sign relation as an example of a construct that appears in this work and let me use it to speak about my own agency in this inquiry. All I need to say about a sign relation at this point is that it is a three-place relation, and therefore can be represented as a relational data-base with three columns, in this case naming the object, the sign, and the interpretant of the relation at each moment in time of the corresponding sign process.
At any given moment of this inquiry I will be participating in a certain sign relation that constitutes the informal context of my activity, the full nature of which I can barely hope to conceptualize in explicitly formal terms. At times, the object of this informal sign relation will itself be a sign relation, typically one that is already formalized or one that I have a better hope of formalizing, but it could conceivably be the original sign relation with which I began.
In such cases, when the object of a sign relation is also a sign relation, the general concept of a sign relation takes on a double duty:
- The less formalized sign relation is used to mediate the present inquiry. As a conceptual construct, it is not yet fully conceived or not yet fully constructed at the moments of inquiry being considered. Perhaps it is better to regard it as a concept under construction. Employed as a contextual apparatus, this sign relation serves an instrumental role in the construal and the study of its designated objective sign relation.
- The more formalized sign relation is mentioned as a substantive object to be contemplated and manipulated by the proceedings of this inquiry. As a conceptual construct, it exemplifies its intended role best if it is already as completely formalized as possible. It is being engaged as a substantive object of inquiry.
I have given this inquiry a reflective or recursive cast, portraying it as an inquiry into inquiry, and one of the consequences of this picture is that every concept employed in the work will take on a divided role, double aspect, or dual purpose. At any moment, the object inquiry of the moment is aimed to take on a formal definition, while the active inquiry need not acknowledge any image that it does not recognize as reflecting itself, nor is it bound by any horizon that does not capture its spirit.
A Formal Permission
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If there are to be synthetic a priori judgments, then reason must be in a position to make connections: connection is a form. Reason must possess the capacity of giving form.
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| — Nietzsche, The Will to Power, [Nie, S530, 288]
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A Formal Invention
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Before there is “thought” (gedacht) there must have been “invention” (gedichtet); the construction of identical cases, of the appearance of sameness, is more primitive than the knowledge of sameness.
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| Nietzsche, The Will to Power, [Nie, S544, 293]
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Recursion in Perpetuity
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Will to truth is a making firm, a making true and durable, an abolition of the false character of things, a reinterpretation of it into beings.
“Truth” is therefore not something there, that might be found or discovered — but something that must be created and that gives a name to a process, or rather to a will to overcome that has in itself no end — introducing truth, as a processus in infinitum, an active determining — not a becoming-conscious of something that is in itself firm and determined.
It is a word for the “will to power”.
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| — Nietzsche, The Will to Power, [Nie, S552, 298]
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\(\cdots\!\)
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Life is founded upon the premise of a belief in enduring and regularly recurring things; the more powerful life is, the wider must be the knowable world to which we, as it were, attribute being. Logicizing, rationalizing, systematizing as expedients of life.
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| — Nietzsche, The Will to Power, [Nie, S552, 298–299]
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\(\cdots\!\)
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Man projects his drive to truth, his “goal” in a certain sense, outside himself as a world that has being, as a metaphysical world, as a “thing-in-itself”, as a world already in existence. His needs as creator invent the world upon which he works, anticipate it; this anticipation (this “belief” in truth) is his support.
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| — Nietzsche, The Will to Power, [Nie, S552, 299]
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\(\cdots\!\)
Processus, Regressus, Progressus
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From time immemorial we have ascribed the value of an action, a character, an existence, to the intention, the purpose for the sake of which one has acted or lived: this age-old idiosyncrasy finally takes a dangerous turn — provided, that is, that the absence of intention and purpose in events comes more and more to the forefront of consciousness.
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| — Nietzsche, The Will to Power, [Nie, S666, 351]
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\(\cdots\!\)
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Thus there seems to be in preparation a universal disvaluation: “Nothing has any meaning” — this melancholy sentence means “All meaning lies in intention, and if intention is altogether lacking, then meaning is altogether lacking, too”.
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| — Nietzsche, The Will to Power, [Nie, S666, 351]
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\(\cdots\!\)
|
In accordance with this valuation, one was constrained to transfer the value of life to a “life after death”, or to the progressive development of ideas or of mankind or of the people or beyond mankind; but with that one had arrived at a progressus in infinitum of purposes: one was at last constrained to make a place for oneself in the “world process” (perhaps with the dysdaemonistic perspective that it was a process into nothingness).
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| — Nietzsche, The Will to Power, [Nie, S666, 351]
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\(\cdots\!\)
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And do you know what “the world” is to me? Shall I show it to you in my mirror? This world: a monster of energy, without beginning, without end; a firm, iron magnitude of force that does not grow bigger or smaller, that does not expend itself but only transforms itself; as a whole, of unalterable size, a household without expenses or losses, but likewise without increase or income; enclosed by “nothingness” as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be “empty” here or there, but rather as force throughout, as a play of forces and waves of forces, at the same time one and many, increasing here and at the same time decreasing there; a sea of forces flowing and rushing together, eternally changing, eternally flooding back, with tremendous years of recurrence, with an ebb and a flood of its forms; out of the simplest forms striving toward the most complex, out of the stillest, most rigid, coldest forms toward the hottest, most turbulent, most self-contradictory, and then again returning home to the simple out of this abundance, out of the play of contradictions back to the joy of concord, still affirming itself in this uniformity of its courses and its years, blessing itself as that which must return eternally, as a becoming that knows no satiety, no disgust, no weariness: this, my Dionysian world of the eternally self-creating, the eternally self-destroying, this mystery world of the twofold voluptuous delight, my “beyond good and evil”, without goal, unless the joy of the circle is itself a goal; without will, unless a ring feels good will toward itself — do you want a name for this world? A solution for all its riddles? A light for you, too, you best-concealed, strongest, most intrepid, most midnightly men? — This world is the will to power — and nothing besides! And you yourselves are also this will to power — and nothing besides!
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| — Nietzsche, The Will to Power, [Nie, S1067, 549–550]
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I have attempted in a narrative form to present an accurate picture of the formalization process as it develops in practice. Of course, accuracy must be distinguished from precision, for there are times when accuracy is better served by a vague outline that captures the manner of the subject than it is by a minute account that misses the mark entirely or catches each detail at the expense of losing the central point. Conveying the traffic between chaos and form under the restraint of an overbearing and excisive taxonomy would have sheared away half the picture and robbed the whole exchange of the lion's share of the duty.
At moments I could do no better than to break into metaphor, but I believe that a certain tolerance for metaphor, especially in the initial stages of formalization, is a necessary capacity for reaching beyond the secure boundaries of what is already comfortable to reason. Plus, a controlled transport of metaphor allows one to draw on the boundless store of ready analogies and germinal morphisms that every natural language provides for free.
Finally, it would leave an unfair impression to delete the characters of narrative and metaphor from the text of the story, and especially after they have had such a hand in creating it.
Even the most precise of established formulations cannot be protected from being reused in ways that initially appear as an abuse of language.
One of the most difficult questions about the development of intelligent systems is how the power of abstraction can arise, beginning from the kinds of formal systems where each symbol has one meaning at most. I think that the natural pathway of this evolution has to go through the obscure territory of ambiguity and metaphor.
A critical phase and a crucial step in the development of intelligent systems, biological or technological, is concerned with achieving a certain power of abstraction, but the real trick is for the budding intelligence to accomplish this without losing a grip on the material contents of the abstract categories, the labels and levels of which this power intercalates and interposes between essence and existence.
If one looks to the surface material of natural languages for signs of how this power of abstraction might arise, one finds a suggestive set of potential precursors in the phenomena of ambiguity, anaphora, and metaphor. Keeping this in mind throughout the project, I aim to pay close attention to the places where the power of abstraction seems to develop, especially in the guises of systematic ambiguity and controlled metaphor.
Paradoxically, and a bit ironically, if one's initial attempt to formalize meaning begins with the aim of stamping out ambiguity, metaphor, and all forms of figurative language use, then one may have precluded all hope of developing a capacity for abstraction at any later stage.
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