Cactus Language • Preliminaries 9

We now have the materials in place to formulate a definition of our subject.

The painted cactus language with paints in the set is the formal language defined as follows.

In the idiom of formal language theory, a string is called a sentence of if and only if it belongs to or simply a sentence if the language is understood.  A sentence of is referred to as a painted and rooted cactus expression on the palette or a cactus expression for short.

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Cactus Language • Preliminaries 7

The array of syntactic operators may be put in more organized form by making a few additional conventions and auxiliary definitions.

The conception of concatenation permits extension to its natural prequel, the corresponding operator on zero operands.

From that beginning the operation of concatenation may be broken into stages by means of the following conceptions.

The precatenation of two strings is defined as follows.

The concatenation of strings may now be given a new definition as the iterated precatenation of strings beginning with and continuing through the remaining strings.

The conception of surcatenation permits extension to its natural prequel, the corresponding operator on zero operands.

From that beginning the operation of surcatenation may be broken into stages by means of the following conceptions.

A subclause in is a string ending with

The subcatenation of a subclause by a string is defined as follows.

The surcatenation of strings may now be given a new definition as the iterated subcatenation of strings beginning with and continuing through the remaining strings.

Notice that the expressions and are defined in such a way that the respective operators and simply ignore, in the manner of constants, whatever sequences of strings may be listed as their ostensible arguments.

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Cactus Language • Preliminaries 6

The definitions of the syntactic connectives can be made a little more succinct by defining the following pair of generic operators on strings.

The concatenation of the sequence of strings is defined recursively as follows.

The surcatenation of the sequence of strings is defined recursively as follows.

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Cactus Language • Preliminaries 4

The informal mechanisms illustrated in the preceding discussion equip us with a description of cactus language adequate to providing conceptual and computational representations for the minimal formal logical system variously known as propositional logic or sentential calculus.

The painted cactus language is actually a parameterized family of languages, consisting of one language for each set of paints.

The alphabet is the disjoint union of the following two sets of symbols.

is the alphabet of markers, the set of punctuation marks, or the collection of syntactic constants common to all the languages   Various ways of representing the elements of are shown in the following display.

is the palette, the alphabet of paints, or the collection of syntactic variables peculiar to the language   The set of signs in may be enumerated as follows.

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Cactus Language • Preliminaries 2

As a temporary notation, let the relationship between a particular sign and a particular object , namely, the fact that denotes or the fact that is denoted by , be symbolized in one of the following two ways.

Now consider the following paradigm.

In the same vein, if we let the sign “blank” denote the sign “ ” then the string of characters inside the first pair of quotation marks will serve as another name for the string of characters inside the second pair of quotation marks.  In other words, “blank” is a higher order sign whose object is the sign “ ” and the string of five characters inside the first pair of quotation marks is a sign at a higher level of signification than the string of one character inside the second pair of quotation marks.  The relation in question can be abbreviated in either one of the following two ways.

Using the raised dot “∙” as a sign to mark the articulation of a quoted string into a sequence of possibly shorter quoted strings, and thus to mark the concatenation of a sequence of quoted strings into a possibly larger quoted string, one can write the following equation.

The above tactic lets us refer to the blank as a type of character and refer to any blank we choose as a token of that type, denoting either in a markèd way, but without the use of quotation marks.  As a blank is just what the name “blank” names, it is possible to represent the denoting of the sign “ ” by the name “blank” in the form of an identity between the named objects, as follows.

Given the above identities it is possible to extend the use of the “∙” sign to mark the articulation of either named or quoted strings into both named and quoted strings.  For example, we have the following equations.

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Cactus Language • Preliminaries 1

Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas.  If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.

Picture two different configurations of such an irregular shape, superimposed on each other in space, like a double exposure photograph.  Of the two images, the only part which coincides is the body.  The two different sets of quills stick out into very different regions of space.  The objective reality we see from within the first position, seemingly so full and spherical, actually agrees with the shifted reality only in the body of common knowledge.  In every direction in which we look at all deeply, the realm of discovered scientific truth could be quite different.  Yet in each of those two different situations, we would have thought the world complete, firmly known, and rather round in its penetration of the space of possible knowledge.

Herbert J. Bernstein • “Idols of Modern Science”

The task before us is to describe the syntax of a family of formal languages intended for use as a sentential calculus, and thus interpreted for the purpose of reasoning about propositions and their logical relations.

To carry out our discussion we need a way of referring to signs as if they were objects like any others, in other words, as the sorts of things which can be named, indicated, described, discussed, and renamed if necessary, which can be placed, arranged, and rearranged within a suitable medium of expression — or else manipulated in the mind — which can be articulated and decomposed into their elementary signs, and which can be strung together in sequences to form complex signs.

Signs having signs as their objects are known as “higher order signs”, a topic which demands an adequate level of formalization, but in due time.  The present discussion needs a quicker way to get into the subject, even if it settles for informal means which cannot be rendered absolutely precise.

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Cactus Language • Overview 3

In the development of Cactus Language to date the following two species of graphs have been instrumental.

• Painted And Rooted Cacti (PARCAI).
• Painted And Rooted Conifers (PARCOI).

It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage.  Partly because the two species are so closely related and partly for the sake of brevity, we’ll always use the genus name “PARC” to denote the corresponding cacti.

To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the fibers or inverse images of the propositions.

Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again.  Moreover, we need to implement both ways of proceeding in computational form.  Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects.  Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets.  Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.

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Differential Propositional Calculus • 37

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

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 type is implied any time we make use of one basis which happens to be included in another basis   When discussing differential relations we usually have in mind the extended alphabet has a special construction or a specific lexical relation with respect to the initial alphabet one which is marked by characteristic types of accents, indices, or inflected forms.

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Differential Propositional Calculus • 36

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

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

The goal is to answer a single question:  What is a propositional tangent functor?  In other words, the 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 we examine the types of transformations we already know as extensions and projections and we use their special cases to illustrate several styles of logical and visual representation which figure in the sequel.

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Differential Propositional Calculus • 35

Example 2. Drives and Their Vicissitudes (concl.)

Applied to the example of ‑gear curves, the indexing scheme results in the data of the next two Tables, showing one period for each orbit.

The states in each orbit are listed as ordered pairs where may be read as a temporal parameter indicating the present time of the state and where is the decimal equivalent of the binary numeral

Grasped more intuitively, the Tables show each state with a subscript equal to the numerator of its rational index, taking for granted the constant denominator   In that way the temporal succession of states can be reckoned by a parallel round‑up rule.  Namely, if is any pair of adjacent digits in the state index then the value of in the next state is

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Differential Propositional Calculus • 34

Example 2. Drives and Their Vicissitudes (cont.)

With a little thought it is possible to devise a canonical indexing scheme for the states in differential logical systems.  A scheme of that order allows for comparing changes of state in universes of discourse that weigh in on different scales of observation.

To that purpose, let us index the states with the dyadic rationals (or the binary fractions) in the half-open interval   Formally and canonically, a state is indexed by a fraction whose denominator is the power of two and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next.

The differential coefficients of the state are the values for where is defined as identical to   To form the binary index of the state the coefficient is read off as the binary digit associated with the place value   Expressed in algebraic terms, the rational index of the state is given by the following equivalent formulations.

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Differential Propositional Calculus • 33

Example 2. Drives and Their Vicissitudes (cont.)

Expressed in the language of drives and gears our next Example may be described as the family of fourth‑gear curves through the fourth extension   Those are the trajectories generated subject to the dynamic law where it’s understood all higher order differences are equal to

Because and all higher differences are fixed, the state vectors vary only with respect to their projections as points of   Thus there is just enough space in a planar venn diagram to plot the orbits and show how they partition the points of   It turns out there are just two possible orbits, of eight points each, as shown in the following Figure.

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Differential Propositional Calculus • 32

I open my scuttle at night and see the far‑sprinkled systems,
And all I see, multiplied as high as I can cipher, edge but
     the rim of the farther systems.

— Walt Whitman • Leaves of Grass

Example 2. Drives and Their Vicissitudes

Before we leave the one‑feature case let’s look at a more substantial example, one which illustrates a general class of curves through the extended feature spaces and affords an opportunity to discuss important themes concerning their structure and dynamics.

As before let   The discussion to follow considers a class of trajectories having the property that for all greater than a fixed value and indulges in the use of a picturesque vocabulary to describe salient classes of those curves.

Given the above finite order condition, there is a highest order non‑zero difference exhibited at each point of any trajectory one may consider.  With respect to any point of the corresponding curve let us call that highest order differential feature the drive at that point.  Curves of constant drive are then referred to as ‑gear curves.

• Note.  The fact that a difference calculus can be developed for boolean functions is well known and was probably familiar to Boole, who was an expert in difference equations before he turned to logic.  And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s.  At the very outset of general purpose mechanized computing we find the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation.

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