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{{DISPLAYTITLE:Cactus Language}}
 
{{DISPLAYTITLE:Cactus Language}}
<pre>
+
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
 
 
 
Inquiry Driven Systems: An Inquiry Into Inquiry
 
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
==The Cactus Patch==
  
| Document History
+
{| align="center" cellpadding="0" cellspacing="0" width="90%"
 
|
 
|
| Subject:  Inquiry Driven Systems:  An Inquiry Into Inquiry
+
<p>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 areasIf our intellectual gaze could shift slightly, it would alter each quill's direction, and suddenly our entire reality would change.</p>
| Contact:  Jon Awbrey <jawbrey@oakland.edu>
+
|-
| Version: Draft 8.70
+
| align="right" | &mdash; Herbert J. Bernstein, &ldquo;Idols of Modern Science&rdquo;, [HJB, 38]
| 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
+
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.
  
1.3.10.8  The Cactus Patch
+
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.
  
| Thus, what looks to us like a sphere of scientific knowledge more accurately
+
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.
| 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:
 
Over the course of this project, I plan to use two species of graphs:
  
1.  "Painted And Rooted Cacti" (PARCAI).
+
# Painted And Rooted Cacti (PARCAI).
 +
# Painted And Rooted Conifers (PARCOI).
  
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 stageAccordingly, 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.
  
For now, it is enough to discuss the former class of data structures,
+
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.
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,
+
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.
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
+
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.
  
1.3.10.8 The Cactus Patch (cont.)
+
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 SubsectionAfter that, the development of the language's more concrete aspects can be seen as a matter of defining two functions:
  
Depending on whether a formal language is called by the type of sign
+
# 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''.
that makes it up or whether it is named after the type of object that
+
# 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.
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,
+
The discussion of syntax brings up a number of associated issues that have to be clarified before going onThese 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).
then the language can begin to acquire a semantic functionIn 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
+
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.
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
+
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.
    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,
+
===The Cactus Language : Syntax===
    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
+
{| align="center" cellpadding="0" cellspacing="0" width="90%"
have to be clarified before going on.  These are questions of "style",
+
|
that is, the sort of description, "grammar", or theory that one finds
+
<p>Picture two different configurations of such an irregular shape, superimposed on each other in space, like a double exposure photographOf 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 knowledgeIn every direction in which we look at all deeply, the realm of discovered scientific truth could be quite differentYet 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.</p>
available or chooses as preferable for a given languageThese issues
+
|-
are discussed in the Subsection after next (Subsection 1.3.10.10).
+
| align="right" | &mdash; Herbert J. Bernstein, "Idols of Modern Science", [HJB, 38]
 
+
|}
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 computationBecause it involves the transformation of signs,
 
it can be recognized as an aspect of semioticsSince 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
+
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 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.
  
1.3.10.9  The Cactus Language: Syntax
+
As a temporary notation, let the relationship between a particular sign <math>s\!</math> and a particular object <math>o\!</math>, namely, the fact that <math>s\!</math> denotes <math>o\!</math> or the fact that <math>o\!</math> is denoted by <math>s\!</math>, be symbolized in one of the following two ways:
  
| Picture two different configurations of such an irregular shape, superimposed
+
{| align="center" cellpadding="8" width="90%"
| 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,
+
<math>\begin{array}{lccc}
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
+
1. & s & \rightarrow & o \\
|
+
\\
| Marcus G. Raskin & Herbert J. Bernstein,
+
2. & o & \leftarrow & s \\
|'New Ways of Knowing:  The Sciences, Society, & Reconstructive Knowledge',
+
\end{array}</math>
| 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:
 
Now consider the following paradigm:
  
1. If       "A"  >->  Ann,
+
{| align="center" cellpadding="8" width="90%"
 +
|
 +
<math>\begin{array}{llccc}
 +
1. &
 +
\operatorname{If} &
 +
^{\backprime\backprime}\operatorname{A}^{\prime\prime} &
 +
\rightarrow &
 +
\operatorname{Ann}, \\
 +
&
 +
\operatorname{that~is}, &
 +
^{\backprime\backprime}\operatorname{A}^{\prime\prime} &
 +
\operatorname{denotes} &
 +
\operatorname{Ann}, \\
 +
&
 +
\operatorname{then} &
 +
\operatorname{A} &
 +
= &
 +
\operatorname{Ann} \\
 +
&
 +
\operatorname{and} &
 +
\operatorname{Ann} &
 +
= &
 +
\operatorname{A}. \\
 +
&
 +
\operatorname{Thus} &
 +
^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} &
 +
\rightarrow &
 +
\operatorname{A}, \\
 +
&
 +
\operatorname{that~is}, &
 +
^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} &
 +
\operatorname{denotes} &
 +
\operatorname{A}. \\
 +
\end{array}</math>
 +
|}
  
    i.e.     "A"  den  Ann,
+
{| align="center" cellpadding="8" width="90%"
 +
|
 +
<math>\begin{array}{llccc}
 +
2. &
 +
\operatorname{If} &
 +
\operatorname{Bob} &
 +
\leftarrow &
 +
^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\
 +
&
 +
\operatorname{that~is}, &
 +
\operatorname{Bob} &
 +
\operatorname{is~denoted~by} &
 +
^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\
 +
&
 +
\operatorname{then} &
 +
\operatorname{Bob} &
 +
= &
 +
\operatorname{B} \\
 +
&
 +
\operatorname{and} &
 +
\operatorname{B} &
 +
= &
 +
\operatorname{Bob}. \\
 +
&
 +
\operatorname{Thus} &
 +
\operatorname{B} &
 +
\leftarrow &
 +
^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}, \\
 +
&
 +
\operatorname{that~is}, &
 +
\operatorname{B} &
 +
\operatorname{is~denoted~by} &
 +
^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}. \\
 +
\end{array}</math>
 +
|}
  
    then      A    =  Ann,
+
When I say that the sign "blank" denotes the sign "&nbsp;", 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 higher order sign whose object is "&nbsp;", 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:
  
    thus      "Ann" >->  A,
+
{| align="center" cellpadding="8" width="90%"
 
 
    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>->  " "
+
<math>\begin{array}{lll}
 +
^{\backprime\backprime}\operatorname{~}^{\prime\prime} &
 +
\leftarrow &
 +
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\
 +
\\
 +
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} &
 +
\rightarrow &
 +
^{\backprime\backprime}\operatorname{~}^{\prime\prime} \\
 +
\end{array}</math>
 +
|}
  
Using the raised dot "·" as a sign to mark the articulation of a
+
Using the raised dot "<math>\cdot</math>" 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:
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:
 
  
 +
{| align="center" cellpadding="8" width="90%"
 
|
 
|
|  " "  <-<  "blank=   "b"·"l"·"a"·"n"·"k"
+
<math>\begin{array}{lllll}
|
+
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
 +
& \leftarrow &
 +
^{\backprime\backprime}\operatorname{blank}^{\prime\prime}
 +
& = &
 +
^{\backprime\backprime}\operatorname{b}^{\prime\prime} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{l}^{\prime\prime} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{a}^{\prime\prime} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{n}^{\prime\prime} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{k}^{\prime\prime} \\
 +
\end{array}</math>
 +
|}
  
This usage allows us to refer to the blank as a type of character, and
+
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 "&nbsp;" by the name "blank" in the form of an identity between the named objects, thus:
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:
 
  
 +
{| align="center" cellpadding="8" width="90%"
 
|
 
|
|  " "  =   blank
+
<math>\begin{array}{lll}
|
+
^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \\
 +
\end{array}</math>
 +
|}
  
With these kinds of identity in mind, it is possible to extend the use of
+
With these kinds of identity in mind, it is possible to extend the use of the "<math>\cdot</math>" sign to mark the articulation of either named or quoted strings into both named and quoted strings.  For example:
the "·" sign to mark the articulation of either named or quoted strings
 
into both named and quoted strings.  For example:
 
  
|   " "       =   " "·" "       =  blank·blank
+
{| align="center" cellpadding="8" width="90%"
 
|
 
|
|  " blank=   " "·"blank=   blank·"blank"
+
<math>\begin{array}{lclcl}
|
+
^{\backprime\backprime}\operatorname{~~}^{\prime\prime}
|  "blank =   "blank"·" "  =   "blank"·blank
+
& = &
 +
^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
 +
& = &
 +
\operatorname{blank} \, \cdot \, \operatorname{blank} \\
 +
\\
 +
^{\backprime\backprime}\operatorname{~blank}^{\prime\prime}
 +
& = &
 +
^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{blank}^{\prime\prime}
 +
& = &
 +
\operatorname{blank} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\
 +
\\
 +
^{\backprime\backprime}\operatorname{blank~}^{\prime\prime}
 +
& = &
 +
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \,
 +
^{\backprime\backprime}\operatorname{~}^{\prime\prime}
 +
& = &
 +
^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \,
 +
\operatorname{blank}
 +
\end{array}</math>
 +
|}
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
A few definitions from formal language theory are required at this point.
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
An ''alphabet'' is a finite set of signs, typically, <math>\mathfrak{A} = \{ \mathfrak{a}_1, \ldots, \mathfrak{a}_n \}.</math>
  
A few definitions from formal language theory are required at this point.
+
A ''string'' over an alphabet <math>\mathfrak{A}</math> is a finite sequence of signs from <math>\mathfrak{A}.</math>
  
An "alphabet" is a finite set of signs, typically, !A! = {a_1, ..., a_n}.
+
The ''length'' of a string is just its length as a sequence of signs.
  
A "string" over an alphabet !A! is a finite sequence of signs from !A!.
+
The ''empty string'' is the unique sequence of length 0.  It is sometimes denoted by an empty pair of quotation marks, <math>^{\backprime\backprime\prime\prime},</math> but more often by the Greek symbols epsilon or lambda.
  
The "length" of a string is just its length as a sequence of signs.
+
A sequence of length <math>k > 0\!</math> is typically presented in the concatenated forms:
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,
+
{| align="center" cellpadding="4" width="90%"
 +
|
 +
<math>s_1 s_2 \ldots s_{k-1} s_k\!</math>
 +
|}
  
 
or
 
or
  
s_1 · s_2 · ... · s_(k-1) · s_k,
+
{| align="center" cellpadding="4" width="90%"
 +
|
 +
<math>s_1 \cdot s_2 \cdot \ldots \cdot s_{k-1} \cdot s_k</math>
 +
|}
  
with s_j in !A!, for all j = 1 to k.
+
with <math>s_j \in \mathfrak{A}</math> for all <math>j = 1 \ldots k.</math>
  
 
Two alternative notations are often useful:
 
Two alternative notations are often useful:
  
1. !e! =   @e@  =   ""   =  the empty string.
+
{| align="center" cellpadding="4" style="text-align:center" width="90%"
 +
|-
 +
| <math>\varepsilon\!</math>
 +
| =
 +
| <math>{}^{\backprime\backprime\prime\prime}\!</math>
 +
| =
 +
| align="left" | the empty string.
 +
|-
 +
| <math>\underline\varepsilon\!</math>
 +
| =
 +
| <math>\{ \varepsilon \}\!</math>
 +
| =
 +
| align="left" | the language consisting of a single empty string.
 +
|}
  
2.  %e%  =  {!e!{""} the language consisting of a single empty string.
+
The ''kleene star'' <math>\mathfrak{A}^*</math> of alphabet <math>\mathfrak{A}</math> is the set of all strings over <math>\mathfrak{A}.</math> In particular, <math>\mathfrak{A}^*</math> includes among its elements the empty string <math>\varepsilon.</math>
  
The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!.
+
The ''kleene plus'' <math>\mathfrak{A}^+</math> of an alphabet <math>\mathfrak{A}</math> is the set of all positive length strings over <math>\mathfrak{A},</math> in other words, everything in <math>\mathfrak{A}^*</math> but the empty string.
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
+
A ''formal language'' <math>\mathfrak{L}</math> over an alphabet <math>\mathfrak{A}</math> is a subset of <math>\mathfrak{A}^*.</math>  In brief, <math>\mathfrak{L} \subseteq \mathfrak{A}^*.</math>  If <math>s\!</math> is a string over <math>\mathfrak{A}</math> and if <math>s\!</math> is an element of <math>\mathfrak{L},</math> then it is customary to call <math>s\!</math> a ''sentence'' of <math>\mathfrak{L}.</math>  Thus, a formal language <math>\mathfrak{L}</math> is defined by specifying its elements, which amounts to saying what it means to be a sentence of <math>\mathfrak{L}.</math>
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!*.
+
One last device turns out to be useful in this connection. If <math>s\!</math> is a string that ends with a sign <math>t,\!</math> then <math>s \cdot t^{-1}</math> is the string that results by ''deleting'' from <math>s\!</math> the terminal <math>t.\!</math>
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.
+
In this context, I make the following distinction:
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:
+
# To ''delete'' an appearance of a sign is to replace it with an appearance of the empty string "".
 +
# To ''erase'' an appearance of a sign is to replace it with an appearance of the blank symbol "&nbsp;".
  
1.  By "deleting" an appearance of a sign,
+
A ''token'' is a particular appearance of a sign.
    I mean replacing it with an appearance
 
    of the empty string "".
 
  
2.  By "erasing" an appearance of a sign,
+
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).
    I mean replacing it with an appearance
 
    of the blank symbol " ".
 
  
A "token" is a particular appearance of a sign.
+
The ''painted cactus language'' <math>\mathfrak{C}</math> is actually a parameterized family of languages, consisting of one language <math>\mathfrak{C}(\mathfrak{P})</math> for each set <math>\mathfrak{P}</math> of ''paints''.
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
The alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> is the disjoint union of two sets of symbols:
  
1.3.10.9  The Cactus Language: Syntax (cont.)
+
<ol style="list-style-type:decimal">
  
The informal mechanisms that have been illustrated in the immediately preceding
+
<li>
discussion are enough to equip the rest of this discussion with a moderately
+
<p><math>\mathfrak{M}</math> is the alphabet of ''measures'', the set of ''punctuation marks'', or the collection of ''syntactic constants'' that is common to all of the languages <math>\mathfrak{C}(\mathfrak{P}).</math>  This set of signs is given as follows:</p>
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
+
<p><math>\begin{array}{lccccccccccc}
family of languages, consisting of one language !C!(!P!) for
+
\mathfrak{M}
each set !P! of "paints".
+
& = &
 +
\{ &
 +
\mathfrak{m}_1 & , &
 +
\mathfrak{m}_2 & , &
 +
\mathfrak{m}_3 & , &
 +
\mathfrak{m}_4 &
 +
\} \\
 +
& = &
 +
\{ &
 +
^{\backprime\backprime} \, \operatorname{~} \, ^{\prime\prime} & , &
 +
^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & , &
 +
^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} & , &
 +
^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} &
 +
\} \\
 +
& = &
 +
\{ &
 +
\operatorname{blank} & , &
 +
\operatorname{links} & , &
 +
\operatorname{comma} & , &
 +
\operatorname{right} &
 +
\} \\
 +
\end{array}</math></p></li>
  
The alphabet !A!  =  !M! |_| !P! is the disjoint union of two sets of symbols:
+
<li>
 +
<p><math>\mathfrak{P}</math> is the ''palette'', the alphabet of ''paints'', or the collection of ''syntactic variables'' that is peculiar to the language <math>\mathfrak{C}(\mathfrak{P}).</math>  This set of signs is given as follows:</p>
  
1.  !M! is the alphabet of "measures", the set of "punctuation marks",
+
<p><math>\mathfrak{P} = \{ \mathfrak{p}_j : j \in J \}.</math></p></li>
    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}
+
</ol>
  
        = {" ", "-(", ",", ")-"}
+
The easiest way to define the language <math>\mathfrak{C}(\mathfrak{P})\!</math> 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 <math>\mathfrak{A}^*\!</math> that are called ''syntactic connectives''. If the strings on which they operate are exclusively sentences of <math>\mathfrak{C}(\mathfrak{P}),\!</math> 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''.
  
        =  {blank, links, comma, right}.
+
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.
  
2.  !P! is the "palette", the alphabet of "paints", or the collection
+
The first step is to define two sets of basic operations on strings of <math>\mathfrak{A}^*.</math>
    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}.
+
<ol style="list-style-type:decimal">
  
The easiest way to define the language !C!(!P!) is to indicate the general sorts
+
<li>
of operations that suffice to construct the greater share of its sentences from
+
<p>The ''concatenation'' of one string <math>s_1\!</math> is just the string <math>s_1.\!</math></p>
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
+
<p>The ''concatenation'' of two strings <math>s_1, s_2\!</math> is the string <math>{s_1 \cdot s_2}.\!</math></p>
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
<p>The ''concatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k\!</math> is the string of the form <math>{s_1 \cdot \ldots \cdot s_k}.\!</math></p></li>
  
Rather than presenting the most concise description of these languages
+
<li>
right from the beginning, it serves comprehension to develop a picture
+
<p>The ''surcatenation'' of one string <math>s_1\!</math> is the string <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
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!*.
+
<p>The ''surcatenation'' of two strings <math>s_1, s_2\!</math> is <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
  
1.  The "concatenation" of one string z_1 is just the string z_1.
+
<p>The ''surcatenation'' of the <math>k\!</math> strings <math>(s_j)_{j = 1}^k</math> is the string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
  
    The "concatenation" of two strings z_1, z_2 is the string z_1 · z_2.
+
</ol>
  
    The "concatenation" of the k strings z_j, for j = 1 to k,
+
These definitions can be made a little more succinct by defining the following sorts of generic operators on strings:
  
    is the string of the form z_1 · ... · z_k.
+
<ol style="list-style-type:decimal">
  
2.  The "surcatenation" of one string z_1 is the string "-(" · z_1 · ")-".
+
<li>The ''concatenation'' <math>\operatorname{Conc}_{j=1}^k</math> of the sequence of <math>k\!</math> strings <math>(s_j)_{j=1}^k</math> is defined recursively as follows:</li>
  
    The "surcatenation" of two strings z_1, z_2 is "-(" · z_1 · "," · z_2 · ")-".
+
<ol style="list-style-type:lower-alpha">
  
    The "surcatenation" of k strings z_j, for j = 1 to k,
+
<li><math>\operatorname{Conc}_{j=1}^1 s_j \ = \ s_1.</math></li>
  
    is the string of the form "-(" · z_1 · "," · ... · "," · z_k · ")-".
+
<li>
 +
<p>For <math>\ell > 1,\!</math></p>
  
These definitions can be made a little more succinct by
+
<p><math>\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Conc}_{j=1}^{\ell - 1} s_j \, \cdot \, s_\ell.</math></p></li>
defining the following sorts of generic operators on strings:
 
  
1.  The "concatenation" Conc^k of the k strings z_j,
+
</ol>
    for j = 1 to k, is defined recursively as follows:
 
  
    a.  Conc^1_j  z_j  = z_1.
+
<li>The ''surcatenation'' <math>\operatorname{Surc}_{j=1}^k</math> of the sequence of <math>k\!</math> strings <math>(s_j)_{j=1}^k</math> is defined recursively as follows:</li>
  
    b.  For k > 1,
+
<ol style="list-style-type:lower-alpha">
  
        Conc^k_j  z_j  = (Conc^(k-1)_j  z_j) · z_k.
+
<li><math>\operatorname{Surc}_{j=1}^1 s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>
  
2.  The "surcatenation" Surc^k of the k strings z_j,
+
<li>
    for j = 1 to k, is defined recursively as follows:
+
<p>For <math>\ell > 1,\!</math></p>
  
    a.  Surc^1_j  z_j  = "-(" · z_1 · ")-".
+
<p><math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
  
    b.  For k > 1,
+
</ol></ol>
  
        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 by making a few additional conventions and auxiliary definitions.
  
The definitions of these syntactic operations can now be organized in a slightly
+
<ol style="list-style-type:decimal">
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
+
<li>
    can be extended to include its natural "prequel":
+
<p>The conception of the <math>k\!</math>-place concatenation operation can be extended to include its natural ''prequel'':</p>
  
    Conc^0 = ""  = the empty string.
+
<p><math>\operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}</math> &nbsp;=&nbsp; the empty string.</p>
  
    Next, the construction of the k-place concatenation can be
+
<p>Next, the construction of the <math>k\!</math>-place concatenation can be broken into stages by means of the following conceptions:</p></li>
    broken into stages by means of the following conceptions:
 
  
    a.  The "precatenation" Prec(z_1, z_2) of the two strings
+
<ol style="list-style-type:lower-alpha">
        z_1, z_2 is the string that is defined as follows:
 
  
        Prec(z_1, z_2) =  z_1 · z_2.
+
<li>
 +
<p>The ''precatenation'' <math>\operatorname{Prec} (s_1, s_2)</math> of the two strings <math>s_1, s_2\!</math> is the string that is defined as follows:</p>
  
    b.  The "concatenation" of the k strings z_1, ..., z_k can now be
+
<p><math>\operatorname{Prec} (s_1, s_2) \ = \ s_1 \cdot s_2.</math></p></li>
        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 = "".
+
<li>
 +
<p>The ''concatenation'' of the sequence of <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> can now be defined as an iterated precatenation over the sequence of <math>k+1\!</math> strings that begins with the string <math>s_0 = \operatorname{Conc}^0 \, = \, ^{\backprime\backprime\prime\prime}</math> and then continues on through the other <math>k\!</math> strings:</p></li>
  
        ii.  For k > 0,
+
<ol style="list-style-type:lower-roman">
  
            Conc^k_j  z_j  = Prec(Conc^(k-1)_j  z_j, z_k).
+
<li>
 +
<p><math>\operatorname{Conc}_{j=0}^0 s_j \ = \ \operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}.</math></p></li>
  
2.  The conception of the k-place surcatenation operation
+
<li>
    can be extended to include its natural "prequel":
+
<p>For <math>\ell > 0,\!</math></p>
  
    Surc^0 =  "-()-".
+
<p><math>\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Prec}(\operatorname{Conc}_{j=0}^{\ell - 1} s_j, s_\ell).</math></p></li>
  
    Finally, the construction of the k-place surcatenation can be
+
</ol></ol>
    broken into stages by means of the following conceptions:
 
  
    a.  A "subclause" in !A!* is a string that ends with a ")-".
+
<li>
 +
<p>The conception of the <math>k\!</math>-place surcatenation operation can be extended to include its natural "prequel":</p>
  
    b.  The "subcatenation" Subc(z_1, z_2)
+
<p><math>\operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math></p>
        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 · ")-".
+
<p>Finally, the construction of the <math>k\!</math>-place surcatenation can be broken into stages by means of the following conceptions:</p>
  
    c.  The "surcatenation" of the k strings z_1, ..., z_k can now be
+
<ol style="list-style-type:lower-alpha">
        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  =  "-()-".
+
<li>
 +
<p>A ''subclause'' in <math>\mathfrak{A}^*</math> is a string that ends with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p></li>
  
        ii.  For k > 0,
+
<li>
 +
<p>The ''subcatenation'' <math>\operatorname{Subc} (s_1, s_2)</math> of a subclause <math>s_1\!</math> by a string <math>s_2\!</math> is the string that is defined as follows:</p>
  
            Surc^k_j  z_j  = Subc(Surc^(k-1)_j  z_j, z_k).
+
<p><math>\operatorname{Subc} (s_1, s_2) \ = \ s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></p>
  
Notice that the expressions Conc^0_j z_j and Surc^0_j z_j
+
<li>
are defined in such a way that the respective operators
+
<p>The ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> can now be defined as an iterated subcatenation over the sequence of <math>k+1\!</math> strings that starts with the string <math>s_0 \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}</math> and then continues on through the other <math>k\!</math> strings:</p></li>
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
+
<ol style="list-style-type:lower-roman">
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
<li>
 +
<p><math>\operatorname{Surc}_{j=0}^0 s_j \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math></p></li>
  
Having defined the basic operations of concatenation and surcatenation
+
<li>
on arbitrary strings, in effect, giving them operational meaning for
+
<p>For <math>\ell > 0,\!</math></p>
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
+
<p><math>\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Subc}(\operatorname{Surc}_{j=0}^{\ell - 1} s_j, s_\ell).</math></p></li>
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
+
</ol></ol></ol>
  
    z_1 is a sentence of !L!, z_2 is a sentence of !L!, and
+
Notice that the expressions <math>\operatorname{Conc}_{j=0}^0 s_j</math> and <math>\operatorname{Surc}_{j=0}^0 s_j</math> are defined in such a way that the respective operators <math>\operatorname{Conc}^0</math> and <math>\operatorname{Surc}^0</math> simply ignore, in the manner of constants, whatever sequences of strings <math>s_j\!</math> may be listed as their ostensible arguments.
  
    z  = z_1 · z_2.
+
Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language <math>\mathfrak{L} = \mathfrak{A}^*,</math> it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.
  
2.  z is the "concatenation" of the k strings z1, ..., z_k in !L!,
+
If <math>\mathfrak{L}</math> is an arbitrary formal language over an alphabet of the sort that
 +
we are talking about, that is, an alphabet of the form <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},</math> then there are a number of basic structural relations that can be defined on the strings of <math>\mathfrak{L}.</math>
  
    if and only if z_j is a sentence of !L!, for all j = 1 to k, and
+
{| align="center" cellpadding="4" width="90%"
 +
| 1. || <math>s\!</math> is the ''concatenation'' of <math>s_1\!</math> and <math>s_2\!</math> in <math>\mathfrak{L}</math> if and only if
 +
|-
 +
| &nbsp; || <math>s_1\!</math> is a sentence of <math>\mathfrak{L},</math> <math>s_2\!</math> is a sentence of <math>\mathfrak{L},</math> and
 +
|-
 +
| &nbsp; || <math>s = s_1 \cdot s_2.</math>
 +
|-
 +
| 2. || <math>s\!</math> is the ''concatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>
 +
|-
 +
| &nbsp; || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>j = 1 \ldots k,</math> and
 +
|-
 +
| &nbsp; || <math>s = \operatorname{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k.</math>
 +
|-
 +
| 3. || <math>s\!</math> is the ''discatenation'' of <math>s_1\!</math> by <math>t\!</math> if and only if
 +
|-
 +
| &nbsp; || <math>s_1\!</math> is a sentence of <math>\mathfrak{L},</math> <math>t\!</math> is an element of <math>\mathfrak{A},</math> and
 +
|-
 +
| &nbsp; || <math>s_1 = s \cdot t.</math>
 +
|-
 +
| &nbsp; || When this is the case, one more commonly writes:
 +
|-
 +
| &nbsp; || <math>s = s_1 \cdot t^{-1}.</math>
 +
|-
 +
| 4. || <math>s\!</math> is a ''subclause'' of <math>\mathfrak{L}</math> if and only if
 +
|-
 +
| &nbsp; || <math>s\!</math> is a sentence of <math>\mathfrak{L}</math> and <math>s\!</math> ends with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
 +
|-
 +
| 5. || <math>s\!</math> is the ''subcatenation'' of <math>s_1\!</math> by <math>s_2\!</math> if and only if
 +
|-
 +
| &nbsp; || <math>s_1\!</math> is a subclause of <math>\mathfrak{L},</math> <math>s_2\!</math> is a sentence of <math>\mathfrak{L},</math> and
 +
|-
 +
| &nbsp; || <math>s = s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
 +
|-
 +
| 6. || <math>s\!</math> is the ''surcatenation'' of the <math>k\!</math> strings <math>s_1, \ldots, s_k\!</math> in <math>\mathfrak{L},</math>
 +
|-
 +
| &nbsp; || if and only if <math>s_j\!</math> is a sentence of <math>\mathfrak{L},</math> for all <math>{j = 1 \ldots k},\!</math> and
 +
|-
 +
| &nbsp; || <math>s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math>
 +
|}
  
    z  =  Conc^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.
  
3.  z is the "discatenation" of z_1 by t if and only if
+
The ''painted cactus language'' with paints in the set <math>\mathfrak{P} = \{ p_j : j \in J \}</math> is the formal language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*</math> that is defined as follows:
  
    z_1 is a sentence of !L!, t is an element of !A!, and
+
{| align="center" cellpadding="4" width="90%"
 +
|-
 +
| PC 1. || The blank symbol <math>m_1\!</math> is a sentence.
 +
|-
 +
| PC 2. || The paint <math>p_j\!</math> is a sentence, for each <math>j\!</math> in <math>J.\!</math>
 +
|-
 +
| PC 3. || <math>\operatorname{Conc}^0</math> and <math>\operatorname{Surc}^0</math> are sentences.
 +
|-
 +
| PC 4. || For each positive integer <math>k,\!</math>
 +
|-
 +
| &nbsp; || if <math>s_1, \ldots, s_k\!</math> are sentences,
 +
|-
 +
| &nbsp; || then <math>\operatorname{Conc}_{j=1}^k s_j</math> is a sentence,
 +
|-
 +
| &nbsp; || and <math>\operatorname{Surc}_{j=1}^k s_j</math> is a sentence.
 +
|}
  
    z_1 = z · t.
+
As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.</math>  An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),\!</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> 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 <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>
  
    When this is the case, one more commonly writes:
+
A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \varnothing.</math>  A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).</math>  This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math>  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.
  
    z = z_1 · t^-1.
+
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&nbsp;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, <math>\operatorname{Conc}^k</math> and <math>\operatorname{Surc}^k,</math> for all <math>k > 0.\!</math> 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.
  
4.  z is a "subclause" of !L! if and only if
+
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''.
  
    z is a sentence of !L! and z ends with a ")-".
+
A notation in the style of <math>S :> T\!</math> is now introduced, to be read among many others in this manifold of ways:
  
5.  z is the "subcatenation" of z_1 by z_2 if and only if
+
{| align="center" cellpadding="4" width="90%"
 +
|-
 +
| <math>S\ \operatorname{covers}\ T</math>
 +
|-
 +
| <math>S\ \operatorname{governs}\ T</math>
 +
|-
 +
| <math>S\ \operatorname{rules}\ T</math>
 +
|-
 +
| <math>S\ \operatorname{subsumes}\ T</math>
 +
|-
 +
| <math>S\ \operatorname{types~over}\ T</math>
 +
|}
  
    z_1 is a subclause of !L!, z_2 is a sentence of !L!, and
+
The form <math>S :> T\!</math> is here recruited for polymorphic employment in at least the following types of roles:
  
    z  =  z_1 · ")-"^(-1) · "," · z_2 · ")-".
+
# To signify that an individually named or quoted string <math>T\!</math> is being typed as a sentence <math>S\!</math> of the language of interest <math>\mathfrak{L}.</math>
 +
# To express the fact or to make the assertion that each member of a specified set of strings <math>T \subseteq \mathfrak{A}^*</math> also belongs to the syntactic category <math>S,\!</math> the one that qualifies a string as being a sentence in the relevant formal language <math>\mathfrak{L}.</math>
 +
# To specify the intension or to signify the intention that every string that fits the conditions of the abstract type <math>T\!</math> must also fall under the grammatical heading of a sentence, as indicated by the type <math>S,\!</math> all within the target language <math>\mathfrak{L}.</math>
  
6.  z is the "surcatenation" of the k strings z_1, ..., z_k in !L!,
+
In these types of situation the letter <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> 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 <math>^{\backprime\backprime} T \, ^{\prime\prime}</math> 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''.
  
    if and only if z_j is a sentence of !L!, for all j = 1 to k, and
+
Combining the singleton set <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \}</math> whose sole member is the initial symbol with the set <math>\mathfrak{Q}</math> that assembles together all of the intermediate symbols results in the set <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}</math> of ''non-terminal symbols''.  Completing the package, the alphabet <math>\mathfrak{A}</math> of the language is also known as the set of ''terminal symbols''.  In this discussion, I will adopt the convention that <math>\mathfrak{Q}</math> is the set of ''intermediate symbols'', but I will often use <math>q\!</math> as a typical variable that ranges over all of the non-terminal symbols, <math>q \in \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}.</math>  Finally, it is convenient to refer to all of the symbols in <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> as the ''augmented alphabet'' of the prospective grammar for the language, and accordingly to describe the strings in <math>( \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> as the ''augmented strings'', in effect, expressing the forms that are superimposed on a language by one of its conceivable grammars.  In certain settings it becomes desirable to separate the augmented strings that contain the symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> from all other sorts of augmented strings.  In these situations the strings in the disjoint union <math>\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*</math> are known as the ''sentential forms'' of the associated grammar.
  
    z  =  Surc^k_j  z_j  =  "-(" · z_1 · "," · ... · "," · z_k · ")-".
+
In forming a grammar for a language statements of the form <math>W :> W',\!</math>
 +
where <math>W\!</math> and <math>W'\!</math> 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 <math>\mathfrak{K}</math> that serves to complete the definition of the formal grammar in question.
  
The converses of these decomposition relations are tantamount to the
+
Correlative with the use of this notation, an expression of the form <math>T <: S,\!</math> read to say that <math>T\!</math> is covered by <math>S,\!</math> can be interpreted to say that <math>T\!</math> is of the type <math>S.\!</math>  Depending on the context, this can be taken in either one of two ways:
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
+
# Treating <math>T\!</math> as a string variable, it means that the individual string <math>T\!</math> is typed as <math>S.\!</math>
 +
# Treating <math>T\!</math> as a type name, it means that any instance of the type <math>T\!</math> also falls under the type <math>S.\!</math>
  
1.3.10.9  The Cactus Language: Syntax (cont.)
+
In accordance with these interpretations, an expression of the form <math>t <: T\!</math> can be read in all of the ways that one typically reads an expression of the form <math>t : T.\!</math>
  
The "painted cactus language" with paints in the
+
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 <math>t <: T\!</math> and <math>T <: S,\!</math> where any of the symbols <math>t, T, S\!</math> can be taken to signify either the tokens or the subtypes of their covering types.
set !P! = {p_j : j in J} is the formal language
 
!L! = !C!(!P!) c !A!* = (!M! |_| !P!)* that is
 
defined as follows:
 
  
PC 1The blank symbol m_1 is a sentence.
+
'''Note.''' 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 the subject matter of the 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.
  
PC 2.  The paint p_j is a sentence, for each j in J.
+
Employing the notion of a covering relation it becomes possible to redescribe the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in the following ways.
  
PC 3.  Conc^0 and Surc^0 are sentences.
+
====Grammar 1====
  
PC 4For each positive integer k,
+
Grammar&nbsp;1 is something of a misnomerIt 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 <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> that comes with the territory of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> it specifies <math>\mathfrak{Q} = \varnothing,</math> in other words, it employs no intermediate symbols, and it embodies the ''covering set'' <math>\mathfrak{K}</math> as listed in the following display.
  
      if    z_1, ..., z_k are sentences,
+
<br>
  
      then Conc^k_j  z_j is a sentence,
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
 +
| align="left" style="border-left:1px solid black;"  width="50%" |
 +
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 1}\!</math>
 +
| align="right" style="border-right:1px solid black;" width="50%" |
 +
<math>\mathfrak{Q} = \varnothing</math>
 +
|-
 +
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
 +
<math>\begin{array}{rcll}
 +
1.
 +
& S
 +
& :>
 +
& m_1 \ = \ ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}
 +
\\
 +
2.
 +
& S
 +
& :>
 +
& p_j, \, \text{for each} \, j \in J
 +
\\
 +
3.
 +
& S
 +
& :>
 +
& \operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}
 +
\\
 +
4.
 +
& S
 +
& :>
 +
& \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
 +
\\
 +
5.
 +
& S
 +
& :>
 +
& S^*
 +
\\
 +
6.
 +
& S
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ( \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, )^* \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\
 +
\end{array}</math>
 +
|}
  
      and  Surc^k_j  z_j is a sentence.
+
<br>
  
As usual, saying that z is a sentence is just a conventional way of
+
In this formulation, the last two lines specify that:
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",
+
<ol style="list-style-type:decimal">
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
+
<li value="5"> The concept of a sentence in <math>\mathfrak{L}</math> covers any concatenation of sentences in <math>\mathfrak{L},</math> in effect, any number of freely chosen sentences that are available to be concatenated one after another.</li>
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
+
<li value="6"> The concept of a sentence in <math>\mathfrak{L}</math> covers any surcatenation of sentences in <math>\mathfrak{L},</math> in effect, any string that opens with a <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime},</math> continues with a sentence, possibly empty, follows with a finite number of phrases of the form <math>^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> and closes with a <math>^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.</math></li>
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,
+
</ol>
to be read among many others in this manifold of ways:
 
  
| S covers T
+
This appears to be just about the most concise description of the cactus language <math>\mathfrak{C} (\mathfrak{P})</math> that one can imagine, but there are 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:
|
 
|  S governs T
 
|
 
|  S rules T
 
|
 
|  S subsumes T
 
|
 
|  S types over T
 
  
The form "S :> T" is here recruited for polymorphic
+
# The invocation of the kleene star operation is not reduced to a manifestly finitary form.
employment in at least the following types of roles:
+
# The type <math>S\!</math> that indicates a sentence is allowed to cover not only itself but also the empty string.
  
1.  To signify that an individually named or quoted string T is
+
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.
    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
+
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.
    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
+
# The ''degree of intermediate organization'' in a grammar.
    string that fits the conditions of the abstract type T must also
+
# The ''distinction between empty and significant strings'', and thus the ''distinction between empty and significant types of strings''.
    fall under the grammatical heading of a sentence, as indicated by
+
# The ''principle of intermediate significance''.  This is a constraint on the grammar that arises from considering the interaction of the first two issues.
    the type name "S", all within the target language !L!.
 
  
In these types of situation the letter "S", that signifies the type of
+
In responding to these issues, it is advisable at first to proceed in a stepwise fashion, all the better 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.
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
+
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 productionsWith respect to this issue, Grammar&nbsp;1 has no intermediate symbols at all, <math>\mathfrak{Q} = \varnothing,</math> and therefore remains at an ostensibly trivial degree of intermediate organizationSome 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.
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
+
One of the troubles that is perceived to affect Grammar&nbsp;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 <math>S :> S^*,\!</math> which implies that:
  
1.3.10.9  The Cactus Language: Syntax (cont.)
+
{| align="center" cellpadding="8" width="90%"
 +
|
 +
<math>\begin{array}{lcccccccccccc}
 +
S
 +
& :>
 +
& S^*
 +
& =
 +
& \underline\varepsilon
 +
& \cup & S
 +
& \cup & S \cdot S
 +
& \cup & S \cdot S \cdot S
 +
& \cup & \ldots \\
 +
\end{array}</math>
 +
|}
  
In forming a grammar for a language, statements of the form W :> W',
+
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 <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math>  In particular, since it implies that <math>S :> \underline\varepsilon,</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},</math> for any formal language <math>\mathfrak{L},</math> the empty string <math>\varepsilon\!</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math>  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.
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
+
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 generateThis amounts, in effect, to ''recognizing a type'', a complex process that involves the following steps:
form "T <: S", read as "T is covered by S", can be interpreted
 
as saying that T is of the type SDepending on the context,
 
this can be taken in either one of two ways:
 
  
1. Treating "T" as a string variable, it means
+
# '''Noticing''' a category of strings that is generated by iteration or recursion.
    that the individual string T is typed as S.
+
# '''Acknowledging''' the fact that it needs to be covered by a non-terminal symbol.
 +
# '''Making a note of it''' by instituting an explicitly-named grammatical category.
  
2Treating "T" as a type name, it means that any
+
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 grammarIn 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.
    instance of the type T also falls under the type S.
 
  
In accordance with these interpretations, an expression like "t <: T" can be
+
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 <math>\mathfrak{C} (\mathfrak{P}),\!</math> one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:
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
+
{| align="center" cellpadding="8" width="90%"
of covering relations. The worst offense is that of allowing symbols to
+
|
stand equivocally either for individual strings or else for their types.
+
<math>\begin{array}{llll}
There is a measure of consistency to this practice, considering the fact
+
1.
that perfectly individual entities are rarely if ever grasped by means of
+
& \operatorname{Conc}_{j=1}^0
signs and finite expressions, which entails that every appearance of an
+
& =
apparent token is only a type of more particular tokens, and meaning in
+
& ^{\backprime\backprime\prime\prime}
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
+
& \operatorname{Conc}_{j=1}^k S_j
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
+
& \operatorname{Prec} (\operatorname{Conc}_{j=1}^{k-1} S_j, S_k)
tokens or the subtypes of their covering types.
+
\\ \\
 +
2.
 +
& \operatorname{Surc}_{j=1}^0
 +
& =
 +
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
 +
\\ \\
 +
& \operatorname{Surc}_{j=1}^k S_j
 +
& =
 +
& \operatorname{Subc} (\operatorname{Surc}_{j=1}^{k-1} S_j, S_k)
 +
\\ \\
 +
\end{array}</math>
 +
|}
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
In order to transform these recursive definitions into grammar rules, one introduces a new pair of intermediate symbols, <math>\operatorname{Conc}</math> and <math>\operatorname{Surc},</math> corresponding to the operations that share the same names but ignoring the inflexions of their individual parameters <math>j\!</math> and <math>k.\!</math>  Recognizing the
 +
type of a sentence by means of the initial symbol <math>S\!</math> and interpreting <math>\operatorname{Conc}</math> and <math>\operatorname{Surc}</math> 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:
  
The combined effect of several typos in my typography
+
{| align="center" cellpadding="8" width="90%"
along with what may be a lack of faith in imagination,
+
|
obliges me to redo a couple of paragraphs from before.
+
<math>\begin{array}{llll}
 +
1.
 +
& \operatorname{Conc}
 +
& :>
 +
& ^{\backprime\backprime\prime\prime}
 +
\\ \\
 +
& \operatorname{Conc}
 +
& :>
 +
& \operatorname{Conc} \cdot S
 +
\\ \\
 +
2.
 +
& \operatorname{Surc}
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
 +
\\ \\
 +
& \operatorname{Surc}
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\ \\
 +
& \operatorname{Surc}
 +
& :>
 +
& \operatorname{Surc} \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\end{array}</math>
 +
|}
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
As given, this particular fragment of the intended grammar contains a couple of features that are desirable to amend.
  
1.3.10.9  The Cactus Language: Syntax (cont.)
+
# Given the covering <math>S :> \operatorname{Conc},</math> the covering rule <math>\operatorname{Conc} :> \operatorname{Conc} \cdot S</math> says no more than the covering rule <math>\operatorname{Conc} :> S \cdot S.</math>  Consequently, all of the information contained in these two covering rules is already covered by the statement that <math>S :> S \cdot S.</math>
 +
# 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 soAccordingly, 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.
  
A notation in the style of "S :> T" is now introduced,
+
====Grammar 2====
to be read among many others in this manifold of ways:
 
  
| S covers T
+
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:
|
 
|  S governs T
 
|
 
|  S rules T
 
|
 
|  S subsumes T
 
|
 
|  S types over T
 
  
The form "S :> T" is here recruited for polymorphic
+
A ''tract'' is a concatenation of a finite sequence of sentences, with a literal comma <math>^{\backprime\backprime} \operatorname{,} ^{\prime\prime}</math> interpolated between each pair of adjacent sentences.  Thus, a typical tract <math>T\!</math> takes the form:
employment in at least the following types of roles:
 
  
1.  To signify that an individually named or quoted string T is
+
{| align="center" cellpadding="8" width="90%"
    being typed as a sentence S of the language of interest !L!.
+
|
 +
<math>\begin{array}{lllllllllll}
 +
T
 +
& =
 +
& S_1
 +
& \cdot
 +
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
 +
& \cdot
 +
& \ldots
 +
& \cdot
 +
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
 +
& \cdot
 +
& S_k
 +
\\
 +
\end{array}</math>
 +
|}
  
2.  To express the fact or to make the assertion that each member
+
A tract must be distinguished from the abstract sequence of sentences, <math>S_1, \ldots, S_k,\!</math> 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.
    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
+
Returning to the case of the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}),</math> 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:
    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
+
<br>
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
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
with the set !Q! that assembles together all of the intermediate symbols
+
| align="left" style="border-left:1px solid black;width="50%" |
results in the set {"S"} |_| !Q! of "non-terminal symbols". Completing
+
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 2}\!</math>
the package, the alphabet !A! of the language is also known as the set
+
| align="right" style="border-right:1px solid black;" width="50%" |
of "terminal symbols".  In this discussion, I will adopt the convention
+
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
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,
+
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
q in {"S"} |_| !Q!.  Finally, it is convenient to refer to all of the
+
<math>\begin{array}{rcll}
symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
+
1.
prospective grammar for the language, and accordingly to describe
+
& S
the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
+
& :>
in effect, expressing the forms that are superimposed on a language
+
& \varepsilon
by one of its conceivable grammars. In certain settings is becomes
+
\\
desirable to separate the augmented strings that contain the symbol
+
2.
"S" from all other sorts of augmented strings. In these situations,
+
& S
the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
+
& :>
as the "sentential forms" of the associated grammar.
+
& m_1
 +
\\
 +
3.
 +
& S
 +
& :>
 +
& p_j, \, \text{for each} \, j \in J
 +
\\
 +
4.
 +
& S
 +
& :>
 +
& S \, \cdot \, S
 +
\\
 +
5.
 +
& S
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\
 +
6.
 +
& T
 +
& :>
 +
& S
 +
\\
 +
7.
 +
& T
 +
& :>
 +
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
 +
\\
 +
\end{array}</math>
 +
|}
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
<br>
  
For some time to come in the discussion that follows,
+
In this rendition, a string of type <math>T\!</math> 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 <math>T\!</math> gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule <math>T :> S\!</math> means that <math>T\!</math> ''inherits'' all of the initial conditions of <math>S,\!</math> namely, <math>T \, :> \, \varepsilon, m_1, p_j.</math>  In accord with these simple beginnings it comes to parse that the rule <math>T \, :> \, T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> with the substitutions <math>T = \varepsilon</math> and <math>S = \varepsilon</math> on the covered side of the rule, bears the germinal implication that <math>T \, :> \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}.</math>
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
+
Grammar&nbsp;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 <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math> 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.
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
====Grammar 3====
  
Employing the notion of a covering relation it becomes possible to
+
Although it is not strictly necessary to do so, it is possible to organize the materials of our developing 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:
redescribe the cactus language !L! = !C!(!P!) in the following way.
 
  
Grammar 1 is something of a misnomer.  It is nowhere near exemplifying
+
A ''rune'' is a string of blanks and paints concatenated togetherThus, a typical rune <math>R\!</math> is a string over <math>\{ m_1 \} \cup \mathfrak{P},</math> possibly the empty string:
any kind of a standard form and it is only intended as a starting point
 
for the initiation of more respectable grammarsSuch 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
+
{| align="center" cellpadding="8" width="90%"
|
+
| <math>R\ \in\ ( \{ m_1 \} \cup \mathfrak{P} )^*</math>
| !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:
+
When there is no possibility of confusion, the letter <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> 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, <math>^{\backprime\backprime} R \, ^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math>  In effect, <math>^{\backprime\backprime} R \, ^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math>  In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>
  
5.  The concept of a sentence in !L! covers any
+
A ''foil'' is a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},\!</math> where <math>T\!</math> is a tract.  Thus, a typical foil <math>F\!</math> has the form:
    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
+
{| align="center" cellpadding="8" width="90%"
    surcatenation of sentences in !L!, in effect,
+
|
    any string that opens with a "-(", continues
+
<math>\begin{array}{*{15}{l}}
    with a sentence, possibly empty, follows with
+
F
    a finite number of phrases of the form "," · S,
+
& =
    and closes with a ")-".
+
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime}
 +
& \cdot
 +
& S_1
 +
& \cdot
 +
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
 +
& \cdot
 +
& \ldots
 +
& \cdot
 +
& ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}
 +
& \cdot
 +
& S_k
 +
& \cdot
 +
& ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\
 +
\end{array}</math>
 +
|}
  
This appears to be just about the most concise description
+
This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math>  Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math>  Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>^{\backprime\backprime} F \, ^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language <math>\mathfrak{C} (\mathfrak{P}).\!</math> 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 <math>^{\backprime\backprime} F \, ^{\prime\prime}.</math>
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 acceptableBriefly stated,
 
these problems turn on the following properties
 
of the presentation:
 
  
1.  The invocation of the kleene star operation
+
<br>
    is not reduced to a manifestly finitary form.
 
  
2. The type of a sentence S is allowed to cover
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
    not only itself but also the empty string.
+
| align="left"  style="border-left:1px solid black;"  width="50%" |
 +
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!</math>
 +
| align="right" style="border-right:1px solid black;" width="50%" |
 +
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
 +
|-
 +
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
 +
<math>\begin{array}{rcll}
 +
1.
 +
& S
 +
& :>
 +
& R
 +
\\
 +
2.
 +
& S
 +
& :>
 +
& F
 +
\\
 +
3.
 +
& S
 +
& :>
 +
& S \, \cdot \, S
 +
\\
 +
4.
 +
& R
 +
& :>
 +
& \varepsilon
 +
\\
 +
5.
 +
& R
 +
& :>
 +
& m_1
 +
\\
 +
6.
 +
& R
 +
& :>
 +
& p_j, \, \text{for each} \, j \in J
 +
\\
 +
7.
 +
& R
 +
& :>
 +
& R \, \cdot \, R
 +
\\
 +
8.
 +
& F
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\
 +
9.
 +
& T
 +
& :>
 +
& S
 +
\\
 +
10.
 +
& T
 +
& :>
 +
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
 +
\\
 +
\end{array}\!</math>
 +
|}
  
I will discuss these issues at first in general, and especially in regard to
+
<br>
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
+
In Grammar&nbsp;3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types.  Rules&nbsp;4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types.  Rule&nbsp;8 characterizes a foil <math>F\!</math> as a string of the form <math>{}^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},\!</math> where <math>T\!</math> is a tract.  The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),\!</math> 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.
  
In the process of developing a grammar for a language, it is possible
+
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.
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.
+
Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),\!</math> in other words, the formal language <math>\operatorname{PARCE}\!</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:
  
2.  The "distinction between empty and significant strings", and thus
+
<ol style="list-style-type:decimal">
    the "distinction between empty and significant types of strings".
 
  
3.  The "principle of intermediate significance".  This is a constraint
+
<li>
    on the grammar that arises from considering the interaction of the
+
<p>The ''emptily painted and rooted cactus expressions'' make up the language <math>\operatorname{EPARCE}</math> that consists of a single empty string as its only sentence. In short:</p>
    first two issues.
 
  
In responding to these issues, it is advisable at first to proceed in
+
<p><math>\operatorname{EPARCE} \ = \ \underline\varepsilon \ = \ \{ \varepsilon \}</math></p></li>
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
+
<li>
intermediate symbols it has and by how they interact with each other by means
+
<p>The ''significantly painted and rooted cactus expressions'' make up the language <math>\operatorname{SPARCE}</math> that consists of everything else, namely, all of the non-empty strings in the language <math>\operatorname{PARCE}.</math>  In sum:</p>
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
+
<p><math>\operatorname{SPARCE} \ = \ \operatorname{PARCE} \setminus \varepsilon</math></p></li>
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 |_| ...
+
</ol>
  
There is nothing wrong with the more expansive pan of the covered equation,
+
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:
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
+
{| align="center" cellpadding="8" width="90%"
the kleene star operation, by reducing it to a purely finitary set of steps,
+
| <math>\operatorname{SPARCE} \ = \ \operatorname{PARCE} \ - \ \operatorname{EPARCE}</math>
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
+
In sum, one has the disjoint union:
    is generated by iteration or recursion.
 
  
2.  Acknowledging the circumstance that the noted category
+
{| align="center" cellpadding="8" width="90%"
    of strings needs to be covered by a non-terminal symbol.
+
| <math>\operatorname{PARCE} \ = \ \operatorname{EPARCE} \ \cup \ \operatorname{SPARCE}</math>
 +
|}
  
3Making a note of it by declaring and instituting
+
For brevity in the present case, and to serve as a generic device in any similar array of situations, let <math>S\!</math> be the type of an arbitrary sentence, possibly empty, and let <math>S'\!</math> be the type of a specifically non-empty sentenceIn addition, let <math>\underline\varepsilon</math> be the type of the empty sentence, in effect, the language
    an explicitly and even expressively named category.
+
<math>\underline\varepsilon = \{ \varepsilon \}</math> that contains a single empty string, and let a plus sign <math>^{\backprime\backprime} + ^{\prime\prime}</math> signify a disjoint union of types. In the most general type of situation, where the type <math>S\!</math> is permitted to include the empty string, one notes the following relation among types:
  
In sum, one introduces a non-terminal symbol for each type of sentence and
+
{| align="center" cellpadding="8" width="90%"
each "part of speech" or sentential component that is generated by means of
+
| <math>S \ = \ \underline\varepsilon \ + \ S'</math>
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
+
With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})</math> that is afforded by Grammar&nbsp;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&nbsp;3, and start again with a single intermediate symbol, as used in Grammar&nbsp;2.  One way of carrying out this strategy leads on to a grammar of the variety that will be articulated next.
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
====Grammar 4====
  
Returning to the case of the cactus language, the process of recognizing an
+
If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar&nbsp;2, then the non-terminal symbols <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> and <math>^{\backprime\backprime} T \, ^{\prime\prime}</math> give rise to the expanded set of non-terminal symbols <math>^{\backprime\backprime} S \, ^{\prime\prime}, \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime},</math> leaving the last three of these to form the new intermediate alphabetGrammar&nbsp;4 has the intermediate alphabet <math>\mathfrak{Q} \, = \, \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \},</math> with the set <math>\mathfrak{K}</math> of covering rules as listed in the next display.
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 partFor 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        =  "".
+
<br>
  
    Conc^k_j S_j Prec(Conc^(k-1)_j S_j, S_k).
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
 +
| align="left" style="border-left:1px solid black;" width="50%" |
 +
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 4}\!</math>
 +
| align="right" style="border-right:1px solid black;" width="50%" |
 +
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \}</math>
 +
|-
 +
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
 +
<math>\begin{array}{rcll}
 +
1.
 +
& S
 +
& :>
 +
& \varepsilon
 +
\\
 +
2.
 +
& S
 +
& :>
 +
& S'
 +
\\
 +
3.
 +
& S'
 +
& :>
 +
& m_1
 +
\\
 +
4.
 +
& S'
 +
& :>
 +
& p_j, \, \text{for each} \, j \in J
 +
\\
 +
5.
 +
& S'
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\
 +
6.
 +
& S'
 +
& :>
 +
& S' \, \cdot \, S'
 +
\\
 +
7.
 +
& T
 +
& :>
 +
& \varepsilon
 +
\\
 +
8.
 +
& T
 +
& :>
 +
& T'
 +
\\
 +
9.
 +
& T'
 +
& :>
 +
& T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S
 +
\\
 +
\end{array}</math>
 +
|}
  
2.  Surc^0        =  "-()-".
+
<br>
  
    Surc^k_j S_j  = Subc(Surc^(k-1)_j S_j, S_k).
+
In this version of a grammar for <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}),</math> the intermediate type <math>T\!</math> is partitioned as <math>T = \underline\varepsilon + T',</math> thereby parsing the intermediate symbol <math>T\!</math> in parallel fashion with the division of its overlying type as <math>S = \underline\varepsilon + S'.</math>  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.
  
In order to transform these recursive definitions into grammar rules,
+
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 stagesConsequently, 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.
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 kRecognizing 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:
 
  
1Conc  :> "".
+
For the sake of future reference, it is convenient to refer to this restriction on intermediate symbols as the ''intermediate significance'' constraintIt can be stated in a compact form as a condition on the relations between non-terminal symbols <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}</math> and sentential forms <math>W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*.</math>
  
    Conc  :> Conc · S.
+
<br>
  
2.  Surc  :> "-()-".
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
 +
| align="center" style="border-left:1px solid black; border-right:1px solid black" |
 +
<math>\text{Condition On Intermediate Significance}\!</math>
 +
|-
 +
| style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
 +
<math>\begin{array}{lccc}
 +
\text{If}
 +
& q
 +
& :>
 +
& W
 +
\\
 +
\text{and}
 +
& W
 +
& =
 +
& \varepsilon
 +
\\
 +
\text{then}
 +
& q
 +
& =
 +
& ^{\backprime\backprime} S \, ^{\prime\prime}
 +
\\
 +
\end{array}</math>
 +
|}
  
    Surc  :> "-(" · S · ")-".
+
<br>
  
    Surc :> Surc ")-"^(-1) · "," · 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, <math>^{\backprime\backprime}\!< \, ^{\prime\prime}.</math>
  
As given, this particular fragment of the intended grammar
+
# The ordering <math>^{\backprime\backprime}\!< \, ^{\prime\prime}</math> on the set of non-terminal symbols, <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},</math> ordains the initial symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> to be strictly prior to every intermediate symbol.  This is tantamount to the axiom that <math>^{\backprime\backprime} S \, ^{\prime\prime} < q,</math> for all <math>q \in \mathfrak{Q}.</math>
contains a couple of features that are desirable to amend.
+
# The ordering <math>^{\backprime\backprime}\!< \, ^{\prime\prime}</math> on the collection of sentential forms, <math>W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*,</math> ordains the empty string to be strictly minor to every other sentential form.  This is stipulated in the axiom that <math>\varepsilon < W,</math> for every non-empty sentential form <math>W.\!</math>
  
1.  Given the covering S :> Conc, the covering rule Conc :> Conc · S
+
Given these two orderings, the constraint in question on intermediate significance can be stated as follows:
    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,
+
<br>
    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
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
introduce an auxiliary type of string, not in general a sentence, but
+
| align="center" style="border-left:1px solid black; border-right:1px solid black" |
a proper component of any sentence that is formed by surcatenation.
+
<math>\text{Condition On Intermediate Significance}\!</math>
Doing this brings one to the following definition:
+
|-
 +
| style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
 +
<math>\begin{array}{lccc}
 +
\text{If}
 +
& q
 +
& :>
 +
& W
 +
\\
 +
\text{and}
 +
& q
 +
& >
 +
& ^{\backprime\backprime} S \, ^{\prime\prime}
 +
\\
 +
\text{then}
 +
& W
 +
& >
 +
& \varepsilon
 +
\\
 +
\end{array}</math>
 +
|}
  
A "tract" is a concatenation of a finite sequence of sentences, with a
+
<br>
literal comma "," interpolated between each pair of adjacent sentences.
 
Thus, a typical tract T takes the form:
 
  
T  =  S_1 · "," · ..· "," · S_k.
+
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 questionIn 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.
  
A tract must be distinguished from the abstract sequence of sentences,
+
====Grammar 5====
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!),
+
With the foregoing array of considerations in mind, one is gradually led to a grammar for <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> in which all of the covering productions have either one of the following two forms:
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
+
{| align="center" cellpadding="8" width="90%"
|
 
| !Q! = {"T"}
 
|
 
| 1.  S  :>  !e!
 
|
 
| 2.  S  :>  m_1
 
|
 
| 3.  S  :>  p_j, for each j in J
 
 
|
 
|
| 4.  S :> S · S
+
<math>\begin{array}{ccll}
|
+
S
| 5.  S  :> "-(" · T · ")-"
+
& :>
|
+
& \varepsilon
| 6.  T  :> S
+
&
|
+
\\
| 7.  T  :>  T · "," · S
+
q
 +
& :>
 +
& W,
 +
& \text{with} \ q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \ \text{and} \ W \in (\mathfrak{Q} \cup \mathfrak{A})^+
 +
\\
 +
\end{array}</math>
 +
|}
  
In this rendition, a string of type T is not in general
+
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 <math>q :> W,\!</math> the replacement string <math>W\!</math> 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 grammarThis 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 <math>S :> \varepsilon,</math> 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.
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 componentsIn 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
+
Grammar&nbsp;5 is a context-free grammar for the painted cactus language that uses <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with <math>\mathfrak{K}</math> as listed in the next display.
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
<br>
  
Grammar 2 achieves a portion of its success through a higher degree of
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
intermediate organization. Roughly speaking, the level of organization
+
| align="left"  style="border-left:1px solid black;" width="50%" |
can be seen as reflected in the cardinality of the intermediate alphabet
+
<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 5}\!</math>
!Q! = {"T"}, but it is clearly not explained by this simple circumstance
+
| align="right" style="border-right:1px solid black;" width="50%" |
alone, since it is taken for granted that the intermediate symbols serve
+
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>
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
+
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
trouble of exploring this aspect of organization and this direction of
+
<math>\begin{array}{rcll}
development a little further. Although it is not strictly necessary
+
1.
to do so, it is possible to organize the materials of the present
+
& S
grammar in a slightly better fashion by recognizing two recurrent
+
& :>
types of strings that appear in the typical cactus expression.
+
& \varepsilon
In doing this, one arrives at the following two definitions:
+
\\
 +
2.
 +
& S
 +
& :>
 +
& S'
 +
\\
 +
3.
 +
& S'
 +
& :>
 +
& m_1
 +
\\
 +
4.
 +
& S'
 +
& :>
 +
& p_j, \, \text{for each} \, j \in J
 +
\\
 +
5.
 +
& S'
 +
& :>
 +
& S' \, \cdot \, S'
 +
\\
 +
6.
 +
& S'
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
 +
\\
 +
7.
 +
& S'
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\
 +
8.
 +
& T
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
 +
\\
 +
9.
 +
& T
 +
& :>
 +
& S'
 +
\\
 +
10.
 +
& T
 +
& :>
 +
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
 +
\\
 +
11.
 +
& T
 +
& :>
 +
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S'
 +
\\
 +
\end{array}</math>
 +
|}
  
A "rune" is a string of blanks and paints concatenated together.
+
<br>
Thus, a typical rune R is a string over {m_1} |_| !P!, possibly
 
the empty string.
 
  
R in  ({m_1} |_| !P!)*.
+
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&nbsp;2, while respecting the principle of intermediate significance, and thus accumulating all the benefits of the context-free format in Grammar&nbsp;5. A plausible synthesis of most of these features is given in Grammar&nbsp;6.
  
When there is no possibility of confusion, the letter "R" can be used
+
====Grammar 6====
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.
+
Grammar&nbsp;6 has the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with the production set <math>\mathfrak{K}</math> as listed in the next display.
Thus, a typical foil F has the form:
 
  
F  =  "-(" · S_1 · "," · ... · "," · S_k · ")-".
+
<br>
  
This is just the surcatenation of the sentences S_1, ..., S_k.
+
{| align="center" cellpadding="12" cellspacing="0" style="border-top:1px solid black" width="90%"
Given the possibility that this sequence of sentences is empty,
+
| align="left" style="border-left:1px solid black;"  width="50%" |
and thus that the tract T is the empty string, the minimum foil
+
<math>{\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}}\!</math>
F is the expression "-()-". Explicitly marking each foil F that
+
| align="right" style="border-right:1px solid black;" width="50%" |
is embodied in a cactus expression is tantamount to recognizing
+
<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}\!</math>
another intermediate symbol, "F" in !Q!, further articulating the
+
|-
structures of sentences and expanding the grammar for the language
+
| colspan="2" style="border-top:1px solid black; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black" |
!C!(!P!). All of the same remarks about the versatile uses of the
+
<math>\begin{array}{rcll}
intermediate symbols, as string variables and as type names, apply
+
1.
again to the letter "F".
+
& S
 +
& :>
 +
& \varepsilon
 +
\\
 +
2.
 +
& S
 +
& :>
 +
& S'
 +
\\
 +
3.
 +
& S'
 +
& :>
 +
& R
 +
\\
 +
4.
 +
& S'
 +
& :>
 +
& F
 +
\\
 +
5.
 +
& S'
 +
& :>
 +
& S' \, \cdot \, S'
 +
\\
 +
6.
 +
& R
 +
& :>
 +
& m_1
 +
\\
 +
7.
 +
& R
 +
& :>
 +
& p_j, \, \text{for each} \, j \in J
 +
\\
 +
8.
 +
& R
 +
& :>
 +
& R \, \cdot \, R
 +
\\
 +
9.
 +
& F
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}
 +
\\
 +
10.
 +
& F
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}
 +
\\
 +
11.
 +
& T
 +
& :>
 +
& ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
 +
\\
 +
12.
 +
& T
 +
& :>
 +
& S'
 +
\\
 +
13.
 +
& T
 +
& :>
 +
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime}
 +
\\
 +
14.
 +
& T
 +
& :>
 +
& T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S'
 +
\\
 +
\end{array}</math>
 +
|}
  
| !C!(!P!).  Grammar 3
+
<br>
|
 
| !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),
+
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.
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
+
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:
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:
+
{| align="center" cellpadding="8" width="90%"
What is a practical strategy for accounting for the empty string in the
+
|-
organization of any formal language that counts it among its sentences?
+
| <math>\operatorname{If}</math>
One answer that presents itself is this:  If the empty string belongs to
+
| the strings <math>S_1, \ldots, S_k\!</math> are sentences,
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
+
| <math>\operatorname{Then}</math>
interesting materials.
+
| their concatenation in the form
 +
|-
 +
| &nbsp;
 +
| <math>\operatorname{Conc}_{j=1}^k S_j \ = \ S_1 \, \cdot \, \ldots \, \cdot \, S_k</math>
 +
|-
 +
| &nbsp;
 +
| is a sentence,
 +
|-
 +
| <math>\operatorname{And}</math>
 +
| their surcatenation in the form
 +
|-
 +
| &nbsp;
 +
| <math>\operatorname{Surc}_{j=1}^k S_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}</math>
 +
|-
 +
| &nbsp;
 +
| is a sentence.
 +
|}
  
Returning to the case of the cactus language !C!(!P!), that is,
+
===Generalities About Formal Grammars===
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:
 
  
1The "emptily painted and rooted cactus expressions"
+
It is fitting to wrap up the foregoing developments by summarizing the notion of a formal grammar that appeared to evolve in the present caseFor 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&ndash;61).
    make up the language EPARCE that consists of
 
    a single empty string as its only sentence.
 
    In short:
 
  
    EPARCE  = {""}.
+
A ''formal grammar'' <math>\mathfrak{G}</math> is given by a four-tuple <math>\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> that takes the following form of description:
  
2.  The "significantly painted and rooted cactus expressions"
+
<ol style="list-style-type:decimal">
    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 \ "".
+
<li><math>^{\backprime\backprime} S \, ^{\prime\prime}</math> is the ''initial'', ''special'', ''start'', or ''sentence'' symbol. Since the letter <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> 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.</li>
  
As a result of marking the distinction between empty and significant sentences,
+
<li><math>\mathfrak{Q} = \{ q_1, \ldots, q_m \}</math> is a finite set of ''intermediate symbols'', all distinct from <math>^{\backprime\backprime} S \, ^{\prime\prime}.</math></li>
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.
+
<li><math>\mathfrak{A} = \{ a_1, \dots, a_n \}</math> is a finite set of ''terminal symbols'', also known as the ''alphabet'' of <math>\mathfrak{G},</math> all distinct from <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> and disjoint from <math>\mathfrak{Q}.</math> Depending on the particular conception of the language <math>\mathfrak{L}</math> that is ''covered'', ''generated'', ''governed'', or ''ruled'' by the grammar <math>\mathfrak{G},</math> that is, whether <math>\mathfrak{L}</math> is conceived to be a set of words, sentences, paragraphs, or more extended structures of discourse, it is usual to describe <math>\mathfrak{A}</math> as the ''alphabet'', ''lexicon'', ''vocabulary'', ''liturgy'', or ''phrase book'' of both the grammar <math>\mathfrak{G}</math> and the language <math>\mathfrak{L}</math> that it regulates.</li>
  
In sum, one has the disjoint union:
+
<li><math>\mathfrak{K}</math> 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''.</li>
  
PARCE  =  EPARCE |_| SPARCE.
+
</ol>
  
For brevity in the present case, and to serve as a generic device
+
To describe the elements of <math>\mathfrak{K}</math> it helps to define some additional terms:
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:
 
  
= %e%  +  S'.
+
<ol style="list-style-type:lower-latin">
  
Consequences of the distinction between empty expressions and
+
<li>The symbols in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A}</math> form the ''augmented alphabet'' of <math>\mathfrak{G}.</math></li>
significant expressions are taken up for discussion next time.
 
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
<li>The symbols in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}</math> are the ''non-terminal symbols'' of <math>\mathfrak{G}.</math></li>
  
1.3.10.9  The Cactus Language:  Syntax (cont.)
+
<li>The symbols in <math>\mathfrak{Q} \cup \mathfrak{A}</math> are the ''non-initial symbols'' of <math>\mathfrak{G}.</math></li>
  
With the distinction between empty and significant expressions in mind,
+
<li>The strings in <math>( \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*</math> are the ''augmented strings'' for <math>\mathfrak{G}.</math></li>
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
+
<li>The strings in <math>\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*</math> are the ''sentential forms'' for <math>\mathfrak{G}.</math></li>
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
+
</ol>
|
 
| !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
+
Each characterization in <math>\mathfrak{K}</math> is an ordered pair of strings <math>(S_1, S_2)\!</math> that takes the following form:
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
+
{| align="center" cellpadding="8" width="90%"
approach to types.  It is reasonable and consistent in its underlying
+
| <math>S_1 \ = \ Q_1 \cdot q \cdot Q_2,</math>
principle, and it provides a rational and a homogeneous strategy toward
+
|-
all parts of speech, but it does require an extra amount of conceptual
+
| <math>S_2 \ = \ Q_1 \cdot W \cdot Q_2.</math>
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
+
In this scheme, <math>S_1\!</math> and <math>S_2\!</math> are members of the augmented strings for <math>\mathfrak{G},</math> more precisely, <math>S_1\!</math> is a non-empty string and a sentential form over <math>\mathfrak{G},</math> while <math>S_2\!</math> is a possibly empty string and also a sentential form over <math>\mathfrak{G}.</math>
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
+
Here also, <math>q\!</math> is a non-terminal symbol, that is, <math>q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},</math> while <math>Q_1, Q_2,\!</math> and <math>W\!</math> are possibly empty strings of non-initial symbols, a fact that can be expressed in the form, <math>Q_1, Q_2, W \in (\mathfrak{Q} \cup \mathfrak{A})^*.</math>
|
 
| If    q :> W
 
|
 
| and   W =  !e!,
 
|
 
| then  q  =  "S".
 
  
If this is beginning to sound like a monotone condition, then it is
+
In practice, the couplets in <math>\mathfrak{K}</math> are used to ''derive'', to ''generate'', or to ''produce'' sentences of the corresponding language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}).</math> The language <math>\mathfrak{L}</math> is then said to be ''governed'', ''licensed'', or ''regulated'' by the grammar <math>\mathfrak{G},</math> a circumstance that is expressed in the form <math>\mathfrak{L} = \langle \mathfrak{G} \rangle.</math>  In order to facilitate this active employment of the grammar, it is conventional to write the abstract characterization <math>(S_1, S_2)\!</math> and the specific characterization <math>(Q_1 \cdot q \cdot Q_2, \ Q_1 \cdot W \cdot Q_2)</math> in the following forms, respectively:
not absurd to sharpen the resemblance and render the likeness more
 
acuteThis 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,
+
{| align="center" cellpadding="8" width="90%"
    q in {"S"} |_| !Q!, ordains the initial symbol "S"
+
|
    to be strictly prior to every intermediate symbol.
+
<math>\begin{array}{lll}
    This is tantamount to the axiom that "S" < q,
+
S_1
    for all q in !Q!.
+
& :>
 +
& S_2
 +
\\
 +
Q_1 \cdot q \cdot Q_2
 +
& :>
 +
& Q_1 \cdot W \cdot Q_2
 +
\\
 +
\end{array}</math>
 +
|}
  
2.  The ordering "<" on the collection of sentential forms,
+
In this usage, the characterization <math>S_1 :> S_2\!</math> is tantamount to a grammatical license to transform a string of the form <math>Q_1 \cdot q \cdot Q_2</math> into a string of the form <math>Q1 \cdot W \cdot Q2,</math> in effect, replacing the non-terminal symbol <math>q\!</math> with the non-initial string <math>W\!</math> in any selected, preserved, and closely adjoining context of the form <math>Q1 \cdot \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])} \cdot Q2.</math>  In this application the notation <math>S_1 :> S_2\!</math> can be read to say that <math>S_1\!</math> ''produces'' <math>S_2\!</math> or that <math>S_1\!</math> ''transforms into'' <math>S_2.\!</math>
    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
+
An ''immediate derivation'' in <math>\mathfrak{G}\!</math> is an ordered pair <math>(W, W^\prime)\!</math> of sentential forms in <math>\mathfrak{G}\!</math> such that:
on intermediate significance can be stated as follows:
 
  
| Condition Of Intermediate Significance
+
{| align="center" cellpadding="8" width="90%"
 
|
 
|
| If    q  :> W
+
<math>\begin{array}{llll}
|
+
W = Q_1 \cdot X \cdot Q_2,
| and   q  >  "S",
+
& W' = Q_1 \cdot Y \cdot Q_2,
|
+
& \text{and}
| then  W  >  !e!.
+
& (X, Y) \in \mathfrak{K}.
 +
\end{array}</math>
 +
|}
  
Achieving a grammar that respects this convention typically requires a more
+
As noted above, it is usual to express the condition <math>(X, Y) \in \mathfrak{K}</math> by writing <math>X :> Y \, \text{in} \, \mathfrak{G}.</math>
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
+
The immediate derivation relation is indicated by saying that <math>W\!</math> ''immediately derives'' <math>W',\!</math> by saying that <math>W'\!</math> is ''immediately derived'' from <math>W\!</math> in <math>\mathfrak{G},</math> and also by writing:
  
1.3.10.9  The Cactus Language: Syntax (cont.)
+
{| align="center" cellpadding="8" width="90%"
 +
| <math>W ::> W'.\!</math>
 +
|}
  
With the array of foregoing considerations in mind,
+
A ''derivation'' in <math>\mathfrak{G}</math> is a finite sequence <math>(W_1, \ldots, W_k)\!</math> of sentential forms over <math>\mathfrak{G}</math> such that each adjacent pair <math>(W_j, W_{j+1})\!</math> of sentential forms in the sequence is an immediate derivation in <math>\mathfrak{G},</math> in other words, such that:
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!
+
{| align="center" cellpadding="8" width="90%"
|
+
| <math>W_j ::> W_{j+1},\ \text{for all}\ j = 1\ \text{to}\ k - 1.</math>
| 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.
+
If there exists a derivation <math>(W_1, \ldots, W_k)\!</math> in <math>\mathfrak{G},</math> one says that <math>W_1\!</math> ''derives'' <math>W_k\!</math> in <math>\mathfrak{G}</math> or that <math>W_k\!</math> is ''derivable'' from <math>W_1\!</math> in <math>\mathfrak{G},</math> and one
The first type of rewrite rule is referred to as a "special production",
+
typically summarizes the derivation by writing:
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
+
{| align="center" cellpadding="8" width="90%"
that uses !Q! = {"S'", "T"}, with !K! as listed in the next display.
+
| <math>W_1 :\!*\!:> W_k.\!</math>
 +
|}
  
| !C!(!P!).  Grammar 5
+
The language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle</math> that is ''generated'' by the formal grammar <math>\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> is the set of strings over the terminal alphabet <math>\mathfrak{A}</math> that are derivable from the initial symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> by way of the intermediate symbols in <math>\mathfrak{Q}</math> according to the characterizations in <math>\mathfrak{K}.</math>  In sum:
|
 
| !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
+
{| align="center" cellpadding="8" width="90%"
|
+
| <math>\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ \, W \in \mathfrak{A}^* \, : \, ^{\backprime\backprime} S \, ^{\prime\prime} \, :\!*\!:> \, W \, \}.</math>
| !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
+
Finally, a string <math>W\!</math> is called a ''word'', a ''sentence'', or so on, of the language generated by <math>\mathfrak{G}</math> if and only if <math>W\!</math> is in <math>\mathfrak{L} (\mathfrak{G}).</math>
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 Cactus Language : Stylistics===
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,
+
{| align="center" cellpadding="0" cellspacing="0" width="90%"
 
|
 
|
| then their concatenation in the form
+
<p>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 &mdash; shifts of gaze &mdash; of every particular discipline and subspecialty.
|
+
</p>
|       Conc^k_j S_j  = S_1 · ... · S_k
+
|-
|
+
| align="right" | &mdash; Herbert J. Bernstein, "Idols of Modern Science", [HJB, 38]
|      is a sentence,
+
|}
|
+
 
| and  their surcatenation in the form
+
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.
|
 
|      Surc^k_j S_j =  "-(" · S_1 · "," · ... · "," · S_k · ")-"
 
|
 
|      is a sentence.
 
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
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.
  
1.3.10.9 The Cactus Language: Syntax (cont.)
+
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 universalFor 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.
  
It is fitting to wrap up the foregoing developments by summarizing the
+
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 freeBut 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.
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 languageThe 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!)
+
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:
that takes the following form of description:
 
  
1.  "S" is the "initial", "special", "start", or "sentence symbol".
+
{| align="center" cellpadding="8" width="90%"
    Since the letter "S" serves this function only in a special setting,
+
| <math>X \Rightarrow Y\ \operatorname{and}\ Y \Rightarrow Z.</math>
    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",
+
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 themThe 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:
    all distinct from "S".
 
  
3.  !A! = {a_1, ..., a_n} is a finite set of "terminal symbols",
+
{| align="center" cellpadding="8" width="90%"
    also known as the "alphabet" of !G!, all distinct from "S" and
+
| <math>X\ \le\ Y\ \le\ Z.</math>
    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
+
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:
    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:
+
{| align="center" cellpadding="8" width="90%"
 +
| <math>Z \Leftarrow Y\ \operatorname{and}\ Y \Leftarrow X.</math>
 +
|}
  
a.  The symbols in {"S"} |_| !Q! |_| !A! form the "augmented alphabet" of !G!.
+
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 themFor 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.
  
b.  The symbols in {"S"} |_| !Q! are the "non-terminal symbols" of !G!.
+
This patch of the ongoing discussion is concerned with describing a particular variety of formal languages, whose typical representative is the painted cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).\!</math>  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?
  
cThe symbols in !Q! |_| !A! are the "non-initial symbols" of !G!.
+
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 constitutionsAs of yet, these sentences subsist in the form of purely abstract, formal, and uninterpreted combinatorial constructions.
  
dThe strings in ({"S"} |_| !Q! |_| !A!)* are the "augmented strings" for G.
+
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 stringsThis calls for a dimension of semantics and a notion of interpretation, topics that are taken up for the case of the cactus language <math>\mathfrak{C} (\mathfrak{P})</math> 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.
  
e.  The strings in {"S"} |_| (!Q! |_| !A!)* are the "sentential forms" for G.
+
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 &mdash; these are questions for an extended future inquiry.
  
Each characterization in !K! is an ordered pair of strings (S_1, S_2)
+
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.
that takes the following form:
 
  
| S_1 =  Q_1 · q · Q_2,
+
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.
|
 
| S_2  =  Q_1 · W · Q_2.
 
  
In this scheme, S_1 and S_2 are members of the augmented strings for !G!,
+
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.
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!,
+
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.
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",
+
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.
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
+
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.
|
 
| Q_1 · q · Q_2  :>  Q_1 · W · Q_2
 
  
In this usage, the characterization S_1 :> S_2 is tantamount to a grammatical
+
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 exerciseThe 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.
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 · ... · Q2Accordingly, 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')
+
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.
of sentential forms in !G! such that:
 
  
| W  = Q_1 · X · Q_2,
+
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 itselfWithin 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.
|
 
| W'  =  Q_1 · Y · Q_2,
 
|
 
| and  (X, Y)  in !K!,
 
|
 
| i.eX :> Y  in !G!.
 
  
This relation is indicated by saying that W "immediately derives" W',
+
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.
that W' is "immediately derived" from W in !G!, and also by writing:
 
  
W ::> W'.
+
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.
  
A "derivation" in !G! is a finite sequence (W_1, ..., W_k)
+
The application of these considerations in the immediate setting is this:  Do not worry too much about what roles the empty string <math>\varepsilon \, = \, ^{\backprime\backprime\prime\prime}</math> and the blank symbol <math>m_1 \, = \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}</math> 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.
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.
+
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.
  
If there exists a derivation (W_1, ..., W_k) in !G!,
+
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.
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.
+
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 questionWhat is the root of this evolution?
  
The language !L! = !L!(!G!) = <!G!> that is "generated"
+
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.
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}.
+
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:
  
Finally, a string W is called a "word", a "sentence", or so on,
+
# The ''additive'' or ''parallel'' styles favor ''sums of products'' <math>(\textstyle\sum\prod)</math> 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).
of the language generated by !G! if and only if W is in !L!(!G!).
+
# The ''multiplicative'' or ''serial'' styles favor ''products of sums'' <math>(\textstyle\prod\sum)</math> 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).
  
Reference
+
There is a curious sort of diagnostic clue 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.
  
| Denning, P.J., Dennis, J.B., Qualitz, J.E.,
+
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.
|'Machines, Languages, and Computation',
 
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
 
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
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.
  
1.3.10.10 The Cactus Language: Stylistics
+
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 symbolExpressed 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:
  
| As a result, we can hardly conceive of how many possibilities there are
+
{| align="center" cellpadding="8" width="90%"
| 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,
+
<math>\begin{matrix}
|'New Ways of Knowing: The Sciences, Society, & Reconstructive Knowledge',
+
q & :> & W_1 \\
| Rowman & Littlefield, Totowa, NJ, 1987.
+
\\
 +
\cdots & \cdots & \cdots \\
 +
\\
 +
q & :> & W_k \\
 +
\end{matrix}</math>
 +
|}
  
This Subsection highlights an issue of "style" that arises in describing
+
It is useful to examine the relationship between the grammatical covering or production relation <math>(:>\!)</math> and the logical relation of implication <math>(\Rightarrow),</math> with one eye to what they have in common and one eye to how they differThe production <math>q :> W\!</math> says that the appearance of the symbol <math>q\!</math> in a sentential form implies the possibility of exchanging it for <math>W.\!</math> Although this sounds like a ''possible implication'', to the extent that ''<math>q\!</math> implies a possible <math>W\!</math>'' or that ''<math>q\!</math> possibly implies <math>W,\!</math>'' 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 <math>^{\backprime\backprime} \, q \Leftarrow W \, ^{\prime\prime}</math> as the best analogue for the sense of the production.
a formal languageIn 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 commonFor 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
+
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 <math>W,\!</math> then it is an admissible hypothesis, according to the theory of the language that is constituted by the formal grammar, that this piece not only fits the type <math>q\!</math> but even comes to be generated under the auspices of the non-terminal symbol <math>^{\backprime\backprime} q ^{\prime\prime}.</math>
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,
+
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 requirementsThe 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 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 universalFor 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
+
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.
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
+
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.
orders, consider the kinds of order that are expressed and exhibited in
 
the following conjunction of implications:
 
  
"X => Y and Y => Z".
+
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.
  
Here, there is a happy conformity between the logical content and the
+
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 appearThis 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 ''sole 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 operatorsFor 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.
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 conjunctionThe logical content is given by the order of
 
propositions in the extended implicational sequence:
 
  
X  =<  Y  =<  Z.
+
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.
  
To see the difference between form and content, or manner and matter,
+
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.
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".
+
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.
  
Any style of declarative programming, also called "logic programming",
+
For example, given the sets <math>P\!</math> and <math>Q,\!</math> overlapping or not, one can define the ''indexed'' or ''marked'' sets <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> amounting to the copy of <math>P\!</math> into the first part of the co-product and the copy of <math>Q\!</math> into the second part of the co-product, in the following manner:
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
+
{| align="center" cellpsadding="8" width="90%"
particular variety of formal languages, whose typical representative
+
|
is the painted cactus language !L! = !C!(!P!).  It is the intention of
+
<math>\begin{array}{lllll}
this work to interpret this language for propositional logic, and thus
+
P_{[1]} & = & (P, 1) & = & \{ (x, 1) : x \in P \}, \\
to use it as a sentential calculus, an order of reasoning that forms an
+
Q_{[2]} & = & (Q, 2) & = & \{ (x, 2) : x \in Q \}. \\
active ingredient and a significant component of all logical reasoning.
+
\end{array}</math>
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
+
Using the coproduct operator (<math>\textstyle\coprod</math>) for this construction, the ''sum'', the ''coproduct'', or the ''disjointed union'' of <math>P\!</math> and <math>Q\!</math> in that order can be represented as the ordinary union of <math>P_{[1]}\!</math> and <math>Q_{[2]}.\!</math>
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,
+
{| align="center" cellpsadding="8" width="90%"
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,
+
<math>\begin{array}{lll}
topics that are taken up for the case of the cactus language !C!(!P!)
+
P \coprod Q & = & P_{[1]} \cup Q_{[2]}. \\
in Subsection 1.3.10.12.  Once a plausible semantics is prescribed for
+
\end{array}</math>
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
+
The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2</math> of the formal languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2</math> without the extra <math>\times\!</math>'s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clearOne 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 informationAs 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.
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 setThey 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 notThus, 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
+
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:
  
1.3.10.10  The Cactus Language: Stylistics (cont.)
+
:{| cellpadding="8"
 +
| <math>^{\backprime\backprime}</math>
 +
| <math>\ldots \times X \times Q \times X \times \ldots</math>
 +
| <math>^{\prime\prime}</math>
 +
|}
  
Perhaps I ought to comment on the differences between the present and
+
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 relationSomewhat sketchily, the strait that corresponds to the stricture just given can be pictured in the following shape:
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 explorationAll 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
+
:{| cellpadding="8"
to encompass the sentence symbol, mainly on account of all the contexts
+
| &nbsp;
where the initial and the intermediate symbols are naturally invoked in
+
| <math>\ldots \times X \times Q \times X \times \ldots</math>
the same breath.  By way of compensating for the usual exclusion of the
+
| &nbsp;
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
+
In this picture <math>Q\!</math> is a certain set and <math>X\!</math> is the universe of discourse that is relevant to a given discussionSince 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:
and the justifications for introducing this distinction, under any
 
name, in the first placeIt 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,
+
{| align="center" cellpadding="8" width="90%"
it looks as if one sought, rather futilely, to avoid the problems of
+
|
recursion by proscribing the main program from calling itself, while
+
<math>\begin{array}{ccccccc}
allowing any subprogram to do so.  But any trouble that is avoidable
+
^{\backprime\backprime} Q ^{\prime\prime}
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,
+
^{\backprime\backprime} X \times Q \times X ^{\prime\prime}
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,
+
^{\backprime\backprime} X \times X \times Q \times X \times X ^{\prime\prime}
or to produce itself, if that turns out to be the best way in the end.
+
& , &
 +
\ldots
 +
\\
 +
\end{array}</math>
 +
|}
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
With respect to what these strictures specify, this leaves all of the following equivalent as straits:
  
1.3.10.10  The Cactus Language:  Stylistics (cont.)
+
{| align="center" cellpadding="8" width="90%"
 +
|
 +
<math>\begin{array}{ccccccc}
 +
Q
 +
& = &
 +
X \times Q \times X
 +
& = &
 +
X \times X \times Q \times X \times X
 +
& = &
 +
\ldots
 +
\\
 +
\end{array}</math>
 +
|}
  
Before I leave this Subsection, I need to say a few things about
+
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.
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,
+
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''.
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
+
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 <math>P\!</math> and <math>Q\!</math> of a universe <math>X.\!</math> Under these conditions, one can use the following sorts of expression as schematic strictures:
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 exerciseThe 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
+
{| align="center" cellpadding="8" width="90%"
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
+
<math>\begin{matrix}
of the circumstance that many special details of language membership
+
^{\backprime\backprime} X ^{\prime\prime} &
are bound, destined, and pre-determined to be glossed over with some
+
^{\backprime\backprime} P ^{\prime\prime} &
measure of indifference, especially when it comes down to the final
+
^{\backprime\backprime} Q ^{\prime\prime} \\
constitution of those equivalence classes of signs that are able to
+
\\
answer for the genuine objects of the whole enterprise of language.
+
^{\backprime\backprime} X \times X ^{\prime\prime} &
When any form of theory, against its initial and its best intentions,
+
^{\backprime\backprime} X \times P ^{\prime\prime} &
leads to this kind of absence of mind that is no longer beneficial in
+
^{\backprime\backprime} X \times Q ^{\prime\prime} \\
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
+
^{\backprime\backprime} P \times X ^{\prime\prime} &
theory are meant to augment.
+
^{\backprime\backprime} P \times P ^{\prime\prime} &
 +
^{\backprime\backprime} P \times Q ^{\prime\prime} \\
 +
\\
 +
^{\backprime\backprime} Q \times X ^{\prime\prime} &
 +
^{\backprime\backprime} Q \times P ^{\prime\prime} &
 +
^{\backprime\backprime} Q \times Q ^{\prime\prime} \\
 +
\end{matrix}</math>
 +
|}
  
The pragmatic theory of sign relations is called for in settings where
+
These strictures and their corresponding straits are stratified according to their amounts of information, or their levels of constraint, as follows:
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
+
{| align="center" cellpadding="8" width="90%"
computes with numbers, but solely with numerals in respect of numbers,
+
|
and numerals themselves are symbols.  Among other things, this renders
+
<math>\begin{array}{lcccc}
all discussion of numeric versus symbolic computation a bit beside the
+
\text{High:}
point, since it is only a question of what kinds of symbols are best for
+
& ^{\backprime\backprime} P \times P ^{\prime\prime}
one's immediate application or for one's selection of ongoing objectives.
+
& ^{\backprime\backprime} P \times Q ^{\prime\prime}
The numerals that everybody knows best are just the canonical symbols,
+
& ^{\backprime\backprime} Q \times P ^{\prime\prime}
the standard signs or the normal terms for numbers, and the process of
+
& ^{\backprime\backprime} Q \times Q ^{\prime\prime}
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.
+
\text{Med:}
 +
& ^{\backprime\backprime} P ^{\prime\prime}
 +
& ^{\backprime\backprime} X \times P ^{\prime\prime}
 +
& ^{\backprime\backprime} P \times X ^{\prime\prime}
 +
\\
 +
\\
 +
\text{Med:}
 +
& ^{\backprime\backprime} Q ^{\prime\prime}
 +
& ^{\backprime\backprime} X \times Q ^{\prime\prime}
 +
& ^{\backprime\backprime} Q \times X ^{\prime\prime}
 +
\\
 +
\\
 +
\text{Low:}
 +
& ^{\backprime\backprime} X ^{\prime\prime}
 +
& ^{\backprime\backprime} X \times X ^{\prime\prime}
 +
\\
 +
\end{array}</math>
 +
|}
  
Having broached the distinction between propositions and sentences, one
+
Within this framework, the more complex strait <math>P \times Q</math> can be expressed
can see its similarity to the distinction between numbers and numerals.
+
in terms of the simpler straits, <math>P \times X</math> and <math>X \times Q.</math>  More specifically, it lends itself to being analyzed as their intersection, in the following way:
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:
+
{| align="center" cellpadding="8" width="90%"
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.
+
<math>\begin{array}{lllll}
As it happens, it is far less important to wonder whether these types
+
P \times Q & = & P \times X & \cap & X \times Q. \\
of formal tokens actually constitute genuine sentences than it is to
+
\end{array}</math>
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.
+
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 <math>P \times Q</math> is described by a conjunction of propositions, namely, <math>P_{[1]} \land Q_{[2]},</math> subject to the following interpretation:
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
+
# <math>P_{[1]}\!</math> asserts that there is an element from the set <math>P\!</math> in the first place of the product.
 +
# <math>Q_{[2]}\!</math> asserts that there is an element from the set <math>Q\!</math> in the second place of the product.
  
1.3.10.10  The Cactus Language:  Stylistics (cont.)
+
The integration of these two pieces of information can be taken in that measure to specify a yet to be fully determined relation.
  
Any description of a language is bound to have alternative descriptions.
+
In a corresponding fashion at the level of the elements, the ordered pair <math>(p, q)\!</math> is described by a conjunction of propositions, namely, <math>p_{[1]} \land q_{[2]},</math> subject to the following interpretation:
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,
+
# <math>p_{[1]}\!</math> says that <math>p\!</math> is in the first place of the product element under construction.
and with the preferences that different observers display toward them,
+
# <math>q_{[2]}\!</math> says that <math>q\!</math> is in the second place of the product element under construction.
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
+
Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> in this way, one shifts the level of the active construction from the tupling of the elements in <math>P\!</math> and <math>Q\!</math> or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1\!</math> and <math>\mathfrak{L}_2\!</math> 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, <math>P_{[1]}\!</math> and <math>Q_{[2]},\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{[1]}</math> and <math>(\mathfrak{L}_2)_{[2]},</math> 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 <math>^{\backprime\backprime} [1] ^{\prime\prime}</math> and <math>^{\backprime\backprime} [2] ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.
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
+
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:
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
+
The <math>j^\text{th}\!</math> ''excerpt'' of a stricture of the form <math>^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime},</math> regarded within a frame of discussion where the number of places is limited to <math>k,\!</math> is the stricture of the form <math>^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, ^{\prime\prime}.</math> In the proper context, this can be written more succinctly as the stricture <math>^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.
    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"
+
The <math>j^\text{th}\!</math> ''extract'' of a strait of the form <math>S_1 \times \ldots \times S_k,\!</math> constrained to a frame of discussion where the number of places is restricted to <math>k,\!</math> is the strait of the form <math>X \times \ldots \times S_j \times \ldots \times X.</math> In the appropriate context, this can be denoted more succinctly by the stricture <math>^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},</math> an assertion that places the <math>j^\text{th}\!</math> set in the <math>j^\text{th}\!</math> place of the product.
    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,
+
In these terms, a stricture of the form <math>^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}</math> can be expressed in terms of simpler strictures, to wit, as a conjunction of its <math>k\!</math> excerpts:
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
+
{| align="center" cellpadding="8" width="90%"
usually possible to tease out the influence of these contrasting traits,
+
|
namely, the "additive" attitude versus the "mutiplicative" tendency that
+
<math>\begin{array}{lll}
go to make up the particular style in question, and even to determine the
+
^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}
dominant inclination or point of view that establishes its perspective on
+
& = &
the target domain.
+
^{\backprime\backprime} \, (S_1)_{[1]} \, ^{\prime\prime}
 +
\, \land \, \ldots \, \land \,
 +
^{\backprime\backprime} \, (S_k)_{[k]} \, ^{\prime\prime}.
 +
\end{array}</math>
 +
|}
  
In each style of formal grammar, the "multiplicative" aspect is present
+
In a similar vein, a strait of the form <math>S_1 \times \ldots \times S_k\!</math> can be expressed in terms of simpler straits, namely, as an intersection of its <math>k\!</math> extracts:
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
+
{| align="center" cellpadding="8" width="90%"
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.
 
 
|
 
|
| ...  ...  ...
+
<math>\begin{array}{lll}
|
+
S_1 \times \ldots \times S_k & = & (S_1)_{[1]} \, \cap \, \ldots \, \cap \, (S_k)_{[k]}.
| q    :>    W_k.
+
\end{array}</math>
 +
|}
  
It is useful to examine the relationship between the grammatical covering
+
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.
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
+
===The Cactus Language : Mechanics===
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
+
{| align="center" cellpadding="0" cellspacing="0" width="90%"
 +
|
 +
<p>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 &mdash; 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.</p>
 +
|-
 +
| align="right" | &mdash; Herbert J. Bernstein, "Idols of Modern Science", [HJB, 38]
 +
|}
  
1.3.10.10  The Cactus Language: Stylistics (cont.)
+
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 programmersIn 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.
  
A moment's reflection on the issue of style, giving due consideration to the
+
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 <math>\{ \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime} \, \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.</math>
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
+
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).
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,
+
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.
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
+
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 nodeAn 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''.
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 combinedWhat 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
+
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 lobeRecalling 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.
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 appearThis 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
+
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).
  
1.3.10.10 The Cactus Language: Stylistics (cont.)
+
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:
  
I am throwing together a wide variety of different operations into each
+
# A PARC is ''planted'' if its list of attachments has just one PARC.
of the bins labeled "additive" and "multiplicative", but it is easy to
+
# A PARC is ''UR'' if its list of attachments has no blanks or paints.
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
+
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''.
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
+
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 "<math>\operatorname{Node}</math>" 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 "<math>\operatorname{Lobe}</math>" 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:
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
+
{| align="center" cellpadding="4" width="90%"
the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the
+
| <math>1.\!</math>
copy of P into the first part of the co-product and the copy of Q into the
+
| <math>\operatorname{Node}^0</math>
second part of the co-product, in the following manner:
+
| <math>=\!</math>
 +
| <math>\operatorname{Node}()</math>
 +
|-
 +
| &nbsp;
 +
| &nbsp;
 +
| <math>=\!</math>
 +
| a node with no attachments.
 +
|-
 +
| &nbsp;
 +
| <math>\operatorname{Node}_{j=1}^k C_j</math>
 +
| <math>=\!</math>
 +
| <math>\operatorname{Node} (C_1, \ldots, C_k)</math>
 +
|-
 +
| &nbsp;
 +
| &nbsp;
 +
| <math>=\!</math>
 +
| a node with the attachments <math>C_1, \ldots, C_k.</math>
 +
|-
 +
| <math>2.\!</math>
 +
| <math>\operatorname{Lobe}^0</math>
 +
| <math>=\!</math>
 +
| <math>\operatorname{Lobe}()</math>
 +
|-
 +
| &nbsp;
 +
| &nbsp;
 +
| <math>=\!</math>
 +
| a lobe with no accoutrements.
 +
|-
 +
| &nbsp;
 +
| <math>\operatorname{Lobe}_{j=1}^k C_j</math>
 +
| <math>=\!</math>
 +
| <math>\operatorname{Lobe} (C_1, \ldots, C_k)</math>
 +
|-
 +
| &nbsp;
 +
| &nbsp;
 +
| <math>=\!</math>
 +
| a lobe with the accoutrements <math>C_1, \ldots, C_k.</math>
 +
|}
  
P_[1]  =  <P, 1> {<x, 1> :  x in P},
+
Working from a structural description of the cactus language, or any suitable formal grammar for <math>\mathfrak{C} (\mathfrak{P}),\!</math> it is possible to give a recursive definition of the function called <math>\operatorname{Parse}</math> that maps each sentence in <math>\operatorname{PARCE} (\mathfrak{P})\!</math> to the corresponding graph in <math>\operatorname{PARC} (\mathfrak{P}).\!</math>  One way to do this proceeds as follows:
  
Q_[2]  =  <Q, 2>  = {<x, 2> :  x in Q}.
+
<ol style="list-style-type:decimal">
  
Using the sign "]_[" for this construction, the "sum", the "co-product",
+
<li>The parse of the concatenation <math>\operatorname{Conc}_{j=1}^k</math> of the <math>k\!</math> sentences <math>(s_j)_{j=1}^k</math> is defined recursively as follows:</li>
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].
+
<ol style="list-style-type:lower-alpha">
  
The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is
+
<li><math>\operatorname{Parse} (\operatorname{Conc}^0) ~=~ \operatorname{Node}^0.</math>
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,
+
<li>
relative to a number of other sets, yet to be specified.  It is assumed
+
<p>For <math>k > 0,\!</math></p>
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 ..."
+
<p><math>\operatorname{Parse} (\operatorname{Conc}_{j=1}^k s_j) ~=~ \operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j).</math></p></li>
  
A "strait" is the object that is specified by a stricture, in effect,
+
</ol>
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 ...
+
<li>The parse of the surcatenation <math>\operatorname{Surc}_{j=1}^k</math> of the <math>k\!</math> sentences <math>(s_j)_{j=1}^k</math> is defined recursively as follows:</li>
  
In this picture, Q is a certain set, and X is the universe of discourse
+
<ol style="list-style-type:lower-alpha">
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",  ...
+
<li><math>\operatorname{Parse} (\operatorname{Surc}^0) ~=~ \operatorname{Lobe}^0.</math>
  
With respect to what these strictures specify, this
+
<li>
leaves all of the following equivalent as straits:
+
<p>For <math>k > 0,\!</math></p>
  
= XxQxX  = XxXxQxXxX  = ...
+
<p><math>\operatorname{Parse} (\operatorname{Surc}_{j=1}^k s_j) ~=~ \operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j).</math></p></li>
  
Within the framework of a particular discussion, it is customary to
+
</ol></ol>
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,
+
For ease of reference, Table&nbsp;13 summarizes the mechanics of these parsing rules.
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
+
<br>
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" ,
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table 13.} ~~ \text{Algorithmic Translation Rules}\!</math>
 +
|- style="height:40px; background:ghostwhite"
 
|
 
|
| "XxX""XxP""XxQ",
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; text-align:center; width:100%"
 +
| width="33%" | <math>\text{Sentence in PARCE}\!</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
 +
| width="33%" | <math>\text{Graph in PARC}\!</math>
 +
|}
 +
|-
 
|
 
|
| "PxX""PxP""PxQ",
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
 +
| width="33%" | <math>\mathrm{Conc}^0\!</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
 +
| width="33%" | <math>\mathrm{Node}^0\!</math>
 +
|-
 +
| width="33%" | <math>\mathrm{Conc}_{j=1}^k s_j\!</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
 +
| width="33%" | <math>\mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j)\!</math>
 +
|}
 +
|-
 
|
 
|
| "QxX""QxP""QxQ".
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
 +
| width="33%" | <math>\mathrm{Surc}^0\!</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
 +
| width="33%" | <math>\mathrm{Lobe}^0\!</math>
 +
|-
 +
| width="33%" | <math>\mathrm{Surc}_{j=1}^k s_j\!</math>
 +
| width="33%" | <math>\xrightarrow{\mathrm{Parse}}\!</math>
 +
| width="33%" | <math>\mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j)\!</math>
 +
|}
 +
|}
  
These strictures and their corresponding straits are stratified according
+
<br>
to their amounts of information, or their levels of constraint, as follows:
 
  
| High: "PxP""PxQ", "QxP", "QxQ".
+
A ''substructure'' of a PARC is defined recursively as follows. Starting at the root node of the cactus <math>C,\!</math> any attachment is a substructure of <math>C.\!</math> If a substructure is a blank or a paint, then it constitutes a minimal substructure, meaning that no further substructures of <math>C\!</math> arise from itIf a substructure is a lobe, then each one of its accoutrements is also a substructure of <math>C,\!</math> and has to be examined for further substructures.
|
 
| Med:    "P" , "XxP", "PxX".
 
|
 
| Med:    "Q" , "XxQ",  "QxX".
 
|
 
| Low:    "X" ,  "XxX".
 
  
Within this framework, the more complex strait PxQ can be expressed
+
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 <math>^{\backprime\backprime} ~ ^{\prime\prime}</math> is treated as a ''primer'', in other words, as a ''clear paint'' or a ''neutral tint''.  In effect, one is letting <math>m_1 = p_0.\!</math> In this frame of discussion, it is useful to make the following distinction:
in terms of the simpler straits, PxX and XxQMore specifically,
 
it lends itself to being analyzed as their intersection, in the
 
following way:
 
  
PxQ  =  PxX |^| XxQ.
+
# To ''delete'' a substructure is to replace it with an empty node, in effect, to reduce the whole structure to a trivial point.
 +
# 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.
  
>From here it is easy to see the relation of concatenation, by virtue of
+
A ''bare PARC'', loosely referred to as a ''bare cactus'', is a PARC on the empty palette <math>\mathfrak{P} = \varnothing.</math> In other veins, a bare cactus can be described in several different ways, depending on how the form arises in practice.
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
+
<ol style="list-style-type:decimal">
    the set P in the first place of the product.
 
  
2.  "Q_<2>" asserts that there is an element from
+
<li>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 <math>\mathfrak{C}^0 = \operatorname{PARCE}^0.</math></li>
    the set Q in the second place of the product.
 
  
The integration of these two pieces of information can be taken
+
<li>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:</li>
in that measure to specify a yet to be fully determined relation.
 
  
In a corresponding fashion at the level of the elements,
+
<ol style="list-style-type:lower-latin">
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
+
<li>A ''bare PARC'' is a PARC whose attachments are limited to blanks and ''bare lobes''.</li>
    of the product element under construction.
 
  
2.  "q_<2>" says that q is in the second place
+
<li>A ''bare lobe'' is a lobe whose accoutrements are limited to bare PARC's.</li>
    of the product element under construction.
 
  
Notice that, in construing the cartesian product of the sets P and Q or the
+
</ol>
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
+
<li>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.</li>
  
1.3.10.10  The Cactus Language:  Stylistics (cont.)
+
</ol>
  
In order to systematize the relations that strictures and straits placed
+
===The Cactus Language : Semantics===
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
+
{| align="center" cellpadding="0" cellspacing="0" width="90%"
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
+
<p>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 &mdash; 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!</p>
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.
+
| align="right" | &mdash; Nietzsche, ''Beyond Good and Evil'', [Nie-2, ¶ 296]
 +
|}
  
The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained
+
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.
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"
+
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.
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>".
+
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 hereThere 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.
  
In a similar vein, a strait of the form S_1 x ... x S_k
+
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.
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>.
+
The task of this Subsection is to specify a ''semantic function'' for the sentences of the cactus language <math>\mathfrak{L} = \mathfrak{C}(\mathfrak{P}),</math> in other words, to define a mapping that "interprets" each sentence of <math>\mathfrak{C}(\mathfrak{P})</math> 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.
  
There is a measure of ambiguity that remains in this formulation,
+
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.
but it is the best that I can do in the present informal context.
 
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
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 <math>k = 0, 1, 2, 3, \ldots.</math>
  
1.3.10.11  The Cactus Language: Mechanics
+
<ol style="list-style-type:decimal">
  
| We are only now beginning to see how this works.  Clearly one of the
+
<li>
| mechanisms for picking a reality is the sociohistorical sense of what
+
<p>The ''blank connective'' of <math>k\!</math> places is signified by the concatenation of the <math>k\!</math> sentences that fill those places.</p>
| 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
+
<p>For the special case of <math>k = 0,\!</math> the blank connective is taken to be an empty string or a blank symbol &mdash; it does not matter which, since both are assigned the same denotation among propositions.</p>
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
+
<p>For the generic case of <math>k > 0,\!</math> the blank connective takes the form <math>s_1 \cdot \ldots \cdot s_k.</math> In the type of data that is called a ''text'', the use of the center dot <math>(\cdot)</math> is generally supplanted by whatever number of spaces and line breaks serve to improve the readability of the resulting text.</p></li>
to the visual imagination, can be described as followsThe 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
+
<li>
a class of cacti that are painted but not rooted, and so it is frequently
+
<p>The ''bound connective'' of <math>k\!</math> places is signified by the surcatenation of the <math>k\!</math> sentences that fill those places.</p>
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
+
<p>For the special case of <math>k = 0,\!</math> the bound connective is taken to be an empty closure &mdash; an expression enjoying one of the forms <math>\underline{(} \underline{)}, \, \underline{(} ~ \underline{)}, \, \underline{(} ~~ \underline{)}, \, \ldots</math> with any number of blank symbols between the parentheses &mdash; all of which are assigned the same logical denotation among propositions.</p>
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
+
<p>For the generic case of <math>k > 0,\!</math> the bound connective takes the form <math>\underline{(} s_1, \ldots, s_k \underline{)}.</math></p></li>
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
+
</ol>
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
+
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.
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
+
<ol style="list-style-type:lower-alpha">
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.
+
<li>There is the literal formal language of strings in <math>\operatorname{PARCE} (\mathfrak{P}),</math> the ''painted and rooted cactus expressions'' that constitute the language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*.</math></li>
  
2.  A PARC is "UR" if its list of attachments has no blanks or paints.
+
<li>There is the figurative formal language of graphs in <math>\operatorname{PARC} (\mathfrak{P}),</math> 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.</li>
  
In short, an UR-planted PARC has a single PARC as its only attachment,
+
</ol>
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
+
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:
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()
+
<ol style="list-style-type:decimal">
  
                  =  a node with no attachments.
+
<li>
 +
<p>The ''conjunction'' <math>\operatorname{Conj}_j^J q_j</math> of a set of propositions, <math>\{ q_j : j \in J \},</math> is a proposition that is true if and only if every one of the <math>q_j\!</math> is true.</p>
  
    Node^k_j  C_j  =  Node(C_1, ..., C_k)
+
<p><math>\operatorname{Conj}_j^J q_j</math> is true &nbsp;<math>\Leftrightarrow</math>&nbsp; <math>q_j\!</math> is true for every <math>j \in J.</math></p></li>
  
                  =  a node with the attachments C_1, ..., C_k.
+
<li>
 +
<p>The ''surjunction'' <math>\operatorname{Surj}_j^J q_j</math> of a set of propositions, <math>\{ q_j : j \in J \},</math> is a proposition that is true if and only if exactly one of the <math>q_j\!</math> is untrue.</p>
  
2. Lobe^0        =  Lobe()
+
<p><math>\operatorname{Surj}_j^J q_j</math> is true &nbsp;<math>\Leftrightarrow</math>&nbsp;  <math>q_j\!</math> is untrue for unique <math>j \in J.</math></p></li>
  
                  =  a lobe with no accoutrements.
+
</ol>
  
    Lobe^k_j  C_j  =  Lobe(C_1, ..., C_k)
+
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.
  
                  = a lobe with the accoutrements C_1, ..., C_k.
+
If <math>J\!</math> is finite, for instance, if <math>J\!</math> consists of the integers in the interval <math>j = 1 ~\text{to}~ k,</math> and if each proposition <math>q_j\!</math> is represented by a sentence <math>s_j,\!</math> then the following strategies of expression are open:
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
<ol style="list-style-type:decimal">
  
1.3.10.11  The Cactus Language: Mechanics (cont.)
+
<li>
 +
<p>The conjunction <math>\operatorname{Conj}_j^J q_j</math> can be represented by a sentence that is constructed by concatenating the <math>s_j\!</math> in the following fashion:</p>
  
Working from a structural description of the cactus language,
+
<p><math>\operatorname{Conj}_j^J q_j ~\leftrightsquigarrow~ s_1 s_2 \ldots s_k.</math></p></li>
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,
+
<li>
    for j = 1 to k, is defined recursively as follows:
+
<p>The surjunction <math>\operatorname{Surj}_j^J q_j</math> can be represented by a sentence that is constructed by surcatenating the <math>s_j\!</math> in the following fashion:</p>
  
    a.  Parse(Conc^0)       =  Node^0.
+
<p><math>\operatorname{Surj}_j^J q_j ~\leftrightsquigarrow~ \underline{(} s_1, s_2, \ldots, s_k \underline{)}.</math></p></li>
  
    b.  For k > 0,
+
</ol>
  
        Parse(Conc^k_j S_j)  =  Node^k_j Parse(S_j).
+
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:
  
2.  The parse of the surcatenation Surc^k of the k sentences S_j,
+
A cactus rooted at a particular node is taken to represent what that node denotes, its logical denotation or its logical interpretation.
    for j = 1 to k, is defined recursively as follows:
 
  
    a.  Parse(Surc^0)       = Lobe^0.
+
# 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 attachmentsThe logical denotation of either a blank symbol or an empty node is the boolean value <math>\underline{1} = \operatorname{true}.</math>  The logical denotation of the paint <math>\mathfrak{p}_j\!</math> is the proposition <math>p_j,\!</math> a proposition that is regarded as ''primitive'', at least, with respect to the level of analysis that is represented in the current instance of <math>\mathfrak{C} (\mathfrak{P}).</math>
 +
# 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 <math>\underline{(} \underline{)},</math> otherwise called a ''needle'', is the boolean value <math>\underline{0} = \operatorname{false}.</math>
  
    b. For k > 0,
+
If one takes the point of view that PARCs and PARCEs amount to a pair of intertranslatable languages for the same domain of objects, then denotation brackets of the form <math>\downharpoonleft \ldots \downharpoonright</math> can be used to indicate the logical denotation <math>\downharpoonleft C_j \downharpoonright</math> of a cactus <math>C_j\!</math> or the logical denotation <math>\downharpoonleft s_j \downharpoonright</math> of a sentence <math>s_j.\!</math>
  
        Parse(Surc^k_j S_j) = Lobe^k_j Parse(S_j).
+
Tables&nbsp;14 and 15 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.
  
For ease of reference, Table 12 summarizes the mechanics of these parsing rules.
+
<br>
  
Table 12. Algorithmic Translation Rules
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
o------------------------o---------o------------------------o
+
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Semantic Translation : Functional Form}\!</math>
|                       | Parse |                       |
+
|- style="height:40px; background:ghostwhite"
| Sentence in PARCE      |   -->   | Graph in PARC          |
+
|
o------------------------o---------o------------------------o
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
|                       |         |                       |
+
| width="20%" | <math>\mathrm{Sentence}\!</math>
| Conc^0                 |   -->   | Node^0                 |
+
| width="20%" | <math>\xrightarrow[\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}]{\mathrm{Parse}}\!</math>
|                       |         |                       |
+
| width="20%" | <math>\mathrm{Graph}\!</math>
| Conc^k_j S_j          |   -->   | Node^k_j Parse(S_j)   |
+
| width="20%" | <math>\xrightarrow[\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}]{\mathrm{Denotation}}\!</math>
|                       |         |                       |
+
| width="20%" | <math>\mathrm{Proposition}\!</math>
| Surc^0                 |   -->   | Lobe^0                 |
+
|}
|                       |         |                       |
+
|-
| Surc^k_j S_j          |   -->   | Lobe^k_j Parse(S_j)   |
+
|
|                       |         |                       |
+
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
o------------------------o---------o------------------------o
+
| width="20%" | <math>s_j\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>C_j\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>q_j\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\mathrm{Conc}^0\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\mathrm{Node}^0\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\underline{1}\!</math>
 +
|-
 +
| width="20%" | <math>\mathrm{Conc}^k_j s_j\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\mathrm{Node}^k_j C_j\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\mathrm{Conj}^k_j q_j\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\mathrm{Surc}^0\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\mathrm{Lobe}^0\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\underline{0}\!</math>
 +
|-
 +
| width="20%" | <math>\mathrm{Surc}^k_j s_j~\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\mathrm{Lobe}^k_j C_j\!</math>
 +
| width="20%" | <math>\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!</math>
 +
| width="20%" | <math>\mathrm{Surj}^k_j q_j\!</math>
 +
|}
 +
|}
  
A "substructure" of a PARC is defined recursively as follows.  Starting
+
<br>
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
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
and erasure operations that respect the structure of the abstract graph.
+
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Semantic Translation : Equational Form}\!</math>
For the purposes of this depiction, a blank symbol " " is treated as
+
|- style="height:40px; background:ghostwhite"
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
+
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:ghostwhite; width:100%"
of discussion, it is useful to make the following distinction:
+
| width="20%" | <math>\downharpoonleft \mathrm{Sentence} \downharpoonright\!</math>
 +
| width="20%" | <math>\stackrel{\mathrm{Parse}}{=}\!</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Graph} \downharpoonright\!</math>
 +
| width="20%" | <math>\stackrel{\mathrm{Denotation}}{=}\!</math>
 +
| width="20%" | <math>\mathrm{Proposition}\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft s_j \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft C_j \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>q_j\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^0 \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Node}^0 \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\underline{1}\!</math>
 +
|-
 +
| width="20%" | <math>\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\mathrm{Conj}^k_j q_j\!</math>
 +
|}
 +
|-
 +
|
 +
{| align="center" border="0" cellpadding="8" cellspacing="0" width="100%"
 +
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^0 \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^0 \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\underline{0}\!</math>
 +
|-
 +
| width="20%" | <math>\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright\!</math>
 +
| width="20%" | <math>=\!</math>
 +
| width="20%" | <math>\mathrm{Surj}^k_j q_j\!</math>
 +
|}
 +
|}
  
1.  To "delete" a substructure is to replace it with an empty node,
+
<br>
    in effect, to reduce the whole structure to a trivial point.
 
  
2To "erase" a substructure is to replace it with a blank symbol,
+
Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their mapsTable&nbsp;14 records the functional associations that connect each domain with the next, taking the triplings of a sentence <math>s_j,\!</math> a cactus <math>C_j,\!</math> and a proposition <math>q_j\!</math> as basic data, and fixing the rest by recursion on these.  Table&nbsp;15 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation 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.
    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
+
A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables.  Let us begin by doing this for the first three cases, <math>k = 0, 1, 2.\!</math>
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
+
A boolean function <math>F^{(0)}\!</math> on <math>0\!</math> variables is just an element of the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.</math> Table&nbsp;16 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 appear in the initial case.
    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
+
<br>
    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
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"
        are limited to blanks and "bare lobes".
+
|+ style="height:30px" | <math>\text{Table 16.} ~~ \text{Boolean Functions on Zero Variables}\!</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | <math>F\!</math>
 +
| width="14%" | <math>F\!</math>
 +
| width="48%" | <math>F()\!</math>
 +
| width="24%" | <math>F\!</math>
 +
|-
 +
| <math>\underline{0}\!</math>
 +
| <math>F_0^{(0)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\texttt{(~)}\!</math>
 +
|-
 +
| <math>\underline{1}\!</math>
 +
| <math>F_1^{(0)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
|}
  
    b.  A "bare lobe" is a lobe whose accoutrements
+
<br>
        are limited to bare PARC's.
 
  
3.  In practice, a bare cactus is usually encountered in the process
+
Column&nbsp;1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.
    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
+
Column&nbsp;2 lists each boolean function in a style of function name <math>F_j^{(k)}\!</math> that is constructed as follows:  The superscript <math>(k)\!</math> gives the dimension of the functional domain, that is, the number of its functional variables, and the subscript <math>j\!</math> is a binary string that recapitulates the functional values, using the obvious translation of boolean values into binary values.
  
1.3.10.12  The Cactus Language:  Semantics
+
Column&nbsp;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.
  
| Alas, and yet what 'are' you, my written and painted thoughts!
+
Column&nbsp;4 shows the usual expressions of these elements in the cactus language, conforming to the practice of omitting the underlines in display formatsHere I illustrate also the convention of using the expression <math>^{\backprime\backprime} ((~)) ^{\prime\prime}</math> as a visible stand-in for the expression of the logical value <math>\operatorname{true},</math> a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.
| 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
+
Table 17 presents the boolean functions on one variable, <math>F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},</math> of which there are precisely four.
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
+
<br>
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
+
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
that at best respects the objects of all of the signs that enter into it,
+
|+ style="height:30px" | <math>\text{Table 17.} ~~ \text{Boolean Functions on One Variable}\!</math>
and so it is the connotative aspect of semiotics that is at stake here.
+
|- style="height:40px; background:ghostwhite"
There is nothing wrong with my saying that I interpret a sentence of a
+
| width="14%" | <math>F\!</math>
formal language as a sign that refers to a function or to a proposition,
+
| width="14%" | <math>F\!</math>
so long as you understand that this reference is likely to be achieved
+
| colspan="2" | <math>F(x)\!</math>
by way of more familiar and perhaps less formal signs that you already
+
| width="24%" | <math>F\!</math>
take to denote those objects.
+
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | &nbsp;
 +
| width="14%" | &nbsp;
 +
| width="24%" | <math>F(\underline{1})</math>
 +
| width="24%" | <math>F(\underline{0})</math>
 +
| width="24%" | &nbsp;
 +
|-
 +
| <math>F_0^{(1)}\!</math>
 +
| <math>F_{00}^{(1)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\texttt{(~)}\!</math>
 +
|-
 +
| <math>F_1^{(1)}\!</math>
 +
| <math>F_{01}^{(1)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{(} x \texttt{)}\!</math>
 +
|-
 +
| <math>F_2^{(1)}\!</math>
 +
| <math>F_{10}^{(1)}~\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>x\!</math>
 +
|-
 +
| <math>F_3^{(1)}\!</math>
 +
| <math>F_{11}^{(1)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
|}
  
On entering a context where a logical interpretation is intended for the
+
<br>
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
+
Here, Column&nbsp;1 codes the contents of Column&nbsp;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 variableThus, one has the synonymous expressions for constant functions that are expressed in the next two chains of equations:
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 functionWhen 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,
+
{| align="center" cellpadding="8" width="90%"
I introduce a family of operators on indicator functions that are
+
|
called "propositional connectives", and I distinguish these from
+
<math>\begin{matrix}
the associated family of syntactic combinations that are called
+
F_0^{(1)}
"sentential connectives", where the relationship between these
+
& = &
two realms of connection is exactly that between objects and
+
F_{00}^{(1)}
their signs.  A propositional connective, as an entity of a
+
& = &
well-defined functional and operational type, can be treated
+
\underline{0} ~:~ \underline\mathbb{B} \to \underline\mathbb{B}
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
+
F_3^{(1)}
is appropriate to the case in question.
+
& = &
 +
F_{11}^{(1)}
 +
& = &
 +
\underline{1} ~:~ \underline\mathbb{B} \to \underline\mathbb{B}
 +
\end{matrix}</math>
 +
|}
  
There are two basic types of connectives, called the "blank connectives"
+
As for the rest, the other two functions are easily recognized as corresponding to the one-place logical connectives, or the monadic operators on <math>\underline\mathbb{B}.</math>  Thus, the function <math>F_1^{(1)} = F_{01}^{(1)}</math> is recognizable as the negation operation, and the function <math>F_2^{(1)} = F_{10}^{(1)}</math> is obviously the identity operation.
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
+
Table&nbsp;18 presents the boolean functions on two variables, <math>F^{(2)} : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> of which there are precisely sixteen.
    concatenation of the k sentences that fill those places.
 
  
    For the special case of k = 0, the "blank connective" is taken to
+
<br>
    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
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:center; width:80%"
    surcatenation of the k sentences that fill those places.
+
|+ style="height:30px" | <math>\text{Table 18.} ~~ \text{Boolean Functions on Two Variables}\!</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | <math>F\!</math>
 +
| width="14%" | <math>F\!</math>
 +
| colspan="4" | <math>F(x, y)\!</math>
 +
| width="24%" | <math>F\!</math>
 +
|- style="height:40px; background:ghostwhite"
 +
| width="14%" | &nbsp;
 +
| width="14%" | &nbsp;
 +
| width="12%" | <math>F(\underline{1}, \underline{1})</math>
 +
| width="12%" | <math>F(\underline{1}, \underline{0})</math>
 +
| width="12%" | <math>F(\underline{0}, \underline{1})</math>
 +
| width="12%" | <math>F(\underline{0}, \underline{0})</math>
 +
| width="24%" | &nbsp;
 +
|-
 +
| <math>F_{0}^{(2)}\!</math>
 +
| <math>F_{0000}^{(2)}~\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\texttt{(~)}\!</math>
 +
|-
 +
| <math>F_{1}^{(2)}\!</math>
 +
| <math>F_{0001}^{(2)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math>
 +
|-
 +
| <math>F_{2}^{(2)}\!</math>
 +
| <math>F_{0010}^{(2)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\texttt{(} x \texttt{)} y\!</math>
 +
|-
 +
| <math>F_{3}^{(2)}\!</math>
 +
| <math>F_{0011}^{(2)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{(} x \texttt{)}\!</math>
 +
|-
 +
| <math>F_{4}^{(2)}\!</math>
 +
| <math>F_{0100}^{(2)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>x \texttt{(} y \texttt{)}\!</math>
 +
|-
 +
| <math>F_{5}^{(2)}\!</math>
 +
| <math>F_{0101}^{(2)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{(} y \texttt{)}\!</math>
 +
|-
 +
| <math>F_{6}^{(2)}\!</math>
 +
| <math>F_{0110}^{(2)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math>
 +
|-
 +
| <math>F_{7}^{(2)}\!</math>
 +
| <math>F_{0111}^{(2)}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{(} x y \texttt{)}\!</math>
 +
|-
 +
| <math>F_{8}^{(2)}\!</math>
 +
| <math>F_{1000}^{(2)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>x y\!</math>
 +
|-
 +
| <math>F_{9}^{(2)}\!</math>
 +
| <math>F_{1001}^{(2)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math>
 +
|-
 +
| <math>F_{10}^{(2)}\!</math>
 +
| <math>F_{1010}^{(2)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>y\!</math>
 +
|-
 +
| <math>F_{11}^{(2)}\!</math>
 +
| <math>F_{1011}^{(2)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{(} x \texttt{(} y \texttt{))}\!</math>
 +
|-
 +
| <math>F_{12}^{(2)}\!</math>
 +
| <math>F_{1100}^{(2)}~\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>x\!</math>
 +
|-
 +
| <math>F_{13}^{(2)}\!</math>
 +
| <math>F_{1101}^{(2)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{((} x \texttt{)} y \texttt{)}\!</math>
 +
|-
 +
| <math>F_{14}^{(2)}\!</math>
 +
| <math>F_{1110}^{(2)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{0}\!</math>
 +
| <math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math>
 +
|-
 +
| <math>F_{15}^{(2)}\!</math>
 +
| <math>F_{1111}^{(2)}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\underline{1}\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
|}
  
    For the special case of k = 0, the "bound connective" is taken to
+
<br>
    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
+
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:
  
1.3.10.12  The Cactus Language: Semantics (cont.)
+
: The constant function <math>\underline{0} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{0}^{(2)}.</math>
  
At this point, there are actually two different "dialects", "scripts",
+
: The constant function <math>\underline{1} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> appears under the name <math>F_{15}^{(2)}.</math>
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 negation and identity of the first variable are <math>F_{3}^{(2)}</math> and <math>F_{12}^{(2)},</math> respectively.
    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 negation and identity of the second variable are <math>F_{5}^{(2)}</math> and <math>F_{10}^{(2)},</math> respectively.
    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
+
: The logical conjunction is given by the function <math>F_{8}^{(2)} (x, y) = x \cdot y.</math>
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},
+
: The logical disjunction is given by the function <math>F_{14}^{(2)} (x, y) = \underline{((} ~x~ \underline{)(} ~y~ \underline{))}.</math>
    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.
+
Functions expressing the ''conditionals'', ''implications'', or ''if-then'' statements are given in the following ways:
  
2.  The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J},
+
: <math>[x \Rightarrow y] = F_{11}^{(2)} (x, y) = \underline{(} ~x~ \underline{(} ~y~ \underline{))} = [\operatorname{not}~ x ~\operatorname{without}~ y].</math>
    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.
+
: <math>[x \Leftarrow y] = F_{13}^{(2)} (x, y) = \underline{((} ~x~ \underline{)} ~y~ \underline{)} = [\operatorname{not}~ y ~\operatorname{without}~ x].</math>
  
If the number of propositions that are being joined together is finite,
+
The function that corresponds to the ''biconditional'', the ''equivalence'', or the ''if and only'' statement is exhibited in the following fashion:
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,
+
: <math>[x \Leftrightarrow y] = [x = y] = F_{9}^{(2)} (x, y) = \underline{((} ~x~,~y~ \underline{))}.</math>
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
+
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'' operation, and geometrically associated with the ''symmetric difference'' of sets.  This function is given by:
    is constructed by concatenating the S_j in the following fashion:
 
  
    Conj^J_j Q_j  <-<   S_1 S_2 ... S_k.
+
: <math>[x \neq y] = [x + y] = F_{6}^{(2)} (x, y) = \underline{(} ~x~,~y~ \underline{)}.</math>
  
2The surjunction Surj^J_j Q_j can be represented by a sentence that
+
Let me now address one last question that may have occurred to someWhat 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 <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, 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 <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  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.
    is constructed by surcatenating the S_j in the following fashion:
 
  
    Surj^J_j Q_j  <-<  -(S_1, S_2, ..., S_k)-.
+
===Stretching Exercises===
  
If one opts for a mode of interpretation that moves more directly from
+
The arrays of boolean connections described above, namely, the boolean functions <math>F^{(k)} : \underline\mathbb{B}^k \to \underline\mathbb{B},</math> for <math>k\!</math> in <math>\{ 0, 1, 2 \},\!</math> supply enough material to demonstrate the use of the stretch operation in a variety of concrete cases.
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
+
For example, suppose that <math>F\!</math> is a connection of the form <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B},</math> that is, any one of the sixteen possibilities in Table&nbsp;18, while <math>p\!</math> and <math>q\!</math> are propositions of the form <math>p, q : X \to \underline\mathbb{B},</math> that is, propositions about things in the universe <math>X,\!</math> or else the indicators of sets contained in <math>X.\!</math>
node denotes, its logical denotation or its logical interpretation.
 
  
1.  The logical denotation of a node is the logical conjunction of that node's
+
Then one has the imagination <math>\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,</math> and the stretch of the connection <math>F\!</math> to <math>\underline{f}\!</math> on <math>X\!</math> amounts to a proposition <math>F^\$ (p, q) : X \to \underline\mathbb{B}</math> that may be read as the ''stretch of <math>F\!</math> to <math>p\!</math> and <math>q.\!</math>'' If one is concerned with many different propositions about things in <math>X,\!</math> or if one is abstractly indifferent to the particular choices for <math>p\!</math> and <math>q,\!</math> then one may detach the operator <math>F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),</math> called the ''stretch of <math>F\!</math> over <math>X,\!</math>'' and consider it in isolation from any concrete application.
    "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
+
When the cactus notation is used to represent boolean functions, a single <math>\$</math> sign at the end of the expression is enough to remind the reader that the connections are meant to be stretched to several propositions on a universe <math>X.\!</math>
    "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
+
For example, take the connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> such that:
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
+
: <math>F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}\!</math>
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
+
The connection in question is a boolean function on the variables <math>x, y\!</math> that returns a value of <math>\underline{1}</math> just when just one of the pair <math>x, y\!</math> is not equal to <math>\underline{1},</math> or what amounts to the same thing, just when just one of the pair <math>x, y\!</math> is equal to <math>\underline{1}.</mathThere is clearly an isomorphism between this connection, viewed as an operation on the boolean domain <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},</math> and the dyadic operation on binary values <math>x, y \in \mathbb{B} = \operatorname{GF}(2)\!</math> that is otherwise known as <math>x + y.\!</math>
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
+
The same connection <math>F : \underline\mathbb{B}^2 \to \underline\mathbb{B}</math> can also be read as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math> If <math>s\!</math> is a sentence that denotes the proposition <math>F,\!</math> then the corresponding assertion says exactly what one states in uttering the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.</math> In such a case, one has <math>\downharpoonleft s \downharpoonright \, = F,</math> and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:
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
+
{| align="center" cellpadding="8" width="90%"
ways of conceptualizing the operations that go to establish their maps.
+
|
Table 13.1 records the functional associations that connect each domain
+
<math>\begin{array}{lll}
with the next, taking the triplings of a sentence S_j, a cactus C_j, and
+
[| \downharpoonleft s \downharpoonright |]
a proposition Q_j as basic data, and fixing the rest by recursion on these.
+
& = & [| F |]
Table 13.2 records these associations in the form of equations, treating
+
\\[6pt]
sentences and graphs as alternative kinds of signs, and generalizing the
+
& = & F^{-1} (\underline{1})
spiny bracket operator to indicate the proposition that either denotes.
+
\\[6pt]
It should be clear at this point that either scheme of translation puts
+
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ s ~\}
the sentences, the graphs, and the propositions that it associates with
+
\\[6pt]
each other roughly in the roles of the signs, the interpretants, and the
+
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) = \underline{1} ~\}
objects, respectively, whose triples define an appropriate sign relation.
+
\\[6pt]
Indeed, the "roughly" can be made "exactly" as soon as the domains of
+
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) ~\}
a suitable sign relation are specified precisely.
+
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} = \underline{1} ~\}
 +
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} ~\}
 +
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{exclusive~or}~ y ~\}
 +
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \operatorname{just~one~true~of}~ x, y ~\}
 +
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{not~equal~to}~ y ~\}
 +
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\}
 +
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \neq y ~\}
 +
\\[6pt]
 +
& = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x + y ~\}.
 +
\end{array}</math>
 +
|}
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
Notice the distinction, that I continue to maintain at this point, between the logical values <math>\{ \operatorname{falsehood}, \operatorname{truth} \}</math> and the algebraic values <math>\{ 0, 1 \}.\!</math>  This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence <math>s\!</math> or the sentence <math>^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}</math> into the context <math>^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},</math> thereby obtaining the corresponding expressions listed above.  It also allows us to assert the proposition <math>F(x, y)\!</math> in a more direct way, without detouring through the equation <math>F(x, y) = \underline{1},</math> since it already has a value in <math>\{ \operatorname{falsehood}, \operatorname{true} \},</math> and thus can be taken as tantamount to an actual sentence.
  
1.3.10.12  The Cactus Language: Semantics (cont.)
+
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 matterIf 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.
  
A good way to illustrate the action of the conjunction and surjunction
+
This covers the properties of the connection <math>F(x, y) = \underline{(}~x~,~y~\underline{)},</math> treated as a proposition about things in the universe <math>X = \underline\mathbb{B}^2.</math>  Staying with this same connection, it is time to demonstrate how it can be "stretched" to form an operator on arbitrary propositions.
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
+
To continue the exercise, let <math>p\!</math> and <math>q\!</math> be arbitrary propositions about things in the universe <math>X,\!</math> that is, maps of the form <math>p, q : X \to \underline\mathbb{B},</math> and suppose that <math>p, q\!</math> are indicator functions of the sets <math>P, Q \subseteq X,</math> respectively.  In other words, we have the following data:
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
+
{| align="center" cellpadding="8" width="90%"
ordinary constant name or under a succinct nickname, respectively.
+
|
 +
<math>\begin{matrix}
 +
p
 +
& = &
 +
\upharpoonleft P \upharpoonright
 +
& : &
 +
X \to \underline\mathbb{B}
 +
\\
 +
\\
 +
q
 +
& = &
 +
\upharpoonleft Q \upharpoonright
 +
& : &
 +
X \to \underline\mathbb{B}
 +
\\
 +
\\
 +
(p, q)
 +
& = &
 +
(\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright)
 +
& : &
 +
(X \to \underline\mathbb{B})^2
 +
\\
 +
\end{matrix}</math>
 +
|}
  
Column 2 lists each boolean function in a style of function name "F^i_j"
+
Then one has an operator <math>F^\$,</math> the stretch of the connection <math>F\!</math> over <math>X,\!</math> and a proposition <math>F^\$ (p, q),</math> the stretch of <math>F\!</math> to <math>(p, q)\!</math> on <math>X,\!</math> with the following properties:
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
+
{| align="center" cellpadding="8" width="90%"
a boolean element appearing in the guise of a function, for each combination
+
|
of its domain values.
+
<math>\begin{array}{ccccl}
 +
F^\$
 +
& = &
 +
\underline{(} \ldots, \ldots \underline{)}^\$
 +
& : &
 +
(X \to \underline\mathbb{B})^2 \to (X \to \underline\mathbb{B})
 +
\\
 +
\\
 +
F^\$ (p, q)
 +
& = &
 +
\underline{(}~p~,~q~\underline{)}^\$
 +
& : &
 +
X \to \underline\mathbb{B}
 +
\\
 +
\end{array}</math>
 +
|}
  
Column 4 shows the usual expressions of these elements in the cactus language,
+
As a result, the application of the proposition <math>F^\$ (p, q)</math> to each <math>x \in X</math> returns a logical value in <math>\underline\mathbb{B},</math> all in accord with the following equations:
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
+
{| align="center" cellpadding="8" width="90%"
o----------o----------o-------------------------------------------o----------o
+
|
| Constant | Function |                    F()                   | Function |
+
<math>\begin{matrix}
o----------o----------o-------------------------------------------o----------o
+
F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \underline\mathbb{B}
|          |          |                                          |          |
+
\\
| %0%      | F^0_0    |                    %0%                    |    ()   |
+
\\
|          |          |                                          |          |
+
\Updownarrow  &  & \Updownarrow
| %1%      | F^0_1    |                    %1%                    |  (())   |
+
\\
|         |          |                                          |          |
+
\\
o----------o----------o-------------------------------------------o----------o
+
F(p(x), q(x))  & = & \underline{(}~p(x)~,~q(x)~\underline{)}  & \in & \underline\mathbb{B}
 +
\\
 +
\end{matrix}</math>
 +
|}
  
Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%,
+
For each choice of propositions <math>p\!</math> and <math>q\!</math> about things in <math>X,\!</math> the stretch of <math>F\!</math> to <math>p\!</math> and <math>q\!</math> on <math>X\!</math> is just another proposition about things in <math>X,\!</math> a simple proposition in its own right, no matter how complex its current expression or its present construction as <math>F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$</math> makes it appear in relation to <math>p\!</math> and <math>q.\!</math> Like any other proposition about things in <math>X,\!</math> it indicates a subset of <math>X,\!</math> namely, the fiber that is variously described in the following ways:
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 variableThus, 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%
+
{| align="center" cellpadding="8" width="90%"
|  
+
|
| F^1_3  = F^1_11  = %1% : %B% -> %B%
+
<math>\begin{array}{lll}
 +
[| F^\$ (p, q) |]
 +
& = & [| \underline{(}~p~,~q~\underline{)}^\$ |]
 +
\\[6pt]
 +
& = & (F^\$ (p, q))^{-1} (\underline{1})
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ F^\$ (p, q)(x) ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ p(x) + q(x) ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ p(x) \neq q(x) ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\}
 +
\\[6pt]
 +
& = & \{~ x \in X ~:~ x \in P + Q ~\}
 +
\\[6pt]
 +
& = & P + Q ~\subseteq~ X
 +
\\[6pt]
 +
& = & [|p|] + [|q|] ~\subseteq~ X
 +
\end{array}</math>
 +
|}
  
As for the rest, the other two functions are easily recognized as corresponding
+
==References==
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
+
* Bernstein, Herbert J. (1987), "Idols of Modern Science and The Reconstruction of Knowledge", pp. 37&ndash;68 in Marcus G. Raskin and Herbert J. Bernstein, ''New Ways of Knowing : The Sciences, Society, and Reconstructive Knowledge'', Rowman and Littlefield, Totowa, NJ, 1987.
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%,
+
* Denning, P.J., Dennis, J.B., and Qualitz, J.E. (1978), ''Machines, Languages, and Computation'', Prentice-Hall, Englewood Cliffs, NJ.
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.
+
* Nietzsche, Friedrich, ''Beyond Good and Evil : Prelude to a Philosophy of the Future'', R.J. Hollingdale (trans.), Michael Tanner (intro.), Penguin Books, London, UK, 1973, 1990.
  
The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15.
+
* Raskin, Marcus G., and Bernstein, Herbert J. (1987, eds.), ''New Ways of Knowing : The Sciences, Society, and Reconstructive Knowledge'', Rowman and Littlefield, Totowa, NJ.
  
The negation and identity of the first variable are F^2_03 and F^2_12, resp.
+
==Document History==
  
The negation and identity of the other variable are F^2_05 and F^2_10, resp.
+
===The Cactus Patch===
  
The logical conjunction is given by the function F^2_08 (x, y=  x · y.
+
<pre>
 +
| Subject:  Inquiry Driven Systems : An Inquiry Into Inquiry
 +
| Contact:  Jon Awbrey
 +
| 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
 +
</pre>
  
The logical disjunction is given by the function F^2_14 (x, y)  = ((x)(y)).
+
===Aug 2000 &bull; Extensions Of Logical Graphs===
  
Functions expressing the "conditionals", "implications",
+
====CG List &bull; Lost Links====
or "if-then" statements are given in the following ways:
 
  
[x => y]  =  F^2_11 (x, y)  =  (x (y))  =  [not x without y].
+
# http://www.virtual-earth.de/CG/cg-list/old/msg03351.html
 +
# http://www.virtual-earth.de/CG/cg-list/old/msg03352.html
 +
# http://www.virtual-earth.de/CG/cg-list/old/msg03353.html
 +
# http://www.virtual-earth.de/CG/cg-list/old/msg03354.html
 +
# http://www.virtual-earth.de/CG/cg-list/old/msg03376.html
 +
# http://www.virtual-earth.de/CG/cg-list/old/msg03379.html
 +
# http://www.virtual-earth.de/CG/cg-list/old/msg03381.html
  
[x <= y]  = F^2_13 (x, y)  = ((x) y)  = [not y without x].
+
====CG List &bull; New Archive====
  
The function that corresponds to the "biconditional",
+
* http://web.archive.org/web/20031104183832/http://mars.virtual-earth.de/pipermail/cg/2000q3/thread.html#3592
the "equivalence", or the "if and only" statement is
+
# http://web.archive.org/web/20030723202219/http://mars.virtual-earth.de/pipermail/cg/2000q3/003592.html
exhibited in the following fashion:
+
# http://web.archive.org/web/20030723202341/http://mars.virtual-earth.de/pipermail/cg/2000q3/003593.html
 +
# &bull;
 +
# http://web.archive.org/web/20030723202516/http://mars.virtual-earth.de/pipermail/cg/2000q3/003595.html
 +
# &bull;
 +
# &bull;
 +
# &bull;
  
[x <=> y]  = [x = y]  = F^2_09 (x, y)  = ((x , y)).
+
====CG List &bull; Old Archive====
  
Finally, there is a boolean function that is logically associated with
+
# &bull;
the "exclusive disjunction", "inequivalence", or "not equals" statement,
+
# http://web.archive.org/web/20020321115639/http://www.virtual-earth.de/CG/cg-list/msg03352.html
algebraically associated with the "binary sum" or "bitsum" operation,
+
# &bull;
and geometrically associated with the "symmetric difference" of sets.
+
# http://web.archive.org/web/20020321120331/http://www.virtual-earth.de/CG/cg-list/msg03354.html
This function is given by:
+
# http://web.archive.org/web/20020321223131/http://www.virtual-earth.de/CG/cg-list/msg03376.html
 +
# &bull;
 +
# http://web.archive.org/web/20020129134132/http://www.virtual-earth.de/CG/cg-list/msg03381.html
  
[x =/= y]  = [x + y]  = F^2_06 (x, y)  = (x , y).
+
===Sep 2000 &bull; Zeroth Order Logic===
  
Table 16. Boolean Functions on Two Variables
+
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/thrd241.html#01246
o----------o----------o-------------------------------------------o----------o
+
* http://web.archive.org/web/20130306202443/http://suo.ieee.org/email/thrd242.html#01406
| 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.
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01246.html
What has happened, in this suggested scheme of functional reasoning,
+
# http://web.archive.org/web/20080905054059/http://suo.ieee.org/email/msg01251.html
to the distinction that is quite pointedly made by careful logicians
+
# http://web.archive.org/web/20070223033521/http://suo.ieee.org/email/msg01380.html
between (1) the connectives called "conditionals" and symbolized by
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01406.html
the signs "->" and "<-", and (2) the assertions called "implications"
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01546.html
and symbolized by the signs "=>" and "<=", and, in a related question:
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01561.html
What has happened to the distinction that is equally insistently made
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01670.html
between (3) the connective called the "biconditional" and signified by
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01966.html
the sign "<->" and (4) the assertion that is called an "equivalence"
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01985.html
and signified by the sign "<=>"?  My answer is this: For my part,
+
# http://web.archive.org/web/20070401102902/http://suo.ieee.org/email/msg01988.html
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
+
===Oct 2000 &bull; All Liar, No Paradox===
  
In this Subsection, I finally bring together many of what may
+
* http://web.archive.org/web/20130306202504/http://suo.ieee.org/email/thrd236.html#01739
have appeared to be wholly independent threads of development,
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01739.html
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
+
===Nov 2000 &bull; Sowa's Top Level Categories===
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
+
====What Language To Use====
various brands of denotational operators can be reviewed here:
 
  
055. http://suo.ieee.org/email/msg07466.html
+
* http://web.archive.org/web/20070218222218/http://suo.ieee.org/email/threads.html#01956
057. http://suo.ieee.org/email/msg07469.html
+
# http://web.archive.org/web/20070320012929/http://suo.ieee.org/email/msg01956.html
  
070.  http://suo.ieee.org/ontology/msg03473.html
+
====Zeroth Order Logic====
071.  http://suo.ieee.org/ontology/msg03479.html
 
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
* http://web.archive.org/web/20070218222218/http://suo.ieee.org/email/threads.html#01966
 +
# http://web.archive.org/web/20070320012940/http://suo.ieee.org/email/msg01966.html
  
1.3.10.13  Stretching Exercises
+
====TLC In KIF====
  
Taking up the preceding arrays of particular connections, namely,
+
* http://web.archive.org/web/20130304163442/http://suo.ieee.org/ontology/thrd110.html#00048
the boolean functions on two or less variables, it possible to
+
# http://web.archive.org/web/20081204195421/http://suo.ieee.org/ontology/msg00048.html
illustrate the use of the stretch operation in a variety of
+
# http://web.archive.org/web/20070320014557/http://suo.ieee.org/ontology/msg00051.html
concrete cases.
 
  
For example, suppose that F is a connection of the form F : %B%^2 -> %B%,
+
===Dec 2000 &bull; Sequential Interactions Generating Hypotheses===
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,
+
* http://web.archive.org/web/20130306202621/http://suo.ieee.org/email/thrd217.html#02607
and the stretch of the connection F to #f# on X amounts to a proposition
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg02607.html
F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg02608.html
the "stretch of F to p and q". If one is concerned with many different
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg03183.html
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,
+
===Jan 2001 &bull; Differential Analytic Turing Automata===
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:
+
====DATA &bull; Arisbe List====
  
F(x, y)  =  F^2_06 (x, y)  =  -(x, y)-.
+
* http://web.archive.org/web/20150107163000/http://stderr.org/pipermail/arisbe/2001-January/thread.html#182
 +
# http://web.archive.org/web/20061013224128/http://stderr.org/pipermail/arisbe/2001-January/000182.html
 +
# http://web.archive.org/web/20061013224814/http://stderr.org/pipermail/arisbe/2001-January/000200.html
  
This connection is the boolean function on a couple of variables x, y
+
====DATA &bull; Ontology List====
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
+
* http://web.archive.org/web/20130304165332/http://suo.ieee.org/ontology/thrd95.html#00596
about things in the universe X = %B%^2. If S is a sentence that denotes
+
# http://web.archive.org/web/20041021223934/http://suo.ieee.org/ontology/msg00596.html
the proposition F, then the corresponding assertion says exactly what one
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg00618.html
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 |]
+
===Mar 2001 &bull; Propositional Equation Reasoning Systems===
  
            = F^(-1)(%1%)
+
====PERS &bull; Arisbe List====
  
            =  {<x, y> in %B%^2  : S}
+
* http://web.archive.org/web/20150107210802/http://stderr.org/pipermail/arisbe/2001-March/thread.html#380
 +
* http://web.archive.org/web/20150107212028/http://stderr.org/pipermail/arisbe/2001-April/thread.html#407
  
            =  {<x, y> in %B%^2  : F(x, y) = %1%}
+
# http://web.archive.org/web/20150107210011/http://stderr.org/pipermail/arisbe/2001-March/000380.html
 +
# http://web.archive.org/web/20050920031758/http://stderr.org/pipermail/arisbe/2001-April/000407.html
 +
# http://web.archive.org/web/20051202010243/http://stderr.org/pipermail/arisbe/2001-April/000409.html
 +
# http://web.archive.org/web/20051202074355/http://stderr.org/pipermail/arisbe/2001-April/000411.html
 +
# http://web.archive.org/web/20051202021217/http://stderr.org/pipermail/arisbe/2001-April/000412.html
 +
# http://web.archive.org/web/20051201225716/http://stderr.org/pipermail/arisbe/2001-April/000413.html
 +
# http://web.archive.org/web/20051202001736/http://stderr.org/pipermail/arisbe/2001-April/000416.html
 +
# http://web.archive.org/web/20051202053817/http://stderr.org/pipermail/arisbe/2001-April/000417.html
 +
# http://web.archive.org/web/20051202013458/http://stderr.org/pipermail/arisbe/2001-April/000421.html
 +
# http://web.archive.org/web/20051202013024/http://stderr.org/pipermail/arisbe/2001-April/000427.html
 +
# http://web.archive.org/web/20051202032812/http://stderr.org/pipermail/arisbe/2001-April/000428.html
 +
# http://web.archive.org/web/20051201225109/http://stderr.org/pipermail/arisbe/2001-April/000430.html
 +
# http://web.archive.org/web/20050908023250/http://stderr.org/pipermail/arisbe/2001-April/000432.html
 +
# http://web.archive.org/web/20051202002952/http://stderr.org/pipermail/arisbe/2001-April/000433.html
 +
# http://web.archive.org/web/20051201220336/http://stderr.org/pipermail/arisbe/2001-April/000434.html
 +
# http://web.archive.org/web/20050906215058/http://stderr.org/pipermail/arisbe/2001-April/000435.html
  
            = {<x, y> in %B%^2  :  F(x, y)}
+
====PERS &bull; Arisbe List &bull; Discussion====
  
            =  {<x, y> in %B%^2  : -(x, y)- = %1%}
+
* http://web.archive.org/web/20150107212028/http://stderr.org/pipermail/arisbe/2001-April/thread.html#397
 +
# http://web.archive.org/web/20150107212003/http://stderr.org/pipermail/arisbe/2001-April/000397.html
  
            = {<x, y> in %B%^2  :  -(x, y)- }
+
====PERS &bull; Ontology List====
  
            =  {<x, y> in %B%^2  : x exclusive-or y}
+
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd74.html#01779
 +
# http://web.archive.org/web/20070326233418/http://suo.ieee.org/ontology/msg01779.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg01897.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02005.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02011.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02014.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02015.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02024.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02046.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02047.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02068.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02102.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02109.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02117.html
 +
# http://web.archive.org/web/20040116001230/http://suo.ieee.org/ontology/msg02125.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02128.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02134.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02138.html
  
            = {<x, y> in %B%^2  :  just one true of x, y}
+
====PERS &bull; SUO List====
  
            =  {<x, y> in %B%^2  : x not equal to y}
+
* http://web.archive.org/web/20130109194711/http://suo.ieee.org/email/thrd187.html#04187
 +
# http://web.archive.org/web/20140423181000/http://suo.ieee.org/email/msg04187.html
 +
# http://web.archive.org/web/20070922193822/http://suo.ieee.org/email/msg04305.html
 +
# http://web.archive.org/web/20071007170752/http://suo.ieee.org/email/msg04413.html
 +
# http://web.archive.org/web/20070121063018/http://suo.ieee.org/email/msg04419.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04422.html
 +
# http://web.archive.org/web/20070305132316/http://suo.ieee.org/email/msg04423.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04432.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04454.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04455.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04476.html
 +
# http://web.archive.org/web/20060718091105/http://suo.ieee.org/email/msg04510.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04517.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04525.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04533.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04536.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04542.html
 +
# http://web.archive.org/web/20050824231950/http://suo.ieee.org/email/msg04546.html
  
            = {<x, y> in %B%^2  :  x <=/=> y}
+
===Jul 2001 &bull; Reflective Extension Of Logical Graphs===
  
            = {<x, y> in %B%^2  :  x =/= y}
+
====RefLog &bull; Arisbe List====
  
            =  {<x, y> in %B%^2  : x + y}.
+
* http://web.archive.org/web/20150109141200/http://stderr.org/pipermail/arisbe/2001-July/thread.html#711
 +
# http://web.archive.org/web/20150109141000/http://stderr.org/pipermail/arisbe/2001-July/000711.html
  
Notice the slight distinction, that I continue to maintain at this point,
+
====RefLog &bull; SUO List====
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
+
* http://web.archive.org/web/20070302133623/http://suo.ieee.org/email/thrd154.html#05694
confusing propositions with sentences and sentences with assertions, then
+
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg05694.html
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)-,
+
===Dec 2001 &bull; Functional Conception Of Quantificational Logic===
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
+
====FunLog &bull; Arisbe List====
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%
+
* http://web.archive.org/web/20141005034441/http://stderr.org/pipermail/arisbe/2001-December/thread.html#1212
|
+
# http://web.archive.org/web/20141005034614/http://stderr.org/pipermail/arisbe/2001-December/001212.html
|  q    =        -{Q}-       :   X -> %B%
+
# http://web.archive.org/web/20141005034615/http://stderr.org/pipermail/arisbe/2001-December/001213.html
|
+
# http://web.archive.org/web/20051202034557/http://stderr.org/pipermail/arisbe/2001-December/001216.html
| <p, q>  =  < -{P}- , -{Q}- >  : (X -> %B%)^2
+
# http://web.archive.org/web/20051202074331/http://stderr.org/pipermail/arisbe/2001-December/001221.html
 +
# http://web.archive.org/web/20051201235028/http://stderr.org/pipermail/arisbe/2001-December/001222.html
 +
# http://web.archive.org/web/20051202052037/http://stderr.org/pipermail/arisbe/2001-December/001223.html
 +
# http://web.archive.org/web/20050827214411/http://stderr.org/pipermail/arisbe/2001-December/001224.html
 +
# http://web.archive.org/web/20051202092500/http://stderr.org/pipermail/arisbe/2001-December/001225.html
 +
# http://web.archive.org/web/20051202051942/http://stderr.org/pipermail/arisbe/2001-December/001226.html
 +
# http://web.archive.org/web/20050425140213/http://stderr.org/pipermail/arisbe/2001-December/001227.html
  
Then one has an operator F^$, the stretch of the connection F over X,
+
====FunLog &bull; Ontology List====
and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with
 
the following properties:
 
  
| F^$        =  -( , )-^$  : (X -> %B%)^2 -> (X -> %B%)
+
* http://web.archive.org/web/20120222171225/http://suo.ieee.org/ontology/thrd38.html#03562
|
+
# http://web.archive.org/web/20110608022546/http://suo.ieee.org/ontology/msg03562.html
| F^$ (p, q)  =  -(p, q)-^$  :   X -> %B%
+
# http://web.archive.org/web/20110608022712/http://suo.ieee.org/ontology/msg03563.html
 +
# http://web.archive.org/web/20110608023312/http://suo.ieee.org/ontology/msg03564.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03565.html
 +
# http://web.archive.org/web/20070812011325/http://suo.ieee.org/ontology/msg03569.html
 +
# http://web.archive.org/web/20110608023228/http://suo.ieee.org/ontology/msg03570.html
 +
# http://web.archive.org/web/20110608022616/http://suo.ieee.org/ontology/msg03568.html
 +
# http://web.archive.org/web/20110608023557/http://suo.ieee.org/ontology/msg03572.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03577.html
 +
# http://web.archive.org/web/20070317021141/http://suo.ieee.org/ontology/msg03578.html
 +
# http://web.archive.org/web/20110608021549/http://suo.ieee.org/ontology/msg03579.html
 +
# http://web.archive.org/web/20110608021332/http://suo.ieee.org/ontology/msg03580.html
 +
# http://web.archive.org/web/20110608020250/http://suo.ieee.org/ontology/msg03581.html
 +
# http://web.archive.org/web/20110608021344/http://suo.ieee.org/ontology/msg03582.html
 +
# http://web.archive.org/web/20110608021557/http://suo.ieee.org/ontology/msg03583.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04247.html
  
As a result, the application of the proposition F^$ (p, q) to each x in X
+
===Dec 2001 &bull; Cactus Language===
yields a logical value in %B%, all in accord with the following equations:
 
  
| F^$ (p, q)(x)  =   -(p, q)-^$ (x)  in  %B%
+
====Cactus Town Cartoons &bull; Arisbe List====
|
 
|  ^                        ^
 
|  |                        |
 
=                         =
 
|  |                        |
 
|  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
+
* http://web.archive.org/web/20141005034441/http://stderr.org/pipermail/arisbe/2001-December/thread.html#1214
F to p and q on X is just another proposition about things in X, a simple
+
# http://web.archive.org/web/20050825005438/http://stderr.org/pipermail/arisbe/2001-December/001214.html
proposition in its own right, no matter how complex its current expression
+
# http://web.archive.org/web/20051202101235/http://stderr.org/pipermail/arisbe/2001-December/001217.html
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)-^$ |]
+
====Cactus Town Cartoons &bull; Ontology List====
  
                  =  (F^$ (p, q))^(-1)(%1%)
+
* http://web.archive.org/web/20120222171225/http://suo.ieee.org/ontology/thrd38.html#03567
 +
# http://web.archive.org/web/20110608023426/http://suo.ieee.org/ontology/msg03567.html
 +
# http://web.archive.org/web/20110608024449/http://suo.ieee.org/ontology/msg03571.html
  
                  = {x in X  :  F^$ (p, q)(x)}
+
===Jan 2002 &bull; Zeroth Order Theories===
  
                  = {x in X  :  -(p, q)-^$ (x)}
+
====ZOT &bull; Arisbe List====
  
                  =  {x in X  : -(p(x), q(x))- }
+
* http://web.archive.org/web/20150109041904/http://stderr.org/pipermail/arisbe/2002-January/thread.html#1293
 +
# http://web.archive.org/web/20150109042401/http://stderr.org/pipermail/arisbe/2002-January/001293.html
 +
# http://web.archive.org/web/20150109042402/http://stderr.org/pipermail/arisbe/2002-January/001294.html
 +
# http://web.archive.org/web/20050503213326/http://stderr.org/pipermail/arisbe/2002-January/001295.html
 +
# http://web.archive.org/web/20050503213330/http://stderr.org/pipermail/arisbe/2002-January/001296.html
 +
# http://web.archive.org/web/20050504070444/http://stderr.org/pipermail/arisbe/2002-January/001299.html
 +
# http://web.archive.org/web/20050504070430/http://stderr.org/pipermail/arisbe/2002-January/001300.html
 +
# http://web.archive.org/web/20050504070700/http://stderr.org/pipermail/arisbe/2002-January/001301.html
 +
# http://web.archive.org/web/20050504070704/http://stderr.org/pipermail/arisbe/2002-January/001302.html
 +
# http://web.archive.org/web/20050504070712/http://stderr.org/pipermail/arisbe/2002-January/001304.html
 +
# http://web.archive.org/web/20050504070717/http://stderr.org/pipermail/arisbe/2002-January/001305.html
 +
# http://web.archive.org/web/20050504070722/http://stderr.org/pipermail/arisbe/2002-January/001306.html
 +
# http://web.archive.org/web/20050504070726/http://stderr.org/pipermail/arisbe/2002-January/001308.html
 +
# http://web.archive.org/web/20050504070730/http://stderr.org/pipermail/arisbe/2002-January/001309.html
 +
# http://web.archive.org/web/20050504070434/http://stderr.org/pipermail/arisbe/2002-January/001310.html
 +
# http://web.archive.org/web/20050504070742/http://stderr.org/pipermail/arisbe/2002-January/001313.html
 +
# http://web.archive.org/web/20050504070746/http://stderr.org/pipermail/arisbe/2002-January/001314.html
 +
# http://web.archive.org/web/20050504070438/http://stderr.org/pipermail/arisbe/2002-January/001315.html
 +
# http://web.archive.org/web/20050504070540/http://stderr.org/pipermail/arisbe/2002-January/001316.html
 +
# http://web.archive.org/web/20050504070750/http://stderr.org/pipermail/arisbe/2002-January/001317.html
  
                  = {x in X  :  p(x) ± q(x)}
+
====ZOT &bull; Arisbe List &bull; Discussion====
  
                  =  {x in X  : p(x) =/= q(x)}
+
* http://web.archive.org/web/20150109041904/http://stderr.org/pipermail/arisbe/2002-January/thread.html#1293
 +
# http://web.archive.org/web/20050503213334/http://stderr.org/pipermail/arisbe/2002-January/001297.html
 +
# http://web.archive.org/web/20050504070656/http://stderr.org/pipermail/arisbe/2002-January/001298.html
 +
# http://web.archive.org/web/20050504070708/http://stderr.org/pipermail/arisbe/2002-January/001303.html
 +
# http://web.archive.org/web/20050504070544/http://stderr.org/pipermail/arisbe/2002-January/001307.html
 +
# http://web.archive.org/web/20050504070734/http://stderr.org/pipermail/arisbe/2002-January/001311.html
 +
# http://web.archive.org/web/20050504070738/http://stderr.org/pipermail/arisbe/2002-January/001312.html
 +
# http://web.archive.org/web/20050504070755/http://stderr.org/pipermail/arisbe/2002-January/001318.html
  
                  = {x in X  :  -{P}- (x) =/= -{Q}- (x)}
+
====ZOT &bull; Ontology List====
  
                  =  {x in X  : x in P <=/=> x in Q}
+
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd35.html#03680
 +
# http://web.archive.org/web/20070323210742/http://suo.ieee.org/ontology/msg03680.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03681.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03682.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03683.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03691.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03693.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03695.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03696.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03701.html
 +
# http://web.archive.org/web/20070329211521/http://suo.ieee.org/ontology/msg03702.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03703.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03706.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03707.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03708.html
 +
# http://web.archive.org/web/20080620074722/http://suo.ieee.org/ontology/msg03712.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03715.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03716.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03717.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03718.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03721.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03722.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03723.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03724.html
  
                  = {x in X  :  x in P-Q or x in Q-P}
+
====ZOT &bull; Ontology List &bull; Discussion====
  
                  =  {x in X  : x in P-Q |_| Q-P}
+
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd35.html#03680
 +
* http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/thrd35.html#03697
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03684.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03685.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03686.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03687.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03689.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03690.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03694.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03697.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03698.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03699.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03700.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03704.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03705.html
 +
# http://web.archive.org/web/20070330093628/http://suo.ieee.org/ontology/msg03709.html
 +
# http://web.archive.org/web/20080705071714/http://suo.ieee.org/ontology/msg03710.html
 +
# http://web.archive.org/web/20080620010020/http://suo.ieee.org/ontology/msg03711.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03713.html
 +
# http://web.archive.org/web/20080620074749/http://suo.ieee.org/ontology/msg03714.html
 +
# http://web.archive.org/web/20061005100254/http://suo.ieee.org/ontology/msg03719.html
 +
# http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03720.html
  
                  = {x in X  :  x in P ± Q}
+
===Mar 2003 &bull; Theme One Program &bull; Logical Cacti===
  
                  =  P ± Q          c  X
+
* http://web.archive.org/web/20150224210000/http://stderr.org/pipermail/inquiry/2003-March/thread.html#102
 +
* http://web.archive.org/web/20150224210000/http://stderr.org/pipermail/inquiry/2003-March/thread.html#114
 +
# http://web.archive.org/web/20081007043317/http://stderr.org/pipermail/inquiry/2003-March/000114.html
 +
# http://web.archive.org/web/20080908075558/http://stderr.org/pipermail/inquiry/2003-March/000115.html
 +
# http://web.archive.org/web/20080908080336/http://stderr.org/pipermail/inquiry/2003-March/000116.html
  
                  = [|p|] ± [|q|]  c  X.
+
===Feb 2005 &bull; Theme One Program &bull; Logical Cacti===
  
Which was to be shown.
+
* http://web.archive.org/web/20150109155110/http://stderr.org/pipermail/inquiry/2005-February/thread.html#2348
 +
* http://web.archive.org/web/20150109155110/http://stderr.org/pipermail/inquiry/2005-February/thread.html#2360
 +
# http://web.archive.org/web/20150109152359/http://stderr.org/pipermail/inquiry/2005-February/002360.html
 +
# http://web.archive.org/web/20150109152401/http://stderr.org/pipermail/inquiry/2005-February/002361.html
 +
# http://web.archive.org/web/20061013233259/http://stderr.org/pipermail/inquiry/2005-February/002362.html
 +
# http://web.archive.org/web/20081121103109/http://stderr.org/pipermail/inquiry/2005-February/002363.html
  
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
[[Category:Artificial Intelligence]]
</pre>
+
[[Category:Charles Sanders Peirce]]
 +
[[Category:Combinatorics]]
 +
[[Category:Computer Science]]
 +
[[Category:Cybernetics]]
 +
[[Category:Equational Reasoning]]
 +
[[Category:Formal Languages]]
 +
[[Category:Formal Systems]]
 +
[[Category:Graph Theory]]
 +
[[Category:Knowledge Representation]]
 +
[[Category:Logic]]
 +
[[Category:Logical Graphs]]
 +
[[Category:Mathematics]]
 +
[[Category:Philosophy]]
 +
[[Category:Semiotics]]
 +
[[Category:Visualization]]

Latest revision as of 20:44, 2 August 2017

Author: Jon Awbrey

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 of Modern Science”, [HJB, 38]

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.

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.

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 of Modern Science", [HJB, 38]

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 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 \(s\!\) and a particular object \(o\!\), namely, the fact that \(s\!\) denotes \(o\!\) or the fact that \(o\!\) is denoted by \(s\!\), be symbolized in one of the following two ways:

\(\begin{array}{lccc} 1. & s & \rightarrow & o \\ \\ 2. & o & \leftarrow & s \\ \end{array}\)

Now consider the following paradigm:

\(\begin{array}{llccc} 1. & \operatorname{If} & ^{\backprime\backprime}\operatorname{A}^{\prime\prime} & \rightarrow & \operatorname{Ann}, \\ & \operatorname{that~is}, & ^{\backprime\backprime}\operatorname{A}^{\prime\prime} & \operatorname{denotes} & \operatorname{Ann}, \\ & \operatorname{then} & \operatorname{A} & = & \operatorname{Ann} \\ & \operatorname{and} & \operatorname{Ann} & = & \operatorname{A}. \\ & \operatorname{Thus} & ^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} & \rightarrow & \operatorname{A}, \\ & \operatorname{that~is}, & ^{\backprime\backprime}\operatorname{Ann}^{\prime\prime} & \operatorname{denotes} & \operatorname{A}. \\ \end{array}\)

\(\begin{array}{llccc} 2. & \operatorname{If} & \operatorname{Bob} & \leftarrow & ^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\ & \operatorname{that~is}, & \operatorname{Bob} & \operatorname{is~denoted~by} & ^{\backprime\backprime}\operatorname{B}^{\prime\prime}, \\ & \operatorname{then} & \operatorname{Bob} & = & \operatorname{B} \\ & \operatorname{and} & \operatorname{B} & = & \operatorname{Bob}. \\ & \operatorname{Thus} & \operatorname{B} & \leftarrow & ^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}, \\ & \operatorname{that~is}, & \operatorname{B} & \operatorname{is~denoted~by} & ^{\backprime\backprime}\operatorname{Bob}^{\prime\prime}. \\ \end{array}\)

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 higher order 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:

\(\begin{array}{lll} ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & \leftarrow & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\ \\ ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} & \rightarrow & ^{\backprime\backprime}\operatorname{~}^{\prime\prime} \\ \end{array}\)

Using the raised dot "\(\cdot\)" 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:

\(\begin{array}{lllll} ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & \leftarrow & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{b}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{l}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{a}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{n}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{k}^{\prime\prime} \\ \end{array}\)

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:

\(\begin{array}{lll} ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \\ \end{array}\)

With these kinds of identity in mind, it is possible to extend the use of the "\(\cdot\)" sign to mark the articulation of either named or quoted strings into both named and quoted strings. For example:

\(\begin{array}{lclcl} ^{\backprime\backprime}\operatorname{~~}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & \operatorname{blank} \, \cdot \, \operatorname{blank} \\ \\ ^{\backprime\backprime}\operatorname{~blank}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{~}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} & = & \operatorname{blank} \, \cdot \, ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \\ \\ ^{\backprime\backprime}\operatorname{blank~}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \, ^{\backprime\backprime}\operatorname{~}^{\prime\prime} & = & ^{\backprime\backprime}\operatorname{blank}^{\prime\prime} \, \cdot \, \operatorname{blank} \end{array}\)

A few definitions from formal language theory are required at this point.

An alphabet is a finite set of signs, typically, \(\mathfrak{A} = \{ \mathfrak{a}_1, \ldots, \mathfrak{a}_n \}.\)

A string over an alphabet \(\mathfrak{A}\) is a finite sequence of signs from \(\mathfrak{A}.\)

The length of a string is just its length as a sequence of signs.

The empty string is the unique sequence of length 0. It is sometimes denoted by an empty pair of quotation marks, \(^{\backprime\backprime\prime\prime},\) but more often by the Greek symbols epsilon or lambda.

A sequence of length \(k > 0\!\) is typically presented in the concatenated forms:

\(s_1 s_2 \ldots s_{k-1} s_k\!\)

or

\(s_1 \cdot s_2 \cdot \ldots \cdot s_{k-1} \cdot s_k\)

with \(s_j \in \mathfrak{A}\) for all \(j = 1 \ldots k.\)

Two alternative notations are often useful:

\(\varepsilon\!\) = \({}^{\backprime\backprime\prime\prime}\!\) = the empty string.
\(\underline\varepsilon\!\) = \(\{ \varepsilon \}\!\) = the language consisting of a single empty string.

The kleene star \(\mathfrak{A}^*\) of alphabet \(\mathfrak{A}\) is the set of all strings over \(\mathfrak{A}.\) In particular, \(\mathfrak{A}^*\) includes among its elements the empty string \(\varepsilon.\)

The kleene plus \(\mathfrak{A}^+\) of an alphabet \(\mathfrak{A}\) is the set of all positive length strings over \(\mathfrak{A},\) in other words, everything in \(\mathfrak{A}^*\) but the empty string.

A formal language \(\mathfrak{L}\) over an alphabet \(\mathfrak{A}\) is a subset of \(\mathfrak{A}^*.\) In brief, \(\mathfrak{L} \subseteq \mathfrak{A}^*.\) If \(s\!\) is a string over \(\mathfrak{A}\) and if \(s\!\) is an element of \(\mathfrak{L},\) then it is customary to call \(s\!\) a sentence of \(\mathfrak{L}.\) Thus, a formal language \(\mathfrak{L}\) is defined by specifying its elements, which amounts to saying what it means to be a sentence of \(\mathfrak{L}.\)

One last device turns out to be useful in this connection. If \(s\!\) is a string that ends with a sign \(t,\!\) then \(s \cdot t^{-1}\) is the string that results by deleting from \(s\!\) the terminal \(t.\!\)

In this context, I make the following distinction:

  1. To delete an appearance of a sign is to replace it with an appearance of the empty string "".
  2. To erase an appearance of a sign is to replace it with an appearance of the blank symbol " ".

A token is a particular appearance of a sign.

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 \(\mathfrak{C}\) is actually a parameterized family of languages, consisting of one language \(\mathfrak{C}(\mathfrak{P})\) for each set \(\mathfrak{P}\) of paints.

The alphabet \(\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}\) is the disjoint union of two sets of symbols:

  1. \(\mathfrak{M}\) is the alphabet of measures, the set of punctuation marks, or the collection of syntactic constants that is common to all of the languages \(\mathfrak{C}(\mathfrak{P}).\) This set of signs is given as follows:

    \(\begin{array}{lccccccccccc} \mathfrak{M} & = & \{ & \mathfrak{m}_1 & , & \mathfrak{m}_2 & , & \mathfrak{m}_3 & , & \mathfrak{m}_4 & \} \\ & = & \{ & ^{\backprime\backprime} \, \operatorname{~} \, ^{\prime\prime} & , & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & , & ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} & , & ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} & \} \\ & = & \{ & \operatorname{blank} & , & \operatorname{links} & , & \operatorname{comma} & , & \operatorname{right} & \} \\ \end{array}\)

  2. \(\mathfrak{P}\) is the palette, the alphabet of paints, or the collection of syntactic variables that is peculiar to the language \(\mathfrak{C}(\mathfrak{P}).\) This set of signs is given as follows:

    \(\mathfrak{P} = \{ \mathfrak{p}_j : j \in J \}.\)

The easiest way to define the language \(\mathfrak{C}(\mathfrak{P})\!\) 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 \(\mathfrak{A}^*\!\) that are called syntactic connectives. If the strings on which they operate are exclusively sentences of \(\mathfrak{C}(\mathfrak{P}),\!\) 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.

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 \(\mathfrak{A}^*.\)

  1. The concatenation of one string \(s_1\!\) is just the string \(s_1.\!\)

    The concatenation of two strings \(s_1, s_2\!\) is the string \({s_1 \cdot s_2}.\!\)

    The concatenation of the \(k\!\) strings \((s_j)_{j = 1}^k\!\) is the string of the form \({s_1 \cdot \ldots \cdot s_k}.\!\)

  2. The surcatenation of one string \(s_1\!\) is the string \(^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

    The surcatenation of two strings \(s_1, s_2\!\) is \(^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

    The surcatenation of the \(k\!\) strings \((s_j)_{j = 1}^k\) is the string of the form \(^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

These definitions can be made a little more succinct by defining the following sorts of generic operators on strings:

  1. The concatenation \(\operatorname{Conc}_{j=1}^k\) of the sequence of \(k\!\) strings \((s_j)_{j=1}^k\) is defined recursively as follows:
    1. \(\operatorname{Conc}_{j=1}^1 s_j \ = \ s_1.\)
    2. For \(\ell > 1,\!\)

      \(\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Conc}_{j=1}^{\ell - 1} s_j \, \cdot \, s_\ell.\)

  2. The surcatenation \(\operatorname{Surc}_{j=1}^k\) of the sequence of \(k\!\) strings \((s_j)_{j=1}^k\) is defined recursively as follows:
    1. \(\operatorname{Surc}_{j=1}^1 s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
    2. For \(\ell > 1,\!\)

      \(\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Surc}_{j=1}^{\ell - 1} s_j \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_\ell \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

The definitions of these syntactic operations can now be organized in a slightly better fashion 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:

    \(\operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}\)  =  the empty string.

    Next, the construction of the \(k\!\)-place concatenation can be broken into stages by means of the following conceptions:

    1. The precatenation \(\operatorname{Prec} (s_1, s_2)\) of the two strings \(s_1, s_2\!\) is the string that is defined as follows:

      \(\operatorname{Prec} (s_1, s_2) \ = \ s_1 \cdot s_2.\)

    2. The concatenation of the sequence of \(k\!\) strings \(s_1, \ldots, s_k\!\) can now be defined as an iterated precatenation over the sequence of \(k+1\!\) strings that begins with the string \(s_0 = \operatorname{Conc}^0 \, = \, ^{\backprime\backprime\prime\prime}\) and then continues on through the other \(k\!\) strings:

      1. \(\operatorname{Conc}_{j=0}^0 s_j \ = \ \operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime}.\)

      2. For \(\ell > 0,\!\)

        \(\operatorname{Conc}_{j=1}^\ell s_j \ = \ \operatorname{Prec}(\operatorname{Conc}_{j=0}^{\ell - 1} s_j, s_\ell).\)

  2. The conception of the \(k\!\)-place surcatenation operation can be extended to include its natural "prequel":

    \(\operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.\)

    Finally, the construction of the \(k\!\)-place surcatenation can be broken into stages by means of the following conceptions:

    1. A subclause in \(\mathfrak{A}^*\) is a string that ends with a \(^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

    2. The subcatenation \(\operatorname{Subc} (s_1, s_2)\) of a subclause \(s_1\!\) by a string \(s_2\!\) is the string that is defined as follows:

      \(\operatorname{Subc} (s_1, s_2) \ = \ s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

    3. The surcatenation of the \(k\!\) strings \(s_1, \ldots, s_k\!\) can now be defined as an iterated subcatenation over the sequence of \(k+1\!\) strings that starts with the string \(s_0 \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}\) and then continues on through the other \(k\!\) strings:

      1. \(\operatorname{Surc}_{j=0}^0 s_j \ = \ \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.\)

      2. For \(\ell > 0,\!\)

        \(\operatorname{Surc}_{j=1}^\ell s_j \ = \ \operatorname{Subc}(\operatorname{Surc}_{j=0}^{\ell - 1} s_j, s_\ell).\)

Notice that the expressions \(\operatorname{Conc}_{j=0}^0 s_j\) and \(\operatorname{Surc}_{j=0}^0 s_j\) are defined in such a way that the respective operators \(\operatorname{Conc}^0\) and \(\operatorname{Surc}^0\) simply ignore, in the manner of constants, whatever sequences of strings \(s_j\!\) may be listed as their ostensible arguments.

Having defined the basic operations of concatenation and surcatenation on arbitrary strings, in effect, giving them operational meaning for the all-inclusive language \(\mathfrak{L} = \mathfrak{A}^*,\) it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

If \(\mathfrak{L}\) is an arbitrary formal language over an alphabet of the sort that we are talking about, that is, an alphabet of the form \(\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},\) then there are a number of basic structural relations that can be defined on the strings of \(\mathfrak{L}.\)

1. \(s\!\) is the concatenation of \(s_1\!\) and \(s_2\!\) in \(\mathfrak{L}\) if and only if
  \(s_1\!\) is a sentence of \(\mathfrak{L},\) \(s_2\!\) is a sentence of \(\mathfrak{L},\) and
  \(s = s_1 \cdot s_2.\)
2. \(s\!\) is the concatenation of the \(k\!\) strings \(s_1, \ldots, s_k\!\) in \(\mathfrak{L},\)
  if and only if \(s_j\!\) is a sentence of \(\mathfrak{L},\) for all \(j = 1 \ldots k,\) and
  \(s = \operatorname{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k.\)
3. \(s\!\) is the discatenation of \(s_1\!\) by \(t\!\) if and only if
  \(s_1\!\) is a sentence of \(\mathfrak{L},\) \(t\!\) is an element of \(\mathfrak{A},\) and
  \(s_1 = s \cdot t.\)
  When this is the case, one more commonly writes:
  \(s = s_1 \cdot t^{-1}.\)
4. \(s\!\) is a subclause of \(\mathfrak{L}\) if and only if
  \(s\!\) is a sentence of \(\mathfrak{L}\) and \(s\!\) ends with a \(^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
5. \(s\!\) is the subcatenation of \(s_1\!\) by \(s_2\!\) if and only if
  \(s_1\!\) is a subclause of \(\mathfrak{L},\) \(s_2\!\) is a sentence of \(\mathfrak{L},\) and
  \(s = s_1 \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_2 \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)
6. \(s\!\) is the surcatenation of the \(k\!\) strings \(s_1, \ldots, s_k\!\) in \(\mathfrak{L},\)
  if and only if \(s_j\!\) is a sentence of \(\mathfrak{L},\) for all \({j = 1 \ldots k},\!\) and
  \(s \ = \ \operatorname{Surc}_{j=1}^k s_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, s_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, s_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

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.

The painted cactus language with paints in the set \(\mathfrak{P} = \{ p_j : j \in J \}\) is the formal language \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*\) 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. \(\operatorname{Conc}^0\) and \(\operatorname{Surc}^0\) are sentences.
PC 4. For each positive integer \(k,\!\)
  if \(s_1, \ldots, s_k\!\) are sentences,
  then \(\operatorname{Conc}_{j=1}^k s_j\) is a sentence,
  and \(\operatorname{Surc}_{j=1}^k s_j\) is a sentence.

As usual, saying that \(s\!\) is a sentence is just a conventional way of stating that the string \(s\!\) belongs to the relevant formal language \(\mathfrak{L}.\) An individual sentence of \(\mathfrak{C} (\mathfrak{P}),\!\) for any palette \(\mathfrak{P},\) is referred to as a painted and rooted cactus expression (PARCE) on the palette \(\mathfrak{P},\) 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 \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P})\) is also described as the set \(\operatorname{PARCE} (\mathfrak{P})\) of PARCE's on the palette \(\mathfrak{P},\) more generically, as the PARCE's that constitute the language \(\operatorname{PARCE}.\)

A bare PARCE, a bit loosely referred to as a bare cactus expression, is a PARCE on the empty palette \(\mathfrak{P} = \varnothing.\) A bare PARCE is a sentence in the bare cactus language, \(\mathfrak{C}^0 = \mathfrak{C} (\varnothing) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\varnothing).\) This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language \(\mathfrak{C} (\mathfrak{P}).\) 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, \(\operatorname{Conc}^k\) and \(\operatorname{Surc}^k,\) 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:

\(S\ \operatorname{covers}\ T\)
\(S\ \operatorname{governs}\ T\)
\(S\ \operatorname{rules}\ T\)
\(S\ \operatorname{subsumes}\ T\)
\(S\ \operatorname{types~over}\ T\)

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 \(\mathfrak{L}.\)
  2. To express the fact or to make the assertion that each member of a specified set of strings \(T \subseteq \mathfrak{A}^*\) also belongs to the syntactic category \(S,\!\) the one that qualifies a string as being a sentence in the relevant formal language \(\mathfrak{L}.\)
  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 \(S,\!\) all within the target language \(\mathfrak{L}.\)

In these types of situation the letter \(^{\backprime\backprime} S \, ^{\prime\prime}\) 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 \(^{\backprime\backprime} T \, ^{\prime\prime}\) 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 \(\{ ^{\backprime\backprime} S \, ^{\prime\prime} \}\) whose sole member is the initial symbol with the set \(\mathfrak{Q}\) that assembles together all of the intermediate symbols results in the set \(\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}\) of non-terminal symbols. Completing the package, the alphabet \(\mathfrak{A}\) of the language is also known as the set of terminal symbols. In this discussion, I will adopt the convention that \(\mathfrak{Q}\) 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, \(q \in \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q}.\) Finally, it is convenient to refer to all of the symbols in \(\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A}\) as the augmented alphabet of the prospective grammar for the language, and accordingly to describe the strings in \(( \{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*\) as the augmented strings, in effect, expressing the forms that are superimposed on a language by one of its conceivable grammars. In certain settings it becomes desirable to separate the augmented strings that contain the symbol \(^{\backprime\backprime} S \, ^{\prime\prime}\) from all other sorts of augmented strings. In these situations the strings in the disjoint union \(\{ ^{\backprime\backprime} S \, ^{\prime\prime} \} \cup (\mathfrak{Q} \cup \mathfrak{A} )^*\) are known as the sentential forms of the associated grammar.

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 \(\mathfrak{K}\) 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 to say that \(T\!\) is covered by \(S,\!\) can be interpreted to say 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 of the form \(t <: T\!\) can be read in all of the ways that one typically reads an expression of the form \(t : T.\!\)

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.

Note. 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 the subject matter of the 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.

Employing the notion of a covering relation it becomes possible to redescribe the cactus language \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P})\) in the following ways.

Grammar 1

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 \(\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}\) that comes with the territory of the cactus language \(\mathfrak{C} (\mathfrak{P}),\!\) it specifies \(\mathfrak{Q} = \varnothing,\) in other words, it employs no intermediate symbols, and it embodies the covering set \(\mathfrak{K}\) as listed in the following display.


\(\mathfrak{C} (\mathfrak{P}) : \text{Grammar 1}\!\)

\(\mathfrak{Q} = \varnothing\)

\(\begin{array}{rcll} 1. & S & :> & m_1 \ = \ ^{\backprime\backprime} \operatorname{~} ^{\prime\prime} \\ 2. & S & :> & p_j, \, \text{for each} \, j \in J \\ 3. & S & :> & \operatorname{Conc}^0 \ = \ ^{\backprime\backprime\prime\prime} \\ 4. & S & :> & \operatorname{Surc}^0 \ = \ ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ 5. & S & :> & S^* \\ 6. & S & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ( \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, )^* \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ \end{array}\)


In this formulation, the last two lines specify that:

  1. The concept of a sentence in \(\mathfrak{L}\) covers any concatenation of sentences in \(\mathfrak{L},\) in effect, any number of freely chosen sentences that are available to be concatenated one after another.
  2. The concept of a sentence in \(\mathfrak{L}\) covers any surcatenation of sentences in \(\mathfrak{L},\) in effect, any string that opens with a \(^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime},\) continues with a sentence, possibly empty, follows with a finite number of phrases of the form \(^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,\) and closes with a \(^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}.\)

This appears to be just about the most concise description of the cactus language \(\mathfrak{C} (\mathfrak{P})\) that one can imagine, but there are 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 \(S\!\) that indicates a sentence 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.

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 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, \(\mathfrak{Q} = \varnothing,\) 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:

\(\begin{array}{lcccccccccccc} S & :> & S^* & = & \underline\varepsilon & \cup & S & \cup & S \cdot S & \cup & S \cdot S \cdot S & \cup & \ldots \\ \end{array}\)

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 a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence \(S.\!\) In particular, since it implies that \(S :> \underline\varepsilon,\) and since \(\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},\) for any formal language \(\mathfrak{L},\) the empty string \(\varepsilon\!\) 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 fact that it needs to be covered by a non-terminal symbol.
  3. Making a note of it by instituting an explicitly-named grammatical 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.

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 \(\mathfrak{C} (\mathfrak{P}),\!\) one has the following definitions of concatenation as iterated precatenation and of surcatenation as iterated subcatenation, respectively:

\(\begin{array}{llll} 1. & \operatorname{Conc}_{j=1}^0 & = & ^{\backprime\backprime\prime\prime} \\ \\ & \operatorname{Conc}_{j=1}^k S_j & = & \operatorname{Prec} (\operatorname{Conc}_{j=1}^{k-1} S_j, S_k) \\ \\ 2. & \operatorname{Surc}_{j=1}^0 & = & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ \\ & \operatorname{Surc}_{j=1}^k S_j & = & \operatorname{Subc} (\operatorname{Surc}_{j=1}^{k-1} S_j, S_k) \\ \\ \end{array}\)

In order to transform these recursive definitions into grammar rules, one introduces a new pair of intermediate symbols, \(\operatorname{Conc}\) and \(\operatorname{Surc},\) 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 \(\operatorname{Conc}\) and \(\operatorname{Surc}\) 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:

\(\begin{array}{llll} 1. & \operatorname{Conc} & :> & ^{\backprime\backprime\prime\prime} \\ \\ & \operatorname{Conc} & :> & \operatorname{Conc} \cdot S \\ \\ 2. & \operatorname{Surc} & :> & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ \\ & \operatorname{Surc} & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ \\ & \operatorname{Surc} & :> & \operatorname{Surc} \, \cdot \, ( \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \, )^{-1} \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \end{array}\)

As given, this particular fragment of the intended grammar contains a couple of features that are desirable to amend.

  1. Given the covering \(S :> \operatorname{Conc},\) the covering rule \(\operatorname{Conc} :> \operatorname{Conc} \cdot S\) says no more than the covering rule \(\operatorname{Conc} :> S \cdot S.\) Consequently, all of the information contained in these two covering rules is already covered by the statement that \(S :> S \cdot S.\)
  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.

Grammar 2

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 \(^{\backprime\backprime} \operatorname{,} ^{\prime\prime}\) interpolated between each pair of adjacent sentences. Thus, a typical tract \(T\!\) takes the form:

\(\begin{array}{lllllllllll} T & = & S_1 & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & \ldots & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & S_k \\ \end{array}\)

A tract must be distinguished from the abstract sequence of sentences, \(S_1, \ldots, S_k,\!\) 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 \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P}),\) 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:


\(\mathfrak{C} (\mathfrak{P}) : \text{Grammar 2}\!\)

\(\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}\)

\(\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & m_1 \\ 3. & S & :> & p_j, \, \text{for each} \, j \in J \\ 4. & S & :> & S \, \cdot \, S \\ 5. & S & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 6. & T & :> & S \\ 7. & T & :> & T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \\ \end{array}\)


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 get 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, \(T \, :> \, \varepsilon, m_1, p_j.\) In accord with these simple beginnings it comes to parse that the rule \(T \, :> \, T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,\) with the substitutions \(T = \varepsilon\) and \(S = \varepsilon\) on the covered side of the rule, bears the germinal implication that \(T \, :> \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}.\)

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 \(\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}\) 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.

Grammar 3

Although it is not strictly necessary to do so, it is possible to organize the materials of our developing 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 \(\{ m_1 \} \cup \mathfrak{P},\) possibly the empty string:

\(R\ \in\ ( \{ m_1 \} \cup \mathfrak{P} )^*\)

When there is no possibility of confusion, the letter \(^{\backprime\backprime} R \, ^{\prime\prime}\) 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, \(^{\backprime\backprime} R \, ^{\prime\prime} \in \mathfrak{Q},\) as a part of a new grammar for \(\mathfrak{C} (\mathfrak{P}).\) In effect, \(^{\backprime\backprime} R \, ^{\prime\prime}\) affords a grammatical recognition for any rune that forms a part of a sentence in \(\mathfrak{C} (\mathfrak{P}).\) 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 \({}^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},\!\) where \(T\!\) is a tract. Thus, a typical foil \(F\!\) has the form:

\(\begin{array}{*{15}{l}} F & = & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & \cdot & S_1 & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & \ldots & \cdot & ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} & \cdot & S_k & \cdot & ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ \end{array}\)

This is just the surcatenation of the sentences \(S_1, \ldots, S_k.\!\) 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 \(^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.\) Explicitly marking each foil \(F\!\) that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, \(^{\backprime\backprime} F \, ^{\prime\prime} \in \mathfrak{Q},\) further articulating the structures of sentences and expanding the grammar for the language \(\mathfrak{C} (\mathfrak{P}).\!\) 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 \(^{\backprime\backprime} F \, ^{\prime\prime}.\)


\(\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!\)

\(\mathfrak{Q} = \{ \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}\)

\(\begin{array}{rcll} 1. & S & :> & R \\ 2. & S & :> & F \\ 3. & S & :> & S \, \cdot \, S \\ 4. & R & :> & \varepsilon \\ 5. & R & :> & m_1 \\ 6. & R & :> & p_j, \, \text{for each} \, j \in J \\ 7. & R & :> & R \, \cdot \, R \\ 8. & F & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 9. & T & :> & S \\ 10. & T & :> & T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \\ \end{array}\!\)


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 \(\varepsilon,\) a blank symbol \(m_1,\!\) a paint \(p_j,\!\) or any concatenation of strings of these three types. Rule 8 characterizes a foil \(F\!\) as a string of the form \({}^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},\!\) 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 \(\mathfrak{C} (\mathfrak{P}),\!\) 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 \(\mathfrak{C} (\mathfrak{P}),\!\) in other words, the formal language \(\operatorname{PARCE}\!\) of painted and rooted cactus expressions, it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:

  1. The emptily painted and rooted cactus expressions make up the language \(\operatorname{EPARCE}\) that consists of a single empty string as its only sentence. In short:

    \(\operatorname{EPARCE} \ = \ \underline\varepsilon \ = \ \{ \varepsilon \}\)

  2. The significantly painted and rooted cactus expressions make up the language \(\operatorname{SPARCE}\) that consists of everything else, namely, all of the non-empty strings in the language \(\operatorname{PARCE}.\) In sum:

    \(\operatorname{SPARCE} \ = \ \operatorname{PARCE} \setminus \varepsilon\)

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:

\(\operatorname{SPARCE} \ = \ \operatorname{PARCE} \ - \ \operatorname{EPARCE}\)

In sum, one has the disjoint union:

\(\operatorname{PARCE} \ = \ \operatorname{EPARCE} \ \cup \ \operatorname{SPARCE}\)

For brevity in the present case, and to serve as a generic device in any similar array of situations, let \(S\!\) be the type of an arbitrary sentence, possibly empty, and let \(S'\!\) be the type of a specifically non-empty sentence. In addition, let \(\underline\varepsilon\) be the type of the empty sentence, in effect, the language \(\underline\varepsilon = \{ \varepsilon \}\) that contains a single empty string, and let a plus sign \(^{\backprime\backprime} + ^{\prime\prime}\) 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:

\(S \ = \ \underline\varepsilon \ + \ S'\)

With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})\) 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.

Grammar 4

If one imposes the distinction between empty and significant types on each non-terminal symbol in Grammar 2, then the non-terminal symbols \(^{\backprime\backprime} S \, ^{\prime\prime}\) and \(^{\backprime\backprime} T \, ^{\prime\prime}\) give rise to the expanded set of non-terminal symbols \(^{\backprime\backprime} S \, ^{\prime\prime}, \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime},\) leaving the last three of these to form the new intermediate alphabet. Grammar 4 has the intermediate alphabet \(\mathfrak{Q} \, = \, \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \},\) with the set \(\mathfrak{K}\) of covering rules as listed in the next display.


\(\mathfrak{C} (\mathfrak{P}) : \text{Grammar 4}\!\)

\(\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime}, \, ^{\backprime\backprime} T' \, ^{\prime\prime} \, \}\)

\(\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & S' \\ 3. & S' & :> & m_1 \\ 4. & S' & :> & p_j, \, \text{for each} \, j \in J \\ 5. & S' & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 6. & S' & :> & S' \, \cdot \, S' \\ 7. & T & :> & \varepsilon \\ 8. & T & :> & T' \\ 9. & T' & :> & T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S \\ \end{array}\)


In this version of a grammar for \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P}),\) the intermediate type \(T\!\) is partitioned as \(T = \underline\varepsilon + T',\) thereby parsing the intermediate symbol \(T\!\) in parallel fashion with the division of its overlying type as \(S = \underline\varepsilon + S'.\) 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 \(q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}\) and sentential forms \(W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*.\)


\(\text{Condition On Intermediate Significance}\!\)

\(\begin{array}{lccc} \text{If} & q & :> & W \\ \text{and} & W & = & \varepsilon \\ \text{then} & q & = & ^{\backprime\backprime} S \, ^{\prime\prime} \\ \end{array}\)


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, \(^{\backprime\backprime}\!< \, ^{\prime\prime}.\)

  1. The ordering \(^{\backprime\backprime}\!< \, ^{\prime\prime}\) on the set of non-terminal symbols, \(q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},\) ordains the initial symbol \(^{\backprime\backprime} S \, ^{\prime\prime}\) to be strictly prior to every intermediate symbol. This is tantamount to the axiom that \(^{\backprime\backprime} S \, ^{\prime\prime} < q,\) for all \(q \in \mathfrak{Q}.\)
  2. The ordering \(^{\backprime\backprime}\!< \, ^{\prime\prime}\) on the collection of sentential forms, \(W \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*,\) ordains the empty string to be strictly minor to every other sentential form. This is stipulated in the axiom that \(\varepsilon < W,\) for every non-empty sentential form \(W.\!\)

Given these two orderings, the constraint in question on intermediate significance can be stated as follows:


\(\text{Condition On Intermediate Significance}\!\)

\(\begin{array}{lccc} \text{If} & q & :> & W \\ \text{and} & q & > & ^{\backprime\backprime} S \, ^{\prime\prime} \\ \text{then} & W & > & \varepsilon \\ \end{array}\)


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.

Grammar 5

With the foregoing array of considerations in mind, one is gradually led to a grammar for \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P})\) in which all of the covering productions have either one of the following two forms:

\(\begin{array}{ccll} S & :> & \varepsilon & \\ q & :> & W, & \text{with} \ q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \ \text{and} \ W \in (\mathfrak{Q} \cup \mathfrak{A})^+ \\ \end{array}\)

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 \(S :> \varepsilon,\) 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 \(\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},\) with \(\mathfrak{K}\) as listed in the next display.


\(\mathfrak{C} (\mathfrak{P}) : \text{Grammar 5}\!\)

\(\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}\)

\(\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & S' \\ 3. & S' & :> & m_1 \\ 4. & S' & :> & p_j, \, \text{for each} \, j \in J \\ 5. & S' & :> & S' \, \cdot \, S' \\ 6. & S' & :> & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ 7. & S' & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 8. & T & :> & ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 9. & T & :> & S' \\ 10. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 11. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S' \\ \end{array}\)


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.

Grammar 6

Grammar 6 has the intermediate alphabet \(\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},\) with the production set \(\mathfrak{K}\) as listed in the next display.


\({\mathfrak{C} (\mathfrak{P}) : \text{Grammar 6}}\!\)

\(\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}\!\)

\(\begin{array}{rcll} 1. & S & :> & \varepsilon \\ 2. & S & :> & S' \\ 3. & S' & :> & R \\ 4. & S' & :> & F \\ 5. & S' & :> & S' \, \cdot \, S' \\ 6. & R & :> & m_1 \\ 7. & R & :> & p_j, \, \text{for each} \, j \in J \\ 8. & R & :> & R \, \cdot \, R \\ 9. & F & :> & ^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime} \\ 10. & F & :> & ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} \\ 11. & T & :> & ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 12. & T & :> & S' \\ 13. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \\ 14. & T & :> & T \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S' \\ \end{array}\)


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:

\(\operatorname{If}\) the strings \(S_1, \ldots, S_k\!\) are sentences,
\(\operatorname{Then}\) their concatenation in the form
  \(\operatorname{Conc}_{j=1}^k S_j \ = \ S_1 \, \cdot \, \ldots \, \cdot \, S_k\)
  is a sentence,
\(\operatorname{And}\) their surcatenation in the form
  \(\operatorname{Surc}_{j=1}^k S_j \ = \ ^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, S_1 \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, \ldots \, \cdot \, ^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} \, \cdot \, S_k \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime}\)
  is a sentence.

Generalities About Formal Grammars

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 \(\mathfrak{G}\) is given by a four-tuple \(\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )\) that takes the following form of description:

  1. \(^{\backprime\backprime} S \, ^{\prime\prime}\) is the initial, special, start, or sentence symbol. Since the letter \(^{\backprime\backprime} S \, ^{\prime\prime}\) 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. \(\mathfrak{Q} = \{ q_1, \ldots, q_m \}\) is a finite set of intermediate symbols, all distinct from \(^{\backprime\backprime} S \, ^{\prime\prime}.\)
  3. \(\mathfrak{A} = \{ a_1, \dots, a_n \}\) is a finite set of terminal symbols, also known as the alphabet of \(\mathfrak{G},\) all distinct from \(^{\backprime\backprime} S \, ^{\prime\prime}\) and disjoint from \(\mathfrak{Q}.\) Depending on the particular conception of the language \(\mathfrak{L}\) that is covered, generated, governed, or ruled by the grammar \(\mathfrak{G},\) that is, whether \(\mathfrak{L}\) is conceived to be a set of words, sentences, paragraphs, or more extended structures of discourse, it is usual to describe \(\mathfrak{A}\) as the alphabet, lexicon, vocabulary, liturgy, or phrase book of both the grammar \(\mathfrak{G}\) and the language \(\mathfrak{L}\) that it regulates.
  4. \(\mathfrak{K}\) 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 \(\mathfrak{K}\) it helps to define some additional terms:

  1. The symbols in \(\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A}\) form the augmented alphabet of \(\mathfrak{G}.\)
  2. The symbols in \(\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q}\) are the non-terminal symbols of \(\mathfrak{G}.\)
  3. The symbols in \(\mathfrak{Q} \cup \mathfrak{A}\) are the non-initial symbols of \(\mathfrak{G}.\)
  4. The strings in \(( \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q} \cup \mathfrak{A} )^*\) are the augmented strings for \(\mathfrak{G}.\)
  5. The strings in \(\{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup (\mathfrak{Q} \cup \mathfrak{A})^*\) are the sentential forms for \(\mathfrak{G}.\)

Each characterization in \(\mathfrak{K}\) is an ordered pair of strings \((S_1, S_2)\!\) that takes the following form:

\(S_1 \ = \ Q_1 \cdot q \cdot Q_2,\)
\(S_2 \ = \ Q_1 \cdot W \cdot Q_2.\)

In this scheme, \(S_1\!\) and \(S_2\!\) are members of the augmented strings for \(\mathfrak{G},\) more precisely, \(S_1\!\) is a non-empty string and a sentential form over \(\mathfrak{G},\) while \(S_2\!\) is a possibly empty string and also a sentential form over \(\mathfrak{G}.\)

Here also, \(q\!\) is a non-terminal symbol, that is, \(q \in \{ \, ^{\backprime\backprime} S \, ^{\prime\prime} \, \} \cup \mathfrak{Q},\) while \(Q_1, Q_2,\!\) and \(W\!\) are possibly empty strings of non-initial symbols, a fact that can be expressed in the form, \(Q_1, Q_2, W \in (\mathfrak{Q} \cup \mathfrak{A})^*.\)

In practice, the couplets in \(\mathfrak{K}\) are used to derive, to generate, or to produce sentences of the corresponding language \(\mathfrak{L} = \mathfrak{L} (\mathfrak{G}).\) The language \(\mathfrak{L}\) is then said to be governed, licensed, or regulated by the grammar \(\mathfrak{G},\) a circumstance that is expressed in the form \(\mathfrak{L} = \langle \mathfrak{G} \rangle.\) In order to facilitate this active employment of the grammar, it is conventional to write the abstract characterization \((S_1, S_2)\!\) and the specific characterization \((Q_1 \cdot q \cdot Q_2, \ Q_1 \cdot W \cdot Q_2)\) in the following forms, respectively:

\(\begin{array}{lll} S_1 & :> & S_2 \\ Q_1 \cdot q \cdot Q_2 & :> & Q_1 \cdot W \cdot Q_2 \\ \end{array}\)

In this usage, the characterization \(S_1 :> S_2\!\) is tantamount to a grammatical license to transform a string of the form \(Q_1 \cdot q \cdot Q_2\) into a string of the form \(Q1 \cdot W \cdot Q2,\) 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 \(Q1 \cdot \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])} \cdot Q2.\) In this application the notation \(S_1 :> S_2\!\) can be read to say that \(S_1\!\) produces \(S_2\!\) or that \(S_1\!\) transforms into \(S_2.\!\)

An immediate derivation in \(\mathfrak{G}\!\) is an ordered pair \((W, W^\prime)\!\) of sentential forms in \(\mathfrak{G}\!\) such that:

\(\begin{array}{llll} W = Q_1 \cdot X \cdot Q_2, & W' = Q_1 \cdot Y \cdot Q_2, & \text{and} & (X, Y) \in \mathfrak{K}. \end{array}\)

As noted above, it is usual to express the condition \((X, Y) \in \mathfrak{K}\) by writing \(X :> Y \, \text{in} \, \mathfrak{G}.\)

The immediate derivation relation is indicated by saying that \(W\!\) immediately derives \(W',\!\) by saying that \(W'\!\) is immediately derived from \(W\!\) in \(\mathfrak{G},\) and also by writing:

\(W ::> W'.\!\)

A derivation in \(\mathfrak{G}\) is a finite sequence \((W_1, \ldots, W_k)\!\) of sentential forms over \(\mathfrak{G}\) such that each adjacent pair \((W_j, W_{j+1})\!\) of sentential forms in the sequence is an immediate derivation in \(\mathfrak{G},\) in other words, such that:

\(W_j ::> W_{j+1},\ \text{for all}\ j = 1\ \text{to}\ k - 1.\)

If there exists a derivation \((W_1, \ldots, W_k)\!\) in \(\mathfrak{G},\) one says that \(W_1\!\) derives \(W_k\!\) in \(\mathfrak{G}\) or that \(W_k\!\) is derivable from \(W_1\!\) in \(\mathfrak{G},\) and one typically summarizes the derivation by writing:

\(W_1 :\!*\!:> W_k.\!\)

The language \(\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle\) that is generated by the formal grammar \(\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )\) is the set of strings over the terminal alphabet \(\mathfrak{A}\) that are derivable from the initial symbol \(^{\backprime\backprime} S \, ^{\prime\prime}\) by way of the intermediate symbols in \(\mathfrak{Q}\) according to the characterizations in \(\mathfrak{K}.\) In sum:

\(\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ \, W \in \mathfrak{A}^* \, : \, ^{\backprime\backprime} S \, ^{\prime\prime} \, :\!*\!:> \, W \, \}.\)

Finally, a string \(W\!\) is called a word, a sentence, or so on, of the language generated by \(\mathfrak{G}\) if and only if \(W\!\) is in \(\mathfrak{L} (\mathfrak{G}).\)

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 of Modern Science", [HJB, 38]

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:

\(X \Rightarrow Y\ \operatorname{and}\ Y \Rightarrow Z.\)

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:

\(X\ \le\ Y\ \le\ Z.\)

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:

\(Z \Leftarrow Y\ \operatorname{and}\ Y \Leftarrow X.\)

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 \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).\!\) 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 \(\mathfrak{C} (\mathfrak{P})\) 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.

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.

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 \(\varepsilon \, = \, ^{\backprime\backprime\prime\prime}\) and the blank symbol \(m_1 \, = \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}\) 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.

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 \((\textstyle\sum\prod)\) 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 \((\textstyle\prod\sum)\) 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 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:

\(\begin{matrix} q & :> & W_1 \\ \\ \cdots & \cdots & \cdots \\ \\ q & :> & W_k \\ \end{matrix}\)

It is useful to examine the relationship between the grammatical covering or production relation \((:>\!)\) and the logical relation of implication \((\Rightarrow),\) 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 \(^{\backprime\backprime} \, q \Leftarrow W \, ^{\prime\prime}\) 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 admissible 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 \(^{\backprime\backprime} q ^{\prime\prime}.\)

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 sole 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.

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 or 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:

\(\begin{array}{lllll} P_{[1]} & = & (P, 1) & = & \{ (x, 1) : x \in P \}, \\ Q_{[2]} & = & (Q, 2) & = & \{ (x, 2) : x \in Q \}. \\ \end{array}\)

Using the coproduct operator (\(\textstyle\coprod\)) for this construction, the sum, the coproduct, or the disjointed union of \(P\!\) and \(Q\!\) in that order can be represented as the ordinary union of \(P_{[1]}\!\) and \(Q_{[2]}.\!\)

\(\begin{array}{lll} P \coprod Q & = & P_{[1]} \cup Q_{[2]}. \\ \end{array}\)

The concatenation \(\mathfrak{L}_1 \cdot \mathfrak{L}_2\) of the formal languages \(\mathfrak{L}_1\!\) and \(\mathfrak{L}_2\!\) is just the cartesian product of sets \(\mathfrak{L}_1 \times \mathfrak{L}_2\) without the extra \(\times\!\)'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:

\(^{\backprime\backprime}\) \(\ldots \times X \times Q \times X \times \ldots\) \(^{\prime\prime}\)

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:

  \(\ldots \times X \times Q \times X \times \ldots\)  

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:

\(\begin{array}{ccccccc} ^{\backprime\backprime} Q ^{\prime\prime} & , & ^{\backprime\backprime} X \times Q \times X ^{\prime\prime} & , & ^{\backprime\backprime} X \times X \times Q \times X \times X ^{\prime\prime} & , & \ldots \\ \end{array}\)

With respect to what these strictures specify, this leaves all of the following equivalent as straits:

\(\begin{array}{ccccccc} Q & = & X \times Q \times X & = & X \times X \times Q \times X \times X & = & \ldots \\ \end{array}\)

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:

\(\begin{matrix} ^{\backprime\backprime} X ^{\prime\prime} & ^{\backprime\backprime} P ^{\prime\prime} & ^{\backprime\backprime} Q ^{\prime\prime} \\ \\ ^{\backprime\backprime} X \times X ^{\prime\prime} & ^{\backprime\backprime} X \times P ^{\prime\prime} & ^{\backprime\backprime} X \times Q ^{\prime\prime} \\ \\ ^{\backprime\backprime} P \times X ^{\prime\prime} & ^{\backprime\backprime} P \times P ^{\prime\prime} & ^{\backprime\backprime} P \times Q ^{\prime\prime} \\ \\ ^{\backprime\backprime} Q \times X ^{\prime\prime} & ^{\backprime\backprime} Q \times P ^{\prime\prime} & ^{\backprime\backprime} Q \times Q ^{\prime\prime} \\ \end{matrix}\)

These strictures and their corresponding straits are stratified according to their amounts of information, or their levels of constraint, as follows:

\(\begin{array}{lcccc} \text{High:} & ^{\backprime\backprime} P \times P ^{\prime\prime} & ^{\backprime\backprime} P \times Q ^{\prime\prime} & ^{\backprime\backprime} Q \times P ^{\prime\prime} & ^{\backprime\backprime} Q \times Q ^{\prime\prime} \\ \\ \text{Med:} & ^{\backprime\backprime} P ^{\prime\prime} & ^{\backprime\backprime} X \times P ^{\prime\prime} & ^{\backprime\backprime} P \times X ^{\prime\prime} \\ \\ \text{Med:} & ^{\backprime\backprime} Q ^{\prime\prime} & ^{\backprime\backprime} X \times Q ^{\prime\prime} & ^{\backprime\backprime} Q \times X ^{\prime\prime} \\ \\ \text{Low:} & ^{\backprime\backprime} X ^{\prime\prime} & ^{\backprime\backprime} X \times X ^{\prime\prime} \\ \end{array}\)

Within this framework, the more complex strait \(P \times Q\) can be expressed in terms of the simpler straits, \(P \times X\) and \(X \times Q.\) More specifically, it lends itself to being analyzed as their intersection, in the following way:

\(\begin{array}{lllll} P \times Q & = & P \times X & \cap & X \times Q. \\ \end{array}\)

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 \(P \times Q\) is described by a conjunction of propositions, namely, \(P_{[1]} \land Q_{[2]},\) 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, \(p_{[1]} \land q_{[2]},\) 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 \(\mathfrak{L}_1\!\) and \(\mathfrak{L}_2\!\) 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 \(\mathfrak{L}_1\!\) and \(\mathfrak{L}_2\!\) 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, \((\mathfrak{L}_1)_{[1]}\) and \((\mathfrak{L}_2)_{[2]},\) 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 \(^{\backprime\backprime} [1] ^{\prime\prime}\) and \(^{\backprime\backprime} [2] ^{\prime\prime}\) can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external mark-up language.

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 \(j^\text{th}\!\) excerpt of a stricture of the form \(^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime},\) regarded within a frame of discussion where the number of places is limited to \(k,\!\) is the stricture of the form \(^{\backprime\backprime} \, X \times \ldots \times S_j \times \ldots \times X \, ^{\prime\prime}.\) In the proper context, this can be written more succinctly as the stricture \(^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},\) an assertion that places the \(j^\text{th}\!\) set in the \(j^\text{th}\!\) place of the product.

The \(j^\text{th}\!\) extract of a strait of the form \(S_1 \times \ldots \times S_k,\!\) constrained to a frame of discussion where the number of places is restricted to \(k,\!\) is the strait of the form \(X \times \ldots \times S_j \times \ldots \times X.\) In the appropriate context, this can be denoted more succinctly by the stricture \(^{\backprime\backprime} \, (S_j)_{[j]} \, ^{\prime\prime},\) an assertion that places the \(j^\text{th}\!\) set in the \(j^\text{th}\!\) place of the product.

In these terms, a stricture of the form \(^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime}\) can be expressed in terms of simpler strictures, to wit, as a conjunction of its \(k\!\) excerpts:

\(\begin{array}{lll} ^{\backprime\backprime} \, S_1 \times \ldots \times S_k \, ^{\prime\prime} & = & ^{\backprime\backprime} \, (S_1)_{[1]} \, ^{\prime\prime} \, \land \, \ldots \, \land \, ^{\backprime\backprime} \, (S_k)_{[k]} \, ^{\prime\prime}. \end{array}\)

In a similar vein, a strait of the form \(S_1 \times \ldots \times S_k\!\) can be expressed in terms of simpler straits, namely, as an intersection of its \(k\!\) extracts:

\(\begin{array}{lll} S_1 \times \ldots \times S_k & = & (S_1)_{[1]} \, \cap \, \ldots \, \cap \, (S_k)_{[k]}. \end{array}\)

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.

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 of Modern Science", [HJB, 38]

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 \(\{ \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime} \, \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.\)

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 "\(\operatorname{Node}\)" 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 "\(\operatorname{Lobe}\)" 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:

\(1.\!\) \(\operatorname{Node}^0\) \(=\!\) \(\operatorname{Node}()\)
    \(=\!\) a node with no attachments.
  \(\operatorname{Node}_{j=1}^k C_j\) \(=\!\) \(\operatorname{Node} (C_1, \ldots, C_k)\)
    \(=\!\) a node with the attachments \(C_1, \ldots, C_k.\)
\(2.\!\) \(\operatorname{Lobe}^0\) \(=\!\) \(\operatorname{Lobe}()\)
    \(=\!\) a lobe with no accoutrements.
  \(\operatorname{Lobe}_{j=1}^k C_j\) \(=\!\) \(\operatorname{Lobe} (C_1, \ldots, C_k)\)
    \(=\!\) a lobe with the accoutrements \(C_1, \ldots, C_k.\)

Working from a structural description of the cactus language, or any suitable formal grammar for \(\mathfrak{C} (\mathfrak{P}),\!\) it is possible to give a recursive definition of the function called \(\operatorname{Parse}\) that maps each sentence in \(\operatorname{PARCE} (\mathfrak{P})\!\) to the corresponding graph in \(\operatorname{PARC} (\mathfrak{P}).\!\) One way to do this proceeds as follows:

  1. The parse of the concatenation \(\operatorname{Conc}_{j=1}^k\) of the \(k\!\) sentences \((s_j)_{j=1}^k\) is defined recursively as follows:
    1. \(\operatorname{Parse} (\operatorname{Conc}^0) ~=~ \operatorname{Node}^0.\)
    2. For \(k > 0,\!\)

      \(\operatorname{Parse} (\operatorname{Conc}_{j=1}^k s_j) ~=~ \operatorname{Node}_{j=1}^k \operatorname{Parse} (s_j).\)

  2. The parse of the surcatenation \(\operatorname{Surc}_{j=1}^k\) of the \(k\!\) sentences \((s_j)_{j=1}^k\) is defined recursively as follows:
    1. \(\operatorname{Parse} (\operatorname{Surc}^0) ~=~ \operatorname{Lobe}^0.\)
    2. For \(k > 0,\!\)

      \(\operatorname{Parse} (\operatorname{Surc}_{j=1}^k s_j) ~=~ \operatorname{Lobe}_{j=1}^k \operatorname{Parse} (s_j).\)

For ease of reference, Table 13 summarizes the mechanics of these parsing rules.


\(\text{Table 13.} ~~ \text{Algorithmic Translation Rules}\!\)
\(\text{Sentence in PARCE}\!\) \(\xrightarrow{\mathrm{Parse}}\!\) \(\text{Graph in PARC}\!\)
\(\mathrm{Conc}^0\!\) \(\xrightarrow{\mathrm{Parse}}\!\) \(\mathrm{Node}^0\!\)
\(\mathrm{Conc}_{j=1}^k s_j\!\) \(\xrightarrow{\mathrm{Parse}}\!\) \(\mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j)\!\)
\(\mathrm{Surc}^0\!\) \(\xrightarrow{\mathrm{Parse}}\!\) \(\mathrm{Lobe}^0\!\)
\(\mathrm{Surc}_{j=1}^k s_j\!\) \(\xrightarrow{\mathrm{Parse}}\!\) \(\mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j)\!\)


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 \(^{\backprime\backprime} ~ ^{\prime\prime}\) is treated as a primer, in other words, as a clear paint or a neutral tint. 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 \(\mathfrak{P} = \varnothing.\) 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 \(\mathfrak{C}^0 = \operatorname{PARCE}^0.\)
  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:
    1. A bare PARC is a PARC whose attachments are limited to blanks and bare lobes.
    2. 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.

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!

— Nietzsche, Beyond Good and Evil, [Nie-2, ¶ 296]

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 \(\mathfrak{L} = \mathfrak{C}(\mathfrak{P}),\) in other words, to define a mapping that "interprets" each sentence of \(\mathfrak{C}(\mathfrak{P})\) 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 \(k = 0, 1, 2, 3, \ldots.\)

  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 \(s_1 \cdot \ldots \cdot s_k.\) In the type of data that is called a text, the use of the center dot \((\cdot)\) is generally 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 empty closure — an expression enjoying one of the forms \(\underline{(} \underline{)}, \, \underline{(} ~ \underline{)}, \, \underline{(} ~~ \underline{)}, \, \ldots\) 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 \(\underline{(} s_1, \ldots, s_k \underline{)}.\)

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.

  1. There is the literal formal language of strings in \(\operatorname{PARCE} (\mathfrak{P}),\) the painted and rooted cactus expressions that constitute the language \(\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*.\)
  2. There is the figurative formal language of graphs in \(\operatorname{PARC} (\mathfrak{P}),\) 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 \(\operatorname{Conj}_j^J q_j\) of a set of propositions, \(\{ q_j : j \in J \},\) is a proposition that is true if and only if every one of the \(q_j\!\) is true.

    \(\operatorname{Conj}_j^J q_j\) is true  \(\Leftrightarrow\)  \(q_j\!\) is true for every \(j \in J.\)

  2. The surjunction \(\operatorname{Surj}_j^J q_j\) of a set of propositions, \(\{ q_j : j \in J \},\) is a proposition that is true if and only if exactly one of the \(q_j\!\) is untrue.

    \(\operatorname{Surj}_j^J q_j\) is true  \(\Leftrightarrow\)  \(q_j\!\) 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 \(J\!\) consists of the integers in the interval \(j = 1 ~\text{to}~ k,\) and if each proposition \(q_j\!\) is represented by a sentence \(s_j,\!\) then the following strategies of expression are open:

  1. The conjunction \(\operatorname{Conj}_j^J q_j\) can be represented by a sentence that is constructed by concatenating the \(s_j\!\) in the following fashion:

    \(\operatorname{Conj}_j^J q_j ~\leftrightsquigarrow~ s_1 s_2 \ldots s_k.\)

  2. The surjunction \(\operatorname{Surj}_j^J q_j\) can be represented by a sentence that is constructed by surcatenating the \(s_j\!\) in the following fashion:

    \(\operatorname{Surj}_j^J q_j ~\leftrightsquigarrow~ \underline{(} s_1, s_2, \ldots, s_k \underline{)}.\)

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 \(\underline{1} = \operatorname{true}.\) The logical denotation of the paint \(\mathfrak{p}_j\!\) 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 \(\mathfrak{C} (\mathfrak{P}).\)
  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 \(\underline{(} \underline{)},\) otherwise called a needle, is the boolean value \(\underline{0} = \operatorname{false}.\)

If one takes the point of view that PARCs and PARCEs amount to a pair of intertranslatable languages for the same domain of objects, then denotation brackets of the form \(\downharpoonleft \ldots \downharpoonright\) can be used to indicate the logical denotation \(\downharpoonleft C_j \downharpoonright\) of a cactus \(C_j\!\) or the logical denotation \(\downharpoonleft s_j \downharpoonright\) of a sentence \(s_j.\!\)

Tables 14 and 15 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.


\(\text{Table 14.} ~~ \text{Semantic Translation : Functional Form}\!\)
\(\mathrm{Sentence}\!\) \(\xrightarrow[\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}]{\mathrm{Parse}}\!\) \(\mathrm{Graph}\!\) \(\xrightarrow[\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}]{\mathrm{Denotation}}\!\) \(\mathrm{Proposition}\!\)
\(s_j\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(C_j\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(q_j\!\)
\(\mathrm{Conc}^0\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\mathrm{Node}^0\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\underline{1}\!\)
\(\mathrm{Conc}^k_j s_j\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\mathrm{Node}^k_j C_j\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\mathrm{Conj}^k_j q_j\!\)
\(\mathrm{Surc}^0\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\mathrm{Lobe}^0\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\underline{0}\!\)
\(\mathrm{Surc}^k_j s_j~\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\mathrm{Lobe}^k_j C_j\!\) \(\xrightarrow{\mathrm{20:44, 2 August 2017 (UTC)20:44, 2 August 2017 (UTC)}}\!\) \(\mathrm{Surj}^k_j q_j\!\)


\(\text{Table 15.} ~~ \text{Semantic Translation : Equational Form}\!\)
\(\downharpoonleft \mathrm{Sentence} \downharpoonright\!\) \(\stackrel{\mathrm{Parse}}{=}\!\) \(\downharpoonleft \mathrm{Graph} \downharpoonright\!\) \(\stackrel{\mathrm{Denotation}}{=}\!\) \(\mathrm{Proposition}\!\)
\(\downharpoonleft s_j \downharpoonright\!\) \(=\!\) \(\downharpoonleft C_j \downharpoonright\!\) \(=\!\) \(q_j\!\)
\(\downharpoonleft \mathrm{Conc}^0 \downharpoonright\!\) \(=\!\) \(\downharpoonleft \mathrm{Node}^0 \downharpoonright\!\) \(=\!\) \(\underline{1}\!\)
\(\downharpoonleft \mathrm{Conc}^k_j s_j \downharpoonright\!\) \(=\!\) \(\downharpoonleft \mathrm{Node}^k_j C_j \downharpoonright\!\) \(=\!\) \(\mathrm{Conj}^k_j q_j\!\)
\(\downharpoonleft \mathrm{Surc}^0 \downharpoonright\!\) \(=\!\) \(\downharpoonleft \mathrm{Lobe}^0 \downharpoonright\!\) \(=\!\) \(\underline{0}\!\)
\(\downharpoonleft \mathrm{Surc}^k_j s_j \downharpoonright\!\) \(=\!\) \(\downharpoonleft \mathrm{Lobe}^k_j C_j \downharpoonright\!\) \(=\!\) \(\mathrm{Surj}^k_j q_j\!\)


Aside from their common topic, the two Tables present slightly different ways of conceptualizing the operations that go to establish their maps. Table 14 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 15 records these associations in the form of equations, treating sentences and graphs as alternative kinds of signs, and generalizing the denotation 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.

A good way to illustrate the action of the conjunction and surjunction operators is to demonstrate how they can be used to construct the boolean functions on any finite number of variables. Let us begin by doing this for the first three cases, \(k = 0, 1, 2.\!\)

A boolean function \(F^{(0)}\!\) on \(0\!\) variables is just an element of the boolean domain \(\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}.\) Table 16 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 appear in the initial case.


\(\text{Table 16.} ~~ \text{Boolean Functions on Zero Variables}\!\)
\(F\!\) \(F\!\) \(F()\!\) \(F\!\)
\(\underline{0}\!\) \(F_0^{(0)}\!\) \(\underline{0}\!\) \(\texttt{(~)}\!\)
\(\underline{1}\!\) \(F_1^{(0)}\!\) \(\underline{1}\!\) \(\texttt{((~))}\!\)


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 \(F_j^{(k)}\!\) that is constructed as follows: The superscript \((k)\!\) 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 underlines in display formats. Here I illustrate also the convention of using the expression \(^{\backprime\backprime} ((~)) ^{\prime\prime}\) as a visible stand-in for the expression of the logical value \(\operatorname{true},\) a value that is minimally represented by a blank expression that tends to elude our giving it much notice in the context of more demonstrative texts.

Table 17 presents the boolean functions on one variable, \(F^{(1)} : \underline\mathbb{B} \to \underline\mathbb{B},\) of which there are precisely four.


\(\text{Table 17.} ~~ \text{Boolean Functions on One Variable}\!\)
\(F\!\) \(F\!\) \(F(x)\!\) \(F\!\)
    \(F(\underline{1})\) \(F(\underline{0})\)  
\(F_0^{(1)}\!\) \(F_{00}^{(1)}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\texttt{(~)}\!\)
\(F_1^{(1)}\!\) \(F_{01}^{(1)}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\texttt{(} x \texttt{)}\!\)
\(F_2^{(1)}\!\) \(F_{10}^{(1)}~\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(x\!\)
\(F_3^{(1)}\!\) \(F_{11}^{(1)}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\texttt{((~))}\!\)


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:

\(\begin{matrix} F_0^{(1)} & = & F_{00}^{(1)} & = & \underline{0} ~:~ \underline\mathbb{B} \to \underline\mathbb{B} \\ \\ F_3^{(1)} & = & F_{11}^{(1)} & = & \underline{1} ~:~ \underline\mathbb{B} \to \underline\mathbb{B} \end{matrix}\)

As for the rest, the other two functions are easily recognized as corresponding to the one-place logical connectives, or the monadic operators on \(\underline\mathbb{B}.\) Thus, the function \(F_1^{(1)} = F_{01}^{(1)}\) is recognizable as the negation operation, and the function \(F_2^{(1)} = F_{10}^{(1)}\) is obviously the identity operation.

Table 18 presents the boolean functions on two variables, \(F^{(2)} : \underline\mathbb{B}^2 \to \underline\mathbb{B},\) of which there are precisely sixteen.


\(\text{Table 18.} ~~ \text{Boolean Functions on Two Variables}\!\)
\(F\!\) \(F\!\) \(F(x, y)\!\) \(F\!\)
    \(F(\underline{1}, \underline{1})\) \(F(\underline{1}, \underline{0})\) \(F(\underline{0}, \underline{1})\) \(F(\underline{0}, \underline{0})\)  
\(F_{0}^{(2)}\!\) \(F_{0000}^{(2)}~\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\texttt{(~)}\!\)
\(F_{1}^{(2)}\!\) \(F_{0001}^{(2)}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\texttt{(} x \texttt{)(} y \texttt{)}\!\)
\(F_{2}^{(2)}\!\) \(F_{0010}^{(2)}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\texttt{(} x \texttt{)} y\!\)
\(F_{3}^{(2)}\!\) \(F_{0011}^{(2)}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\texttt{(} x \texttt{)}\!\)
\(F_{4}^{(2)}\!\) \(F_{0100}^{(2)}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(x \texttt{(} y \texttt{)}\!\)
\(F_{5}^{(2)}\!\) \(F_{0101}^{(2)}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\texttt{(} y \texttt{)}\!\)
\(F_{6}^{(2)}\!\) \(F_{0110}^{(2)}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\texttt{(} x \texttt{,} y \texttt{)}\!\)
\(F_{7}^{(2)}\!\) \(F_{0111}^{(2)}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\texttt{(} x y \texttt{)}\!\)
\(F_{8}^{(2)}\!\) \(F_{1000}^{(2)}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(x y\!\)
\(F_{9}^{(2)}\!\) \(F_{1001}^{(2)}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\texttt{((} x \texttt{,} y \texttt{))}\!\)
\(F_{10}^{(2)}\!\) \(F_{1010}^{(2)}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(y\!\)
\(F_{11}^{(2)}\!\) \(F_{1011}^{(2)}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\texttt{(} x \texttt{(} y \texttt{))}\!\)
\(F_{12}^{(2)}\!\) \(F_{1100}^{(2)}~\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{0}\!\) \(x\!\)
\(F_{13}^{(2)}\!\) \(F_{1101}^{(2)}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\underline{1}\!\) \(\texttt{((} x \texttt{)} y \texttt{)}\!\)
\(F_{14}^{(2)}\!\) \(F_{1110}^{(2)}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{0}\!\) \(\texttt{((} x \texttt{)(} y \texttt{))}\!\)
\(F_{15}^{(2)}\!\) \(F_{1111}^{(2)}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\underline{1}\!\) \(\texttt{((~))}\!\)


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 \(\underline{0} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}\) appears under the name \(F_{0}^{(2)}.\)
The constant function \(\underline{1} ~:~ \underline\mathbb{B}^2 \to \underline\mathbb{B}\) appears under the name \(F_{15}^{(2)}.\)
The negation and identity of the first variable are \(F_{3}^{(2)}\) and \(F_{12}^{(2)},\) respectively.
The negation and identity of the second variable are \(F_{5}^{(2)}\) and \(F_{10}^{(2)},\) respectively.
The logical conjunction is given by the function \(F_{8}^{(2)} (x, y) = x \cdot y.\)
The logical disjunction is given by the function \(F_{14}^{(2)} (x, y) = \underline{((} ~x~ \underline{)(} ~y~ \underline{))}.\)

Functions expressing the conditionals, implications, or if-then statements are given in the following ways:

\[[x \Rightarrow y] = F_{11}^{(2)} (x, y) = \underline{(} ~x~ \underline{(} ~y~ \underline{))} = [\operatorname{not}~ x ~\operatorname{without}~ y].\]

\[[x \Leftarrow y] = F_{13}^{(2)} (x, y) = \underline{((} ~x~ \underline{)} ~y~ \underline{)} = [\operatorname{not}~ y ~\operatorname{without}~ x].\]

The function that corresponds to the biconditional, the equivalence, or the if and only statement is exhibited in the following fashion:

\[[x \Leftrightarrow y] = [x = y] = F_{9}^{(2)} (x, y) = \underline{((} ~x~,~y~ \underline{))}.\]

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 operation, and geometrically associated with the symmetric difference of sets. This function is given by:

\[[x \neq y] = [x + y] = F_{6}^{(2)} (x, y) = \underline{(} ~x~,~y~ \underline{)}.\]

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 \((\rightarrow)\) and \((\leftarrow),\) and (2) the assertions called implications and symbolized by the signs \((\Rightarrow)\) and \((\Leftarrow)\), 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 \((\leftrightarrow)\) and (4) the assertion that is called an equivalence and signified by the sign \((\Leftrightarrow)\)? 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.

Stretching Exercises

The arrays of boolean connections described above, namely, the boolean functions \(F^{(k)} : \underline\mathbb{B}^k \to \underline\mathbb{B},\) for \(k\!\) in \(\{ 0, 1, 2 \},\!\) supply enough material to demonstrate the use of the stretch operation in a variety of concrete cases.

For example, suppose that \(F\!\) is a connection of the form \(F : \underline\mathbb{B}^2 \to \underline\mathbb{B},\) that is, any one of the sixteen possibilities in Table 18, while \(p\!\) and \(q\!\) are propositions of the form \(p, q : X \to \underline\mathbb{B},\) that is, propositions about things in the universe \(X,\!\) or else the indicators of sets contained in \(X.\!\)

Then one has the imagination \(\underline{f} = (f_1, f_2) = (p, q) : (X \to \underline\mathbb{B})^2,\) and the stretch of the connection \(F\!\) to \(\underline{f}\!\) on \(X\!\) amounts to a proposition \(F^\$ (p, q) : X \to \underline\mathbb{B}\) that may be read 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 may detach the operator \(F^\$ : (X \to \underline\mathbb{B}))^2 \to (X \to \underline\mathbb{B})),\) 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 the reader that the connections are meant to be stretched to several propositions on a universe \(X.\!\)

For example, take the connection \(F : \underline\mathbb{B}^2 \to \underline\mathbb{B}\) such that:

\[F(x, y) ~=~ F_{6}^{(2)} (x, y) ~=~ \underline{(}~x~,~y~\underline{)}\!\]

The connection in question is a boolean function on the variables \(x, y\!\) that returns a value of \(\underline{1}\) just when just one of the pair \(x, y\!\) is not equal to \(\underline{1},\) or what amounts to the same thing, just when just one of the pair \(x, y\!\) is equal to \(\underline{1}.\) There is clearly an isomorphism between this connection, viewed as an operation on the boolean domain \(\underline\mathbb{B} = \{ \underline{0}, \underline{1} \},\) and the dyadic operation on binary values \(x, y \in \mathbb{B} = \operatorname{GF}(2)\!\) that is otherwise known as \(x + y.\!\)

The same connection \(F : \underline\mathbb{B}^2 \to \underline\mathbb{B}\) can also be read as a proposition about things in the universe \(X = \underline\mathbb{B}^2.\) If \(s\!\) is a sentence that denotes the proposition \(F,\!\) then the corresponding assertion says exactly what one states in uttering the sentence \(^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}.\) In such a case, one has \(\downharpoonleft s \downharpoonright \, = F,\) and all of the following expressions are ordinarily taken as equivalent descriptions of the same set:

\(\begin{array}{lll} [| \downharpoonleft s \downharpoonright |] & = & [| F |] \\[6pt] & = & F^{-1} (\underline{1}) \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ s ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) = \underline{1} ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ F(x, y) ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} = \underline{1} ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \underline{(}~x~,~y~\underline{)} ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{exclusive~or}~ y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ \operatorname{just~one~true~of}~ x, y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x ~\operatorname{not~equal~to}~ y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \nLeftrightarrow y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x \neq y ~\} \\[6pt] & = & \{~ (x, y) \in \underline\mathbb{B}^2 ~:~ x + y ~\}. \end{array}\)

Notice the distinction, that I continue to maintain at this point, between the logical values \(\{ \operatorname{falsehood}, \operatorname{truth} \}\) and the algebraic values \(\{ 0, 1 \}.\!\) This makes it legitimate to write a sentence directly into the righthand side of a set-builder expression, for instance, weaving the sentence \(s\!\) or the sentence \(^{\backprime\backprime} \, x ~\operatorname{is~not~equal~to}~ y \, ^{\prime\prime}\) into the context \(^{\backprime\backprime} \, \{ (x, y) \in \underline{B}^2 : \ldots \} \, ^{\prime\prime},\) thereby obtaining the corresponding expressions listed above. It also allows us to assert the proposition \(F(x, y)\!\) in a more direct way, without detouring through the equation \(F(x, y) = \underline{1},\) since it already has a value in \(\{ \operatorname{falsehood}, \operatorname{true} \},\) 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 \(F(x, y) = \underline{(}~x~,~y~\underline{)},\) treated as a proposition about things in the universe \(X = \underline\mathbb{B}^2.\) Staying with this same connection, it is time to demonstrate how it can be "stretched" to form 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 \(p, q : X \to \underline\mathbb{B},\) and suppose that \(p, q\!\) are indicator functions of the sets \(P, Q \subseteq X,\) respectively. In other words, we have the following data:

\(\begin{matrix} p & = & \upharpoonleft P \upharpoonright & : & X \to \underline\mathbb{B} \\ \\ q & = & \upharpoonleft Q \upharpoonright & : & X \to \underline\mathbb{B} \\ \\ (p, q) & = & (\upharpoonleft P \upharpoonright, \upharpoonleft Q \upharpoonright) & : & (X \to \underline\mathbb{B})^2 \\ \end{matrix}\)

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:

\(\begin{array}{ccccl} F^\$ & = & \underline{(} \ldots, \ldots \underline{)}^\$ & : & (X \to \underline\mathbb{B})^2 \to (X \to \underline\mathbb{B}) \\ \\ F^\$ (p, q) & = & \underline{(}~p~,~q~\underline{)}^\$ & : & X \to \underline\mathbb{B} \\ \end{array}\)

As a result, the application of the proposition \(F^\$ (p, q)\) to each \(x \in X\) returns a logical value in \(\underline\mathbb{B},\) all in accord with the following equations:

\(\begin{matrix} F^\$ (p, q)(x) & = & \underline{(}~p~,~q~\underline{)}^\$ (x) & \in & \underline\mathbb{B} \\ \\ \Updownarrow & & \Updownarrow \\ \\ F(p(x), q(x)) & = & \underline{(}~p(x)~,~q(x)~\underline{)} & \in & \underline\mathbb{B} \\ \end{matrix}\)

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 \(F^\$ (p, q) = \underline{(}~p~,~q~\underline{)}^\$\) 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:

\(\begin{array}{lll} [| F^\$ (p, q) |] & = & [| \underline{(}~p~,~q~\underline{)}^\$ |] \\[6pt] & = & (F^\$ (p, q))^{-1} (\underline{1}) \\[6pt] & = & \{~ x \in X ~:~ F^\$ (p, q)(x) ~\} \\[6pt] & = & \{~ x \in X ~:~ \underline{(}~p~,~q~\underline{)}^\$ (x) ~\} \\[6pt] & = & \{~ x \in X ~:~ \underline{(}~p(x)~,~q(x)~\underline{)} ~\} \\[6pt] & = & \{~ x \in X ~:~ p(x) + q(x) ~\} \\[6pt] & = & \{~ x \in X ~:~ p(x) \neq q(x) ~\} \\[6pt] & = & \{~ x \in X ~:~ \upharpoonleft P \upharpoonright (x) ~\neq~ \upharpoonleft Q \upharpoonright (x) ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P ~\nLeftrightarrow~ x \in Q ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P\!-\!Q ~\operatorname{or}~ x \in Q\!-\!P ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P\!-\!Q ~\cup~ Q\!-\!P ~\} \\[6pt] & = & \{~ x \in X ~:~ x \in P + Q ~\} \\[6pt] & = & P + Q ~\subseteq~ X \\[6pt] & = & [|p|] + [|q|] ~\subseteq~ X \end{array}\)

References

  • Bernstein, Herbert J. (1987), "Idols of Modern Science and The Reconstruction of Knowledge", pp. 37–68 in Marcus G. Raskin and Herbert J. Bernstein, New Ways of Knowing : The Sciences, Society, and Reconstructive Knowledge, Rowman and Littlefield, Totowa, NJ, 1987.
  • Denning, P.J., Dennis, J.B., and Qualitz, J.E. (1978), Machines, Languages, and Computation, Prentice-Hall, Englewood Cliffs, NJ.
  • Nietzsche, Friedrich, Beyond Good and Evil : Prelude to a Philosophy of the Future, R.J. Hollingdale (trans.), Michael Tanner (intro.), Penguin Books, London, UK, 1973, 1990.
  • Raskin, Marcus G., and Bernstein, Herbert J. (1987, eds.), New Ways of Knowing : The Sciences, Society, and Reconstructive Knowledge, Rowman and Littlefield, Totowa, NJ.

Document History

The Cactus Patch

| Subject:  Inquiry Driven Systems : An Inquiry Into Inquiry
| Contact:  Jon Awbrey
| 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

Aug 2000 • Extensions Of Logical Graphs

CG List • Lost Links

  1. http://www.virtual-earth.de/CG/cg-list/old/msg03351.html
  2. http://www.virtual-earth.de/CG/cg-list/old/msg03352.html
  3. http://www.virtual-earth.de/CG/cg-list/old/msg03353.html
  4. http://www.virtual-earth.de/CG/cg-list/old/msg03354.html
  5. http://www.virtual-earth.de/CG/cg-list/old/msg03376.html
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CG List • New Archive

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CG List • Old Archive

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  3. http://web.archive.org/web/20020321223131/http://www.virtual-earth.de/CG/cg-list/msg03376.html
  4. http://web.archive.org/web/20020129134132/http://www.virtual-earth.de/CG/cg-list/msg03381.html

Sep 2000 • Zeroth Order Logic

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01246.html
  2. http://web.archive.org/web/20080905054059/http://suo.ieee.org/email/msg01251.html
  3. http://web.archive.org/web/20070223033521/http://suo.ieee.org/email/msg01380.html
  4. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01406.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01546.html
  6. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01561.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01670.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01966.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01985.html
  10. http://web.archive.org/web/20070401102902/http://suo.ieee.org/email/msg01988.html

Oct 2000 • All Liar, No Paradox

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg01739.html

Nov 2000 • Sowa's Top Level Categories

What Language To Use

  1. http://web.archive.org/web/20070320012929/http://suo.ieee.org/email/msg01956.html

Zeroth Order Logic

  1. http://web.archive.org/web/20070320012940/http://suo.ieee.org/email/msg01966.html

TLC In KIF

  1. http://web.archive.org/web/20081204195421/http://suo.ieee.org/ontology/msg00048.html
  2. http://web.archive.org/web/20070320014557/http://suo.ieee.org/ontology/msg00051.html

Dec 2000 • Sequential Interactions Generating Hypotheses

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg02607.html
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  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg03183.html

Jan 2001 • Differential Analytic Turing Automata

DATA • Arisbe List

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DATA • Ontology List

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Mar 2001 • Propositional Equation Reasoning Systems

PERS • Arisbe List

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  11. http://web.archive.org/web/20051202032812/http://stderr.org/pipermail/arisbe/2001-April/000428.html
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PERS • Arisbe List • Discussion

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PERS • Ontology List

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  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02005.html
  4. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02011.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02014.html
  6. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02015.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02024.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02046.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02047.html
  10. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02068.html
  11. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02102.html
  12. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02109.html
  13. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02117.html
  14. http://web.archive.org/web/20040116001230/http://suo.ieee.org/ontology/msg02125.html
  15. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02128.html
  16. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02134.html
  17. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg02138.html

PERS • SUO List

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  3. http://web.archive.org/web/20071007170752/http://suo.ieee.org/email/msg04413.html
  4. http://web.archive.org/web/20070121063018/http://suo.ieee.org/email/msg04419.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04422.html
  6. http://web.archive.org/web/20070305132316/http://suo.ieee.org/email/msg04423.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04432.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04454.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04455.html
  10. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04476.html
  11. http://web.archive.org/web/20060718091105/http://suo.ieee.org/email/msg04510.html
  12. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04517.html
  13. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04525.html
  14. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04533.html
  15. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04536.html
  16. http://web.archive.org/web/20070705085032/http://suo.ieee.org/email/msg04542.html
  17. http://web.archive.org/web/20050824231950/http://suo.ieee.org/email/msg04546.html

Jul 2001 • Reflective Extension Of Logical Graphs

RefLog • Arisbe List

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RefLog • SUO List

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Dec 2001 • Functional Conception Of Quantificational Logic

FunLog • Arisbe List

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  3. http://web.archive.org/web/20051202034557/http://stderr.org/pipermail/arisbe/2001-December/001216.html
  4. http://web.archive.org/web/20051202074331/http://stderr.org/pipermail/arisbe/2001-December/001221.html
  5. http://web.archive.org/web/20051201235028/http://stderr.org/pipermail/arisbe/2001-December/001222.html
  6. http://web.archive.org/web/20051202052037/http://stderr.org/pipermail/arisbe/2001-December/001223.html
  7. http://web.archive.org/web/20050827214411/http://stderr.org/pipermail/arisbe/2001-December/001224.html
  8. http://web.archive.org/web/20051202092500/http://stderr.org/pipermail/arisbe/2001-December/001225.html
  9. http://web.archive.org/web/20051202051942/http://stderr.org/pipermail/arisbe/2001-December/001226.html
  10. http://web.archive.org/web/20050425140213/http://stderr.org/pipermail/arisbe/2001-December/001227.html

FunLog • Ontology List

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  2. http://web.archive.org/web/20110608022712/http://suo.ieee.org/ontology/msg03563.html
  3. http://web.archive.org/web/20110608023312/http://suo.ieee.org/ontology/msg03564.html
  4. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03565.html
  5. http://web.archive.org/web/20070812011325/http://suo.ieee.org/ontology/msg03569.html
  6. http://web.archive.org/web/20110608023228/http://suo.ieee.org/ontology/msg03570.html
  7. http://web.archive.org/web/20110608022616/http://suo.ieee.org/ontology/msg03568.html
  8. http://web.archive.org/web/20110608023557/http://suo.ieee.org/ontology/msg03572.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03577.html
  10. http://web.archive.org/web/20070317021141/http://suo.ieee.org/ontology/msg03578.html
  11. http://web.archive.org/web/20110608021549/http://suo.ieee.org/ontology/msg03579.html
  12. http://web.archive.org/web/20110608021332/http://suo.ieee.org/ontology/msg03580.html
  13. http://web.archive.org/web/20110608020250/http://suo.ieee.org/ontology/msg03581.html
  14. http://web.archive.org/web/20110608021344/http://suo.ieee.org/ontology/msg03582.html
  15. http://web.archive.org/web/20110608021557/http://suo.ieee.org/ontology/msg03583.html
  16. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg04247.html

Dec 2001 • Cactus Language

Cactus Town Cartoons • Arisbe List

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Cactus Town Cartoons • Ontology List

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Jan 2002 • Zeroth Order Theories

ZOT • Arisbe List

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  4. http://web.archive.org/web/20050503213330/http://stderr.org/pipermail/arisbe/2002-January/001296.html
  5. http://web.archive.org/web/20050504070444/http://stderr.org/pipermail/arisbe/2002-January/001299.html
  6. http://web.archive.org/web/20050504070430/http://stderr.org/pipermail/arisbe/2002-January/001300.html
  7. http://web.archive.org/web/20050504070700/http://stderr.org/pipermail/arisbe/2002-January/001301.html
  8. http://web.archive.org/web/20050504070704/http://stderr.org/pipermail/arisbe/2002-January/001302.html
  9. http://web.archive.org/web/20050504070712/http://stderr.org/pipermail/arisbe/2002-January/001304.html
  10. http://web.archive.org/web/20050504070717/http://stderr.org/pipermail/arisbe/2002-January/001305.html
  11. http://web.archive.org/web/20050504070722/http://stderr.org/pipermail/arisbe/2002-January/001306.html
  12. http://web.archive.org/web/20050504070726/http://stderr.org/pipermail/arisbe/2002-January/001308.html
  13. http://web.archive.org/web/20050504070730/http://stderr.org/pipermail/arisbe/2002-January/001309.html
  14. http://web.archive.org/web/20050504070434/http://stderr.org/pipermail/arisbe/2002-January/001310.html
  15. http://web.archive.org/web/20050504070742/http://stderr.org/pipermail/arisbe/2002-January/001313.html
  16. http://web.archive.org/web/20050504070746/http://stderr.org/pipermail/arisbe/2002-January/001314.html
  17. http://web.archive.org/web/20050504070438/http://stderr.org/pipermail/arisbe/2002-January/001315.html
  18. http://web.archive.org/web/20050504070540/http://stderr.org/pipermail/arisbe/2002-January/001316.html
  19. http://web.archive.org/web/20050504070750/http://stderr.org/pipermail/arisbe/2002-January/001317.html

ZOT • Arisbe List • Discussion

  1. http://web.archive.org/web/20050503213334/http://stderr.org/pipermail/arisbe/2002-January/001297.html
  2. http://web.archive.org/web/20050504070656/http://stderr.org/pipermail/arisbe/2002-January/001298.html
  3. http://web.archive.org/web/20050504070708/http://stderr.org/pipermail/arisbe/2002-January/001303.html
  4. http://web.archive.org/web/20050504070544/http://stderr.org/pipermail/arisbe/2002-January/001307.html
  5. http://web.archive.org/web/20050504070734/http://stderr.org/pipermail/arisbe/2002-January/001311.html
  6. http://web.archive.org/web/20050504070738/http://stderr.org/pipermail/arisbe/2002-January/001312.html
  7. http://web.archive.org/web/20050504070755/http://stderr.org/pipermail/arisbe/2002-January/001318.html

ZOT • Ontology List

  1. http://web.archive.org/web/20070323210742/http://suo.ieee.org/ontology/msg03680.html
  2. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03681.html
  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03682.html
  4. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03683.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03691.html
  6. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03693.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03695.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03696.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03701.html
  10. http://web.archive.org/web/20070329211521/http://suo.ieee.org/ontology/msg03702.html
  11. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03703.html
  12. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03706.html
  13. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03707.html
  14. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03708.html
  15. http://web.archive.org/web/20080620074722/http://suo.ieee.org/ontology/msg03712.html
  16. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03715.html
  17. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03716.html
  18. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03717.html
  19. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03718.html
  20. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03721.html
  21. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03722.html
  22. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03723.html
  23. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03724.html

ZOT • Ontology List • Discussion

  1. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03684.html
  2. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03685.html
  3. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03686.html
  4. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03687.html
  5. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03689.html
  6. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03690.html
  7. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03694.html
  8. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03697.html
  9. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03698.html
  10. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03699.html
  11. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03700.html
  12. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03704.html
  13. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03705.html
  14. http://web.archive.org/web/20070330093628/http://suo.ieee.org/ontology/msg03709.html
  15. http://web.archive.org/web/20080705071714/http://suo.ieee.org/ontology/msg03710.html
  16. http://web.archive.org/web/20080620010020/http://suo.ieee.org/ontology/msg03711.html
  17. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03713.html
  18. http://web.archive.org/web/20080620074749/http://suo.ieee.org/ontology/msg03714.html
  19. http://web.archive.org/web/20061005100254/http://suo.ieee.org/ontology/msg03719.html
  20. http://web.archive.org/web/20070705085032/http://suo.ieee.org/ontology/msg03720.html

Mar 2003 • Theme One Program • Logical Cacti

  1. http://web.archive.org/web/20081007043317/http://stderr.org/pipermail/inquiry/2003-March/000114.html
  2. http://web.archive.org/web/20080908075558/http://stderr.org/pipermail/inquiry/2003-March/000115.html
  3. http://web.archive.org/web/20080908080336/http://stderr.org/pipermail/inquiry/2003-March/000116.html

Feb 2005 • Theme One Program • Logical Cacti

  1. http://web.archive.org/web/20150109152359/http://stderr.org/pipermail/inquiry/2005-February/002360.html
  2. http://web.archive.org/web/20150109152401/http://stderr.org/pipermail/inquiry/2005-February/002361.html
  3. http://web.archive.org/web/20061013233259/http://stderr.org/pipermail/inquiry/2005-February/002362.html
  4. http://web.archive.org/web/20081121103109/http://stderr.org/pipermail/inquiry/2005-February/002363.html