Changes

MyWikiBiz, Author Your Legacy — Friday March 14, 2025
Jump to navigationJump to search

Directory:Jon Awbrey/Papers/Cactus Language (view source)

Revision as of 19:01, 22 December 2008

152,075 bytes added ,  19:01, 22 December 2008
import text file
Line 1: Line 1:  
{{DISPLAYTITLE:Cactus Language}}
 
{{DISPLAYTITLE:Cactus Language}}
 +
<pre>
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
Inquiry Driven Systems:  An Inquiry Into Inquiry
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
| Document History
 +
|
 +
| Subject:  Inquiry Driven Systems:  An Inquiry Into Inquiry
 +
| Contact:  Jon Awbrey <jawbrey@oakland.edu>
 +
| Version:  Draft 8.70
 +
| Created:  23 Jun 1996
 +
| Revised:  06 Jan 2002
 +
| Advisor:  M.A. Zohdy
 +
| Setting:  Oakland University, Rochester, Michigan, USA
 +
| Excerpt:  Section 1.3.10 (Recurring Themes)
 +
| Excerpt:  Subsections 1.3.10.8 - 1.3.10.13
 +
|
 +
| http://members.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.8  The Cactus Patch
 +
 +
| 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.
 +
|
 +
| Herbert J. Bernstein, "Idols", page 38.
 +
|
 +
| Herbert J. Bernstein,
 +
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
 +
|
 +
| Marcus G. Raskin & Herbert J. Bernstein,
 +
|'New Ways of Knowing:  The Sciences, Society, & Reconstructive Knowledge',
 +
| Rowman & Littlefield, Totowa, NJ, 1987.
 +
 +
In this and the four subsections that follow, I describe a calculus for
 +
representing propositions as sentences, in other words, as syntactically
 +
defined sequences of signs, and for manipulating these sentences chiefly
 +
in the light of their semantically defined contents, in other words, with
 +
respect to their logical values as propositions.  In their computational
 +
representation, the expressions of this calculus parse into a class of
 +
tree-like data structures called "painted cacti".  This is a family of
 +
graph-theoretic data structures that can be observed to have especially
 +
nice properties, turning out to be not only useful from a computational
 +
standpoint but also quite interesting from a theoretical point of view.
 +
The rest of this subsection serves to motivate the development of this
 +
calculus and treats a number of general issues that surround the topic.
 +
 +
In order to facilitate the use of propositions as indicator functions
 +
it helps to acquire a flexible notation for referring to propositions
 +
in that light, for interpreting sentences in a corresponding role, and
 +
for negotiating the requirements of mutual sense between the two domains.
 +
If none of the formalisms that are readily available or in common use are
 +
able to meet all of the design requirements that come to mind, then it is
 +
necessary to contemplate the design of a new language that is especially
 +
tailored to the purpose.  In the present application, there is a pressing
 +
need to devise a general calculus for composing propositions, computing
 +
their values on particular arguments, and inverting their indications to
 +
arrive at the sets of things in the universe that are indicated by them.
 +
 +
For computational purposes, it is convenient to have a middle ground or
 +
an intermediate language for negotiating between the koine of sentences
 +
regarded as strings of literal characters and the realm of propositions
 +
regarded as objects of logical value, even if this renders it necessary
 +
to introduce an artificial medium of exchange between these two domains.
 +
If one envisions these computations to be carried out in any organized
 +
fashion, and ultimately or partially by means of the familiar sorts of
 +
machines, then the strings that express these logical propositions are
 +
likely to find themselves parsed into tree-like data structures at some
 +
stage of the game.  With regard to their abstract structures as graphs,
 +
there are several species of graph-theoretic data structures that can be
 +
used to accomplish this job in a reasonably effective and efficient way.
 +
 +
Over the course of this project, I plan to use two species of graphs:
 +
 +
1.  "Painted And Rooted Cacti" (PARCAI).
 +
 +
2.  "Painted And Rooted Conifers" (PARCOI).
 +
 +
For now, it is enough to discuss the former class of data structures,
 +
leaving the consideration of the latter class to a part of the project
 +
where their distinctive features are key to developments at that stage.
 +
Accordingly, within the context of the current patch of discussion, or
 +
until it becomes necessary to attach further notice to the conceivable
 +
varieties of parse graphs, the acronym "PARC" is sufficient to indicate
 +
the pertinent genus of abstract graphs that are under consideration.
 +
 +
By way of making these tasks feasible to carry out on a regular basis,
 +
a prospective language designer is required not only to supply a fluent
 +
medium for the expression of propositions, but further to accompany the
 +
assertions of their sentences with a canonical mechanism for teasing out
 +
the fibers of their indicator functions.  Accordingly, with regard to a
 +
body of conceivable propositions, one needs to furnish a standard array
 +
of techniques for following the threads of their indications from their
 +
objective universe to their values for the mind and back again, that is,
 +
for tracing the clues that sentences provide from the universe of their
 +
objects to the signs of their values, and, in turn, from signs to objects.
 +
Ultimately, one seeks to render propositions so functional as indicators
 +
of sets and so essential for examining the equality of sets that they can
 +
constitute a veritable criterion for the practical conceivability of sets.
 +
Tackling this task requires me to introduce a number of new definitions
 +
and a collection of additional notational devices, to which I now turn.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.8  The Cactus Patch (cont.)
 +
 +
Depending on whether a formal language is called by the type of sign
 +
that makes it up or whether it is named after the type of object that
 +
its signs are intended to denote, one may refer to this cactus language
 +
as a "sentential calculus" or as a "propositional calculus", respectively.
 +
 +
When the syntactic definition of the language is well enough understood,
 +
then the language can begin to acquire a semantic function.  In natural
 +
circumstances, the syntax and the semantics are likely to be engaged in
 +
a process of co-evolution, whether in ontogeny or in phylogeny, that is,
 +
the two developments probably form parallel sides of a single bootstrap.
 +
But this is not always the easiest way, at least, at first, to formally
 +
comprehend the nature of their action or the power of their interaction.
 +
 +
According to the customary mode of formal reconstruction, the language
 +
is first presented in terms of its syntax, in other words, as a formal
 +
language of strings called "sentences", amounting to a particular subset
 +
of the possible strings that can be formed on a finite alphabet of signs.
 +
A syntactic definition of the "cactus language", one that proceeds along
 +
purely formal lines, is carried out in the next Subsection.  After that,
 +
the development of the language's more concrete aspects can be seen as
 +
a matter of defining two functions:
 +
 +
1.  The first is a function that takes each sentence of the language
 +
    into a computational data structure, to be exact, a tree-like
 +
    parse graph called a "painted cactus".
 +
 +
2.  The second is a function that takes each sentence of the language,
 +
    or its interpolated parse graph, into a logical proposition, in effect,
 +
    ending up with an indicator function as the object denoted by the sentence.
 +
 +
The discussion of syntax brings up a number of associated issues that
 +
have to be clarified before going on.  These are questions of "style",
 +
that is, the sort of description, "grammar", or theory that one finds
 +
available or chooses as preferable for a given language.  These issues
 +
are discussed in the Subsection after next (Subsection 1.3.10.10).
 +
 +
There is an aspect of syntax that is so schematic in its basic character
 +
that it can be conveyed by computational data structures, so algorithmic
 +
in its uses that it can be automated by routine mechanisms, and so fixed
 +
in its nature that its practical exploitation can be served by the usual
 +
devices of computation.  Because it involves the transformation of signs,
 +
it can be recognized as an aspect of semiotics.  Since it can be carried
 +
out in abstraction from meaning, it is not up to the level of semantics,
 +
much less a complete pragmatics, though it does incline to the pragmatic
 +
aspects of computation that are auxiliary to and incidental to the human
 +
use of language.  Therefore, I refer to this aspect of formal language
 +
use as the "algorithmics" or the "mechanics" of language processing.
 +
A mechanical conversion of the "cactus language" into its associated
 +
data structures is discussed in Subsection 1.3.10.11.
 +
 +
In the usual way of proceeding on formal grounds, meaning is added by giving
 +
each "grammatical sentence", or each syntactically distinguished string, an
 +
interpretation as a logically meaningful sentence, in effect, equipping or
 +
providing each abstractly well-formed sentence with a logical proposition
 +
for it to denote.  A semantic interpretation of the "cactus language" is
 +
carried out in Subsection 1.3.10.12.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax
 +
 +
| 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", page 38.
 +
|
 +
| Herbert J. Bernstein,
 +
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
 +
|
 +
| Marcus G. Raskin & Herbert J. Bernstein,
 +
|'New Ways of Knowing:  The Sciences, Society, & Reconstructive Knowledge',
 +
| Rowman & Littlefield, Totowa, NJ, 1987.
 +
 +
In this Subsection, I describe the syntax of a family of formal languages
 +
that I intend to use as a sentential calculus, and thus to interpret for
 +
the purpose of reasoning about propositions and their logical relations.
 +
In order to carry out the discussion, I need a way of referring to signs
 +
as if they were objects like any others, in other words, as the sorts of
 +
things that are subject to being named, indicated, described, discussed,
 +
and renamed if necessary, that can be placed, arranged, and rearranged
 +
within a suitable medium of expression, or else manipulated in the mind,
 +
that can be articulated and decomposed into their elementary signs, and
 +
that can be strung together in sequences to form complex signs.  Signs
 +
that have signs as their objects are called "higher order" (HO) signs,
 +
and this is a topic that demands an apt formalization, but in due time.
 +
The present discussion requires a quicker way to get into this subject,
 +
even if it takes informal means that cannot be made absolutely precise.
 +
 +
As a temporary notation, let the relationship between a particular sign z
 +
and a particular object o, namely, the fact that z denotes o or the fact
 +
that o is denoted by z, be symbolized in one of the following two ways:
 +
 +
1.  z  >->  o,
 +
 +
    z  den  o.
 +
 +
2.  o  <-<  z,
 +
 +
    o  ned  z.
 +
 +
Now consider the following paradigm:
 +
 +
1.  If        "A"  >->  Ann,
 +
 +
    i.e.      "A"  den  Ann,
 +
 +
    then      A    =  Ann,
 +
 +
    thus      "Ann"  >->  A,
 +
 +
    i.e.      "Ann"  den  A.
 +
 +
2.  If        Bob  <-<  "B",
 +
 +
    i.e.      Bob  ned  "B",
 +
 +
    then      Bob  =    B,
 +
 +
    thus      B  <-<  "Bob",
 +
 +
    i.e.      B  ned  "Bob".
 +
 +
When I say that the sign "blank" denotes the sign " ",
 +
it means that the string of characters inside the first
 +
pair of quotation marks can be used as another name for
 +
the string of characters inside the second pair of quotes.
 +
In other words, "blank" is a HO sign whose object is " ",
 +
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.  This relationship can be abbreviated in
 +
either one of the following ways:
 +
 +
|  " "      <-<  "blank"
 +
|
 +
|  "blank"  >->  " "
 +
 +
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:
 +
 +
|
 +
|  " "  <-<  "blank"  =  "b"·"l"·"a"·"n"·"k"
 +
|
 +
 +
This usage allows us to refer to the blank as a type of character, and
 +
also to refer any blank we choose as a token of this type, referring to
 +
either of them in a marked way, but without the use of quotation marks,
 +
as I just did.  Now, since a blank is just what the name "blank" names,
 +
it is possible to represent the denotation of the sign " " by the name
 +
"blank" in the form of an identity between the named objects, thus:
 +
 +
|
 +
|  " "  =  blank
 +
|
 +
 +
With these kinds of identity in mind, 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:
 +
 +
|  "  "      =  " "·" "      =  blank·blank
 +
|
 +
|  " blank"  =  " "·"blank"  =  blank·"blank"
 +
|
 +
|  "blank "  =  "blank"·" "  =  "blank"·blank
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
A few definitions from formal language theory are required at this point.
 +
 +
An "alphabet" is a finite set of signs, typically, !A! = {a_1, ..., a_n}.
 +
 +
A "string" over an alphabet !A! is a finite sequence of signs from !A!.
 +
 +
The "length" of a string is just its length as a sequence of signs.
 +
A sequence of length 0 yields the "empty string", here presented as "".
 +
A sequence of length k > 0 is typically presented in the concatenated forms:
 +
 +
s_1 s_2 ... s_(k-1) s_k,
 +
 +
or
 +
 +
s_1 · s_2 · ... · s_(k-1) · s_k,
 +
 +
with s_j in !A!, for all j = 1 to k.
 +
 +
Two alternative notations are often useful:
 +
 +
1.  !e!  =  @e@  =  ""  =  the empty string.
 +
 +
2.  %e%  =  {!e!}  =  {""}  =  the language consisting of a single empty string.
 +
 +
The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!.
 +
In particular, !A!* includes among its elements the empty string !e!.
 +
 +
The "surplus" !A!^+ of an alphabet !A! is the set of all positive length
 +
strings over !A!, in other words, everything in !A!* but the empty string.
 +
 +
A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*.
 +
If z is a string over !A! and if z is an element of !L!, then it is
 +
customary to call z a "sentence" of !L!.  Thus, a formal language !L!
 +
is defined by specifying its elements, which amounts to saying what it
 +
means to be a sentence of !L!.
 +
 +
One last device turns out to be useful in this connection.
 +
If z is a string that ends with a sign t, then z · t^-1 is
 +
the string that results by "deleting" from z the terminal t.
 +
 +
In this context, I make the following distinction:
 +
 +
1.  By "deleting" an appearance of a sign,
 +
    I mean replacing it with an appearance
 +
    of the empty string "".
 +
 +
2.  By "erasing" an appearance of a sign,
 +
    I mean replacing it with an appearance
 +
    of the blank symbol " ".
 +
 +
A "token" is a particular appearance of a sign.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
The informal mechanisms that have been illustrated in the immediately preceding
 +
discussion are enough to equip the rest of this discussion with a moderately
 +
exact description of the so-called "cactus language" that I intend to use
 +
in both my conceptual and my computational representations of the minimal
 +
formal logical system that is variously known to sundry communities of
 +
interpretation as "propositional logic", "sentential calculus", or
 +
more inclusively, "zeroth order logic" (ZOL).
 +
 +
The "painted cactus language" !C! is actually a parameterized
 +
family of languages, consisting of one language !C!(!P!) for
 +
each set !P! of "paints".
 +
 +
The alphabet !A!  =  !M! |_| !P! is the disjoint union of two sets of symbols:
 +
 +
1.  !M! is the alphabet of "measures", the set of "punctuation marks",
 +
    or the collection of "syntactic constants" that is common to all
 +
    of the languages !C!(!P!).  This set of signs is given as follows:
 +
 +
    !M!  =  {m_1, m_2, m_3, m_4}
 +
 +
        =  {" ", "-(", ",", ")-"}
 +
 +
        =  {blank, links, comma, right}.
 +
 +
2.  !P! is the "palette", the alphabet of "paints", or the collection
 +
    of "syntactic variables" that is peculiar to the language !C!(!P!).
 +
    This set of signs is given as follows:
 +
 +
    !P!  =  {p_j  :  j in J}.
 +
 +
The easiest way to define the language !C!(!P!) is to indicate the general sorts
 +
of operations that suffice to construct the greater share of its sentences from
 +
the specified few of its sentences that require a special election.  In accord
 +
with this manner of proceeding, I introduce a family of operations on strings
 +
of !A!* that are called "syntactic connectives".  If the strings on which
 +
they operate are exclusively sentences of !C!(!P!), then these operations
 +
are tantamount to "sentential connectives", and if the syntactic sentences,
 +
considered as abstract strings of meaningless signs, are given a semantics
 +
in which they denote propositions, considered as indicator functions over
 +
some universe, then these operations amount to "propositional connectives".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
Rather than presenting the most concise description of these languages
 +
right from the beginning, it serves comprehension to develop a picture
 +
of their forms in gradual stages, starting from the most natural ways
 +
of viewing their elements, if somewhat at a distance, and working
 +
through the most easily grasped impressions of their structures,
 +
if not always the sharpest acquaintances with their details.
 +
 +
The first step is to define two sets of basic operations on strings of !A!*.
 +
 +
1.  The "concatenation" of one string z_1 is just the string z_1.
 +
 +
    The "concatenation" of two strings z_1, z_2 is the string z_1 · z_2.
 +
 +
    The "concatenation" of the k strings z_j, for j = 1 to k,
 +
 +
    is the string of the form z_1 · ... · z_k.
 +
 +
2.  The "surcatenation" of one string z_1 is the string "-(" · z_1 · ")-".
 +
 +
    The "surcatenation" of two strings z_1, z_2 is "-(" · z_1 · "," · z_2 · ")-".
 +
 +
    The "surcatenation" of k strings z_j, for j = 1 to k,
 +
 +
    is the string of the form "-(" · z_1 · "," · ... · "," · z_k · ")-".
 +
 +
These definitions can be made a little more succinct by
 +
defining the following sorts of generic operators on strings:
 +
 +
1.  The "concatenation" Conc^k of the k strings z_j,
 +
    for j = 1 to k, is defined recursively as follows:
 +
 +
    a.  Conc^1_j  z_j  =  z_1.
 +
 +
    b.  For k > 1,
 +
 +
        Conc^k_j  z_j  =  (Conc^(k-1)_j  z_j) · z_k.
 +
 +
2.  The "surcatenation" Surc^k of the k strings z_j,
 +
    for j = 1 to k, is defined recursively as follows:
 +
 +
    a.  Surc^1_j  z_j  =  "-(" · z_1 · ")-".
 +
 +
    b.  For k > 1,
 +
 +
        Surc^k_j  z_j  =  (Surc^(k-1)_j  z_j) · ")-"^(-1) · "," · z_k · ")-".
 +
 +
The definitions of these syntactic operations can now be organized in a slightly
 +
better fashion, for both conceptual and computational purposes, by making a few
 +
additional conventions and auxiliary definitions.
 +
 +
1.  The conception of the k-place concatenation operation
 +
    can be extended to include its natural "prequel":
 +
 +
    Conc^0  =  ""  =  the empty string.
 +
 +
    Next, the construction of the k-place concatenation can be
 +
    broken into stages by means of the following conceptions:
 +
 +
    a.  The "precatenation" Prec(z_1, z_2) of the two strings
 +
        z_1, z_2 is the string that is defined as follows:
 +
 +
        Prec(z_1, z_2)  =  z_1 · z_2.
 +
 +
    b.  The "concatenation" of the k strings z_1, ..., z_k can now be
 +
        defined as an iterated precatenation over the sequence of k+1
 +
        strings that begins with the string z_0 = Conc^0 = "" and then
 +
        continues on through the other k strings:
 +
 +
        i.  Conc^0_j  z_j  =  Conc^0  =  "".
 +
 +
        ii.  For k > 0,
 +
 +
            Conc^k_j  z_j  =  Prec(Conc^(k-1)_j  z_j, z_k).
 +
 +
2.  The conception of the k-place surcatenation operation
 +
    can be extended to include its natural "prequel":
 +
 +
    Surc^0  =  "-()-".
 +
 +
    Finally, the construction of the k-place surcatenation can be
 +
    broken into stages by means of the following conceptions:
 +
 +
    a.  A "subclause" in !A!* is a string that ends with a ")-".
 +
 +
    b.  The "subcatenation" Subc(z_1, z_2)
 +
        of a subclause z_1 by a string z_2 is
 +
        the string that is defined as follows:
 +
 +
        Subc(z_1, z_2)  =  z_1 · ")-"^(-1) · "," · z_2 · ")-".
 +
 +
    c.  The "surcatenation" of the k strings z_1, ..., z_k can now be
 +
        defined as an iterated subcatenation over the sequence of k+1
 +
        strings that starts with the string z_0 = Surc^0 = "-()-" and
 +
        then continues on through the other k strings:
 +
 +
        i.  Surc^0_j  z_j  =  Surc^0  =  "-()-".
 +
 +
        ii.  For k > 0,
 +
 +
            Surc^k_j  z_j  =  Subc(Surc^(k-1)_j  z_j, z_k).
 +
 +
Notice that the expressions Conc^0_j z_j and Surc^0_j z_j
 +
are defined in such a way that the respective operators
 +
Conc^0 and Surc^0 basically "ignore", in the manner of
 +
constants, whatever sequences of strings z_j may be
 +
listed as their ostensible arguments.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
Having defined the basic operations of concatenation and surcatenation
 +
on arbitrary strings, in effect, giving them operational meaning for
 +
the all-inclusive language !L! = !A!*, it is time to adjoin the
 +
notion of a more discriminating grammaticality, in other words,
 +
a more properly restrictive concept of a sentence.
 +
 +
If !L! is an arbitrary formal language over an alphabet of the sort that
 +
we are talking about, that is, an alphabet of the form !A! = !M! |_| !P!,
 +
then there are a number of basic structural relations that can be defined
 +
on the strings of !L!.
 +
 +
1.  z is the "concatenation" of z_1 and z_2 in !L! if and only if
 +
 +
    z_1 is a sentence of !L!, z_2 is a sentence of !L!, and
 +
 +
    z  =  z_1 · z_2.
 +
 +
2.  z is the "concatenation" of the k strings z1, ..., z_k in !L!,
 +
 +
    if and only if z_j is a sentence of !L!, for all j = 1 to k, and
 +
 +
    z  =  Conc^k_j  z_j  =  z_1 · ... · z_k.
 +
 +
3.  z is the "discatenation" of z_1 by t if and only if
 +
 +
    z_1 is a sentence of !L!, t is an element of !A!, and
 +
 +
    z_1  =  z · t.
 +
 +
    When this is the case, one more commonly writes:
 +
 +
    z  =  z_1 · t^-1.
 +
 +
4.  z is a "subclause" of !L! if and only if
 +
 +
    z is a sentence of !L! and z ends with a ")-".
 +
 +
5.  z is the "subcatenation" of z_1 by z_2 if and only if
 +
 +
    z_1 is a subclause of !L!, z_2 is a sentence of !L!, and
 +
 +
    z  =  z_1 · ")-"^(-1) · "," · z_2 · ")-".
 +
 +
6.  z is the "surcatenation" of the k strings z_1, ..., z_k in !L!,
 +
 +
    if and only if z_j is a sentence of !L!, for all j = 1 to k, and
 +
 +
    z  =  Surc^k_j  z_j  =  "-(" · z_1 · "," · ... · "," · z_k · ")-".
 +
 +
The converses of these decomposition relations are tantamount to the
 +
corresponding forms of composition operations, making it possible for
 +
these complementary forms of analysis and synthesis to articulate the
 +
structures of strings and sentences in two directions.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
The "painted cactus language" with paints in the
 +
set !P! = {p_j : j in J} is the formal language
 +
!L! = !C!(!P!) c !A!* = (!M! |_| !P!)* that is
 +
defined as follows:
 +
 +
PC 1.  The blank symbol m_1 is a sentence.
 +
 +
PC 2.  The paint p_j is a sentence, for each j in J.
 +
 +
PC 3.  Conc^0 and Surc^0 are sentences.
 +
 +
PC 4.  For each positive integer k,
 +
 +
      if    z_1, ..., z_k are sentences,
 +
 +
      then  Conc^k_j  z_j is a sentence,
 +
 +
      and  Surc^k_j  z_j is a sentence.
 +
 +
As usual, saying that z is a sentence is just a conventional way of
 +
stating that the string z belongs to the relevant formal language !L!.
 +
An individual sentence of !C!(!P!), for any palette !P!, is referred to
 +
as a "painted and rooted cactus expression" (PARCE) on the palette !P!,
 +
or a "cactus expression", for short.  Anticipating the forms that the
 +
parse graphs of these PARCE's will take, to be described in the next
 +
Subsection, the language !L! = !C!(!P!) is also described as the
 +
set PARCE(!P!) of PARCE's on the palette !P!, more generically,
 +
as the PARCE's that constitute the language PARCE.
 +
 +
A "bare" PARCE, a bit loosely referred to as a "bare cactus expression",
 +
is a PARCE on the empty palette !P! = {}.  A bare PARCE is a sentence
 +
in the "bare cactus language", !C!^0 = !C!({}) = PARCE^0 = PARCE({}).
 +
This set of strings, regarded as a formal language in its own right,
 +
is a sublanguage of every cactus language !C!(!P!).  A bare cactus
 +
expression is commonly encountered in practice when one has occasion
 +
to start with an arbitrary PARCE and then finds a reason to delete or
 +
to erase all of its paints.
 +
 +
Only one thing remains to cast this description of the cactus language
 +
into a form that is commonly found acceptable.  As presently formulated,
 +
the principle PC 4 appears to be attempting to define an infinite number
 +
of new concepts all in a single step, at least, it appears to invoke the
 +
indefinitely long sequences of operators, Conc^k and Surc^k, for all k > 0.
 +
As a general rule, one prefers to have an effectively finite description of
 +
conceptual objects, and this means restricting the description to a finite
 +
number of schematic principles, each of which involves a finite number of
 +
schematic effects, that is, a finite number of schemata that explicitly
 +
relate conditions to results.
 +
 +
A start in this direction, taking steps toward an effective description
 +
of the cactus language, a finitary conception of its membership conditions,
 +
and a bounded characterization of a typical sentence in the language, can be
 +
made by recasting the present description of these expressions into the pattern
 +
of what is called, more or less roughly, a "formal grammar".
 +
 +
A notation in the style of "S :> T" is now introduced,
 +
to be read among many others in this manifold of ways:
 +
 +
|  S covers T
 +
|
 +
|  S governs T
 +
|
 +
|  S rules T
 +
|
 +
|  S subsumes T
 +
|
 +
|  S types over T
 +
 +
The form "S :> T" is here recruited for polymorphic
 +
employment in at least the following types of roles:
 +
 +
1.  To signify that an individually named or quoted string T is
 +
    being typed as a sentence S of the language of interest !L!.
 +
 +
2.  To express the fact or to make the assertion that each member
 +
    of a specified set of strings T c !A!* also belongs to the
 +
    syntactic category S, the one that qualifies a string as
 +
    being a sentence in the relevant formal language !L!.
 +
 +
3.  To specify the intension or to signify the intention that every
 +
    string that fits the conditions of the abstract type T must also
 +
    fall under the grammatical heading of a sentence, as indicated by
 +
    the type name "S", all within the target language !L!.
 +
 +
In these types of situation the letter "S", that signifies the type of
 +
a sentence in the language of interest, is called the "initial symbol"
 +
or the "sentence symbol" of a candidate formal grammar for the language,
 +
while any number of letters like "T", signifying other types of strings
 +
that are necessary to a reasonable account or a rational reconstruction
 +
of the sentences that belong to the language, are collectively referred
 +
to as "intermediate symbols".
 +
 +
Combining the singleton set {"S"} whose sole member is the initial symbol
 +
with the set !Q! that assembles together all of the intermediate symbols
 +
results in the set {"S"} |_| !Q! of "non-terminal symbols".  Completing
 +
the package, the alphabet !A! of the language is also known as the set
 +
of "terminal symbols".  In this discussion, I will adopt the convention
 +
that !Q! is the set of intermediate symbols, but I will often use "q"
 +
as a typical variable that ranges over all of the non-terminal symbols,
 +
q in {"S"} |_| !Q!.  Finally, it is convenient to refer to all of the
 +
symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
 +
prospective grammar for the language, and accordingly to describe
 +
the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
 +
in effect, expressing the forms that are superimposed on a language
 +
by one of its conceivable grammars.  In certain settings is becomes
 +
desirable to separate the augmented strings that contain the symbol
 +
"S" from all other sorts of augmented strings.  In these situations,
 +
the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
 +
as the "sentential forms" of the associated grammar.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
In forming a grammar for a language, statements of the form W :> W',
 +
where W and W' are augmented strings or sentential forms of specified
 +
types that depend on the style of the grammar that is being sought, are
 +
variously known as "characterizations", "covering rules", "productions",
 +
"rewrite rules", "subsumptions", "transformations", or "typing rules".
 +
These are collected together into a set !K! that serves to complete
 +
the definition of the formal grammar in question.
 +
 +
Correlative with the use of this notation, an expression of the
 +
form "T <: S", read as "T is covered by S", can be interpreted
 +
as saying that T is of the type S.  Depending on the context,
 +
this can be taken in either one of two ways:
 +
 +
1.  Treating "T" as a string variable, it means
 +
    that the individual string T is typed as S.
 +
 +
2.  Treating "T" as a type name, it means that any
 +
    instance of the type T also falls under the type S.
 +
 +
In accordance with these interpretations, an expression like "t <: T" can be
 +
read in all of the ways that one typically reads an expression like "t : T".
 +
 +
There are several abuses of notation that commonly tolerated in the use
 +
of covering relations.  The worst offense is that of allowing symbols to
 +
stand equivocally either for individual strings or else for their types.
 +
There is a measure of consistency to this practice, considering the fact
 +
that perfectly individual entities are rarely if ever grasped by means of
 +
signs and finite expressions, which entails that every appearance of an
 +
apparent token is only a type of more particular tokens, and meaning in
 +
the end that there is never any recourse but to the sort of discerning
 +
interpretation that can decide just how each sign is intended.  In view
 +
of all this, I continue to permit expressions like "t <: T" and "T <: S",
 +
where any of the symbols "t", "T", "S" can be taken to signify either the
 +
tokens or the subtypes of their covering types.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
The combined effect of several typos in my typography
 +
along with what may be a lack of faith in imagination,
 +
obliges me to redo a couple of paragraphs from before.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
A notation in the style of "S :> T" is now introduced,
 +
to be read among many others in this manifold of ways:
 +
 +
|  S covers T
 +
|
 +
|  S governs T
 +
|
 +
|  S rules T
 +
|
 +
|  S subsumes T
 +
|
 +
|  S types over T
 +
 +
The form "S :> T" is here recruited for polymorphic
 +
employment in at least the following types of roles:
 +
 +
1.  To signify that an individually named or quoted string T is
 +
    being typed as a sentence S of the language of interest !L!.
 +
 +
2.  To express the fact or to make the assertion that each member
 +
    of a specified set of strings T c !A!* also belongs to the
 +
    syntactic category S, the one that qualifies a string as
 +
    being a sentence in the relevant formal language !L!.
 +
 +
3.  To specify the intension or to signify the intention that every
 +
    string that fits the conditions of the abstract type T must also
 +
    fall under the grammatical heading of a sentence, as indicated by
 +
    the type name "S", all within the target language !L!.
 +
 +
In these types of situation the letter "S", that signifies the type of
 +
a sentence in the language of interest, is called the "initial symbol"
 +
or the "sentence symbol" of a candidate formal grammar for the language,
 +
while any number of letters like "T", signifying other types of strings
 +
that are necessary to a reasonable account or a rational reconstruction
 +
of the sentences that belong to the language, are collectively referred
 +
to as "intermediate symbols".
 +
 +
Combining the singleton set {"S"} whose sole member is the initial symbol
 +
with the set !Q! that assembles together all of the intermediate symbols
 +
results in the set {"S"} |_| !Q! of "non-terminal symbols".  Completing
 +
the package, the alphabet !A! of the language is also known as the set
 +
of "terminal symbols".  In this discussion, I will adopt the convention
 +
that !Q! is the set of intermediate symbols, but I will often use "q"
 +
as a typical variable that ranges over all of the non-terminal symbols,
 +
q in {"S"} |_| !Q!.  Finally, it is convenient to refer to all of the
 +
symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
 +
prospective grammar for the language, and accordingly to describe
 +
the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
 +
in effect, expressing the forms that are superimposed on a language
 +
by one of its conceivable grammars.  In certain settings is becomes
 +
desirable to separate the augmented strings that contain the symbol
 +
"S" from all other sorts of augmented strings.  In these situations,
 +
the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
 +
as the "sentential forms" of the associated grammar.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
For some time to come in the discussion that follows,
 +
although I will continue to focus on the cactus language
 +
as my principal object example, my more general purpose will
 +
be to develop and to demonstrate the subject materials and the
 +
technical methodology of the theory of formal languages and grammars.
 +
I will do this by taking up a particular method of "stepwise refinement"
 +
and using it to extract a rigorous formal grammar for the cactus language,
 +
starting with little more than a rough description of the target language
 +
and applying a systematic analysis to develop a sequence of increasingly
 +
more effective and more exact approximations to the desired grammar.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
Employing the notion of a covering relation it becomes possible to
 +
redescribe the cactus language !L! = !C!(!P!) in the following way.
 +
 +
Grammar 1 is something of a misnomer.  It is nowhere near exemplifying
 +
any kind of a standard form and it is only intended as a starting point
 +
for the initiation of more respectable grammars.  Such as it is, it uses
 +
the terminal alphabet !A! = !M! |_| !P! that comes with the territory of
 +
the cactus language !C!(!P!), it specifies !Q! = {}, in other words, it
 +
employs no intermediate symbols, and it embodies the "covering set" !K!
 +
as listed in the following display.
 +
 +
| !C!(!P!).  Grammar 1
 +
|
 +
| !Q! = {}
 +
|
 +
| 1.  S  :>  m_1  =  " "
 +
|
 +
| 2.  S  :>  p_j, for each j in J
 +
|
 +
| 3.  S  :>  Conc^0  =  ""
 +
|
 +
| 4.  S  :>  Surc^0  =  "-()-"
 +
|
 +
| 5.  S  :>  S*
 +
|
 +
| 6.  S  :>  "-(" · S · ("," · S)* · ")-"
 +
 +
In this formulation, the last two lines specify that:
 +
 +
5.  The concept of a sentence in !L! covers any
 +
    concatenation of sentences in !L!, in effect,
 +
    any number of freely chosen sentences that are
 +
    available to be concatenated one after another.
 +
 +
6.  The concept of a sentence in !L! covers any
 +
    surcatenation of sentences in !L!, in effect,
 +
    any string that opens with a "-(", continues
 +
    with a sentence, possibly empty, follows with
 +
    a finite number of phrases of the form "," · S,
 +
    and closes with a ")-".
 +
 +
This appears to be just about the most concise description
 +
of the cactus language !C!(!P!) that one can imagine, but
 +
there exist a couple of problems that are commonly felt
 +
to afflict this style of presentation and to make it
 +
less than completely acceptable.  Briefly stated,
 +
these problems turn on the following properties
 +
of the presentation:
 +
 +
1.  The invocation of the kleene star operation
 +
    is not reduced to a manifestly finitary form.
 +
 +
2.  The type of a sentence S is allowed to cover
 +
    not only itself but also the empty string.
 +
 +
I will discuss these issues at first in general, and especially in regard to
 +
how the two features interact with one another, and then I return to address
 +
in further detail the questions that they engender on their individual bases.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
In the process of developing a grammar for a language, it is possible
 +
to notice a number of organizational, pragmatic, and stylistic questions,
 +
whose moment to moment answers appear to decide the ongoing direction of the
 +
grammar that develops and the impact of whose considerations work in tandem
 +
to determine, or at least to influence, the sort of grammar that turns out.
 +
The issues that I can see arising at this point I can give the following
 +
prospective names, putting off the discussion of their natures and the
 +
treatment of their details to the points in the development of the
 +
present example where they evolve their full import.
 +
 +
1.  The "degree of intermediate organization" in a grammar.
 +
 +
2.  The "distinction between empty and significant strings", and thus
 +
    the "distinction between empty and significant types of strings".
 +
 +
3.  The "principle of intermediate significance".  This is a constraint
 +
    on the grammar that arises from considering the interaction of the
 +
    first two issues.
 +
 +
In responding to these issues, it is advisable at first to proceed in
 +
a stepwise fashion, all the better thereby to accommodate the chances
 +
of pursuing a series of parallel developments in the grammar, to allow
 +
for the possibility of reversing many steps in its development, indeed,
 +
to take into account the near certain necessity of having to revisit,
 +
to revise, and to reverse many decisions about how to proceed toward
 +
an optimal description or a satisfactory grammar for the language.
 +
Doing all this means exploring the effects of various alterations
 +
and innovations as independently from each other as possible.
 +
 +
The degree of intermediate organization in a grammar is measured by how many
 +
intermediate symbols it has and by how they interact with each other by means
 +
of its productions.  With respect to this issue, Grammar 1 has no intermediate
 +
symbols at all, !Q! = {}, and therefore remains at an ostensibly trivial degree
 +
of intermediate organization.  Some additions to the list of intermediate symbols
 +
are practically obligatory in order to arrive at any reasonable grammar at all,
 +
other inclusions appear to have a more optional character, though obviously
 +
useful from the standpoints of clarity and ease of comprehension.
 +
 +
One of the troubles that is perceived to affect Grammar 1 is that it wastes
 +
so much of the available potential for efficient description in recounting
 +
over and over again the simple fact that the empty string is present in
 +
the language.  This arises in part from the statement that S :> S*,
 +
which implies that:
 +
 +
S  :>  S*  =  %e% |_| S |_| S · S |_| S · S · S |_| ...
 +
 +
There is nothing wrong with the more expansive pan of the covered equation,
 +
since it follows straightforwardly from the definition of the kleene star
 +
operation, but the covering statement, to the effect that S :> S*, is not
 +
necessarily a very productive piece of information, to the extent that it
 +
does always tell us very much about the language that is being supposed to
 +
fall under the type of a sentence S.  In particular, since it implies that
 +
S :> %e%, and since %e%·!L!  =  !L!·%e%  =  !L!, for any formal language !L!,
 +
the empty string !e! = "" is counted over and over in every term of the union,
 +
and every non-empty sentence under S appears again and again in every term of
 +
the union that follows the initial appearance of S.  As a result, this style
 +
of characterization has to be classified as "true but not very informative".
 +
If at all possible, one prefers to partition the language of interest into
 +
a disjoint union of subsets, thereby accounting for each sentence under
 +
its proper term, and one whose place under the sum serves as a useful
 +
parameter of its character or its complexity.  In general, this form
 +
of description is not always possible to achieve, but it is usually
 +
worth the trouble to actualize it whenever it is.
 +
 +
Suppose that one tries to deal with this problem by eliminating each use of
 +
the kleene star operation, by reducing it to a purely finitary set of steps,
 +
or by finding an alternative way to cover the sublanguage that it is used to
 +
generate.  This amounts, in effect, to "recognizing a type", a complex process
 +
that involves the following steps:
 +
 +
1.  Noticing a category of strings that
 +
    is generated by iteration or recursion.
 +
 +
2.  Acknowledging the circumstance that the noted category
 +
    of strings needs to be covered by a non-terminal symbol.
 +
 +
3.  Making a note of it by declaring and instituting
 +
    an explicitly and even expressively named category.
 +
 +
In sum, one introduces a non-terminal symbol for each type of sentence and
 +
each "part of speech" or sentential component that is generated by means of
 +
iteration or recursion under the ruling constraints of the grammar.  In order
 +
to do this one needs to analyze the iteration of each grammatical operation in
 +
a way that is analogous to a mathematically inductive definition, but further in
 +
a way that is not forced explicitly to recognize a distinct and separate type of
 +
expression merely to account for and to recount every increment in the parameter
 +
of iteration.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
Returning to the case of the cactus language, the process of recognizing an
 +
iterative type or a recursive type can be illustrated in the following way.
 +
The operative phrases in the simplest sort of recursive definition are its
 +
initial part and its generic part.  For the cactus language !C!(!P!), one
 +
has the following definitions of concatenation as iterated precatenation
 +
and of surcatenation as iterated subcatenation, respectively:
 +
 +
1.  Conc^0        =  "".
 +
 +
    Conc^k_j S_j  =  Prec(Conc^(k-1)_j S_j, S_k).
 +
 +
2.  Surc^0        =  "-()-".
 +
 +
    Surc^k_j S_j  =  Subc(Surc^(k-1)_j S_j, S_k).
 +
 +
In order to transform these recursive definitions into grammar rules,
 +
one introduces a new pair of intermediate symbols, "Conc" and "Surc",
 +
corresponding to the operations that share the same names but ignoring
 +
the inflexions of their individual parameters j and k.  Recognizing the
 +
type of a sentence by means of the initial symbol "S", and interpreting
 +
"Conc" and "Surc" as names for the types of strings that are generated
 +
by concatenation and by surcatenation, respectively, one arrives at
 +
the following transformation of the ruling operator definitions
 +
into the form of covering grammar rules:
 +
 +
1.  Conc  :>  "".
 +
 +
    Conc  :>  Conc · S.
 +
 +
2.  Surc  :>  "-()-".
 +
 +
    Surc  :>  "-(" · S · ")-".
 +
 +
    Surc  :>  Surc ")-"^(-1) · "," · S · ")-".
 +
 +
As given, this particular fragment of the intended grammar
 +
contains a couple of features that are desirable to amend.
 +
 +
1.  Given the covering S :> Conc, the covering rule Conc :> Conc · S
 +
    says no more than the covering rule Conc :> S · S.  Consequently,
 +
    all of the information contained in these two covering rules is
 +
    already covered by the statement that S :> S · S.
 +
 +
2.  A grammar rule that invokes a notion of decatenation, deletion, erasure,
 +
    or any other sort of retrograde production, is frequently considered to
 +
    be lacking in elegance, and a there is a style of critique for grammars
 +
    that holds it preferable to avoid these types of operations if it is at
 +
    all possible to do so.  Accordingly, contingent on the prescriptions of
 +
    the informal rule in question, and pursuing the stylistic dictates that
 +
    are writ in the realm of its aesthetic regime, it becomes necessary for
 +
    us to backtrack a little bit, to temporarily withdraw the suggestion of
 +
    employing these elliptical types of operations, but without, of course,
 +
    eliding the record of doing so.
 +
 +
One way to analyze the surcatenation of any number of sentences is to
 +
introduce an auxiliary type of string, not in general a sentence, but
 +
a proper component of any sentence that is formed by surcatenation.
 +
Doing this brings one to the following definition:
 +
 +
A "tract" is a concatenation of a finite sequence of sentences, with a
 +
literal comma "," interpolated between each pair of adjacent sentences.
 +
Thus, a typical tract T takes the form:
 +
 +
T  =  S_1 · "," · ...  · "," · S_k.
 +
 +
A tract must be distinguished from the abstract sequence of sentences,
 +
S_1, ..., S_k, where the commas that appear to come to mind, as if being
 +
called up to separate the successive sentences of the sequence, remain as
 +
partially abstract conceptions, or as signs that retain a disengaged status
 +
on the borderline between the text and the mind.  In effect, the types of
 +
commas that appear to follow in the abstract sequence continue to exist
 +
as conceptual abstractions and fail to be cognized in a wholly explicit
 +
fashion, whether as concrete tokens in the object language, or as marks
 +
in the strings of signs that are able to engage one's parsing attention.
 +
 +
Returning to the case of the painted cactus language !L! = !C!(!P!),
 +
it is possible to put the currently assembled pieces of a grammar
 +
together in the light of the presently adopted canons of style,
 +
to arrive a more refined analysis of the fact that the concept
 +
of a sentence covers any concatenation of sentences and any
 +
surcatenation of sentences, and so to obtain the following
 +
form of a grammar:
 +
 +
| !C!(!P!).  Grammar 2
 +
|
 +
| !Q! = {"T"}
 +
|
 +
| 1.  S  :>  !e!
 +
|
 +
| 2.  S  :>  m_1
 +
|
 +
| 3.  S  :>  p_j, for each j in J
 +
|
 +
| 4.  S  :>  S · S
 +
|
 +
| 5.  S  :>  "-(" · T · ")-"
 +
|
 +
| 6.  T  :>  S
 +
|
 +
| 7.  T  :>  T · "," · S
 +
 +
In this rendition, a string of type T is not in general
 +
a sentence itself but a proper "part of speech", that is,
 +
a strictly "lesser" component of a sentence in any suitable
 +
ordering of sentences and their components.  In order to see
 +
how the grammatical category T gets off the ground, that is,
 +
to detect its minimal strings and to discover how its ensuing
 +
generations gets started from these, it is useful to observe
 +
that the covering rule T :> S means that T "inherits" all of
 +
the initial conditions of S, namely, T  :>  !e!, m_1, p_j.
 +
In accord with these simple beginnings it comes to parse
 +
that the rule T :> T · "," · S, with the substitutions
 +
T = !e! and S = !e! on the covered side of the rule,
 +
bears the germinal implication that T :> ",".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
Grammar 2 achieves a portion of its success through a higher degree of
 +
intermediate organization.  Roughly speaking, the level of organization
 +
can be seen as reflected in the cardinality of the intermediate alphabet
 +
!Q! = {"T"}, but it is clearly not explained by this simple circumstance
 +
alone, since it is taken for granted that the intermediate symbols serve
 +
a purpose, a purpose that is easily recognizable but that may not be so
 +
easy to pin down and to specify exactly.  Nevertheless, it is worth the
 +
trouble of exploring this aspect of organization and this direction of
 +
development a little further.  Although it is not strictly necessary
 +
to do so, it is possible to organize the materials of the present
 +
grammar in a slightly better fashion by recognizing two recurrent
 +
types of strings that appear in the typical cactus expression.
 +
In doing this, one arrives at the following two definitions:
 +
 +
A "rune" is a string of blanks and paints concatenated together.
 +
Thus, a typical rune R is a string over {m_1} |_| !P!, possibly
 +
the empty string.
 +
 +
R  in  ({m_1} |_| !P!)*.
 +
 +
When there is no possibility of confusion, the letter "R" can be used
 +
either as a string variable that ranges over the set of runes or else
 +
as a type name for the class of runes.  The latter reading amounts to
 +
the enlistment of a fresh intermediate symbol, "R" in !Q!, as a part
 +
of a new grammar for !C!(!P!).  In effect, "R" affords a grammatical
 +
recognition for any rune that forms a part of a sentence in !C!(!P!).
 +
In situations where these variant usages are likely to be confused,
 +
the types of strings can be indicated by means of expressions like
 +
"r <: R" and "W <: R".
 +
 +
A "foil" is a string of the form "-(" · T · ")-", where T is a tract.
 +
Thus, a typical foil F has the form:
 +
 +
F  =  "-(" · S_1 · "," · ... · "," · S_k · ")-".
 +
 +
This is just the surcatenation of the sentences S_1, ..., S_k.
 +
Given the possibility that this sequence of sentences is empty,
 +
and thus that the tract T is the empty string, the minimum foil
 +
F is the expression "-()-".  Explicitly marking each foil F that
 +
is embodied in a cactus expression is tantamount to recognizing
 +
another intermediate symbol, "F" in !Q!, further articulating the
 +
structures of sentences and expanding the grammar for the language
 +
!C!(!P!).  All of the same remarks about the versatile uses of the
 +
intermediate symbols, as string variables and as type names, apply
 +
again to the letter "F".
 +
 +
| !C!(!P!).  Grammar 3
 +
|
 +
| !Q! = {"F", "R", "T"}
 +
|
 +
|  1.  S  :>  R
 +
|
 +
|  2.  S  :>  F
 +
|
 +
|  3.  S  :>  S · S
 +
|
 +
|  4.  R  :>  !e!
 +
|
 +
|  5.  R  :>  m_1
 +
|
 +
|  6.  R  :>  p_j, for each j in J
 +
|
 +
|  7.  R  :>  R · R
 +
|
 +
|  8.  F  :>  "-(" · T · ")-"
 +
|
 +
|  9.  T  :>  S
 +
|
 +
| 10.  T  :>  T · "," · S
 +
 +
In Grammar 3, the first three Rules say that a sentence (a string of type S),
 +
is a rune (a string of type R), a foil (a string of type F), or an arbitrary
 +
concatenation of strings of these two types.  Rules 4 through 7 specify that
 +
a rune R is an empty string !e! = "", a blank symbol m_1 = " ", a paint p_j,
 +
for j in J, or any concatenation of strings of these three types.  Rule 8
 +
characterizes a foil F as a string of the form "-(" · T · ")-", where T is
 +
a tract.  The last two Rules say that a tract T is either a sentence S or
 +
else the concatenation of a tract, a comma, and a sentence, in that order.
 +
 +
At this point in the succession of grammars for !C!(!P!), the explicit
 +
uses of indefinite iterations, like the kleene star operator, are now
 +
completely reduced to finite forms of concatenation, but the problems
 +
that some styles of analysis have with allowing non-terminal symbols
 +
to cover both themselves and the empty string are still present.
 +
 +
Any degree of reflection on this difficulty raises the general question:
 +
What is a practical strategy for accounting for the empty string in the
 +
organization of any formal language that counts it among its sentences?
 +
One answer that presents itself is this:  If the empty string belongs to
 +
a formal language, it suffices to count it once at the beginning of the
 +
formal account that enumerates its sentences and then to move on to more
 +
interesting materials.
 +
 +
Returning to the case of the cactus language !C!(!P!), that is,
 +
the formal language of "painted and rooted cactus expressions",
 +
it serves the purpose of efficient accounting to partition the
 +
language PARCE into the following couple of sublanguages:
 +
 +
1.  The "emptily painted and rooted cactus expressions"
 +
    make up the language EPARCE that consists of
 +
    a single empty string as its only sentence.
 +
    In short:
 +
 +
    EPARCE  =  {""}.
 +
 +
2.  The "significantly painted and rooted cactus expressions"
 +
    make up the language SPARCE that consists of everything else,
 +
    namely, all of the non-empty strings in the language PARCE.
 +
    In sum:
 +
 +
    SPARCE  =  PARCE \ "".
 +
 +
As a result of marking the distinction between empty and significant sentences,
 +
that is, by categorizing each of these three classes of strings as an entity
 +
unto itself and by conceptualizing the whole of its membership as falling
 +
under a distinctive symbol, one obtains an equation of sets that connects
 +
the three languages being marked:
 +
 +
SPARCE  =  PARCE - EPARCE.
 +
 +
In sum, one has the disjoint union:
 +
 +
PARCE  =  EPARCE |_| SPARCE.
 +
 +
For brevity in the present case, and to serve as a generic device
 +
in any similar array of situations, let the symbol "S" be used to
 +
signify the type of an arbitrary sentence, possibly empty, whereas
 +
the symbol "S'" is reserved to designate the type of a specifically
 +
non-empty sentence.  In addition, let the symbol "%e%" be employed
 +
to indicate the type of the empty sentence, in effect, the language
 +
%e% = {""} that contains a single empty string, and let a plus sign
 +
"+" signify a disjoint union of types.  In the most general type of
 +
situation, where the type S is permitted to include the empty string,
 +
one notes the following relation among types:
 +
 +
S  =  %e%  +  S'.
 +
 +
Consequences of the distinction between empty expressions and
 +
significant expressions are taken up for discussion next time.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
With the distinction between empty and significant expressions in mind,
 +
I return to the grasp of the cactus language !L! = !C!(!P!) = PARCE(!P!)
 +
that is afforded by Grammar 2, and, taking that as a point of departure,
 +
explore other avenues of possible improvement in the comprehension of
 +
these expressions.  In order to observe the effects of this alteration
 +
as clearly as possible, in isolation from any other potential factors,
 +
it is useful to strip away the higher levels intermediate organization
 +
that are present in Grammar 3, and start again with a single intermediate
 +
symbol, as used in Grammar 2.  One way of carrying out this strategy leads
 +
on to a grammar of the variety that will be articulated next.
 +
 +
If one imposes the distinction between empty and significant types on
 +
each non-terminal symbol in Grammar 2, then the non-terminal symbols
 +
"S" and "T" give rise to the non-terminal symbols "S", "S'", "T", "T'",
 +
leaving the last three of these to form the new intermediate alphabet.
 +
Grammar 4 has the intermediate alphabet !Q! = {"S'", "T", "T'"}, with
 +
the set !K! of covering production rules as listed in the next display.
 +
 +
| !C!(!P!).  Grammar 4
 +
|
 +
| !Q! = {"S'", "T", "T'"}
 +
|
 +
| 1.  S  :>  !e!
 +
|
 +
| 2.  S  :>  S'
 +
|
 +
| 3.  S'  :>  m_1
 +
|
 +
| 4.  S'  :>  p_j, for each j in J
 +
|
 +
| 5.  S'  :>  "-(" · T · ")-"
 +
|
 +
| 6.  S'  :>  S' · S'
 +
|
 +
| 7.  T  :>  !e!
 +
|
 +
| 8.  T  :>  T'
 +
|
 +
| 9.  T'  :>  T · "," · S
 +
 +
In this version of a grammar for !L! = !C!(!P!), the intermediate type T
 +
is partitioned as T = %e% + T', thereby parsing the intermediate symbol T
 +
in parallel fashion with the division of its overlying type as S = %e% + S'.
 +
This is an option that I will choose to close off for now, but leave it open
 +
to consider at a later point.  Thus, it suffices to give a brief discussion
 +
of what it involves, in the process of moving on to its chief alternative.
 +
 +
There does not appear to be anything radically wrong with trying this
 +
approach to types.  It is reasonable and consistent in its underlying
 +
principle, and it provides a rational and a homogeneous strategy toward
 +
all parts of speech, but it does require an extra amount of conceptual
 +
overhead, in that every non-trivial type has to be split into two parts
 +
and comprehended in two stages.  Consequently, in view of the largely
 +
practical difficulties of making the requisite distinctions for every
 +
intermediate symbol, it is a common convention, whenever possible, to
 +
restrict intermediate types to covering exclusively non-empty strings.
 +
 +
For the sake of future reference, it is convenient to refer to this restriction
 +
on intermediate symbols as the "intermediate significance" constraint.  It can
 +
be stated in a compact form as a condition on the relations between non-terminal
 +
symbols q in {"S"} |_| !Q! and sentential forms W in {"S"} |_| (!Q! |_| !A!)*.
 +
 +
| Condition On Intermediate Significance
 +
|
 +
| If    q  :>  W
 +
|
 +
| and  W  =  !e!,
 +
|
 +
| then  q  =  "S".
 +
 +
If this is beginning to sound like a monotone condition, then it is
 +
not absurd to sharpen the resemblance and render the likeness more
 +
acute.  This is done by declaring a couple of ordering relations,
 +
denoting them under variant interpretations by the same sign "<".
 +
 +
1.  The ordering "<" on the set of non-terminal symbols,
 +
    q in {"S"} |_| !Q!, ordains the initial symbol "S"
 +
    to be strictly prior to every intermediate symbol.
 +
    This is tantamount to the axiom that "S" < q,
 +
    for all q in !Q!.
 +
 +
2.  The ordering "<" on the collection of sentential forms,
 +
    W in {"S"} |_| (!Q! |_| !A!)*, ordains the empty string
 +
    to be strictly minor to every other sentential form.
 +
    This is stipulated in the axiom that !e! < W,
 +
    for every non-empty sentential form W.
 +
 +
Given these two orderings, the constraint in question
 +
on intermediate significance can be stated as follows:
 +
 +
| Condition Of Intermediate Significance
 +
|
 +
| If    q  :>  W
 +
|
 +
| and  q  >  "S",
 +
|
 +
| then  W  >  !e!.
 +
 +
Achieving a grammar that respects this convention typically requires a more
 +
detailed account of the initial setting of a type, both with regard to the
 +
type of context that incites its appearance and also with respect to the
 +
minimal strings that arise under the type in question.  In order to find
 +
covering productions that satisfy the intermediate significance condition,
 +
one must be prepared to consider a wider variety of calling contexts or
 +
inciting situations that can be noted to surround each recognized type,
 +
and also to enumerate a larger number of the smallest cases that can
 +
be observed to fall under each significant type.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
With the array of foregoing considerations in mind,
 +
one is gradually led to a grammar for !L! = !C!(!P!)
 +
in which all of the covering productions have either
 +
one of the following two forms:
 +
 +
| S  :>  !e!
 +
|
 +
| q  :>  W,  with  q in {"S"} |_| !Q!,  and  W in (!Q! |_| !A!)^+
 +
 +
A grammar that fits into this mold is called a "context-free" grammar.
 +
The first type of rewrite rule is referred to as a "special production",
 +
while the second type of rewrite rule is called an "ordinary production".
 +
An "ordinary derivation" is one that employs only ordinary productions.
 +
In ordinary productions, those that have the form q :> W, the replacement
 +
string W is never the empty string, and so the lengths of the augmented
 +
strings or the sentential forms that follow one another in an ordinary
 +
derivation, on account of using the ordinary types of rewrite rules,
 +
never decrease at any stage of the process, up to and including the
 +
terminal string that is finally generated by the grammar.  This type
 +
of feature is known as the "non-contracting property" of productions,
 +
derivations, and grammars.  A grammar is said to have the property if
 +
all of its covering productions, with the possible exception of S :> e,
 +
are non-contracting.  In particular, context-free grammars are special
 +
cases of non-contracting grammars.  The presence of the non-contracting
 +
property within a formal grammar makes the length of the augmented string
 +
available as a parameter that can figure into mathematical inductions and
 +
motivate recursive proofs, and this handle on the generative process makes
 +
it possible to establish the kinds of results about the generated language
 +
that are not easy to achieve in more general cases, nor by any other means
 +
even in these brands of special cases.
 +
 +
Grammar 5 is a context-free grammar for the painted cactus language
 +
that uses !Q! = {"S'", "T"}, with !K! as listed in the next display.
 +
 +
| !C!(!P!).  Grammar 5
 +
|
 +
| !Q! = {"S'", "T"}
 +
|
 +
|  1. S  :>  !e!
 +
|
 +
|  2.  S  :>  S'
 +
|
 +
|  3.  S'  :>  m_1
 +
|
 +
|  4.  S'  :>  p_j, for each j in J
 +
|
 +
|  5.  S'  :>  S' · S'
 +
|
 +
|  6.  S'  :>  "-()-"
 +
|
 +
|  7.  S'  :>  "-(" · T · ")-"
 +
|
 +
|  8.  T  :>  ","
 +
|
 +
|  9.  T  :>  S'
 +
|
 +
| 10.  T  :>  T · ","
 +
|
 +
| 11.  T  :>  T · "," · S'
 +
 +
Finally, it is worth trying to bring together the advantages of these
 +
diverse styles of grammar, to whatever extent that they are compatible.
 +
To do this, a prospective grammar must be capable of maintaining a high
 +
level of intermediate organization, like that arrived at in Grammar 2,
 +
while respecting the principle of intermediate significance, and thus
 +
accumulating all the benefits of the context-free format in Grammar 5.
 +
A plausible synthesis of most of these features is given in Grammar 6.
 +
 +
| !C!(!P!).  Grammar 6
 +
|
 +
| !Q! = {"S'", "R", "F", "T"}
 +
|
 +
|  1.  S  :>  !e!
 +
|
 +
|  2.  S  :>  S'
 +
|
 +
|  3.  S'  :>  R
 +
|
 +
|  4.  S'  :>  F
 +
|
 +
|  5.  S'  :>  S' · S'
 +
|
 +
|  6.  R  :>  m_1
 +
|
 +
|  7.  R  :>  p_j, for each j in J
 +
|
 +
|  8.  R  :>  R · R
 +
|
 +
|  9.  F  :>  "-()-"
 +
|
 +
| 10.  F  :>  "-(" · T · ")-"
 +
|
 +
| 11.  T  :>  ","
 +
|
 +
| 12.  T  :>  S'
 +
|
 +
| 13.  T  :>  T · ","
 +
|
 +
| 14.  T  :>  T · "," · S'
 +
 +
The preceding development provides a typical example of how an initially
 +
effective and conceptually succinct description of a formal language, but
 +
one that is terse to the point of allowing its prospective interpreter to
 +
waste exorbitant amounts of energy in trying to unravel its implications,
 +
can be converted into a form that is more efficient from the operational
 +
point of view, even if slightly more ungainly in regard to its elegance.
 +
 +
The basic idea behind all of this machinery remains the same:  Besides
 +
the select body of formulas that are introduced as boundary conditions,
 +
it merely institutes the following general rule:
 +
 +
| If    the strings S_1, ..., S_k are sentences,
 +
|
 +
| then  their concatenation in the form
 +
|
 +
|      Conc^k_j S_j  =  S_1 · ... · S_k
 +
|
 +
|      is a sentence,
 +
|
 +
| and  their surcatenation in the form
 +
|
 +
|      Surc^k_j S_j  =  "-(" · S_1 · "," · ... · "," · S_k · ")-"
 +
|
 +
|      is a sentence.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.9  The Cactus Language:  Syntax (cont.)
 +
 +
It is fitting to wrap up the foregoing developments by summarizing the
 +
notion of a formal grammar that appeared to evolve in the present case.
 +
For the sake of future reference and the chance of a wider application,
 +
it is also useful to try to extract the scheme of a formalization that
 +
potentially holds for any formal language.  The following presentation
 +
of the notion of a formal grammar is adapted, with minor modifications,
 +
from the treatment in (DDQ, 60-61).
 +
 +
A "formal grammar" !G! is given by a four-tuple !G! = ("S", !Q!, !A!, !K!)
 +
that takes the following form of description:
 +
 +
1.  "S" is the "initial", "special", "start", or "sentence symbol".
 +
    Since the letter "S" serves this function only in a special setting,
 +
    its employment in this role need not create any confusion with its
 +
    other typical uses as a string variable or as a sentence variable.
 +
 +
2.  !Q! = {q_1, ..., q_m} is a finite set of "intermediate symbols",
 +
    all distinct from "S".
 +
 +
3.  !A! = {a_1, ..., a_n} is a finite set of "terminal symbols",
 +
    also known as the "alphabet" of !G!, all distinct from "S" and
 +
    disjoint from !Q!.  Depending on the particular conception of the
 +
    language !L! that is "covered", "generated", "governed", or "ruled"
 +
    by the grammar !G!, that is, whether !L! is conceived to be a set of
 +
    words, sentences, paragraphs, or more extended structures of discourse,
 +
    it is usual to describe !A! as the "alphabet", "lexicon", "vocabulary",
 +
    "liturgy", or "phrase book" of both the grammar !G! and the language !L!
 +
    that it regulates.
 +
 +
4.  !K! is a finite set of "characterizations".  Depending on how they
 +
    come into play, these are variously described as "covering rules",
 +
    "formations", "productions", "rewrite rules", "subsumptions",
 +
    "transformations", or "typing rules".
 +
 +
To describe the elements of !K! it helps to define some additional terms:
 +
 +
a.  The symbols in {"S"} |_| !Q! |_| !A! form the "augmented alphabet" of !G!.
 +
 +
b.  The symbols in {"S"} |_| !Q! are the "non-terminal symbols" of !G!.
 +
 +
c.  The symbols in !Q! |_| !A! are the "non-initial symbols" of !G!.
 +
 +
d.  The strings in ({"S"} |_| !Q! |_| !A!)*  are the "augmented strings" for G.
 +
 +
e.  The strings in {"S"} |_| (!Q! |_| !A!)* are the "sentential forms" for G.
 +
 +
Each characterization in !K! is an ordered pair of strings (S_1, S_2)
 +
that takes the following form:
 +
 +
| S_1  =  Q_1 · q · Q_2,
 +
|
 +
| S_2  =  Q_1 · W · Q_2.
 +
 +
In this scheme, S_1 and S_2 are members of the augmented strings for !G!,
 +
more precisely, S_1 is a non-empty string and a sentential form over !G!,
 +
while S_2 is a possibly empty string and also a sentential form over !G!.
 +
 +
Here also, q is a non-terminal symbol, that is, q is in {"S"} |_| !Q!,
 +
while Q_1, Q_2, and W are possibly empty strings of non-initial symbols,
 +
a fact that can be expressed in the form:  Q_1, Q_2, W in (!Q! |_| !A!)*.
 +
 +
In practice, the ordered pairs of strings in !K! are used to "derive",
 +
to "generate", or to "produce" sentences of the language !L! = <!G!>
 +
that is then said to be "governed" or "regulated" by the grammar !G!.
 +
In order to facilitate this active employment of the grammar, it is
 +
conventional to write the characterization (S_1, S_2) in either one
 +
of the next two forms, where the more generic form is followed by
 +
the more specific form:
 +
 +
| S_1            :>  S_2
 +
|
 +
| Q_1 · q · Q_2  :>  Q_1 · W · Q_2
 +
 +
In this usage, the characterization S_1 :> S_2 is tantamount to a grammatical
 +
license to transform a string of the form Q_1 · q · Q_2 into a string of the
 +
form Q1 · W · Q2, in effect, replacing the non-terminal symbol q with the
 +
non-initial string W in any selected, preserved, and closely adjoining
 +
context of the form Q1 · ... · Q2.  Accordingly, in this application
 +
the notation "S_1 :> S_2" can be read as "S_1 produces S_2" or as
 +
"S_1 transforms into S_2".
 +
 +
An "immediate derivation" in !G! is an ordered pair (W, W')
 +
of sentential forms in !G! such that:
 +
 +
| W  =  Q_1 · X · Q_2,
 +
|
 +
| W'  =  Q_1 · Y · Q_2,
 +
|
 +
| and  (X, Y)  in !K!,
 +
|
 +
| i.e.  X :> Y  in !G!.
 +
 +
This relation is indicated by saying that W "immediately derives" W',
 +
that W' is "immediately derived" from W in !G!, and also by writing:
 +
 +
W  ::>  W'.
 +
 +
A "derivation" in !G! is a finite sequence (W_1, ..., W_k)
 +
of sentential forms over !G! such that each adjacent pair
 +
(W_j, W_(j+1)) of sentential forms in the sequence is an
 +
immediate derivation in !G!, in other words, such that:
 +
 +
W_j  ::>  W_(j+1),  for all j = 1 to k-1.
 +
 +
If there exists a derivation (W_1, ..., W_k) in !G!,
 +
one says that W_1 "derives" W_k in !G!, conversely,
 +
that W_k is "derivable" from W_1 in !G!, and one
 +
typically summarizes the derivation by writing:
 +
 +
W_1  :*:>  W_k.
 +
 +
The language !L! = !L!(!G!) = <!G!> that is "generated"
 +
by the formal grammar !G! = ("S", !Q!, !A!, !K!) is the
 +
set of strings over the terminal alphabet !A! that are
 +
derivable from the initial symbol "S" by way of the
 +
intermediate symbols in !Q! according to the
 +
characterizations in K.  In sum:
 +
 +
!L!(!G!)  =  <!G!>  =  {W in !A!*  :  "S" :*:> W}.
 +
 +
Finally, a string W is called a "word", a "sentence", or so on,
 +
of the language generated by !G! if and only if W is in !L!(!G!).
 +
 +
Reference
 +
 +
| Denning, P.J., Dennis, J.B., Qualitz, J.E.,
 +
|'Machines, Languages, and Computation',
 +
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.10  The Cactus Language:  Stylistics
 +
 +
| As a result, we can hardly conceive of how many possibilities there are
 +
| for what we call objective reality.  Our sharp quills of knowledge are so
 +
| narrow and so concentrated in particular directions that with science there
 +
| are myriads of totally different real worlds, each one accessible from the
 +
| next simply by slight alterations -- shifts of gaze -- of every particular
 +
| discipline and subspecialty.
 +
|
 +
| Herbert J. Bernstein, "Idols", page 38.
 +
|
 +
| Herbert J. Bernstein,
 +
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
 +
|
 +
| Marcus G. Raskin & Herbert J. Bernstein,
 +
|'New Ways of Knowing:  The Sciences, Society, & Reconstructive Knowledge',
 +
| Rowman & Littlefield, Totowa, NJ, 1987.
 +
 +
This Subsection highlights an issue of "style" that arises in describing
 +
a formal language.  In broad terms, I use the word "style" to refer to a
 +
loosely specified class of formal systems, typically ones that have a set
 +
of distinctive features in common.  For instance, a style of proof system
 +
usually dictates one or more rules of inference that are acknowledged as
 +
conforming to that style.  In the present context, the word "style" is a
 +
natural choice to characterize the varieties of formal grammars, or any
 +
other sorts of formal systems that can be contemplated for deriving the
 +
sentences of a formal language.
 +
 +
In looking at what seems like an incidental issue, the discussion arrives
 +
at a critical point.  The question is:  What decides the issue of style?
 +
Taking a given language as the object of discussion, what factors enter
 +
into and determine the choice of a style for its presentation, that is,
 +
a particular way of arranging and selecting the materials that come to
 +
be involved in a description, a grammar, or a theory of the language?
 +
To what degree is the determination accidental, empirical, pragmatic,
 +
rhetorical, or stylistic, and to what extent is the choice essential,
 +
logical, and necessary?  For that matter, what determines the order
 +
of signs in a word, a sentence, a text, or a discussion?  All of
 +
the corresponding parallel questions about the character of this
 +
choice can be posed with regard to the constituent part as well
 +
as with regard to the main constitution of the formal language.
 +
 +
In order to answer this sort of question, at any level of articulation,
 +
one has to inquire into the type of distinction that it invokes, between
 +
arrangements and orders that are essential, logical, and necessary and
 +
orders and arrangements that are accidental, rhetorical, and stylistic.
 +
As a rough guide to its comprehension, a "logical order", if it resides
 +
in the subject at all, can be approached by considering all of the ways
 +
of saying the same things, in all of the languages that are capable of
 +
saying roughly the same things about that subject.  Of course, the "all"
 +
that appears in this rule of thumb has to be interpreted as a fittingly
 +
qualified sort of universal.  For all practical purposes, it simply means
 +
"all of the ways that a person can think of" and "all of the languages
 +
that a person can conceive of", with all things being relative to the
 +
particular moment of investigation.  For all of these reasons, the rule
 +
must stand as little more than a rough idea of how to approach its object.
 +
 +
If it is demonstrated that a given formal language can be presented in
 +
any one of several styles of formal grammar, then the choice of a format
 +
is accidental, optional, and stylistic to the very extent that it is free.
 +
But if it can be shown that a particular language cannot be successfully
 +
presented in a particular style of grammar, then the issue of style is
 +
no longer free and rhetorical, but becomes to that very degree essential,
 +
necessary, and obligatory, in other words, a question of the objective
 +
logical order that can be found to reside in the object language.
 +
 +
As a rough illustration of the difference between logical and rhetorical
 +
orders, consider the kinds of order that are expressed and exhibited in
 +
the following conjunction of implications:
 +
 +
"X => Y  and  Y => Z".
 +
 +
Here, there is a happy conformity between the logical content and the
 +
rhetorical form, indeed, to such a degree that one hardly notices the
 +
difference between them.  The rhetorical form is given by the order
 +
of sentences in the two implications and the order of implications
 +
in the conjunction.  The logical content is given by the order of
 +
propositions in the extended implicational sequence:
 +
 +
X  =<  Y  =<  Z.
 +
 +
To see the difference between form and content, or manner and matter,
 +
it is enough to observe a few of the ways that the expression can be
 +
varied without changing its meaning, for example:
 +
 +
"Z <= Y  and  Y <= X".
 +
 +
Any style of declarative programming, also called "logic programming",
 +
depends on a capacity, as embodied in a programming language or other
 +
formal system, to describe the relation between problems and solutions
 +
in logical terms.  A recurring problem in building this capacity is in
 +
bridging the gap between ostensibly non-logical orders and the logical
 +
orders that are used to describe and to represent them.  For instance,
 +
to mention just a couple of the most pressing cases, and the ones that
 +
are currently proving to be the most resistant to a complete analysis,
 +
one has the orders of dynamic evolution and rhetorical transition that
 +
manifest themselves in the process of inquiry and in the communication
 +
of its results.
 +
 +
This patch of the ongoing discussion is concerned with describing a
 +
particular variety of formal languages, whose typical representative
 +
is the painted cactus language !L! = !C!(!P!).  It is the intention of
 +
this work to interpret this language for propositional logic, and thus
 +
to use it as a sentential calculus, an order of reasoning that forms an
 +
active ingredient and a significant component of all logical reasoning.
 +
To describe this language, the standard devices of formal grammars and
 +
formal language theory are more than adequate, but this only raises the
 +
next question:  What sorts of devices are exactly adequate, and fit the
 +
task to a "T"?  The ultimate desire is to turn the tables on the order
 +
of description, and so begins a process of eversion that evolves to the
 +
point of asking:  To what extent can the language capture the essential
 +
features and laws of its own grammar and describe the active principles
 +
of its own generation?  In other words:  How well can the language be
 +
described by using the language itself to do so?
 +
 +
In order to speak to these questions, I have to express what a grammar says
 +
about a language in terms of what a language can say on its own.  In effect,
 +
it is necessary to analyze the kinds of meaningful statements that grammars
 +
are capable of making about languages in general and to relate them to the
 +
kinds of meaningful statements that the syntactic "sentences" of the cactus
 +
language might be interpreted as making about the very same topics.  So far
 +
in the present discussion, the sentences of the cactus language do not make
 +
any meaningful statements at all, much less any meaningful statements about
 +
languages and their constitutions.  As of yet, these sentences subsist in the
 +
form of purely abstract, formal, and uninterpreted combinatorial constructions.
 +
 +
Before the capacity of a language to describe itself can be evaluated,
 +
the missing link to meaning has to be supplied for each of its strings.
 +
This calls for a dimension of semantics and a notion of interpretation,
 +
topics that are taken up for the case of the cactus language !C!(!P!)
 +
in Subsection 1.3.10.12.  Once a plausible semantics is prescribed for
 +
this language it will be possible to return to these questions and to
 +
address them in a meaningful way.
 +
 +
The prominent issue at this point is the distinct placements of formal
 +
languages and formal grammars with respect to the question of meaning.
 +
The sentences of a formal language are merely the abstract strings of
 +
abstract signs that happen to belong to a certain set.  They do not by
 +
themselves make any meaningful statements at all, not without mounting
 +
a separate effort of interpretation, but the rules of a formal grammar
 +
make meaningful statements about a formal language, to the extent that
 +
they say what strings belong to it and what strings do not.  Thus, the
 +
formal grammar, a formalism that appears to be even more skeletal than
 +
the formal language, still has bits and pieces of meaning attached to it.
 +
In a sense, the question of meaning is factored into two parts, structure
 +
and value, leaving the aspect of value reduced in complexity and subtlety
 +
to the simple question of belonging.  Whether this single bit of meaningful
 +
value is enough to encompass all of the dimensions of meaning that we require,
 +
and whether it can be compounded to cover the complexity that actually exists
 +
in the realm of meaning -- these are questions for an extended future inquiry.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.10  The Cactus Language:  Stylistics (cont.)
 +
 +
Perhaps I ought to comment on the differences between the present and
 +
the standard definition of a formal grammar, since I am attempting to
 +
strike a compromise with several alternative conventions of usage, and
 +
thus to leave certain options open for future exploration.  All of the
 +
changes are minor, in the sense that they are not intended to alter the
 +
classes of languages that are able to be generated, but only to clear up
 +
various ambiguities and sundry obscurities that affect their conception.
 +
 +
Primarily, the conventional scope of non-terminal symbols was expanded
 +
to encompass the sentence symbol, mainly on account of all the contexts
 +
where the initial and the intermediate symbols are naturally invoked in
 +
the same breath.  By way of compensating for the usual exclusion of the
 +
sentence symbol from the non-terminal class, an equivalent distinction
 +
was introduced in the fashion of a distinction between the initial and
 +
the intermediate symbols, and this serves its purpose in all of those
 +
contexts where the two kind of symbols need to be treated separately.
 +
 +
At the present point, I remain a bit worried about the motivations
 +
and the justifications for introducing this distinction, under any
 +
name, in the first place.  It is purportedly designed to guarantee
 +
that the process of derivation at least gets started in a definite
 +
direction, while the real questions have to do with how it all ends.
 +
The excuses of efficiency and expediency that I offered as plausible
 +
and sufficient reasons for distinguishing between empty and significant
 +
sentences are likely to be ephemeral, if not entirely illusory, since
 +
intermediate symbols are still permitted to characterize or to cover
 +
themselves, not to mention being allowed to cover the empty string,
 +
and so the very types of traps that one exerts oneself to avoid at
 +
the outset are always there to afflict the process at all of the
 +
intervening times.
 +
 +
If one reflects on the form of grammar that is being prescribed here,
 +
it looks as if one sought, rather futilely, to avoid the problems of
 +
recursion by proscribing the main program from calling itself, while
 +
allowing any subprogram to do so.  But any trouble that is avoidable
 +
in the part is also avoidable in the main, while any trouble that is
 +
inevitable in the part is also inevitable in the main.  Consequently,
 +
I am reserving the right to change my mind at a later stage, perhaps
 +
to permit the initial symbol to characterize, to cover, to regenerate,
 +
or to produce itself, if that turns out to be the best way in the end.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.10  The Cactus Language:  Stylistics (cont.)
 +
 +
Before I leave this Subsection, I need to say a few things about
 +
the manner in which the abstract theory of formal languages and
 +
the pragmatic theory of sign relations interact with each other.
 +
 +
Formal language theory can seem like an awfully picky subject at times,
 +
treating every symbol as a thing in itself the way it does, sorting out
 +
the nominal types of symbols as objects in themselves, and singling out
 +
the passing tokens of symbols as distinct entities in their own rights.
 +
It has to continue doing this, if not for any better reason than to aid
 +
in clarifying the kinds of languages that people are accustomed to use,
 +
to assist in writing computer programs that are capable of parsing real
 +
sentences, and to serve in designing programming languages that people
 +
would like to become accustomed to use.  As a matter of fact, the only
 +
time that formal language theory becomes too picky, or a bit too myopic
 +
in its focus, is when it leads one to think that one is dealing with the
 +
thing itself and not just with the sign of it, in other words, when the
 +
people who use the tools of formal language theory forget that they are
 +
dealing with the mere signs of more interesting objects and not with the
 +
objects of ultimate interest in and of themselves.
 +
 +
As a result, there a number of deleterious effects that can arise from
 +
the extreme pickiness of formal language theory, arising, as is often the
 +
case, when formal theorists forget the practical context of theorization.
 +
It frequently happens that the exacting task of defining the membership
 +
of a formal language leads one to think that this object and this object
 +
alone is the justifiable end of the whole exercise.  The distractions of
 +
this mediate objective render one liable to forget that one's penultimate
 +
interest lies always with various kinds of equivalence classes of signs,
 +
not entirely or exclusively with their more meticulous representatives.
 +
 +
When this happens, one typically goes on working oblivious to the fact
 +
that many details about what transpires in the meantime do not matter
 +
at all in the end, and one is likely to remain in blissful ignorance
 +
of the circumstance that many special details of language membership
 +
are bound, destined, and pre-determined to be glossed over with some
 +
measure of indifference, especially when it comes down to the final
 +
constitution of those equivalence classes of signs that are able to
 +
answer for the genuine objects of the whole enterprise of language.
 +
When any form of theory, against its initial and its best intentions,
 +
leads to this kind of absence of mind that is no longer beneficial in
 +
all of its main effects, the situation calls for an antidotal form of
 +
theory, one that can restore the presence of mind that all forms of
 +
theory are meant to augment.
 +
 +
The pragmatic theory of sign relations is called for in settings where
 +
everything that can be named has many other names, that is to say, in
 +
the usual case.  Of course, one would like to replace this superfluous
 +
multiplicity of signs with an organized system of canonical signs, one
 +
for each object that needs to be denoted, but reducing the redundancy
 +
too far, beyond what is necessary to eliminate the factor of "noise" in
 +
the language, that is, to clear up its effectively useless distractions,
 +
can destroy the very utility of a typical language, which is intended to
 +
provide a ready means to express a present situation, clear or not, and
 +
to describe an ongoing condition of experience in just the way that it
 +
seems to present itself.  Within this fleshed out framework of language,
 +
moreover, the process of transforming the manifestations of a sign from
 +
its ordinary appearance to its canonical aspect is the whole problem of
 +
computation in a nutshell.
 +
 +
It is a well-known truth, but an often forgotten fact, that nobody
 +
computes with numbers, but solely with numerals in respect of numbers,
 +
and numerals themselves are symbols.  Among other things, this renders
 +
all discussion of numeric versus symbolic computation a bit beside the
 +
point, since it is only a question of what kinds of symbols are best for
 +
one's immediate application or for one's selection of ongoing objectives.
 +
The numerals that everybody knows best are just the canonical symbols,
 +
the standard signs or the normal terms for numbers, and the process of
 +
computation is a matter of getting from the arbitrarily obscure signs
 +
that the data of a situation are capable of throwing one's way to the
 +
indications of its character that are clear enough to motivate action.
 +
 +
Having broached the distinction between propositions and sentences, one
 +
can see its similarity to the distinction between numbers and numerals.
 +
What are the implications of the foregoing considerations for reasoning
 +
about propositions and for the realm of reckonings in sentential logic?
 +
If the purpose of a sentence is just to denote a proposition, then the
 +
proposition is just the object of whatever sign is taken for a sentence.
 +
This means that the computational manifestation of a piece of reasoning
 +
about propositions amounts to a process that takes place entirely within
 +
a language of sentences, a procedure that can rationalize its account by
 +
referring to the denominations of these sentences among propositions.
 +
 +
The application of these considerations in the immediate setting is this:
 +
Do not worry too much about what roles the empty string "" and the blank
 +
symbol " " are supposed to play in a given species of formal languages.
 +
As it happens, it is far less important to wonder whether these types
 +
of formal tokens actually constitute genuine sentences than it is to
 +
decide what equivalence classes it makes sense to form over all of
 +
the sentences in the resulting language, and only then to bother
 +
about what equivalence classes these limiting cases of sentences
 +
are most conveniently taken to represent.
 +
 +
These concerns about boundary conditions betray a more general issue.
 +
Already by this point in discussion the limits of the purely syntactic
 +
approach to a language are beginning to be visible.  It is not that one
 +
cannot go a whole lot further by this road in the analysis of a particular
 +
language and in the study of languages in general, but when it comes to the
 +
questions of understanding the purpose of a language, of extending its usage
 +
in a chosen direction, or of designing a language for a particular set of uses,
 +
what matters above all else are the "pragmatic equivalence classes" of signs that
 +
are demanded by the application and intended by the designer, and not so much the
 +
peculiar characters of the signs that represent these classes of practical meaning.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.10  The Cactus Language:  Stylistics (cont.)
 +
 +
Any description of a language is bound to have alternative descriptions.
 +
More precisely, a circumscribed description of a formal language, as any
 +
effectively finite description is bound to be, is certain to suggest the
 +
equally likely existence and the possible utility of other descriptions.
 +
A single formal grammar describes but a single formal language, but any
 +
formal language is described by many different formal grammars, not all
 +
of which afford the same grasp of its structure, provide an equivalent
 +
comprehension of its character, or yield an interchangeable view of its
 +
aspects.  Consequently, even with respect to the same formal language,
 +
different formal grammars are typically better for different purposes.
 +
 +
With the distinctions that evolve among the different styles of grammar,
 +
and with the preferences that different observers display toward them,
 +
there naturally comes the question:  What is the root of this evolution?
 +
 +
One dimension of variation in the styles of formal grammars can be seen
 +
by treating the union of languages, and especially the disjoint union of
 +
languages, as a "sum", by treating the concatenation of languages as a
 +
"product", and then by distinguishing the styles of analysis that favor
 +
"sums of products" from those that favor "products of sums" as their
 +
canonical forms of description.  If one examines the relation between
 +
languages and grammars carefully enough to see the presence and the
 +
influence of these different styles, and when one comes to appreciate
 +
the ways that different styles of grammars can be used with different
 +
degrees of success for different purposes, then one begins to see the
 +
possibility that alternative styles of description can be based on
 +
altogether different linguistic and logical operations.
 +
 +
It possible to trace this divergence of styles to an even more primitive
 +
division, one that distinguishes the "additive" or the "parallel" styles
 +
from the "multiplicative" or the "serial" styles.  The issue is somewhat
 +
confused by the fact that an "additive" analysis is typically expressed
 +
in the form of a "series", in other words, a disjoint union of sets or a
 +
linear sum of their independent effects.  But it is easy enough to sort
 +
this out if one observes the more telling connection between "parallel"
 +
and "independent".  Another way to keep the right associations straight
 +
is to employ the term "sequential" in preference to the more misleading
 +
term "serial".  Whatever one calls this broad division of styles, the
 +
scope and sweep of their dimensions of variation can be delineated in
 +
the following way:
 +
 +
1.  The "additive" or "parallel" styles favor "sums of products" as
 +
    canonical forms of expression, pulling sums, unions, co-products,
 +
    and logical disjunctions to the outermost layers of analysis and
 +
    synthesis, while pushing products, intersections, concatenations,
 +
    and logical conjunctions to the innermost levels of articulation
 +
    and generation.  In propositional logic, this style leads to the
 +
    "disjunctive normal form" (DNF).
 +
 +
2.  The "multiplicative" or "serial" styles favor "products of sums"
 +
    as canonical forms of expression, pulling products, intersections,
 +
    concatenations, and logical conjunctions to the outermost layers of
 +
    analysis and synthesis, while pushing sums, unions, co-products,
 +
    and logical disjunctions to the innermost levels of articulation
 +
    and generation.  In propositional logic, this style leads to the
 +
    "conjunctive normal form" (CNF).
 +
 +
There is a curious sort of diagnostic clue, a veritable shibboleth,
 +
that often serves to reveal the dominance of one mode or the other
 +
within an individual thinker's cognitive style.  Examined on the
 +
question of what constitutes the "natural numbers", an "additive"
 +
thinker tends to start the sequence at 0, while a "multiplicative"
 +
thinker tends to regard it as beginning at 1.
 +
 +
In any style of description, grammar, or theory of a language, it is
 +
usually possible to tease out the influence of these contrasting traits,
 +
namely, the "additive" attitude versus the "mutiplicative" tendency that
 +
go to make up the particular style in question, and even to determine the
 +
dominant inclination or point of view that establishes its perspective on
 +
the target domain.
 +
 +
In each style of formal grammar, the "multiplicative" aspect is present
 +
in the sequential concatenation of signs, both in the augmented strings
 +
and in the terminal strings.  In settings where the non-terminal symbols
 +
classify types of strings, the concatenation of the non-terminal symbols
 +
signifies the cartesian product over the corresponding sets of strings.
 +
 +
In the context-free style of formal grammar, the "additive" aspect is
 +
easy enough to spot.  It is signaled by the parallel covering of many
 +
augmented strings or sentential forms by the same non-terminal symbol.
 +
Expressed in active terms, this calls for the independent rewriting
 +
of that non-terminal symbol by a number of different successors,
 +
as in the following scheme:
 +
 +
| q    :>    W_1.
 +
|
 +
| ...  ...  ...
 +
|
 +
| q    :>    W_k.
 +
 +
It is useful to examine the relationship between the grammatical covering
 +
or production relation ":>" and the logical relation of implication "=>",
 +
with one eye to what they have in common and one eye to how they differ.
 +
The production "q :> W" says that the appearance of the symbol "q" in
 +
a sentential form implies the possibility of exchanging it for "W".
 +
Although this sounds like a "possible implication", to the extent
 +
that "q implies a possible W" or that "q possibly implies W", the
 +
qualifiers "possible" and "possibly" are the critical elements in
 +
these statements, and they are crucial to the meaning of what is
 +
actually being implied.  In effect, these qualifications reverse
 +
the direction of implication, yielding "q <= W" as the best
 +
analogue for the sense of the production.
 +
 +
One way to sum this up is to say that non-terminal symbols have the
 +
significance of hypotheses.  The terminal strings form the empirical
 +
matter of a language, while the non-terminal symbols mark the patterns
 +
or the types of substrings that can be noticed in the profusion of data.
 +
If one observes a portion of a terminal string that falls into the pattern
 +
of the sentential form W, then it is an admissable hypothesis, according to
 +
the theory of the language that is constituted by the formal grammar, that
 +
this piece not only fits the type q but even comes to be generated under
 +
the auspices of the non-terminal symbol "q".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.10  The Cactus Language:  Stylistics (cont.)
 +
 +
A moment's reflection on the issue of style, giving due consideration to the
 +
received array of stylistic choices, ought to inspire at least the question:
 +
"Are these the only choices there are?"  In the present setting, there are
 +
abundant indications that other options, more differentiated varieties of
 +
description and more integrated ways of approaching individual languages,
 +
are likely to be conceivable, feasible, and even more ultimately viable.
 +
If a suitably generic style, one that incorporates the full scope of
 +
logical combinations and operations, is broadly available, then it
 +
would no longer be necessary, or even apt, to argue in universal
 +
terms about "which style is best", but more useful to investigate
 +
how we might adapt the local styles to the local requirements.
 +
The medium of a generic style would yield a viable compromise
 +
between "additive" and "multiplicative" canons, and render the
 +
choice between "parallel" and "serial" a false alternative,
 +
at least, when expressed in the globally exclusive terms
 +
that are currently most commonly adopted to pose it.
 +
 +
One set of indications comes from the study of machines, languages, and
 +
computation, especially the theories of their structures and relations.
 +
The forms of composition and decomposition that are generally known as
 +
"parallel" and "serial" are merely the extreme special cases, in variant
 +
directions of specialization, of a more generic form, usually called the
 +
"cascade" form of combination.  This is a well-known fact in the theories
 +
that deal with automata and their associated formal languages, but its
 +
implications do not seem to be widely appreciated outside these fields.
 +
In particular, it dispells the need to choose one extreme or the other,
 +
since most of the natural cases are likely to exist somewhere in between.
 +
 +
Another set of indications appears in algebra and category theory,
 +
where forms of composition and decomposition related to the cascade
 +
combination, namely, the "semi-direct product" and its special case,
 +
the "wreath product", are encountered at higher levels of generality
 +
than the cartesian products of sets or the direct products of spaces.
 +
 +
In these domains of operation, one finds it necessary to consider also
 +
the "co-product" of sets and spaces, a construction that artificially
 +
creates a disjoint union of sets, that is, a union of spaces that are
 +
being treated as independent.  It does this, in effect, by "indexing",
 +
"coloring", or "preparing" the otherwise possibly overlapping domains
 +
that are being combined.  What renders this a "chimera" or a "hybrid"
 +
form of combination is the fact that this indexing is tantamount to a
 +
cartesian product of a singleton set, namely, the conventional "index",
 +
"color", or "affix" in question, with the individual domain that is
 +
entering as a factor, a term, or a participant in the final result.
 +
 +
One of the insights that arises out of Peirce's logical work is that
 +
the set operations of complementation, intersection, and union, along
 +
with the logical operations of negation, conjunction, and disjunction
 +
that operate in isomorphic tandem with them, are not as fundamental as
 +
they first appear.  This is because all of them can be constructed from
 +
or derived from a smaller set of operations, in fact, taking the logical
 +
side of things, from either one of two "solely sufficient" operators,
 +
called "amphecks" by Peirce, "strokes" by those who re-discovered them
 +
later, and known in computer science as the NAND and the NNOR operators.
 +
For this reason, that is, by virtue of their precedence in the orders
 +
of construction and derivation, these operations have to be regarded
 +
as the simplest and the most primitive in principle, even if they are
 +
scarcely recognized as lying among the more familiar elements of logic.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.10  The Cactus Language:  Stylistics (cont.)
 +
 +
I am throwing together a wide variety of different operations into each
 +
of the bins labeled "additive" and "multiplicative", but it is easy to
 +
observe a natural organization and even some relations approaching
 +
isomorphisms among and between the members of each class.
 +
 +
The relation between logical disjunction and set-theoretic union and the
 +
relation between logical conjunction and set-theoretic intersection ought
 +
to be clear enough for the purposes of the immediately present context.
 +
In any case, all of these relations are scheduled to receive a thorough
 +
examination in a subsequent discussion (Subsection 1.3.10.13).  But the
 +
relation of a set-theoretic union to a category-theoretic co-product and
 +
the relation of a set-theoretic intersection to a syntactic concatenation
 +
deserve a closer look at this point.
 +
 +
The effect of a co-product as a "disjointed union", in other words, that
 +
creates an object tantamount to a disjoint union of sets in the resulting
 +
co-product even if some of these sets intersect non-trivially and even if
 +
some of them are identical "in reality", can be achieved in several ways.
 +
The most usual conception is that of making a "separate copy", for each
 +
part of the intended co-product, of the set that is intended to go there.
 +
Often one thinks of the set that is assigned to a particular part of the
 +
co-product as being distinguished by a particular "color", in other words,
 +
by the attachment of a distinct "index", "label", or "tag", being a marker
 +
that is inherited by and passed on to every element of the set in that part.
 +
A concrete image of this construction can be achieved by imagining that each
 +
set and each element of each set is placed in an ordered pair with the sign
 +
of its color, index, label, or tag.  One describes this as the "injection"
 +
of each set into the corresponding "part" of the co-product.
 +
 +
For example, given the sets P and Q, overlapping or not, one can define
 +
the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the
 +
copy of P into the first part of the co-product and the copy of Q into the
 +
second part of the co-product, in the following manner:
 +
 +
P_[1]  =  <P, 1>  =  {<x, 1>  :  x in P},
 +
 +
Q_[2]  =  <Q, 2>  =  {<x, 2>  :  x in Q}.
 +
 +
Using the sign "]_[" for this construction, the "sum", the "co-product",
 +
or the "disjointed union" of P and Q in that order can be represented as
 +
the ordinary disjoint union of P_[1] and Q_[2].
 +
 +
P ]_[ Q  =  P_[1] |_| Q_[2].
 +
 +
The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is
 +
just the cartesian product of sets L_1 x L_2 without the extra x's,
 +
but the relation of cartesian products to set-theoretic intersections
 +
and thus to logical conjunctions is far from being clear.  One way of
 +
seeing a type of relation is to focus on the information that is needed
 +
to specify each construction, and thus to reflect on the signs that are
 +
used to carry this information.  As a first approach to the topic of
 +
information, according to a strategy that seeks to be as elementary
 +
and as informal as possible, I introduce the following set of ideas,
 +
intended to be taken in a very provisional way.
 +
 +
A "stricture" is a specification of a certain set in a certain place,
 +
relative to a number of other sets, yet to be specified.  It is assumed
 +
that one knows enough to tell if two strictures are equivalent as pieces
 +
of information, but any more determinate indications, like names for the
 +
places that are mentioned in the stricture, or bounds on the number of
 +
places that are involved, are regarded as being extraneous impositions,
 +
outside the proper concern of the definition, no matter how convenient
 +
they are found to be for a particular discussion.  As a schematic form
 +
of illustration, a stricture can be pictured in the following shape:
 +
 +
"... x X x Q x X x ..."
 +
 +
A "strait" is the object that is specified by a stricture, in effect,
 +
a certain set in a certain place of an otherwise yet to be specified
 +
relation.  Somewhat sketchily, the strait that corresponds to the
 +
stricture just given can be pictured in the following shape:
 +
 +
... x X x Q x X x ...
 +
 +
In this picture, Q is a certain set, and X is the universe of discourse
 +
that is relevant to a given discussion.  Since a stricture does not, by
 +
itself, contain a sufficient amount of information to specify the number
 +
of sets that it intends to set in place, or even to specify the absolute
 +
location of the set that its does set in place, it appears to place an
 +
unspecified number of unspecified sets in a vague and uncertain strait.
 +
Taken out of its interpretive context, the residual information that a
 +
stricture can convey makes all of the following potentially equivalent
 +
as strictures:
 +
 +
"Q",  "XxQxX",  "XxXxQxXxX",  ...
 +
 +
With respect to what these strictures specify, this
 +
leaves all of the following equivalent as straits:
 +
 +
Q  =  XxQxX  =  XxXxQxXxX  =  ...
 +
 +
Within the framework of a particular discussion, it is customary to
 +
set a bound on the number of places and to limit the variety of sets
 +
that are regarded as being under active consideration, and it is also
 +
convenient to index the places of the indicated relations, and of their
 +
encompassing cartesian products, in some fixed way.  But the whole idea
 +
of a stricture is to specify a strait that is capable of extending through
 +
and beyond any fixed frame of discussion.  In other words, a stricture is
 +
conceived to constrain a strait at a certain point, and then to leave it
 +
literally embedded, if tacitly expressed, in a yet to be fully specified
 +
relation, one that involves an unspecified number of unspecified domains.
 +
 +
A quantity of information is a measure of constraint.  In this respect,
 +
a set of comparable strictures is ordered on account of the information
 +
that each one conveys, and a system of comparable straits is ordered in
 +
accord with the amount of information that it takes to pin each one of
 +
them down.  Strictures that are more constraining and straits that are
 +
more constrained are placed at higher levels of information than those
 +
that are less so, and entities that involve more information are said
 +
to have a greater "complexity" in comparison with those entities that
 +
involve less information, that are said to have a greater "simplicity".
 +
 +
In order to create a concrete example, let me now institute a frame of
 +
discussion where the number of places in a relation is bounded at two,
 +
and where the variety of sets under active consideration is limited to
 +
the typical subsets P and Q of a universe X.  Under these conditions,
 +
one can use the following sorts of expression as schematic strictures:
 +
 +
|  "X" ,  "P" ,  "Q" ,
 +
|
 +
| "XxX",  "XxP",  "XxQ",
 +
|
 +
| "PxX",  "PxP",  "PxQ",
 +
|
 +
| "QxX",  "QxP",  "QxQ".
 +
 +
These strictures and their corresponding straits are stratified according
 +
to their amounts of information, or their levels of constraint, as follows:
 +
 +
| High:  "PxP",  "PxQ",  "QxP",  "QxQ".
 +
|
 +
| Med:    "P" ,  "XxP",  "PxX".
 +
|
 +
| Med:    "Q" ,  "XxQ",  "QxX".
 +
|
 +
| Low:    "X" ,  "XxX".
 +
 +
Within this framework, the more complex strait PxQ can be expressed
 +
in terms of the simpler straits, PxX and XxQ.  More specifically,
 +
it lends itself to being analyzed as their intersection, in the
 +
following way:
 +
 +
PxQ  =  PxX |^| XxQ.
 +
 +
>From here it is easy to see the relation of concatenation, by virtue of
 +
these types of intersection, to the logical conjunction of propositions.
 +
The cartesian product PxQ is described by a conjunction of propositions,
 +
namely, "P_<1> and Q_<2>", subject to the following interpretation:
 +
 +
1.  "P_<1>" asserts that there is an element from
 +
    the set P in the first place of the product.
 +
 +
2.  "Q_<2>" asserts that there is an element from
 +
    the set Q in the second place of the product.
 +
 +
The integration of these two pieces of information can be taken
 +
in that measure to specify a yet to be fully determined relation.
 +
 +
In a corresponding fashion at the level of the elements,
 +
the ordered pair <p, q> is described by a conjunction
 +
of propositions, namely, "p_<1> and q_<2>", subject
 +
to the following interpretation:
 +
 +
1.  "p_<1>" says that p is in the first place
 +
    of the product element under construction.
 +
 +
2.  "q_<2>" says that q is in the second place
 +
    of the product element under construction.
 +
 +
Notice that, in construing the cartesian product of the sets P and Q or the
 +
concatenation of the languages L_1 and L_2 in this way, one shifts the level
 +
of the active construction from the tupling of the elements in P and Q or the
 +
concatenation of the strings that are internal to the languages L_1 and L_2 to
 +
the concatenation of the external signs that it takes to indicate these sets or
 +
these languages, in other words, passing to a conjunction of indexed propositions,
 +
"P_<1> and Q_<2>", or to a conjunction of assertions, "L_1_<1> and L_2_<2>", that
 +
marks the sets or the languages in question for insertion in the indicated places
 +
of a product set or a product language, respectively.  In effect, the subscripting
 +
by the indices "<1>" and "<2>" can be recognized as a special case of concatenation,
 +
albeit through the posting of editorial remarks from an external "mark-up" language.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.10  The Cactus Language:  Stylistics (cont.)
 +
 +
In order to systematize the relations that strictures and straits placed
 +
at higher levels of complexity, constraint, information, and organization
 +
have with those that are placed at the associated lower levels, I introduce
 +
the following pair of definitions:
 +
 +
The j^th "excerpt" of a stricture of the form "S_1 x ... x S_k", regarded
 +
within a frame of discussion where the number of places is limited to k,
 +
is the stricture of the form "X x ... x S_j x ... x X".  In the proper
 +
context, this can be written more succinctly as the stricture "S_j_<j>",
 +
an assertion that places the j^th set in the j^th place of the product.
 +
 +
The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained
 +
to a frame of discussion where the number of places is restricted to k,
 +
is the strait of the form X x ... x S_j x ... x X.  In the appropriate
 +
context, this can be denoted more succinctly by the stricture "S_j_<j>",
 +
an assertion that places the j^th set in the j^th place of the product.
 +
 +
In these terms, a stricture of the form "S_1 x ... x S_k"
 +
can be expressed in terms of simpler strictures, to wit,
 +
as a conjunction of its k excerpts:
 +
 +
"S_1 x ... x S_k"  =  "S_1_<1>" &  ...  & "S_k_<k>".
 +
 +
In a similar vein, a strait of the form S_1 x ... x S_k
 +
can be expressed in terms of simpler straits, namely,
 +
as an intersection of its k extracts:
 +
 +
S_1 x ... x S_k    =    S_1_<1> |^| ... |^| S_k_<k>.
 +
 +
There is a measure of ambiguity that remains in this formulation,
 +
but it is the best that I can do in the present informal context.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.11  The Cactus Language:  Mechanics
 +
 +
| We are only now beginning to see how this works.  Clearly one of the
 +
| mechanisms for picking a reality is the sociohistorical sense of what
 +
| is important -- which research program, with all its particularity of
 +
| knowledge, seems most fundamental, most productive, most penetrating.
 +
| The very judgments which make us push narrowly forward simultaneously
 +
| make us forget how little we know.  And when we look back at history,
 +
| where the lesson is plain to find, we often fail to imagine ourselves
 +
| in a parallel situation.  We ascribe the differences in world view
 +
| to error, rather than to unexamined but consistent and internally
 +
| justified choice.
 +
|
 +
| Herbert J. Bernstein, "Idols", page 38.
 +
|
 +
| Herbert J. Bernstein,
 +
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
 +
|
 +
| Marcus G. Raskin & Herbert J. Bernstein,
 +
|'New Ways of Knowing:  The Sciences, Society, & Reconstructive Knowledge',
 +
| Rowman & Littlefield, Totowa, NJ, 1987.
 +
 +
In this Subsection, I discuss the "mechanics" of parsing the
 +
cactus language into the corresponding class of computational
 +
data structures.  This provides each sentence of the language
 +
with a translation into a computational form that articulates
 +
its syntactic structure and prepares it for automated modes of
 +
processing and evaluation.  For this purpose, it is necessary
 +
to describe the target data structures at a fairly high level
 +
of abstraction only, ignoring the details of address pointers
 +
and record structures and leaving the more operational aspects
 +
of implementation to the imagination of prospective programmers.
 +
In this way, I can put off to another stage of elaboration and
 +
refinement the description of the program that constructs these
 +
pointers and operates on these graph-theoretic data structures.
 +
 +
The structure of a "painted cactus", insofar as it presents itself
 +
to the visual imagination, can be described as follows.  The overall
 +
structure, as given by its underlying graph, falls within the species
 +
of graph that is commonly known as a "rooted cactus", and the only novel
 +
feature that it adds to this is that each of its nodes can be "painted"
 +
with a finite sequence of "paints", chosen from a "palette" that is given
 +
by the parametric set {" "} |_| !P!  =  {m_1} |_| {p_1, ..., p_k}.
 +
 +
It is conceivable, from a purely graph-theoretical point of view, to have
 +
a class of cacti that are painted but not rooted, and so it is frequently
 +
necessary, for the sake of precision, to more exactly pinpoint the target
 +
species of graphical structure as a "painted and rooted cactus" (PARC).
 +
 +
A painted cactus, as a rooted graph, has a distinguished "node" that is
 +
called its "root".  By starting from the root and working recursively,
 +
the rest of its structure can be described in the following fashion.
 +
 +
Each "node" of a PARC consists of a graphical "point" or "vertex" plus
 +
a finite sequence of "attachments", described in relative terms as the
 +
attachments "at" or "to" that node.  An empty sequence of attachments
 +
defines the "empty node".  Otherwise, each attachment is one of three
 +
kinds:  a blank, a paint, or a type of PARC that is called a "lobe".
 +
 +
Each "lobe" of a PARC consists of a directed graphical "cycle" plus a
 +
finite sequence of "accoutrements", described in relative terms as the
 +
accoutrements "of" or "on" that lobe.  Recalling the circumstance that
 +
every lobe that comes under consideration comes already attached to a
 +
particular node, exactly one vertex of the corresponding cycle is the
 +
vertex that comes from that very node.  The remaining vertices of the
 +
cycle have their definitions filled out according to the accoutrements
 +
of the lobe in question.  An empty sequence of accoutrements is taken
 +
to be tantamount to a sequence that contains a single empty node as its
 +
unique accoutrement, and either one of these ways of approaching it can
 +
be regarded as defining a graphical structure that is called a "needle"
 +
or a "terminal edge".  Otherwise, each accoutrement of a lobe is itself
 +
an arbitrary PARC.
 +
 +
Although this definition of a lobe in terms of its intrinsic structural
 +
components is logically sufficient, it is also useful to characterize the
 +
structure of a lobe in comparative terms, that is, to view the structure
 +
that typifies a lobe in relation to the structures of other PARC's and to
 +
mark the inclusion of this special type within the general run of PARC's.
 +
This approach to the question of types results in a form of description
 +
that appears to be a bit more analytic, at least, in mnemonic or prima
 +
facie terms, if not ultimately more revealing.  Working in this vein,
 +
a "lobe" can be characterized as a special type of PARC that is called
 +
an "unpainted root plant" (UR-plant).
 +
 +
An "UR-plant" is a PARC of a simpler sort, at least, with respect to the
 +
recursive ordering of structures that is being followed here.  As a type,
 +
it is defined by the presence of two properties, that of being "planted"
 +
and that of having an "unpainted root".  These are defined as follows:
 +
 +
1.  A PARC is "planted" if its list of attachments has just one PARC.
 +
 +
2.  A PARC is "UR" if its list of attachments has no blanks or paints.
 +
 +
In short, an UR-planted PARC has a single PARC as its only attachment,
 +
and since this attachment is prevented from being a blank or a paint,
 +
the single attachment at its root has to be another sort of structure,
 +
that which we call a "lobe".
 +
 +
To express the description of a PARC in terms of its nodes, each node
 +
can be specified in the fashion of a functional expression, letting a
 +
citation of the generic function name "Node" be followed by a list of
 +
arguments that enumerates the attachments of the node in question, and
 +
letting a citation of the generic function name "Lobe" be followed by a
 +
list of arguments that details the accoutrements of the lobe in question.
 +
Thus, one can write expressions of the following forms:
 +
 +
1.  Node^0        =  Node()
 +
 +
                  =  a node with no attachments.
 +
 +
    Node^k_j  C_j  =  Node(C_1, ..., C_k)
 +
 +
                  =  a node with the attachments C_1, ..., C_k.
 +
 +
2.  Lobe^0        =  Lobe()
 +
 +
                  =  a lobe with no accoutrements.
 +
 +
    Lobe^k_j  C_j  =  Lobe(C_1, ..., C_k)
 +
 +
                  =  a lobe with the accoutrements C_1, ..., C_k.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.11  The Cactus Language:  Mechanics (cont.)
 +
 +
Working from a structural description of the cactus language,
 +
or any suitable formal grammar for !C!(!P!), it is possible to
 +
give a recursive definition of the function called "Parse" that
 +
maps each sentence in PARCE(!P!) to the corresponding graph in
 +
PARC(!P!).  One way to do this proceeds as follows:
 +
 +
1.  The parse of the concatenation Conc^k of the k sentences S_j,
 +
    for j = 1 to k, is defined recursively as follows:
 +
 +
    a.  Parse(Conc^0)        =  Node^0.
 +
 +
    b.  For k > 0,
 +
 +
        Parse(Conc^k_j S_j)  =  Node^k_j Parse(S_j).
 +
 +
2.  The parse of the surcatenation Surc^k of the k sentences S_j,
 +
    for j = 1 to k, is defined recursively as follows:
 +
 +
    a.  Parse(Surc^0)        =  Lobe^0.
 +
 +
    b.  For k > 0,
 +
 +
        Parse(Surc^k_j S_j)  =  Lobe^k_j Parse(S_j).
 +
 +
For ease of reference, Table 12 summarizes the mechanics of these parsing rules.
 +
 +
Table 12.  Algorithmic Translation Rules
 +
o------------------------o---------o------------------------o
 +
|                        |  Parse  |                        |
 +
| Sentence in PARCE      |  -->  | Graph in PARC          |
 +
o------------------------o---------o------------------------o
 +
|                        |        |                        |
 +
| Conc^0                |  -->  | Node^0                |
 +
|                        |        |                        |
 +
| Conc^k_j  S_j          |  -->  | Node^k_j  Parse(S_j)  |
 +
|                        |        |                        |
 +
| Surc^0                |  -->  | Lobe^0                |
 +
|                        |        |                        |
 +
| Surc^k_j  S_j          |  -->  | Lobe^k_j  Parse(S_j)  |
 +
|                        |        |                        |
 +
o------------------------o---------o------------------------o
 +
 +
A "substructure" of a PARC is defined recursively as follows.  Starting
 +
at the root node of the cactus C, any attachment is a substructure of C.
 +
If a substructure is a blank or a paint, then it constitutes a minimal
 +
substructure, meaning that no further substructures of C arise from it.
 +
If a substructure is a lobe, then each one of its accoutrements is also
 +
a substructure of C, and has to be examined for further substructures.
 +
 +
The concept of substructure can be used to define varieties of deletion
 +
and erasure operations that respect the structure of the abstract graph.
 +
For the purposes of this depiction, a blank symbol " " is treated as
 +
a "primer", in other words, as a "clear paint", a "neutral tint", or
 +
a "white wash".  In effect, one is letting m_1 = p_0.  In this frame
 +
of discussion, it is useful to make the following distinction:
 +
 +
1.  To "delete" a substructure is to replace it with an empty node,
 +
    in effect, to reduce the whole structure to a trivial point.
 +
 +
2.  To "erase" a substructure is to replace it with a blank symbol,
 +
    in effect, to paint it out of the picture or to overwrite it.
 +
 +
A "bare" PARC, loosely referred to as a "bare cactus", is a PARC on the
 +
empty palette !P! = {}.  In other veins, a bare cactus can be described
 +
in several different ways, depending on how the form arises in practice.
 +
 +
1.  Leaning on the definition of a bare PARCE, a bare PARC can be
 +
    described as the kind of a parse graph that results from parsing
 +
    a bare cactus expression, in other words, as the kind of a graph
 +
    that issues from the requirements of processing a sentence of
 +
    the bare cactus language !C!^0 = PARCE^0.
 +
 +
2.  To express it more in its own terms, a bare PARC can be defined
 +
    by tracing the recursive definition of a generic PARC, but then
 +
    by detaching an independent form of description from the source
 +
    of that analogy.  The method is sufficiently sketched as follows:
 +
 +
    a.  A "bare PARC" is a PARC whose attachments
 +
        are limited to blanks and "bare lobes".
 +
 +
    b.  A "bare lobe" is a lobe whose accoutrements
 +
        are limited to bare PARC's.
 +
 +
3.  In practice, a bare cactus is usually encountered in the process
 +
    of analyzing or handling an arbitrary PARC, the circumstances of
 +
    which frequently call for deleting or erasing all of its paints.
 +
    In particular, this generally makes it easier to observe the
 +
    various properties of its underlying graphical structure.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.12  The Cactus Language:  Semantics
 +
 +
| Alas, and yet what 'are' you, my written and painted thoughts!
 +
| It is not long ago that you were still so many-coloured,
 +
| young and malicious, so full of thorns and hidden
 +
| spices you made me sneeze and laugh -- and now?
 +
| You have already taken off your novelty and
 +
| some of you, I fear, are on the point of
 +
| becoming truths:  they already look so
 +
| immortal, so pathetically righteous,
 +
| so boring!
 +
|
 +
| Friedrich Nietzsche, 'Beyond Good and Evil', Paragraph 296.
 +
|
 +
| Friedrich Nietzsche,
 +
|'Beyond Good and Evil:  Prelude to a Philosophy of the Future',
 +
| trans. by R.J. Hollingdale, intro. by Michael Tanner,
 +
| Penguin Books, London, UK, 1973, 1990.
 +
 +
In this Subsection, I describe a particular semantics for the
 +
painted cactus language, telling what meanings I aim to attach
 +
to its bare syntactic forms.  This supplies an "interpretation"
 +
for this parametric family of formal languages, but it is good
 +
to remember that it forms just one of many such interpretations
 +
that are conceivable and even viable.  In deed, the distinction
 +
between the object domain and the sign domain can be observed in
 +
the fact that many languages can be deployed to depict the same
 +
set of objects and that any language worth its salt is bound to
 +
to give rise to many different forms of interpretive saliency.
 +
 +
In formal settings, it is common to speak of "interpretation" as if it
 +
created a direct connection between the signs of a formal language and
 +
the objects of the intended domain, in other words, as if it determined
 +
the denotative component of a sign relation.  But a closer attention to
 +
what goes on reveals that the process of interpretation is more indirect,
 +
that what it does is to provide each sign of a prospectively meaningful
 +
source language with a translation into an already established target
 +
language, where "already established" means that its relationship to
 +
pragmatic objects is taken for granted at the moment in question.
 +
 +
With this in mind, it is clear that interpretation is an affair of signs
 +
that at best respects the objects of all of the signs that enter into it,
 +
and so it is the connotative aspect of semiotics that is at stake here.
 +
There is nothing wrong with my saying that I interpret a sentence of a
 +
formal language as a sign that refers to a function or to a proposition,
 +
so long as you understand that this reference is likely to be achieved
 +
by way of more familiar and perhaps less formal signs that you already
 +
take to denote those objects.
 +
 +
On entering a context where a logical interpretation is intended for the
 +
sentences of a formal language there are a few conventions that make it
 +
easier to make the translation from abstract syntactic forms to their
 +
intended semantic senses.  Although these conventions are expressed in
 +
unnecessarily colorful terms, from a purely abstract point of view, they
 +
do provide a useful array of connotations that help to negotiate what is
 +
otherwise a difficult transition.  This terminology is introduced as the
 +
need for it arises in the process of interpreting the cactus language.
 +
 +
The task of this Subsection is to specify a "semantic function" for
 +
the sentences of the cactus language !L! = !C!(!P!), in other words,
 +
to define a mapping that "interprets" each sentence of !C!(!P!) as
 +
a sentence that says something, as a sentence that bears a meaning,
 +
in short, as a sentence that denotes a proposition, and thus as a
 +
sign of an indicator function.  When the syntactic sentences of a
 +
formal language are given a referent significance in logical terms,
 +
for example, as denoting propositions or indicator functions, then
 +
each form of syntactic combination takes on a corresponding form
 +
of logical significance.
 +
 +
By way of providing a logical interpretation for the cactus language,
 +
I introduce a family of operators on indicator functions that are
 +
called "propositional connectives", and I distinguish these from
 +
the associated family of syntactic combinations that are called
 +
"sentential connectives", where the relationship between these
 +
two realms of connection is exactly that between objects and
 +
their signs.  A propositional connective, as an entity of a
 +
well-defined functional and operational type, can be treated
 +
in every way as a logical or a mathematical object, and thus
 +
as the type of object that can be denoted by the corresponding
 +
form of syntactic entity, namely, the sentential connective that
 +
is appropriate to the case in question.
 +
 +
There are two basic types of connectives, called the "blank connectives"
 +
and the "bound connectives", respectively, with one connective of each
 +
type for each natural number k = 0, 1, 2, 3, ... .
 +
 +
1.  The "blank connective" of k places is signified by the
 +
    concatenation of the k sentences that fill those places.
 +
 +
    For the special case of k = 0, the "blank connective" is taken to
 +
    be an empty string or a blank symbol -- it does not matter which,
 +
    since both are assigned the same denotation among propositions.
 +
    For the generic case of k > 0, the "blank connective" takes
 +
    the form "S_1 · ... · S_k".  In the type of data that is
 +
    called a "text", the raised dots "·" are usually omitted,
 +
    supplanted by whatever number of spaces and line breaks
 +
    serve to improve the readability of the resulting text.
 +
 +
2.  The "bound connective" of k places is signified by the
 +
    surcatenation of the k sentences that fill those places.
 +
 +
    For the special case of k = 0, the "bound connective" is taken to
 +
    be an expression of the form "-()-", "-( )-", "-(  )-", and so on,
 +
    with any number of blank symbols between the parentheses, all of
 +
    which are assigned the same logical denotation among propositions.
 +
    For the generic case of k > 0, the "bound connective" takes the
 +
    form "-(S_1, ..., S_k)-".
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.12  The Cactus Language:  Semantics (cont.)
 +
 +
At this point, there are actually two different "dialects", "scripts",
 +
or "modes" of presentation for the cactus language that need to be
 +
interpreted, in other words, that need to have a semantic function
 +
defined on their domains.
 +
 +
a.  There is the literal formal language of strings in PARCE(!P!),
 +
    the "painted and rooted cactus expressions" that constitute
 +
    the langauge !L! = !C!(!P!) c !A!* = (!M! |_| !P!)*.
 +
 +
b.  There is the figurative formal language of graphs in PARC(!P!),
 +
    the "painted and rooted cacti" themselves, a parametric family
 +
    of graphs or a species of computational data structures that
 +
    is graphically analogous to the language of literal strings.
 +
 +
Of course, these two modalities of formal language, like written and
 +
spoken natural languages, are meant to have compatible interpretations,
 +
and so it is usually sufficient to give just the meanings of either one.
 +
All that remains is to provide a "codomain" or a "target space" for the
 +
intended semantic function, in other words, to supply a suitable range
 +
of logical meanings for the memberships of these languages to map into.
 +
Out of the many interpretations that are formally possible to arrange,
 +
one way of doing this proceeds by making the following definitions:
 +
 +
1.  The "conjunction" Conj^J_j Q_j of a set of propositions, {Q_j : j in J},
 +
    is a proposition that is true if and only if each one of the Q_j is true.
 +
 +
    Conj^J_j Q_j is true  <=>  Q_j is true for every j in J.
 +
 +
2.  The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J},
 +
    is a proposition that is true if and only if just one of the Q_j is untrue.
 +
 +
    Surj^J_j Q_j is true  <=>  Q_j is untrue for unique j in J.
 +
 +
If the number of propositions that are being joined together is finite,
 +
then the conjunction and the surjunction can be represented by means of
 +
sentential connectives, incorporating the sentences that represent these
 +
propositions into finite strings of symbols.
 +
 +
If J is finite, for instance, if J constitutes the interval j = 1 to k,
 +
and if each proposition Q_j is represented by a sentence S_j, then the
 +
following strategies of expression are open:
 +
 +
1.  The conjunction Conj^J_j Q_j can be represented by a sentence that
 +
    is constructed by concatenating the S_j in the following fashion:
 +
 +
    Conj^J_j Q_j  <-<  S_1 S_2 ... S_k.
 +
 +
2.  The surjunction Surj^J_j Q_j can be represented by a sentence that
 +
    is constructed by surcatenating the S_j in the following fashion:
 +
 +
    Surj^J_j Q_j  <-<  -(S_1, S_2, ..., S_k)-.
 +
 +
If one opts for a mode of interpretation that moves more directly from
 +
the parse graph of a sentence to the potential logical meaning of both
 +
the PARC and the PARCE, then the following specifications are in order:
 +
 +
A cactus rooted at a particular node is taken to represent what that
 +
node denotes, its logical denotation or its logical interpretation.
 +
 +
1.  The logical denotation of a node is the logical conjunction of that node's
 +
    "arguments", which are defined as the logical denotations of that node's
 +
    attachments.  The logical denotation of either a blank symbol or an empty
 +
    node is the boolean value %1% = "true".  The logical denotation of the
 +
    paint p_j is the proposition P_j, a proposition that is regarded as
 +
    "primitive", at least, with respect to the level of analysis that
 +
    is represented in the current instance of !C!(!P!).
 +
 +
2.  The logical denotation of a lobe is the logical surjunction of that lobe's
 +
    "arguments", which are defined as the logical denotations of that lobe's
 +
    accoutrements.  As a corollary, the logical denotation of the parse graph
 +
    of "-()-", otherwise called a "needle", is the boolean value %0% = "false".
 +
 +
If one takes the point of view that PARC's and PARCE's amount to a
 +
pair of intertranslatable languages for the same domain of objects,
 +
then the "spiny bracket" notation, as in "-[C_j]-" or "-[S_j]-",
 +
can be used on either domain of signs to indicate the logical
 +
denotation of a cactus C_j or the logical denotation of
 +
a sentence S_j, respectively.
 +
 +
Tables 13.1 and 13.2 summarize the relations that serve to connect the
 +
formal language of sentences with the logical language of propositions.
 +
Between these two realms of expression there is a family of graphical
 +
data structures that arise in parsing the sentences and that serve to
 +
facilitate the performance of computations on the indicator functions.
 +
The graphical language supplies an intermediate form of representation
 +
between the formal sentences and the indicator functions, and the form
 +
of mediation that it provides is very useful in rendering the possible
 +
connections between the other two languages conceivable in fact, not to
 +
mention in carrying out the necessary translations on a practical basis.
 +
These Tables include this intermediate domain in their Central Columns.
 +
Between their First and Middle Columns they illustrate the mechanics of
 +
parsing the abstract sentences of the cactus language into the graphical
 +
data structures of the corresponding species.  Between their Middle and
 +
Final Columns they summarize the semantics of interpreting the graphical
 +
forms of representation for the purposes of reasoning with propositions.
 +
 +
Table 13.1  Semantic Translations:  Functional Form
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  | Par |                  | Den |                  |
 +
| Sentence          | --> | Graph            | --> | Proposition      |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  |    |                  |    |                  |
 +
| S_j              | --> | C_j              | --> | Q_j              |
 +
|                  |    |                  |    |                  |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  |    |                  |    |                  |
 +
| Conc^0            | --> | Node^0            | --> | %1%              |
 +
|                  |    |                  |    |                  |
 +
| Conc^k_j  S_j    | --> | Node^k_j  C_j    | --> | Conj^k_j  Q_j    |
 +
|                  |    |                  |    |                  |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  |    |                  |    |                  |
 +
| Surc^0            | --> | Lobe^0            | --> | %0%              |
 +
|                  |    |                  |    |                  |
 +
| Surc^k_j  S_j    | --> | Lobe^k_j  C_j    | --> | Surj^k_j  Q_j    |
 +
|                  |    |                  |    |                  |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
 +
Table 13.2  Semantic Translations:  Equational Form
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  | Par |                  | Den |                  |
 +
| -[Sentence]-      |  =  | -[Graph]-        |  =  | Proposition      |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  |    |                  |    |                  |
 +
| -[S_j]-          |  =  | -[C_j]-          |  =  | Q_j              |
 +
|                  |    |                  |    |                  |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  |    |                  |    |                  |
 +
| -[Conc^0]-        |  =  | -[Node^0]-        |  =  | %1%              |
 +
|                  |    |                  |    |                  |
 +
| -[Conc^k_j  S_j]- |  =  | -[Node^k_j  C_j]- |  =  | Conj^k_j  Q_j    |
 +
|                  |    |                  |    |                  |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
|                  |    |                  |    |                  |
 +
| -[Surc^0]-        |  =  | -[Lobe^0]-        |  =  | %0%              |
 +
|                  |    |                  |    |                  |
 +
| -[Surc^k_j  S_j]- |  =  | -[Lobe^k_j  C_j]- |  =  | Surj^k_j  Q_j    |
 +
|                  |    |                  |    |                  |
 +
o-------------------o-----o-------------------o-----o-------------------o
 +
 +
Aside from their common topic, the two Tables present slightly different
 +
ways of conceptualizing the operations that go to establish their maps.
 +
Table 13.1 records the functional associations that connect each domain
 +
with the next, taking the triplings of a sentence S_j, a cactus C_j, and
 +
a proposition Q_j as basic data, and fixing the rest by recursion on these.
 +
Table 13.2 records these associations in the form of equations, treating
 +
sentences and graphs as alternative kinds of signs, and generalizing the
 +
spiny bracket operator to indicate the proposition that either denotes.
 +
It should be clear at this point that either scheme of translation puts
 +
the sentences, the graphs, and the propositions that it associates with
 +
each other roughly in the roles of the signs, the interpretants, and the
 +
objects, respectively, whose triples define an appropriate sign relation.
 +
Indeed, the "roughly" can be made "exactly" as soon as the domains of
 +
a suitable sign relation are specified precisely.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.12  The Cactus Language:  Semantics (cont.)
 +
 +
A good way to illustrate the action of the conjunction and surjunction
 +
operators is to demonstate how they can be used to construct all of the
 +
boolean functions on k variables, just now, let us say, for k = 0, 1, 2.
 +
 +
A boolean function on 0 variables is just a boolean constant F^0 in the
 +
boolean domain %B% = {%0%, %1%}.  Table 14 shows several different ways
 +
of referring to these elements, just for the sake of consistency using
 +
the same format that will be used in subsequent Tables, no matter how
 +
degenerate it tends to appears in the immediate case:
 +
 +
Column 1 lists each boolean element or boolean function under its
 +
ordinary constant name or under a succinct nickname, respectively.
 +
 +
Column 2 lists each boolean function in a style of function name "F^i_j"
 +
that is constructed as follows:  The superscript "i" gives the dimension
 +
of the functional domain, that is, the number of its functional variables,
 +
and the subscript "j" is a binary string that recapitulates the functional
 +
values, using the obvious translation of boolean values into binary values.
 +
 +
Column 3 lists the functional values for each boolean function, or possibly
 +
a boolean element appearing in the guise of a function, for each combination
 +
of its domain values.
 +
 +
Column 4 shows the usual expressions of these elements in the cactus language,
 +
conforming to the practice of omitting the strike-throughs in display formats.
 +
Here I illustrate also the useful convention of sending the expression "(())"
 +
as a visible stand-in for the expression of a constantly "true" truth value,
 +
one that would otherwise be represented by a blank expression, and tend to
 +
elude our giving it much notice in the context of more demonstrative texts.
 +
 +
Table 14.  Boolean Functions on Zero Variables
 +
o----------o----------o-------------------------------------------o----------o
 +
| Constant | Function |                    F()                    | Function |
 +
o----------o----------o-------------------------------------------o----------o
 +
|          |          |                                          |          |
 +
| %0%      | F^0_0    |                    %0%                    |    ()    |
 +
|          |          |                                          |          |
 +
| %1%      | F^0_1    |                    %1%                    |  (())  |
 +
|          |          |                                          |          |
 +
o----------o----------o-------------------------------------------o----------o
 +
 +
Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%,
 +
of which there are precisely four.  Here, Column 1 codes the contents of
 +
Column 2 in a more concise form, compressing the lists of boolean values,
 +
recorded as bits in the subscript string, into their decimal equivalents.
 +
Naturally, the boolean constants reprise themselves in this new setting
 +
as constant functions on one variable.  Thus, one has the synonymous
 +
expressions for constant functions that are expressed in the next
 +
two chains of equations:
 +
 +
| F^1_0  =  F^1_00  =  %0% : %B% -> %B%
 +
|
 +
| F^1_3  =  F^1_11  =  %1% : %B% -> %B%
 +
 +
As for the rest, the other two functions are easily recognized as corresponding
 +
to the one-place logical connectives, or the monadic operators on %B%.  Thus,
 +
the function F^1_1  =  F^1_01 is recognizable as the negation operation, and
 +
the function F^1_2  =  F^1_10 is obviously the identity operation.
 +
 +
Table 15.  Boolean Functions on One Variable
 +
o----------o----------o-------------------------------------------o----------o
 +
| Function | Function |                  F(x)                    | Function |
 +
o----------o----------o---------------------o---------------------o----------o
 +
|          |          |      F(%0%)        |      F(%1%)        |          |
 +
o----------o----------o---------------------o---------------------o----------o
 +
|          |          |                    |                    |          |
 +
| F^1_0    | F^1_00  |        %0%        |        %0%        |  ( )    |
 +
|          |          |                    |                    |          |
 +
| F^1_1    | F^1_01  |        %0%        |        %1%        |  (x)    |
 +
|          |          |                    |                    |          |
 +
| F^1_2    | F^1_10  |        %1%        |        %0%        |    x    |
 +
|          |          |                    |                    |          |
 +
| F^1_3    | F^1_11  |        %1%        |        %1%        |  (( ))  |
 +
|          |          |                    |                    |          |
 +
o----------o----------o---------------------o---------------------o----------o
 +
 +
Table 16 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%,
 +
of which there are precisely sixteen in number.  As before, all of the boolean
 +
functions of fewer variables are subsumed in this Table, though under a set of
 +
alternative names and possibly different interpretations.  Just to acknowledge
 +
a few of the more notable pseudonyms:
 +
 +
The constant function %0% : %B%^2 -> %B% appears under the name of F^2_00.
 +
 +
The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15.
 +
 +
The negation and identity of the first variable are F^2_03 and F^2_12, resp.
 +
 +
The negation and identity of the other variable are F^2_05 and F^2_10, resp.
 +
 +
The logical conjunction is given by the function F^2_08 (x, y)  =  x · y.
 +
 +
The logical disjunction is given by the function F^2_14 (x, y)  =  ((x)(y)).
 +
 +
Functions expressing the "conditionals", "implications",
 +
or "if-then" statements are given in the following ways:
 +
 +
[x => y]  =  F^2_11 (x, y)  =  (x (y))  =  [not x without y].
 +
 +
[x <= y]  =  F^2_13 (x, y)  =  ((x) y)  =  [not y without x].
 +
 +
The function that corresponds to the "biconditional",
 +
the "equivalence", or the "if and only" statement is
 +
exhibited in the following fashion:
 +
 +
[x <=> y]  =  [x = y]  =  F^2_09 (x, y)  =  ((x , y)).
 +
 +
Finally, there is a boolean function that is logically associated with
 +
the "exclusive disjunction", "inequivalence", or "not equals" statement,
 +
algebraically associated with the "binary sum" or "bitsum" operation,
 +
and geometrically associated with the "symmetric difference" of sets.
 +
This function is given by:
 +
 +
[x =/= y]  =  [x + y]  =  F^2_06 (x, y)  =  (x , y).
 +
 +
Table 16.  Boolean Functions on Two Variables
 +
o----------o----------o-------------------------------------------o----------o
 +
| Function | Function |                  F(x, y)                  | Function |
 +
o----------o----------o----------o----------o----------o----------o----------o
 +
|          |          | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% |          |
 +
o----------o----------o----------o----------o----------o----------o----------o
 +
|          |          |          |          |          |          |          |
 +
| F^2_00  | F^2_0000 |  %0%    |  %0%    |  %0%    |  %0%    |    ()    |
 +
|          |          |          |          |          |          |          |
 +
| F^2_01  | F^2_0001 |  %0%    |  %0%    |  %0%    |  %1%    |  (x)(y)  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_02  | F^2_0010 |  %0%    |  %0%    |  %1%    |  %0%    |  (x) y  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_03  | F^2_0011 |  %0%    |  %0%    |  %1%    |  %1%    |  (x)    |
 +
|          |          |          |          |          |          |          |
 +
| F^2_04  | F^2_0100 |  %0%    |  %1%    |  %0%    |  %0%    |  x (y)  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_05  | F^2_0101 |  %0%    |  %1%    |  %0%    |  %1%    |    (y)  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_06  | F^2_0110 |  %0%    |  %1%    |  %1%    |  %0%    |  (x, y)  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_07  | F^2_0111 |  %0%    |  %1%    |  %1%    |  %1%    |  (x  y)  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_08  | F^2_1000 |  %1%    |  %0%    |  %0%    |  %0%    |  x  y  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_09  | F^2_1001 |  %1%    |  %0%    |  %0%    |  %1%    | ((x, y)) |
 +
|          |          |          |          |          |          |          |
 +
| F^2_10  | F^2_1010 |  %1%    |  %0%    |  %1%    |  %0%    |      y  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_11  | F^2_1011 |  %1%    |  %0%    |  %1%    |  %1%    |  (x (y)) |
 +
|          |          |          |          |          |          |          |
 +
| F^2_12  | F^2_1100 |  %1%    |  %1%    |  %0%    |  %0%    |  x      |
 +
|          |          |          |          |          |          |          |
 +
| F^2_13  | F^2_1101 |  %1%    |  %1%    |  %0%    |  %1%    | ((x) y)  |
 +
|          |          |          |          |          |          |          |
 +
| F^2_14  | F^2_1110 |  %1%    |  %1%    |  %1%    |  %0%    | ((x)(y)) |
 +
|          |          |          |          |          |          |          |
 +
| F^2_15  | F^2_1111 |  %1%    |  %1%    |  %1%    |  %1%    |  (())  |
 +
|          |          |          |          |          |          |          |
 +
o----------o----------o----------o----------o----------o----------o----------o
 +
 +
Let me now address one last question that may have occurred to some.
 +
What has happened, in this suggested scheme of functional reasoning,
 +
to the distinction that is quite pointedly made by careful logicians
 +
between (1) the connectives called "conditionals" and symbolized by
 +
the signs "->" and "<-", and (2) the assertions called "implications"
 +
and symbolized by the signs "=>" and "<=", and, in a related question:
 +
What has happened to the distinction that is equally insistently made
 +
between (3) the connective called the "biconditional" and signified by
 +
the sign "<->" and (4) the assertion that is called an "equivalence"
 +
and signified by the sign "<=>"?  My answer is this:  For my part,
 +
I am deliberately avoiding making these distinctions at the level
 +
of syntax, preferring to treat them instead as distinctions in
 +
the use of boolean functions, turning on whether the function
 +
is mentioned directly and used to compute values on arguments,
 +
or whether its inverse is being invoked to indicate the fibers
 +
of truth or untruth under the propositional function in question.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
In this Subsection, I finally bring together many of what may
 +
have appeared to be wholly independent threads of development,
 +
in the hope of paying off a percentage of my promissory notes,
 +
even if a goodly number my creditors have no doubt long since
 +
forgotten, if not exactly forgiven the debentures in question.
 +
 +
For ease of reference, I repeat here a couple of the
 +
definitions that are needed again in this discussion.
 +
 +
| A "boolean connection" of degree k, also known as a "boolean function"
 +
| on k variables, is a map of the form F : %B%^k -> %B%.  In other words,
 +
| a boolean connection of degree k is a proposition about things in the
 +
| universe of discourse X = %B%^k.
 +
|
 +
| An "imagination" of degree k on X is a k-tuple of propositions
 +
| about things in the universe X.  By way of displaying the kinds
 +
| of notation that are used to express this idea, the imagination
 +
| #f# = <f_1, ..., f_k> is can be given as a sequence of indicator
 +
| functions f_j : X -> %B%, for j = 1 to k.  All of these features
 +
| of the typical imagination #f# can be summed up in either one of
 +
| two ways:  either in the form of a membership statement, stating
 +
| words to the effect that #f# belongs to the space (X -> %B%)^k,
 +
| or in the form of the type declaration that #f# : (X -> %B%)^k,
 +
| though perhaps the latter specification is slightly more precise
 +
| than the former.
 +
 +
The definition of the "stretch" operation and the uses of the
 +
various brands of denotational operators can be reviewed here:
 +
 +
055.  http://suo.ieee.org/email/msg07466.html
 +
057.  http://suo.ieee.org/email/msg07469.html
 +
 +
070.  http://suo.ieee.org/ontology/msg03473.html
 +
071.  http://suo.ieee.org/ontology/msg03479.html
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
 +
1.3.10.13  Stretching Exercises
 +
 +
Taking up the preceding arrays of particular connections, namely,
 +
the boolean functions on two or less variables, it possible to
 +
illustrate the use of the stretch operation in a variety of
 +
concrete cases.
 +
 +
For example, suppose that F is a connection of the form F : %B%^2 -> %B%,
 +
that is, any one of the sixteen possibilities in Table 16, while p and q
 +
are propositions of the form p, q : X -> %B%, that is, propositions about
 +
things in the universe X, or else the indicators of sets contained in X.
 +
 +
Then one has the imagination #f# = <f_1, f_2> = <p, q> : (X -> %B%)^2,
 +
and the stretch of the connection F to #f# on X amounts to a proposition
 +
F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as
 +
the "stretch of F to p and q".  If one is concerned with many different
 +
propositions about things in X, or if one is abstractly indifferent to
 +
the particular choices for p and q, then one can detach the operator
 +
F^$ : (X -> %B%)^2 -> (X -> %B%), called the "stretch of F over X",
 +
and consider it in isolation from any concrete application.
 +
 +
When the "cactus notation" is used to represent boolean functions,
 +
a single "$" sign at the end of the expression is enough to remind
 +
a reader that the connections are meant to be stretched to several
 +
propositions on a universe X.
 +
 +
For instance, take the connection F : %B%^2 -> %B% such that:
 +
 +
F(x, y)  =  F^2_06 (x, y)  =  -(x, y)-.
 +
 +
This connection is the boolean function on a couple of variables x, y
 +
that yields a value of %1% if and only if just one of x, y is not %1%,
 +
that is, if and only if just one of x, y is %1%.  There is clearly an
 +
isomorphism between this connection, viewed as an operation on the
 +
boolean domain %B% = {%0%, %1%}, and the dyadic operation on binary
 +
values x, y in !B! = GF(2) that is otherwise known as "x + y".
 +
 +
The same connection F : %B%^2 -> %B% can also be read as a proposition
 +
about things in the universe X = %B%^2.  If S is a sentence that denotes
 +
the proposition F, then the corresponding assertion says exactly what one
 +
otherwise states by uttering "x is not equal to y".  In such a case, one
 +
has -[S]- = F, and all of the following expressions are ordinarily taken
 +
as equivalent descriptions of the same set:
 +
 +
[| -[S]- |]  =  [| F |]
 +
 +
            =  F^(-1)(%1%)
 +
 +
            =  {<x, y> in %B%^2  :  S}
 +
 +
            =  {<x, y> in %B%^2  :  F(x, y) = %1%}
 +
 +
            =  {<x, y> in %B%^2  :  F(x, y)}
 +
 +
            =  {<x, y> in %B%^2  :  -(x, y)- = %1%}
 +
 +
            =  {<x, y> in %B%^2  :  -(x, y)- }
 +
 +
            =  {<x, y> in %B%^2  :  x exclusive-or y}
 +
 +
            =  {<x, y> in %B%^2  :  just one true of x, y}
 +
 +
            =  {<x, y> in %B%^2  :  x not equal to y}
 +
 +
            =  {<x, y> in %B%^2  :  x <=/=> y}
 +
 +
            =  {<x, y> in %B%^2  :  x =/= y}
 +
 +
            =  {<x, y> in %B%^2  :  x + y}.
 +
 +
Notice the slight distinction, that I continue to maintain at this point,
 +
between the logical values {false, true} and the algebraic values {0, 1}.
 +
This makes it legitimate to write a sentence directly into the right side
 +
of the set-builder expression, for instance, weaving the sentence S or the
 +
sentence "x is not equal to y" into the context "{<x, y> in %B%^2 : ... }",
 +
thereby obtaining the corresponding expressions listed above, while the
 +
proposition F(x, y) can also be asserted more directly without equating
 +
it to %1%, since it already has a value in {false, true}, and thus can
 +
be taken as tantamount to an actual sentence.
 +
 +
If the appropriate safeguards can be kept in mind, avoiding all danger of
 +
confusing propositions with sentences and sentences with assertions, then
 +
the marks of these distinctions need not be forced to clutter the account
 +
of the more substantive indications, that is, the ones that really matter.
 +
If this level of understanding can be achieved, then it may be possible
 +
to relax these restrictions, along with the absolute dichotomy between
 +
algebraic and logical values, which tends to inhibit the flexibility
 +
of interpretation.
 +
 +
This covers the properties of the connection F(x, y) = -(x, y)-,
 +
treated as a proposition about things in the universe X = %B%^2.
 +
Staying with this same connection, it is time to demonstrate how
 +
it can be "stretched" into an operator on arbitrary propositions.
 +
 +
To continue the exercise, let p and q be arbitrary propositions about
 +
things in the universe X, that is, maps of the form p, q : X -> %B%,
 +
and suppose that p, q are indicator functions of the sets P, Q c X,
 +
respectively.  In other words, one has the following set of data:
 +
 +
|  p    =        -{P}-        :  X -> %B%
 +
|
 +
|  q    =        -{Q}-        :  X -> %B%
 +
|
 +
| <p, q>  =  < -{P}- , -{Q}- >  :  (X -> %B%)^2
 +
 +
Then one has an operator F^$, the stretch of the connection F over X,
 +
and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with
 +
the following properties:
 +
 +
| F^$        =  -( , )-^$  :  (X -> %B%)^2 -> (X -> %B%)
 +
|
 +
| F^$ (p, q)  =  -(p, q)-^$  :  X -> %B%
 +
 +
As a result, the application of the proposition F^$ (p, q) to each x in X
 +
yields a logical value in %B%, all in accord with the following equations:
 +
 +
| F^$ (p, q)(x)  =  -(p, q)-^$ (x)  in  %B%
 +
|
 +
|  ^                        ^
 +
|  |                        |
 +
|  =                        =
 +
|  |                        |
 +
|  v                        v
 +
|
 +
| F(p(x), q(x))  =  -(p(x), q(x))-  in  %B%
 +
 +
For each choice of propositions p and q about things in X, the stretch of
 +
F to p and q on X is just another proposition about things in X, a simple
 +
proposition in its own right, no matter how complex its current expression
 +
or its present construction as F^$ (p, q) = -(p, q)^$ makes it appear in
 +
relation to p and q.  Like any other proposition about things in X, it
 +
indicates a subset of X, namely, the fiber that is variously described
 +
in the following ways:
 +
 +
[| F^$ (p, q) |]  =  [| -(p, q)-^$ |]
 +
 +
                  =  (F^$ (p, q))^(-1)(%1%)
 +
 +
                  =  {x in X  :  F^$ (p, q)(x)}
 +
 +
                  =  {x in X  :  -(p, q)-^$ (x)}
 +
 +
                  =  {x in X  :  -(p(x), q(x))- }
 +
 +
                  =  {x in X  :  p(x) ± q(x)}
 +
 +
                  =  {x in X  :  p(x) =/= q(x)}
 +
 +
                  =  {x in X  :  -{P}- (x) =/= -{Q}- (x)}
 +
 +
                  =  {x in X  :  x in P <=/=> x in Q}
 +
 +
                  =  {x in X  :  x in P-Q or x in Q-P}
 +
 +
                  =  {x in X  :  x in P-Q |_| Q-P}
 +
 +
                  =  {x in X  :  x in P ± Q}
 +
 +
                  =  P ± Q          c  X
 +
 +
                  =  [|p|] ± [|q|]  c  X.
 +
 +
Which was to be shown.
 +
 +
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 +
</pre>
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/75536"

Navigation menu