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Differential Logic and Dynamic Systems • Part 2

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Author: Jon Awbrey



Transformations of Discourse

 

It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the fons et origo of an unfathomable transformation.

 
— Robert Musil, The Man Without Qualities, [Mus, 39]

In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.

My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.

As a first step I discuss the kinds of transformations that we already know as extensions and projections, and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.

Foreshadowing Transformations : Extensions and Projections of Discourse

 

And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.

 
— Gaston Leroux, The Phantom of the Opera, [Ler, 126]

Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type is implied any time that we make use of one alphabet that happens to be included in another alphabet When we are discussing differential issues we usually have in mind that the extended alphabet has a special construction or a specific lexical relation with respect to the initial alphabet one that is marked by characteristic types of accents, indices, or inflected forms.

Extension from 1 to 2 Dimensions

Figure 18-a lays out the angular form of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an areal view of each universe of discourse.

Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif

Figure 18-b shows the differential extension from to in a bundle of boxes form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a proposition at a point, in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.

Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif

Figure 18-c shows the same extension in a compact style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.

Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif

Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or digraph form of representation. (Notice that my definition of a digraph allows for loops or slings at individual points, in addition to arcs or arrows between the points.)

Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif

Extension from 2 to 4 Dimensions

Figure 19-a lays out the areal view or the angular form of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.

Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif

Figure 19-b shows the differential extension from to in the bundle of boxes form of venn diagram.

Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif

As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.

Figure 19-c illustrates the extension from 2 to 4 dimensions in the compact style of venn diagram. Here, just the changes with respect to the center cell are shown.

Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif

Figure 19-d gives the digraph form of representation for the differential extension where the 4 nodes marked with a circle are the cells respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.

Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif

Thematization of Functions : And a Declaration of Independence for Variables

And as imagination bodies forth
The forms of things unknown, the poet's pen
Turns them to shapes, and gives to airy nothing
A local habitation and a name.

A Midsummer Night's Dream, 5.1.18

In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.

Thematization : Venn Diagrams

 

The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.

 
— Walt Whitman, Leaves of Grass, [Whi, 11–12]

Figure 20-i traces the first couple of steps in this order of thematic progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition in the universe The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition a distinctive functional name Second, one has come to think explicitly about the target domain that contains the functional values of as when writing

Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif

In Figure 20-ii the proposition is viewed explicitly as a transformation from one universe of discourse to another.

Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif
o-------------------------------o     o-------------------------------o
|                               |     |                               |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|      /       \ /       \      |     |      /       \ /       \      |
|     /         o         \     |     |     /         o         \     |
|    /         /`\         \    |     |    /         /`\         \    |
|   o         o```o         o   |     |   o         o```o         o   |
|   |    u    |```|    v    |   |     |   |    u    |```|    v    |   |
|   o         o```o         o   |     |   o         o```o         o   |
|    \         \`/         /    |     |    \         \`/         /    |
|     \         o         /     |     |     \         o         /     |
|      \       / \       /      |     |      \       / \       /      |
|       o-----o   o-----o       |     |       o-----o   o-----o       |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
 \                             /       \                             /
   \                         /           \                         /
     \                     /               \          J          /
       \                 /                   \                 /
         \             /                       \             /
o----------\---------/----------o     o----------\---------/----------o
|            \     /            |     |            \     /            |
|              \ /              |     |              \ /              |
|         o-----@-----o         |     |         o-----@-----o         |
|        /`````````````\        |     |        /`````````````\        |
|       /```````````````\       |     |       /```````````````\       |
|      /`````````````````\      |     |      /`````````````````\      |
|     o```````````````````o     |     |     o```````````````````o     |
|     |```````````````````|     |     |     |```````````````````|     |
|     |```````` J ````````|     |     |     |```````` x ````````|     |
|     |```````````````````|     |     |     |```````````````````|     |
|     o```````````````````o     |     |     o```````````````````o     |
|      \`````````````````/      |     |      \`````````````````/      |
|       \```````````````/       |     |       \```````````````/       |
|        \`````````````/        |     |        \`````````````/        |
|         o-----------o         |     |         o-----------o         |
|                               |     |                               |
|                               |     |                               |
o-------------------------------o     o-------------------------------o
             J = u v                             x = J<u, v>

Figure 20-ii.  Thematization of Conjunction (Stage 2)

In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function to serve as the name of its dependent variable does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.

The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing and thereby assigns a concrete type to the abstract codomain To make this induction of variables more formal one can append subscripts, as in to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name with the variable name Thus we may think of as the cache variable corresponding to the function or the symbol considered as a contingent variable.

In Figure 20-iii we arrive at a stage where the functional equations and are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse and respectively. Subject to the cautions already noted, the function name can be reinterpreted as the name of a feature and the equation can be read as the logical equivalence To give it a generic name let us call this newly expressed, collateral proposition the thematization or the thematic extension of the original proposition

Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif

The first venn diagram represents the thematization of the conjunction with shading in the appropriate regions of the universe Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.

In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name are resolved by introducing a new variable name to take the place of and the region that represents this fresh featured is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name to the proposition we now give the name to its thematization Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function

From now on, the terms thematic extension and thematization will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from to we introduce a class of operators symbolized by the Greek letter writing in the present instance. The operator in the present situation bearing the type provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.

Figure 21 shows how the thematic extension operator acts on two further examples, the disjunction and the equality Referring to the disjunction as and the equality as we may express the thematic extensions as and

Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif

Thematization : Truth Tables

 

That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.

 
— Walt Whitman, Leaves of Grass, [Whi, 19]

Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.

A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions and



Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using and as function names and creating new variables and to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the -dimensional universes of and to the -dimensional universes of and The top halves of the Tables replicate the truth table patterns for and in the form and The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for and under the copies for and At this stage, the columns for and are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions and


 

 


All the data are now in place to give the truth tables for and All that remains to be done is to permute the rows and change the roles of and from dependent to independent variables. In Tables 24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples and in binary numerical order, suitable for viewing as the arguments of the maps and Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions and to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables and are now treated as independent variables.


 
 

 


An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables 25-i and 25-ii sort the rows in a different order, in effect treating and as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form and makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable is true then exhibits the pattern of the original and when is false then exhibits the pattern of its negation



 
 
 
 
 

 


Finally, Tables 26-i and 26-ii compare the tacit extensions and with the thematic extensions of the same types, as applied to the propositions and respectively.



Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form and Column 5 simplifies these equations into the form of algebraic expressions. As always, refers to exclusive disjunction and each appearing in the last two Columns refers to the corresponding variable name


   
     
     


In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables 28 and 29 present ordinary truth tables for the functions and for the corresponding thematizations


               
               
               
               


               
               
               
               
               
               
               
               


Propositional Transformations

 

If only the word ‘artificial’ were associated with the idea of art, or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that logical refers to artificial thought.

 
— John Dewey, How We Think, [Dew, 56–57]

In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.

Alias and Alibi Transformations

There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:

  1. A perspectival or alias transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.
  2. A transitional or alibi transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.

(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)

Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.

In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.

Transformations of General Type

 

Es ist passiert, “it just sort of happened”, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.

 
— Robert Musil, The Man Without Qualities, [Mus, 34]

Consider the situation illustrated in Figure 30, where the alphabets and are used to label basic features in two different logical universes, and

             o-------------------------------------------------------o
             | U                                                     |
             |                                                       |
             |             o-----------o   o-----------o             |
             |            /             \ /             \            |
             |           /               o               \           |
             |          /               / \               \          |
             |         /               /   \               \         |
             |        o               o     o               o        |
             |        |               |     |               |        |
             |        |       u       |     |       v       |        |
             |        |               |     |               |        |
             |        o               o     o               o        |
             |         \               \   /               /         |
             |          \               \ /               /          |
             |           \               o               /           |
             |            \             / \             /            |
             |             o-----------o   o-----------o             |
             |                                                       |
             |                                                       |
             o---------------------------o---------------------------o
            / \                         / \                         / \
           /   \                       /   \                       /   \
          /     \                     /     \                     /     \
         /       \                   /       \                   /       \
        /         \                 /         \                 /         \
       /           \               /           \               /           \
      /             \             /             \             /             \
     /               \           /               \           /               \
    /                 \         /                 \         /                 \
   /                   \       /                   \       /                   \
  /                     \     /                     \     /                     \
 /                       \   /                       \   /                       \
o-------------------------o o-------------------------o o-------------------------o
| U                       | | U                       | | U                       |
|      o---o   o---o      | |      o---o   o---o      | |      o---o   o---o      |
|     /     \ /     \     | |     /     \ /     \     | |     /     \ /     \     |
|    /       o       \    | |    /       o       \    | |    /       o       \    |
|   /       / \       \   | |   /       / \       \   | |   /       / \       \   |
|  o       o   o       o  | |  o       o   o       o  | |  o       o   o       o  |
|  |   u   |   |   v   |  | |  |   u   |   |   v   |  | |  |   u   |   |   v   |  |
|  o       o   o       o  | |  o       o   o       o  | |  o       o   o       o  |
|   \       \ /       /   | |   \       \ /       /   | |   \       \ /       /   |
|    \       o       /    | |    \       o       /    | |    \       o       /    |
|     \     / \     /     | |     \     / \     /     | |     \     / \     /     |
|      o---o   o---o      | |      o---o   o---o      | |      o---o   o---o      |
|                         | |                         | |                         |
o-------------------------o o-------------------------o o-------------------------o
 \                        |  \                       /  |                        /
  \                       |   \                     /   |                       /
   \                      |    \                   /    |                      /
    \                     |     \                 /     |                     /
     \       g            |      \       f       /      |            h       /
      \                   |       \             /       |                   /
       \                  |        \           /        |                  /
        \                 |         \         /         |                 /
         \                |          \       /          |                /
          \    o----------|-----------\-----/-----------|----------o    /
           \   | X        |            \   /            |          |   /
            \  |          |             \ /             |          |  /
             \ |          |        o-----o-----o        |          | /
              \|          |       /             \       |          |/
               \          |      /               \      |          /
               |\         |     /                 \     |         /|
               | \        |    /                   \    |        / |
               |  \       |   /                     \   |       /  |
               |   \      |  o           x           o  |      /   |
               |    \     |  |                       |  |     /    |
               |     \    |  |                       |  |    /     |
               |      \   |  |                       |  |   /      |
               |       \  |  |                       |  |  /       |
               |        \ |  |                       |  | /        |
               |         \|  |                       |  |/         |
               |          o--o--------o     o--------o--o          |
               |         /    \        \   /        /    \         |
               |        /      \        \ /        /      \        |
               |       /        \        o        /        \       |
               |      /          \      / \      /          \      |
               |     /            \    /   \    /            \     |
               |    o              o--o-----o--o              o    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    |        y        |     |        z        |    |
               |    |                 |     |                 |    |
               |    |                 |     |                 |    |
               |    o                 o     o                 o    |
               |     \                 \   /                 /     |
               |      \                 \ /                 /      |
               |       \                 o                 /       |
               |        \               / \               /        |
               |         \             /   \             /         |
               |          o-----------o     o-----------o          |
               |                                                   |
               |                                                   |
               o---------------------------------------------------o
                \                                                 /
                  \                                             /
                    \                                         /
                      \                                     /
                        \                                 /
                          \            p , q            /
                            \                         /
                              \                     /
                                \                 /
                                  \             /
                                    \         /
                                      \     /
                                        \ /
                                         o

Figure 30.  Generic Frame of a Logical Transformation

Enter the picture, as we usually do, in the middle of things, with features like that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps Then we discover that the simple features are really more complex than we thought at first, and it becomes useful to regard them as functions of other features that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.

A particular transformation may be expressed by a system of equations, as shown below. Here, is defined by its component maps where each component map in is a proposition of type



Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions in one universe of discourse and the special collection of simple propositions on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.

Analytic Expansions : Operators and Functors

 

Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.

 
— C.S. Peirce, “The Maxim of Pragmatism”, CP 5.438

Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.

Operators on Propositions and Transformations

The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to “get the drift” of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.

The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure 31 illustrates the typical situation.

o---------------------------------------o
|                                       |
|                                       |
|      U%           F           X%      |
|         o------------------>o         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|     !W! |                   | !W!     |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         v                   v         |
|         o------------------>o         |
|   !W!U%         !W!F          !W!X%   |
|                                       |
|                                       |
o---------------------------------------o
Figure 31.  Operator Diagram (1)

In this Figure stands for a generic operator in this case one that takes a logical transformation of type into a logical transformation of type Thus, the operator must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like and and for logical transformations like

Note. Strictly speaking, an operator like works between two whole categories of universes and transformations, which we call the source and the target categories of Given this setting, specifies for each universe in its source category a definite universe in its target category, and to each transformation in its source category it assigns a unique transformation in its target category. Naturally, this only works if takes the source and the target of the map over to the source and the target of the map With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation and thus we can take it for granted that the assignment of universes under is defined appropriately at the source and target ends of It is not always the case, though, that we need to use the particular names (like and ) that assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator what universes they are.

In Figure 31 the maps and are displayed horizontally, the way one normally orients functional arrows in a written text, and rolls the map downward into the images that are associated with In Figure 32 the same information is redrawn so that the maps and flow down the page, and unfurls the map rightward into domains that are the eminent purview of

o---------------------------------------o
|                                       |
|                                       |
|      U%          !W!          !W!U%   |
|         o------------------>o         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|      F  |                   | !W!F    |
|         |                   |         |
|         |                   |         |
|         |                   |         |
|         v                   v         |
|         o------------------>o         |
|      X%          !W!          !W!X%   |
|                                       |
|                                       |
o---------------------------------------o
Figure 32.  Operator Diagram (2)

The latter arrangement, as exhibited in Figure 32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.

Differential Analysis of Propositions and Transformations

  The resultant metaphysical problem now is this: Does the man go round the squirrel or not?  
— William James, Pragmatism, [Jam, 43]

The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators that act on propositions or on transformations to yield the corresponding operator maps The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.

  • Remark on Strategy. At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the trope extension, to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of contingency spaces. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.

We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:



If we assume that the source universe and the target universe have finite dimensions and respectively, then each operator is encompassed by the same abstract type:



Since the range features of the operator result can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator can be separated accordingly into two components, in the form Given a fixed context of source and target universes, is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context has the form:



On the other hand, the operator is specific to each In this context always has the form:



In the types just assigned to and and by implication to their results and we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:



Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.

In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the “sans serif” operators and their “serified” components which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results and can be fixed in our frame of reference by tethering the operative adjective for or to the anchoring epithet “map”, in conformity with an already standard practice.

The Secant Operator : E
 

Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.

 
— William James, Pragmatism, [Jam, 46]

Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted which receives the principal investment of analytic attention, and on the constituent parts of which derive their shares of significance as developed by the analysis. In the sequel, we refer to as the secant operator, taking it for granted that a context has been chosen that defines its type. The secant operator has the component description and its active ingredient is known as the enlargement operator. (Here, we name after the literal ancestor of the shift operator in the calculus of finite differences, defined so that for any suitable function though of course the logical analogue that we take up here must have a rather different definition.)

U%          $E$      $E$U%        $E$U%        $E$U%
   o------------------>o============o============o
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
F  |                   | $E$F   =   | $d$^0.F  + | $r$^0.F
   |                   |            |            |
   |                   |            |            |
   |                   |            |            |
   v                   v            v            v
   o------------------>o============o============o
X%          $E$      $E$X%        $E$X%        $E$X%

Figure 33-i.  Analytic Diagram (1)
U%          $E$      $E$U%        $E$U%        $E$U%        $E$U%
   o------------------>o============o============o============o
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
F  |                   | $E$F   =   | $d$^0.F  + | $d$^1.F  + | $r$^1.F
   |                   |            |            |            |
   |                   |            |            |            |
   |                   |            |            |            |
   v                   v            v            v            v
   o------------------>o============o============o============o
X%          $E$      $E$X%        $E$X%        $E$X%        $E$X%

Figure 33-ii.  Analytic Diagram (2)

In its action on universes yields the same result as a fact that can be expressed in equational form by writing for any universe Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure 30.

Acting on a transformation from universe to universe the operator determines a transformation from to The map forms the main body of evidence to be investigated in performing a differential analysis of Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the “big picture”, it is critically important to emphasize that the map is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation until we can lay out the full “parts diagram” of along the lines of the generic frame in Figure 30.

Working within the confines of propositional calculus, it is possible to give an elementary definition of by means of a system of propositional equations, as we now describe.

Given a transformation

of concrete type

the transformation

of concrete type

is defined by means of the following system of logical equations:



It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over variables denoted by:

In this light, it should be clear that the system of equations defining embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type

The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing for any map This is tantamount to regarding as a complex operator, with a form of application that distributes each component of the operator to work on each component of the operand, as follows:

Quite a lot of “thematic infrastructure” or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of thematic frame, an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of

The generic notations in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing When the analysis is halted at a partial stage of development, notations like may be used to summarize the contributions to that remain to be analyzed. The Figure illustrates a convention that makes in effect, the sum of all differentials of order strictly greater than

We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.

The Radius Operator : e
 

And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.

 
— William James, Pragmatism, [Jam, 46]

The operator identified as in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for in the appropriately extended context. Construed in terms of its broadest components, is equivalent to the doubly tacit extension operator in recognition of which let us redub it as Pursuing a geometric analogy, we may refer to as the radius operator. The operation intended by all of these forms is defined by the following equation:

which is tantamount to the system of equations below.



The Phantom of the Operators : η
 

I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, which was playing the most perfect music!

 
— Gaston Leroux, The Phantom of the Opera, [Ler, 81]

We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.

Given a transformation we often have call to consider a family of related transformations, all having the form:

The operator is introduced to deal with the simplest one of these maps:

which is defined by the following equations:



In effect, the operator is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator Operating independently, achieves precisely the same results that the second in accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of and in this setting combines the aims of both the tacit and the thematic extensions, and because reflects in regard to little more than the application of a differential twist, a mere turn of phrase, we refer to as the trope extension operator.

The Chord Operator : D
 

What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.

 
— William James, Pragmatism, [Jam, 45]

Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.

This is the operator that is referred to as in the initial stage of analysis (Figure 33-i) and that is expanded as in the subsequent step (Figure 33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the chord operator and denote it In the more casual terms that are here introduced, is defined as the remainder of and and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise and the bar of exigency

The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write calling the difference operator and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord is not one that need be lost at any stage of development. At the stage of play it can always be reconstituted in the following form:



The Tangent Operator : T
 

They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.

 
— William James, Pragmatism, [Jam, 300]

The operator tagged as in the analytic diagram (Figure 33) is called the tangent operator and is usually denoted in this text as or Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a tangent functor. According to the custom adopted here, we dissect it as where is the operator that yields the first order differential when applied to a transformation and whose name is legion.

Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor and attend to it chiefly as it bears on the first order differential in the analytic expansion of In this situation we often refer to the extended universes and under the equivalent designations and respectively. The purpose of the tangent functor is to extract the tangent map at each point of and the tangent map tells us not only what the transformation is doing at each point of the universe but also what is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.

U%          $T$      $T$U%        $T$U%
   o------------------>o============o
   |                   |            |
   |                   |            |
   |                   |            |
   |                   |            |
F  |                   | $T$F   =   | <!e!, d> F
   |                   |            |
   |                   |            |
   |                   |            |
   v                   v            v
   o------------------>o============o
X%          $T$      $T$X%        $T$X%

Figure 34.  Tangent Functor Diagram
  • NB. There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators in so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.