Cactus Language
MyWikiBiz, Author Your Legacy — Sunday December 01, 2024
Jump to navigationJump to searcho~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Inquiry Driven Systems: An Inquiry Into Inquiry
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| Document History
|
| Subject: Inquiry Driven Systems: An Inquiry Into Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 8.70
| Created: 23 Jun 1996
| Revised: 06 Jan 2002
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Section 1.3.10 (Recurring Themes)
| Excerpt: Subsections 1.3.10.8 - 1.3.10.13
|
| http://members.door.net/arisbe/menu/library/aboutcsp/awbrey/inquiry.htm
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.8 The Cactus Patch
| Thus, what looks to us like a sphere of scientific knowledge more accurately
| should be represented as the inside of a highly irregular and spiky object,
| like a pincushion or porcupine, with very sharp extensions in certain
| directions, and virtually no knowledge in immediately adjacent areas.
| If our intellectual gaze could shift slightly, it would alter each
| quill's direction, and suddenly our entire reality would change.
|
| Herbert J. Bernstein, "Idols", page 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing: The Sciences, Society, & Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.
In this and the four subsections that follow, I describe a calculus for
representing propositions as sentences, in other words, as syntactically
defined sequences of signs, and for manipulating these sentences chiefly
in the light of their semantically defined contents, in other words, with
respect to their logical values as propositions. In their computational
representation, the expressions of this calculus parse into a class of
tree-like data structures called "painted cacti". This is a family of
graph-theoretic data structures that can be observed to have especially
nice properties, turning out to be not only useful from a computational
standpoint but also quite interesting from a theoretical point of view.
The rest of this subsection serves to motivate the development of this
calculus and treats a number of general issues that surround the topic.
In order to facilitate the use of propositions as indicator functions
it helps to acquire a flexible notation for referring to propositions
in that light, for interpreting sentences in a corresponding role, and
for negotiating the requirements of mutual sense between the two domains.
If none of the formalisms that are readily available or in common use are
able to meet all of the design requirements that come to mind, then it is
necessary to contemplate the design of a new language that is especially
tailored to the purpose. In the present application, there is a pressing
need to devise a general calculus for composing propositions, computing
their values on particular arguments, and inverting their indications to
arrive at the sets of things in the universe that are indicated by them.
For computational purposes, it is convenient to have a middle ground or
an intermediate language for negotiating between the koine of sentences
regarded as strings of literal characters and the realm of propositions
regarded as objects of logical value, even if this renders it necessary
to introduce an artificial medium of exchange between these two domains.
If one envisions these computations to be carried out in any organized
fashion, and ultimately or partially by means of the familiar sorts of
machines, then the strings that express these logical propositions are
likely to find themselves parsed into tree-like data structures at some
stage of the game. With regard to their abstract structures as graphs,
there are several species of graph-theoretic data structures that can be
used to accomplish this job in a reasonably effective and efficient way.
Over the course of this project, I plan to use two species of graphs:
1. "Painted And Rooted Cacti" (PARCAI).
2. "Painted And Rooted Conifers" (PARCOI).
For now, it is enough to discuss the former class of data structures,
leaving the consideration of the latter class to a part of the project
where their distinctive features are key to developments at that stage.
Accordingly, within the context of the current patch of discussion, or
until it becomes necessary to attach further notice to the conceivable
varieties of parse graphs, the acronym "PARC" is sufficient to indicate
the pertinent genus of abstract graphs that are under consideration.
By way of making these tasks feasible to carry out on a regular basis,
a prospective language designer is required not only to supply a fluent
medium for the expression of propositions, but further to accompany the
assertions of their sentences with a canonical mechanism for teasing out
the fibers of their indicator functions. Accordingly, with regard to a
body of conceivable propositions, one needs to furnish a standard array
of techniques for following the threads of their indications from their
objective universe to their values for the mind and back again, that is,
for tracing the clues that sentences provide from the universe of their
objects to the signs of their values, and, in turn, from signs to objects.
Ultimately, one seeks to render propositions so functional as indicators
of sets and so essential for examining the equality of sets that they can
constitute a veritable criterion for the practical conceivability of sets.
Tackling this task requires me to introduce a number of new definitions
and a collection of additional notational devices, to which I now turn.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.8 The Cactus Patch (cont.)
Depending on whether a formal language is called by the type of sign
that makes it up or whether it is named after the type of object that
its signs are intended to denote, one may refer to this cactus language
as a "sentential calculus" or as a "propositional calculus", respectively.
When the syntactic definition of the language is well enough understood,
then the language can begin to acquire a semantic function. In natural
circumstances, the syntax and the semantics are likely to be engaged in
a process of co-evolution, whether in ontogeny or in phylogeny, that is,
the two developments probably form parallel sides of a single bootstrap.
But this is not always the easiest way, at least, at first, to formally
comprehend the nature of their action or the power of their interaction.
According to the customary mode of formal reconstruction, the language
is first presented in terms of its syntax, in other words, as a formal
language of strings called "sentences", amounting to a particular subset
of the possible strings that can be formed on a finite alphabet of signs.
A syntactic definition of the "cactus language", one that proceeds along
purely formal lines, is carried out in the next Subsection. After that,
the development of the language's more concrete aspects can be seen as
a matter of defining two functions:
1. The first is a function that takes each sentence of the language
into a computational data structure, to be exact, a tree-like
parse graph called a "painted cactus".
2. The second is a function that takes each sentence of the language,
or its interpolated parse graph, into a logical proposition, in effect,
ending up with an indicator function as the object denoted by the sentence.
The discussion of syntax brings up a number of associated issues that
have to be clarified before going on. These are questions of "style",
that is, the sort of description, "grammar", or theory that one finds
available or chooses as preferable for a given language. These issues
are discussed in the Subsection after next (Subsection 1.3.10.10).
There is an aspect of syntax that is so schematic in its basic character
that it can be conveyed by computational data structures, so algorithmic
in its uses that it can be automated by routine mechanisms, and so fixed
in its nature that its practical exploitation can be served by the usual
devices of computation. Because it involves the transformation of signs,
it can be recognized as an aspect of semiotics. Since it can be carried
out in abstraction from meaning, it is not up to the level of semantics,
much less a complete pragmatics, though it does incline to the pragmatic
aspects of computation that are auxiliary to and incidental to the human
use of language. Therefore, I refer to this aspect of formal language
use as the "algorithmics" or the "mechanics" of language processing.
A mechanical conversion of the "cactus language" into its associated
data structures is discussed in Subsection 1.3.10.11.
In the usual way of proceeding on formal grounds, meaning is added by giving
each "grammatical sentence", or each syntactically distinguished string, an
interpretation as a logically meaningful sentence, in effect, equipping or
providing each abstractly well-formed sentence with a logical proposition
for it to denote. A semantic interpretation of the "cactus language" is
carried out in Subsection 1.3.10.12.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax
| Picture two different configurations of such an irregular shape, superimposed
| on each other in space, like a double exposure photograph. Of the two images,
| the only part which coincides is the body. The two different sets of quills
| stick out into very different regions of space. The objective reality we
| see from within the first position, seemingly so full and spherical,
| actually agrees with the shifted reality only in the body of common
| knowledge. In every direction in which we look at all deeply, the
| realm of discovered scientific truth could be quite different.
| Yet in each of those two different situations, we would have
| thought the world complete, firmly known, and rather round
| in its penetration of the space of possible knowledge.
|
| Herbert J. Bernstein, "Idols", page 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing: The Sciences, Society, & Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.
In this Subsection, I describe the syntax of a family of formal languages
that I intend to use as a sentential calculus, and thus to interpret for
the purpose of reasoning about propositions and their logical relations.
In order to carry out the discussion, I need a way of referring to signs
as if they were objects like any others, in other words, as the sorts of
things that are subject to being named, indicated, described, discussed,
and renamed if necessary, that can be placed, arranged, and rearranged
within a suitable medium of expression, or else manipulated in the mind,
that can be articulated and decomposed into their elementary signs, and
that can be strung together in sequences to form complex signs. Signs
that have signs as their objects are called "higher order" (HO) signs,
and this is a topic that demands an apt formalization, but in due time.
The present discussion requires a quicker way to get into this subject,
even if it takes informal means that cannot be made absolutely precise.
As a temporary notation, let the relationship between a particular sign z
and a particular object o, namely, the fact that z denotes o or the fact
that o is denoted by z, be symbolized in one of the following two ways:
1. z >-> o,
z den o.
2. o <-< z,
o ned z.
Now consider the following paradigm:
1. If "A" >-> Ann,
i.e. "A" den Ann,
then A = Ann,
thus "Ann" >-> A,
i.e. "Ann" den A.
2. If Bob <-< "B",
i.e. Bob ned "B",
then Bob = B,
thus B <-< "Bob",
i.e. B ned "Bob".
When I say that the sign "blank" denotes the sign " ",
it means that the string of characters inside the first
pair of quotation marks can be used as another name for
the string of characters inside the second pair of quotes.
In other words, "blank" is a HO sign whose object is " ",
and the string of five characters inside the first pair of
quotation marks is a sign at a higher level of signification
than the string of one character inside the second pair of
quotation marks. This relationship can be abbreviated in
either one of the following ways:
| " " <-< "blank"
|
| "blank" >-> " "
Using the raised dot "·" as a sign to mark the articulation of a
quoted string into a sequence of possibly shorter quoted strings,
and thus to mark the concatenation of a sequence of quoted strings
into a possibly larger quoted string, one can write:
|
| " " <-< "blank" = "b"·"l"·"a"·"n"·"k"
|
This usage allows us to refer to the blank as a type of character, and
also to refer any blank we choose as a token of this type, referring to
either of them in a marked way, but without the use of quotation marks,
as I just did. Now, since a blank is just what the name "blank" names,
it is possible to represent the denotation of the sign " " by the name
"blank" in the form of an identity between the named objects, thus:
|
| " " = blank
|
With these kinds of identity in mind, it is possible to extend the use of
the "·" sign to mark the articulation of either named or quoted strings
into both named and quoted strings. For example:
| " " = " "·" " = blank·blank
|
| " blank" = " "·"blank" = blank·"blank"
|
| "blank " = "blank"·" " = "blank"·blank
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
A few definitions from formal language theory are required at this point.
An "alphabet" is a finite set of signs, typically, !A! = {a_1, ..., a_n}.
A "string" over an alphabet !A! is a finite sequence of signs from !A!.
The "length" of a string is just its length as a sequence of signs.
A sequence of length 0 yields the "empty string", here presented as "".
A sequence of length k > 0 is typically presented in the concatenated forms:
s_1 s_2 ... s_(k-1) s_k,
or
s_1 · s_2 · ... · s_(k-1) · s_k,
with s_j in !A!, for all j = 1 to k.
Two alternative notations are often useful:
1. !e! = @e@ = "" = the empty string.
2. %e% = {!e!} = {""} = the language consisting of a single empty string.
The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!.
In particular, !A!* includes among its elements the empty string !e!.
The "surplus" !A!^+ of an alphabet !A! is the set of all positive length
strings over !A!, in other words, everything in !A!* but the empty string.
A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*.
If z is a string over !A! and if z is an element of !L!, then it is
customary to call z a "sentence" of !L!. Thus, a formal language !L!
is defined by specifying its elements, which amounts to saying what it
means to be a sentence of !L!.
One last device turns out to be useful in this connection.
If z is a string that ends with a sign t, then z · t^-1 is
the string that results by "deleting" from z the terminal t.
In this context, I make the following distinction:
1. By "deleting" an appearance of a sign,
I mean replacing it with an appearance
of the empty string "".
2. By "erasing" an appearance of a sign,
I mean replacing it with an appearance
of the blank symbol " ".
A "token" is a particular appearance of a sign.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
The informal mechanisms that have been illustrated in the immediately preceding
discussion are enough to equip the rest of this discussion with a moderately
exact description of the so-called "cactus language" that I intend to use
in both my conceptual and my computational representations of the minimal
formal logical system that is variously known to sundry communities of
interpretation as "propositional logic", "sentential calculus", or
more inclusively, "zeroth order logic" (ZOL).
The "painted cactus language" !C! is actually a parameterized
family of languages, consisting of one language !C!(!P!) for
each set !P! of "paints".
The alphabet !A! = !M! |_| !P! is the disjoint union of two sets of symbols:
1. !M! is the alphabet of "measures", the set of "punctuation marks",
or the collection of "syntactic constants" that is common to all
of the languages !C!(!P!). This set of signs is given as follows:
!M! = {m_1, m_2, m_3, m_4}
= {" ", "-(", ",", ")-"}
= {blank, links, comma, right}.
2. !P! is the "palette", the alphabet of "paints", or the collection
of "syntactic variables" that is peculiar to the language !C!(!P!).
This set of signs is given as follows:
!P! = {p_j : j in J}.
The easiest way to define the language !C!(!P!) is to indicate the general sorts
of operations that suffice to construct the greater share of its sentences from
the specified few of its sentences that require a special election. In accord
with this manner of proceeding, I introduce a family of operations on strings
of !A!* that are called "syntactic connectives". If the strings on which
they operate are exclusively sentences of !C!(!P!), then these operations
are tantamount to "sentential connectives", and if the syntactic sentences,
considered as abstract strings of meaningless signs, are given a semantics
in which they denote propositions, considered as indicator functions over
some universe, then these operations amount to "propositional connectives".
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
Rather than presenting the most concise description of these languages
right from the beginning, it serves comprehension to develop a picture
of their forms in gradual stages, starting from the most natural ways
of viewing their elements, if somewhat at a distance, and working
through the most easily grasped impressions of their structures,
if not always the sharpest acquaintances with their details.
The first step is to define two sets of basic operations on strings of !A!*.
1. The "concatenation" of one string z_1 is just the string z_1.
The "concatenation" of two strings z_1, z_2 is the string z_1 · z_2.
The "concatenation" of the k strings z_j, for j = 1 to k,
is the string of the form z_1 · ... · z_k.
2. The "surcatenation" of one string z_1 is the string "-(" · z_1 · ")-".
The "surcatenation" of two strings z_1, z_2 is "-(" · z_1 · "," · z_2 · ")-".
The "surcatenation" of k strings z_j, for j = 1 to k,
is the string of the form "-(" · z_1 · "," · ... · "," · z_k · ")-".
These definitions can be made a little more succinct by
defining the following sorts of generic operators on strings:
1. The "concatenation" Conc^k of the k strings z_j,
for j = 1 to k, is defined recursively as follows:
a. Conc^1_j z_j = z_1.
b. For k > 1,
Conc^k_j z_j = (Conc^(k-1)_j z_j) · z_k.
2. The "surcatenation" Surc^k of the k strings z_j,
for j = 1 to k, is defined recursively as follows:
a. Surc^1_j z_j = "-(" · z_1 · ")-".
b. For k > 1,
Surc^k_j z_j = (Surc^(k-1)_j z_j) · ")-"^(-1) · "," · z_k · ")-".
The definitions of these syntactic operations can now be organized in a slightly
better fashion, for both conceptual and computational purposes, by making a few
additional conventions and auxiliary definitions.
1. The conception of the k-place concatenation operation
can be extended to include its natural "prequel":
Conc^0 = "" = the empty string.
Next, the construction of the k-place concatenation can be
broken into stages by means of the following conceptions:
a. The "precatenation" Prec(z_1, z_2) of the two strings
z_1, z_2 is the string that is defined as follows:
Prec(z_1, z_2) = z_1 · z_2.
b. The "concatenation" of the k strings z_1, ..., z_k can now be
defined as an iterated precatenation over the sequence of k+1
strings that begins with the string z_0 = Conc^0 = "" and then
continues on through the other k strings:
i. Conc^0_j z_j = Conc^0 = "".
ii. For k > 0,
Conc^k_j z_j = Prec(Conc^(k-1)_j z_j, z_k).
2. The conception of the k-place surcatenation operation
can be extended to include its natural "prequel":
Surc^0 = "-()-".
Finally, the construction of the k-place surcatenation can be
broken into stages by means of the following conceptions:
a. A "subclause" in !A!* is a string that ends with a ")-".
b. The "subcatenation" Subc(z_1, z_2)
of a subclause z_1 by a string z_2 is
the string that is defined as follows:
Subc(z_1, z_2) = z_1 · ")-"^(-1) · "," · z_2 · ")-".
c. The "surcatenation" of the k strings z_1, ..., z_k can now be
defined as an iterated subcatenation over the sequence of k+1
strings that starts with the string z_0 = Surc^0 = "-()-" and
then continues on through the other k strings:
i. Surc^0_j z_j = Surc^0 = "-()-".
ii. For k > 0,
Surc^k_j z_j = Subc(Surc^(k-1)_j z_j, z_k).
Notice that the expressions Conc^0_j z_j and Surc^0_j z_j
are defined in such a way that the respective operators
Conc^0 and Surc^0 basically "ignore", in the manner of
constants, whatever sequences of strings z_j may be
listed as their ostensible arguments.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
Having defined the basic operations of concatenation and surcatenation
on arbitrary strings, in effect, giving them operational meaning for
the all-inclusive language !L! = !A!*, it is time to adjoin the
notion of a more discriminating grammaticality, in other words,
a more properly restrictive concept of a sentence.
If !L! is an arbitrary formal language over an alphabet of the sort that
we are talking about, that is, an alphabet of the form !A! = !M! |_| !P!,
then there are a number of basic structural relations that can be defined
on the strings of !L!.
1. z is the "concatenation" of z_1 and z_2 in !L! if and only if
z_1 is a sentence of !L!, z_2 is a sentence of !L!, and
z = z_1 · z_2.
2. z is the "concatenation" of the k strings z1, ..., z_k in !L!,
if and only if z_j is a sentence of !L!, for all j = 1 to k, and
z = Conc^k_j z_j = z_1 · ... · z_k.
3. z is the "discatenation" of z_1 by t if and only if
z_1 is a sentence of !L!, t is an element of !A!, and
z_1 = z · t.
When this is the case, one more commonly writes:
z = z_1 · t^-1.
4. z is a "subclause" of !L! if and only if
z is a sentence of !L! and z ends with a ")-".
5. z is the "subcatenation" of z_1 by z_2 if and only if
z_1 is a subclause of !L!, z_2 is a sentence of !L!, and
z = z_1 · ")-"^(-1) · "," · z_2 · ")-".
6. z is the "surcatenation" of the k strings z_1, ..., z_k in !L!,
if and only if z_j is a sentence of !L!, for all j = 1 to k, and
z = Surc^k_j z_j = "-(" · z_1 · "," · ... · "," · z_k · ")-".
The converses of these decomposition relations are tantamount to the
corresponding forms of composition operations, making it possible for
these complementary forms of analysis and synthesis to articulate the
structures of strings and sentences in two directions.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
The "painted cactus language" with paints in the
set !P! = {p_j : j in J} is the formal language
!L! = !C!(!P!) c !A!* = (!M! |_| !P!)* that is
defined as follows:
PC 1. The blank symbol m_1 is a sentence.
PC 2. The paint p_j is a sentence, for each j in J.
PC 3. Conc^0 and Surc^0 are sentences.
PC 4. For each positive integer k,
if z_1, ..., z_k are sentences,
then Conc^k_j z_j is a sentence,
and Surc^k_j z_j is a sentence.
As usual, saying that z is a sentence is just a conventional way of
stating that the string z belongs to the relevant formal language !L!.
An individual sentence of !C!(!P!), for any palette !P!, is referred to
as a "painted and rooted cactus expression" (PARCE) on the palette !P!,
or a "cactus expression", for short. Anticipating the forms that the
parse graphs of these PARCE's will take, to be described in the next
Subsection, the language !L! = !C!(!P!) is also described as the
set PARCE(!P!) of PARCE's on the palette !P!, more generically,
as the PARCE's that constitute the language PARCE.
A "bare" PARCE, a bit loosely referred to as a "bare cactus expression",
is a PARCE on the empty palette !P! = {}. A bare PARCE is a sentence
in the "bare cactus language", !C!^0 = !C!({}) = PARCE^0 = PARCE({}).
This set of strings, regarded as a formal language in its own right,
is a sublanguage of every cactus language !C!(!P!). A bare cactus
expression is commonly encountered in practice when one has occasion
to start with an arbitrary PARCE and then finds a reason to delete or
to erase all of its paints.
Only one thing remains to cast this description of the cactus language
into a form that is commonly found acceptable. As presently formulated,
the principle PC 4 appears to be attempting to define an infinite number
of new concepts all in a single step, at least, it appears to invoke the
indefinitely long sequences of operators, Conc^k and Surc^k, for all k > 0.
As a general rule, one prefers to have an effectively finite description of
conceptual objects, and this means restricting the description to a finite
number of schematic principles, each of which involves a finite number of
schematic effects, that is, a finite number of schemata that explicitly
relate conditions to results.
A start in this direction, taking steps toward an effective description
of the cactus language, a finitary conception of its membership conditions,
and a bounded characterization of a typical sentence in the language, can be
made by recasting the present description of these expressions into the pattern
of what is called, more or less roughly, a "formal grammar".
A notation in the style of "S :> T" is now introduced,
to be read among many others in this manifold of ways:
| S covers T
|
| S governs T
|
| S rules T
|
| S subsumes T
|
| S types over T
The form "S :> T" is here recruited for polymorphic
employment in at least the following types of roles:
1. To signify that an individually named or quoted string T is
being typed as a sentence S of the language of interest !L!.
2. To express the fact or to make the assertion that each member
of a specified set of strings T c !A!* also belongs to the
syntactic category S, the one that qualifies a string as
being a sentence in the relevant formal language !L!.
3. To specify the intension or to signify the intention that every
string that fits the conditions of the abstract type T must also
fall under the grammatical heading of a sentence, as indicated by
the type name "S", all within the target language !L!.
In these types of situation the letter "S", that signifies the type of
a sentence in the language of interest, is called the "initial symbol"
or the "sentence symbol" of a candidate formal grammar for the language,
while any number of letters like "T", signifying other types of strings
that are necessary to a reasonable account or a rational reconstruction
of the sentences that belong to the language, are collectively referred
to as "intermediate symbols".
Combining the singleton set {"S"} whose sole member is the initial symbol
with the set !Q! that assembles together all of the intermediate symbols
results in the set {"S"} |_| !Q! of "non-terminal symbols". Completing
the package, the alphabet !A! of the language is also known as the set
of "terminal symbols". In this discussion, I will adopt the convention
that !Q! is the set of intermediate symbols, but I will often use "q"
as a typical variable that ranges over all of the non-terminal symbols,
q in {"S"} |_| !Q!. Finally, it is convenient to refer to all of the
symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
prospective grammar for the language, and accordingly to describe
the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
in effect, expressing the forms that are superimposed on a language
by one of its conceivable grammars. In certain settings is becomes
desirable to separate the augmented strings that contain the symbol
"S" from all other sorts of augmented strings. In these situations,
the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
as the "sentential forms" of the associated grammar.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
In forming a grammar for a language, statements of the form W :> W',
where W and W' are augmented strings or sentential forms of specified
types that depend on the style of the grammar that is being sought, are
variously known as "characterizations", "covering rules", "productions",
"rewrite rules", "subsumptions", "transformations", or "typing rules".
These are collected together into a set !K! that serves to complete
the definition of the formal grammar in question.
Correlative with the use of this notation, an expression of the
form "T <: S", read as "T is covered by S", can be interpreted
as saying that T is of the type S. Depending on the context,
this can be taken in either one of two ways:
1. Treating "T" as a string variable, it means
that the individual string T is typed as S.
2. Treating "T" as a type name, it means that any
instance of the type T also falls under the type S.
In accordance with these interpretations, an expression like "t <: T" can be
read in all of the ways that one typically reads an expression like "t : T".
There are several abuses of notation that commonly tolerated in the use
of covering relations. The worst offense is that of allowing symbols to
stand equivocally either for individual strings or else for their types.
There is a measure of consistency to this practice, considering the fact
that perfectly individual entities are rarely if ever grasped by means of
signs and finite expressions, which entails that every appearance of an
apparent token is only a type of more particular tokens, and meaning in
the end that there is never any recourse but to the sort of discerning
interpretation that can decide just how each sign is intended. In view
of all this, I continue to permit expressions like "t <: T" and "T <: S",
where any of the symbols "t", "T", "S" can be taken to signify either the
tokens or the subtypes of their covering types.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
The combined effect of several typos in my typography
along with what may be a lack of faith in imagination,
obliges me to redo a couple of paragraphs from before.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
A notation in the style of "S :> T" is now introduced,
to be read among many others in this manifold of ways:
| S covers T
|
| S governs T
|
| S rules T
|
| S subsumes T
|
| S types over T
The form "S :> T" is here recruited for polymorphic
employment in at least the following types of roles:
1. To signify that an individually named or quoted string T is
being typed as a sentence S of the language of interest !L!.
2. To express the fact or to make the assertion that each member
of a specified set of strings T c !A!* also belongs to the
syntactic category S, the one that qualifies a string as
being a sentence in the relevant formal language !L!.
3. To specify the intension or to signify the intention that every
string that fits the conditions of the abstract type T must also
fall under the grammatical heading of a sentence, as indicated by
the type name "S", all within the target language !L!.
In these types of situation the letter "S", that signifies the type of
a sentence in the language of interest, is called the "initial symbol"
or the "sentence symbol" of a candidate formal grammar for the language,
while any number of letters like "T", signifying other types of strings
that are necessary to a reasonable account or a rational reconstruction
of the sentences that belong to the language, are collectively referred
to as "intermediate symbols".
Combining the singleton set {"S"} whose sole member is the initial symbol
with the set !Q! that assembles together all of the intermediate symbols
results in the set {"S"} |_| !Q! of "non-terminal symbols". Completing
the package, the alphabet !A! of the language is also known as the set
of "terminal symbols". In this discussion, I will adopt the convention
that !Q! is the set of intermediate symbols, but I will often use "q"
as a typical variable that ranges over all of the non-terminal symbols,
q in {"S"} |_| !Q!. Finally, it is convenient to refer to all of the
symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
prospective grammar for the language, and accordingly to describe
the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
in effect, expressing the forms that are superimposed on a language
by one of its conceivable grammars. In certain settings is becomes
desirable to separate the augmented strings that contain the symbol
"S" from all other sorts of augmented strings. In these situations,
the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
as the "sentential forms" of the associated grammar.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
For some time to come in the discussion that follows,
although I will continue to focus on the cactus language
as my principal object example, my more general purpose will
be to develop and to demonstrate the subject materials and the
technical methodology of the theory of formal languages and grammars.
I will do this by taking up a particular method of "stepwise refinement"
and using it to extract a rigorous formal grammar for the cactus language,
starting with little more than a rough description of the target language
and applying a systematic analysis to develop a sequence of increasingly
more effective and more exact approximations to the desired grammar.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
Employing the notion of a covering relation it becomes possible to
redescribe the cactus language !L! = !C!(!P!) in the following way.
Grammar 1 is something of a misnomer. It is nowhere near exemplifying
any kind of a standard form and it is only intended as a starting point
for the initiation of more respectable grammars. Such as it is, it uses
the terminal alphabet !A! = !M! |_| !P! that comes with the territory of
the cactus language !C!(!P!), it specifies !Q! = {}, in other words, it
employs no intermediate symbols, and it embodies the "covering set" !K!
as listed in the following display.
| !C!(!P!). Grammar 1
|
| !Q! = {}
|
| 1. S :> m_1 = " "
|
| 2. S :> p_j, for each j in J
|
| 3. S :> Conc^0 = ""
|
| 4. S :> Surc^0 = "-()-"
|
| 5. S :> S*
|
| 6. S :> "-(" · S · ("," · S)* · ")-"
In this formulation, the last two lines specify that:
5. The concept of a sentence in !L! covers any
concatenation of sentences in !L!, in effect,
any number of freely chosen sentences that are
available to be concatenated one after another.
6. The concept of a sentence in !L! covers any
surcatenation of sentences in !L!, in effect,
any string that opens with a "-(", continues
with a sentence, possibly empty, follows with
a finite number of phrases of the form "," · S,
and closes with a ")-".
This appears to be just about the most concise description
of the cactus language !C!(!P!) that one can imagine, but
there exist a couple of problems that are commonly felt
to afflict this style of presentation and to make it
less than completely acceptable. Briefly stated,
these problems turn on the following properties
of the presentation:
1. The invocation of the kleene star operation
is not reduced to a manifestly finitary form.
2. The type of a sentence S is allowed to cover
not only itself but also the empty string.
I will discuss these issues at first in general, and especially in regard to
how the two features interact with one another, and then I return to address
in further detail the questions that they engender on their individual bases.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
In the process of developing a grammar for a language, it is possible
to notice a number of organizational, pragmatic, and stylistic questions,
whose moment to moment answers appear to decide the ongoing direction of the
grammar that develops and the impact of whose considerations work in tandem
to determine, or at least to influence, the sort of grammar that turns out.
The issues that I can see arising at this point I can give the following
prospective names, putting off the discussion of their natures and the
treatment of their details to the points in the development of the
present example where they evolve their full import.
1. The "degree of intermediate organization" in a grammar.
2. The "distinction between empty and significant strings", and thus
the "distinction between empty and significant types of strings".
3. The "principle of intermediate significance". This is a constraint
on the grammar that arises from considering the interaction of the
first two issues.
In responding to these issues, it is advisable at first to proceed in
a stepwise fashion, all the better thereby to accommodate the chances
of pursuing a series of parallel developments in the grammar, to allow
for the possibility of reversing many steps in its development, indeed,
to take into account the near certain necessity of having to revisit,
to revise, and to reverse many decisions about how to proceed toward
an optimal description or a satisfactory grammar for the language.
Doing all this means exploring the effects of various alterations
and innovations as independently from each other as possible.
The degree of intermediate organization in a grammar is measured by how many
intermediate symbols it has and by how they interact with each other by means
of its productions. With respect to this issue, Grammar 1 has no intermediate
symbols at all, !Q! = {}, and therefore remains at an ostensibly trivial degree
of intermediate organization. Some additions to the list of intermediate symbols
are practically obligatory in order to arrive at any reasonable grammar at all,
other inclusions appear to have a more optional character, though obviously
useful from the standpoints of clarity and ease of comprehension.
One of the troubles that is perceived to affect Grammar 1 is that it wastes
so much of the available potential for efficient description in recounting
over and over again the simple fact that the empty string is present in
the language. This arises in part from the statement that S :> S*,
which implies that:
S :> S* = %e% |_| S |_| S · S |_| S · S · S |_| ...
There is nothing wrong with the more expansive pan of the covered equation,
since it follows straightforwardly from the definition of the kleene star
operation, but the covering statement, to the effect that S :> S*, is not
necessarily a very productive piece of information, to the extent that it
does always tell us very much about the language that is being supposed to
fall under the type of a sentence S. In particular, since it implies that
S :> %e%, and since %e%·!L! = !L!·%e% = !L!, for any formal language !L!,
the empty string !e! = "" is counted over and over in every term of the union,
and every non-empty sentence under S appears again and again in every term of
the union that follows the initial appearance of S. As a result, this style
of characterization has to be classified as "true but not very informative".
If at all possible, one prefers to partition the language of interest into
a disjoint union of subsets, thereby accounting for each sentence under
its proper term, and one whose place under the sum serves as a useful
parameter of its character or its complexity. In general, this form
of description is not always possible to achieve, but it is usually
worth the trouble to actualize it whenever it is.
Suppose that one tries to deal with this problem by eliminating each use of
the kleene star operation, by reducing it to a purely finitary set of steps,
or by finding an alternative way to cover the sublanguage that it is used to
generate. This amounts, in effect, to "recognizing a type", a complex process
that involves the following steps:
1. Noticing a category of strings that
is generated by iteration or recursion.
2. Acknowledging the circumstance that the noted category
of strings needs to be covered by a non-terminal symbol.
3. Making a note of it by declaring and instituting
an explicitly and even expressively named category.
In sum, one introduces a non-terminal symbol for each type of sentence and
each "part of speech" or sentential component that is generated by means of
iteration or recursion under the ruling constraints of the grammar. In order
to do this one needs to analyze the iteration of each grammatical operation in
a way that is analogous to a mathematically inductive definition, but further in
a way that is not forced explicitly to recognize a distinct and separate type of
expression merely to account for and to recount every increment in the parameter
of iteration.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
Returning to the case of the cactus language, the process of recognizing an
iterative type or a recursive type can be illustrated in the following way.
The operative phrases in the simplest sort of recursive definition are its
initial part and its generic part. For the cactus language !C!(!P!), one
has the following definitions of concatenation as iterated precatenation
and of surcatenation as iterated subcatenation, respectively:
1. Conc^0 = "".
Conc^k_j S_j = Prec(Conc^(k-1)_j S_j, S_k).
2. Surc^0 = "-()-".
Surc^k_j S_j = Subc(Surc^(k-1)_j S_j, S_k).
In order to transform these recursive definitions into grammar rules,
one introduces a new pair of intermediate symbols, "Conc" and "Surc",
corresponding to the operations that share the same names but ignoring
the inflexions of their individual parameters j and k. Recognizing the
type of a sentence by means of the initial symbol "S", and interpreting
"Conc" and "Surc" as names for the types of strings that are generated
by concatenation and by surcatenation, respectively, one arrives at
the following transformation of the ruling operator definitions
into the form of covering grammar rules:
1. Conc :> "".
Conc :> Conc · S.
2. Surc :> "-()-".
Surc :> "-(" · S · ")-".
Surc :> Surc ")-"^(-1) · "," · S · ")-".
As given, this particular fragment of the intended grammar
contains a couple of features that are desirable to amend.
1. Given the covering S :> Conc, the covering rule Conc :> Conc · S
says no more than the covering rule Conc :> S · S. Consequently,
all of the information contained in these two covering rules is
already covered by the statement that S :> S · S.
2. A grammar rule that invokes a notion of decatenation, deletion, erasure,
or any other sort of retrograde production, is frequently considered to
be lacking in elegance, and a there is a style of critique for grammars
that holds it preferable to avoid these types of operations if it is at
all possible to do so. Accordingly, contingent on the prescriptions of
the informal rule in question, and pursuing the stylistic dictates that
are writ in the realm of its aesthetic regime, it becomes necessary for
us to backtrack a little bit, to temporarily withdraw the suggestion of
employing these elliptical types of operations, but without, of course,
eliding the record of doing so.
One way to analyze the surcatenation of any number of sentences is to
introduce an auxiliary type of string, not in general a sentence, but
a proper component of any sentence that is formed by surcatenation.
Doing this brings one to the following definition:
A "tract" is a concatenation of a finite sequence of sentences, with a
literal comma "," interpolated between each pair of adjacent sentences.
Thus, a typical tract T takes the form:
T = S_1 · "," · ... · "," · S_k.
A tract must be distinguished from the abstract sequence of sentences,
S_1, ..., S_k, where the commas that appear to come to mind, as if being
called up to separate the successive sentences of the sequence, remain as
partially abstract conceptions, or as signs that retain a disengaged status
on the borderline between the text and the mind. In effect, the types of
commas that appear to follow in the abstract sequence continue to exist
as conceptual abstractions and fail to be cognized in a wholly explicit
fashion, whether as concrete tokens in the object language, or as marks
in the strings of signs that are able to engage one's parsing attention.
Returning to the case of the painted cactus language !L! = !C!(!P!),
it is possible to put the currently assembled pieces of a grammar
together in the light of the presently adopted canons of style,
to arrive a more refined analysis of the fact that the concept
of a sentence covers any concatenation of sentences and any
surcatenation of sentences, and so to obtain the following
form of a grammar:
| !C!(!P!). Grammar 2
|
| !Q! = {"T"}
|
| 1. S :> !e!
|
| 2. S :> m_1
|
| 3. S :> p_j, for each j in J
|
| 4. S :> S · S
|
| 5. S :> "-(" · T · ")-"
|
| 6. T :> S
|
| 7. T :> T · "," · S
In this rendition, a string of type T is not in general
a sentence itself but a proper "part of speech", that is,
a strictly "lesser" component of a sentence in any suitable
ordering of sentences and their components. In order to see
how the grammatical category T gets off the ground, that is,
to detect its minimal strings and to discover how its ensuing
generations gets started from these, it is useful to observe
that the covering rule T :> S means that T "inherits" all of
the initial conditions of S, namely, T :> !e!, m_1, p_j.
In accord with these simple beginnings it comes to parse
that the rule T :> T · "," · S, with the substitutions
T = !e! and S = !e! on the covered side of the rule,
bears the germinal implication that T :> ",".
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
Grammar 2 achieves a portion of its success through a higher degree of
intermediate organization. Roughly speaking, the level of organization
can be seen as reflected in the cardinality of the intermediate alphabet
!Q! = {"T"}, but it is clearly not explained by this simple circumstance
alone, since it is taken for granted that the intermediate symbols serve
a purpose, a purpose that is easily recognizable but that may not be so
easy to pin down and to specify exactly. Nevertheless, it is worth the
trouble of exploring this aspect of organization and this direction of
development a little further. Although it is not strictly necessary
to do so, it is possible to organize the materials of the present
grammar in a slightly better fashion by recognizing two recurrent
types of strings that appear in the typical cactus expression.
In doing this, one arrives at the following two definitions:
A "rune" is a string of blanks and paints concatenated together.
Thus, a typical rune R is a string over {m_1} |_| !P!, possibly
the empty string.
R in ({m_1} |_| !P!)*.
When there is no possibility of confusion, the letter "R" can be used
either as a string variable that ranges over the set of runes or else
as a type name for the class of runes. The latter reading amounts to
the enlistment of a fresh intermediate symbol, "R" in !Q!, as a part
of a new grammar for !C!(!P!). In effect, "R" affords a grammatical
recognition for any rune that forms a part of a sentence in !C!(!P!).
In situations where these variant usages are likely to be confused,
the types of strings can be indicated by means of expressions like
"r <: R" and "W <: R".
A "foil" is a string of the form "-(" · T · ")-", where T is a tract.
Thus, a typical foil F has the form:
F = "-(" · S_1 · "," · ... · "," · S_k · ")-".
This is just the surcatenation of the sentences S_1, ..., S_k.
Given the possibility that this sequence of sentences is empty,
and thus that the tract T is the empty string, the minimum foil
F is the expression "-()-". Explicitly marking each foil F that
is embodied in a cactus expression is tantamount to recognizing
another intermediate symbol, "F" in !Q!, further articulating the
structures of sentences and expanding the grammar for the language
!C!(!P!). All of the same remarks about the versatile uses of the
intermediate symbols, as string variables and as type names, apply
again to the letter "F".
| !C!(!P!). Grammar 3
|
| !Q! = {"F", "R", "T"}
|
| 1. S :> R
|
| 2. S :> F
|
| 3. S :> S · S
|
| 4. R :> !e!
|
| 5. R :> m_1
|
| 6. R :> p_j, for each j in J
|
| 7. R :> R · R
|
| 8. F :> "-(" · T · ")-"
|
| 9. T :> S
|
| 10. T :> T · "," · S
In Grammar 3, the first three Rules say that a sentence (a string of type S),
is a rune (a string of type R), a foil (a string of type F), or an arbitrary
concatenation of strings of these two types. Rules 4 through 7 specify that
a rune R is an empty string !e! = "", a blank symbol m_1 = " ", a paint p_j,
for j in J, or any concatenation of strings of these three types. Rule 8
characterizes a foil F as a string of the form "-(" · T · ")-", where T is
a tract. The last two Rules say that a tract T is either a sentence S or
else the concatenation of a tract, a comma, and a sentence, in that order.
At this point in the succession of grammars for !C!(!P!), the explicit
uses of indefinite iterations, like the kleene star operator, are now
completely reduced to finite forms of concatenation, but the problems
that some styles of analysis have with allowing non-terminal symbols
to cover both themselves and the empty string are still present.
Any degree of reflection on this difficulty raises the general question:
What is a practical strategy for accounting for the empty string in the
organization of any formal language that counts it among its sentences?
One answer that presents itself is this: If the empty string belongs to
a formal language, it suffices to count it once at the beginning of the
formal account that enumerates its sentences and then to move on to more
interesting materials.
Returning to the case of the cactus language !C!(!P!), that is,
the formal language of "painted and rooted cactus expressions",
it serves the purpose of efficient accounting to partition the
language PARCE into the following couple of sublanguages:
1. The "emptily painted and rooted cactus expressions"
make up the language EPARCE that consists of
a single empty string as its only sentence.
In short:
EPARCE = {""}.
2. The "significantly painted and rooted cactus expressions"
make up the language SPARCE that consists of everything else,
namely, all of the non-empty strings in the language PARCE.
In sum:
SPARCE = PARCE \ "".
As a result of marking the distinction between empty and significant sentences,
that is, by categorizing each of these three classes of strings as an entity
unto itself and by conceptualizing the whole of its membership as falling
under a distinctive symbol, one obtains an equation of sets that connects
the three languages being marked:
SPARCE = PARCE - EPARCE.
In sum, one has the disjoint union:
PARCE = EPARCE |_| SPARCE.
For brevity in the present case, and to serve as a generic device
in any similar array of situations, let the symbol "S" be used to
signify the type of an arbitrary sentence, possibly empty, whereas
the symbol "S'" is reserved to designate the type of a specifically
non-empty sentence. In addition, let the symbol "%e%" be employed
to indicate the type of the empty sentence, in effect, the language
%e% = {""} that contains a single empty string, and let a plus sign
"+" signify a disjoint union of types. In the most general type of
situation, where the type S is permitted to include the empty string,
one notes the following relation among types:
S = %e% + S'.
Consequences of the distinction between empty expressions and
significant expressions are taken up for discussion next time.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
With the distinction between empty and significant expressions in mind,
I return to the grasp of the cactus language !L! = !C!(!P!) = PARCE(!P!)
that is afforded by Grammar 2, and, taking that as a point of departure,
explore other avenues of possible improvement in the comprehension of
these expressions. In order to observe the effects of this alteration
as clearly as possible, in isolation from any other potential factors,
it is useful to strip away the higher levels intermediate organization
that are present in Grammar 3, and start again with a single intermediate
symbol, as used in Grammar 2. One way of carrying out this strategy leads
on to a grammar of the variety that will be articulated next.
If one imposes the distinction between empty and significant types on
each non-terminal symbol in Grammar 2, then the non-terminal symbols
"S" and "T" give rise to the non-terminal symbols "S", "S'", "T", "T'",
leaving the last three of these to form the new intermediate alphabet.
Grammar 4 has the intermediate alphabet !Q! = {"S'", "T", "T'"}, with
the set !K! of covering production rules as listed in the next display.
| !C!(!P!). Grammar 4
|
| !Q! = {"S'", "T", "T'"}
|
| 1. S :> !e!
|
| 2. S :> S'
|
| 3. S' :> m_1
|
| 4. S' :> p_j, for each j in J
|
| 5. S' :> "-(" · T · ")-"
|
| 6. S' :> S' · S'
|
| 7. T :> !e!
|
| 8. T :> T'
|
| 9. T' :> T · "," · S
In this version of a grammar for !L! = !C!(!P!), the intermediate type T
is partitioned as T = %e% + T', thereby parsing the intermediate symbol T
in parallel fashion with the division of its overlying type as S = %e% + S'.
This is an option that I will choose to close off for now, but leave it open
to consider at a later point. Thus, it suffices to give a brief discussion
of what it involves, in the process of moving on to its chief alternative.
There does not appear to be anything radically wrong with trying this
approach to types. It is reasonable and consistent in its underlying
principle, and it provides a rational and a homogeneous strategy toward
all parts of speech, but it does require an extra amount of conceptual
overhead, in that every non-trivial type has to be split into two parts
and comprehended in two stages. Consequently, in view of the largely
practical difficulties of making the requisite distinctions for every
intermediate symbol, it is a common convention, whenever possible, to
restrict intermediate types to covering exclusively non-empty strings.
For the sake of future reference, it is convenient to refer to this restriction
on intermediate symbols as the "intermediate significance" constraint. It can
be stated in a compact form as a condition on the relations between non-terminal
symbols q in {"S"} |_| !Q! and sentential forms W in {"S"} |_| (!Q! |_| !A!)*.
| Condition On Intermediate Significance
|
| If q :> W
|
| and W = !e!,
|
| then q = "S".
If this is beginning to sound like a monotone condition, then it is
not absurd to sharpen the resemblance and render the likeness more
acute. This is done by declaring a couple of ordering relations,
denoting them under variant interpretations by the same sign "<".
1. The ordering "<" on the set of non-terminal symbols,
q in {"S"} |_| !Q!, ordains the initial symbol "S"
to be strictly prior to every intermediate symbol.
This is tantamount to the axiom that "S" < q,
for all q in !Q!.
2. The ordering "<" on the collection of sentential forms,
W in {"S"} |_| (!Q! |_| !A!)*, ordains the empty string
to be strictly minor to every other sentential form.
This is stipulated in the axiom that !e! < W,
for every non-empty sentential form W.
Given these two orderings, the constraint in question
on intermediate significance can be stated as follows:
| Condition Of Intermediate Significance
|
| If q :> W
|
| and q > "S",
|
| then W > !e!.
Achieving a grammar that respects this convention typically requires a more
detailed account of the initial setting of a type, both with regard to the
type of context that incites its appearance and also with respect to the
minimal strings that arise under the type in question. In order to find
covering productions that satisfy the intermediate significance condition,
one must be prepared to consider a wider variety of calling contexts or
inciting situations that can be noted to surround each recognized type,
and also to enumerate a larger number of the smallest cases that can
be observed to fall under each significant type.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
With the array of foregoing considerations in mind,
one is gradually led to a grammar for !L! = !C!(!P!)
in which all of the covering productions have either
one of the following two forms:
| S :> !e!
|
| q :> W, with q in {"S"} |_| !Q!, and W in (!Q! |_| !A!)^+
A grammar that fits into this mold is called a "context-free" grammar.
The first type of rewrite rule is referred to as a "special production",
while the second type of rewrite rule is called an "ordinary production".
An "ordinary derivation" is one that employs only ordinary productions.
In ordinary productions, those that have the form q :> W, the replacement
string W is never the empty string, and so the lengths of the augmented
strings or the sentential forms that follow one another in an ordinary
derivation, on account of using the ordinary types of rewrite rules,
never decrease at any stage of the process, up to and including the
terminal string that is finally generated by the grammar. This type
of feature is known as the "non-contracting property" of productions,
derivations, and grammars. A grammar is said to have the property if
all of its covering productions, with the possible exception of S :> e,
are non-contracting. In particular, context-free grammars are special
cases of non-contracting grammars. The presence of the non-contracting
property within a formal grammar makes the length of the augmented string
available as a parameter that can figure into mathematical inductions and
motivate recursive proofs, and this handle on the generative process makes
it possible to establish the kinds of results about the generated language
that are not easy to achieve in more general cases, nor by any other means
even in these brands of special cases.
Grammar 5 is a context-free grammar for the painted cactus language
that uses !Q! = {"S'", "T"}, with !K! as listed in the next display.
| !C!(!P!). Grammar 5
|
| !Q! = {"S'", "T"}
|
| 1. S :> !e!
|
| 2. S :> S'
|
| 3. S' :> m_1
|
| 4. S' :> p_j, for each j in J
|
| 5. S' :> S' · S'
|
| 6. S' :> "-()-"
|
| 7. S' :> "-(" · T · ")-"
|
| 8. T :> ","
|
| 9. T :> S'
|
| 10. T :> T · ","
|
| 11. T :> T · "," · S'
Finally, it is worth trying to bring together the advantages of these
diverse styles of grammar, to whatever extent that they are compatible.
To do this, a prospective grammar must be capable of maintaining a high
level of intermediate organization, like that arrived at in Grammar 2,
while respecting the principle of intermediate significance, and thus
accumulating all the benefits of the context-free format in Grammar 5.
A plausible synthesis of most of these features is given in Grammar 6.
| !C!(!P!). Grammar 6
|
| !Q! = {"S'", "R", "F", "T"}
|
| 1. S :> !e!
|
| 2. S :> S'
|
| 3. S' :> R
|
| 4. S' :> F
|
| 5. S' :> S' · S'
|
| 6. R :> m_1
|
| 7. R :> p_j, for each j in J
|
| 8. R :> R · R
|
| 9. F :> "-()-"
|
| 10. F :> "-(" · T · ")-"
|
| 11. T :> ","
|
| 12. T :> S'
|
| 13. T :> T · ","
|
| 14. T :> T · "," · S'
The preceding development provides a typical example of how an initially
effective and conceptually succinct description of a formal language, but
one that is terse to the point of allowing its prospective interpreter to
waste exorbitant amounts of energy in trying to unravel its implications,
can be converted into a form that is more efficient from the operational
point of view, even if slightly more ungainly in regard to its elegance.
The basic idea behind all of this machinery remains the same: Besides
the select body of formulas that are introduced as boundary conditions,
it merely institutes the following general rule:
| If the strings S_1, ..., S_k are sentences,
|
| then their concatenation in the form
|
| Conc^k_j S_j = S_1 · ... · S_k
|
| is a sentence,
|
| and their surcatenation in the form
|
| Surc^k_j S_j = "-(" · S_1 · "," · ... · "," · S_k · ")-"
|
| is a sentence.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.9 The Cactus Language: Syntax (cont.)
It is fitting to wrap up the foregoing developments by summarizing the
notion of a formal grammar that appeared to evolve in the present case.
For the sake of future reference and the chance of a wider application,
it is also useful to try to extract the scheme of a formalization that
potentially holds for any formal language. The following presentation
of the notion of a formal grammar is adapted, with minor modifications,
from the treatment in (DDQ, 60-61).
A "formal grammar" !G! is given by a four-tuple !G! = ("S", !Q!, !A!, !K!)
that takes the following form of description:
1. "S" is the "initial", "special", "start", or "sentence symbol".
Since the letter "S" serves this function only in a special setting,
its employment in this role need not create any confusion with its
other typical uses as a string variable or as a sentence variable.
2. !Q! = {q_1, ..., q_m} is a finite set of "intermediate symbols",
all distinct from "S".
3. !A! = {a_1, ..., a_n} is a finite set of "terminal symbols",
also known as the "alphabet" of !G!, all distinct from "S" and
disjoint from !Q!. Depending on the particular conception of the
language !L! that is "covered", "generated", "governed", or "ruled"
by the grammar !G!, that is, whether !L! is conceived to be a set of
words, sentences, paragraphs, or more extended structures of discourse,
it is usual to describe !A! as the "alphabet", "lexicon", "vocabulary",
"liturgy", or "phrase book" of both the grammar !G! and the language !L!
that it regulates.
4. !K! is a finite set of "characterizations". Depending on how they
come into play, these are variously described as "covering rules",
"formations", "productions", "rewrite rules", "subsumptions",
"transformations", or "typing rules".
To describe the elements of !K! it helps to define some additional terms:
a. The symbols in {"S"} |_| !Q! |_| !A! form the "augmented alphabet" of !G!.
b. The symbols in {"S"} |_| !Q! are the "non-terminal symbols" of !G!.
c. The symbols in !Q! |_| !A! are the "non-initial symbols" of !G!.
d. The strings in ({"S"} |_| !Q! |_| !A!)* are the "augmented strings" for G.
e. The strings in {"S"} |_| (!Q! |_| !A!)* are the "sentential forms" for G.
Each characterization in !K! is an ordered pair of strings (S_1, S_2)
that takes the following form:
| S_1 = Q_1 · q · Q_2,
|
| S_2 = Q_1 · W · Q_2.
In this scheme, S_1 and S_2 are members of the augmented strings for !G!,
more precisely, S_1 is a non-empty string and a sentential form over !G!,
while S_2 is a possibly empty string and also a sentential form over !G!.
Here also, q is a non-terminal symbol, that is, q is in {"S"} |_| !Q!,
while Q_1, Q_2, and W are possibly empty strings of non-initial symbols,
a fact that can be expressed in the form: Q_1, Q_2, W in (!Q! |_| !A!)*.
In practice, the ordered pairs of strings in !K! are used to "derive",
to "generate", or to "produce" sentences of the language !L! = <!G!>
that is then said to be "governed" or "regulated" by the grammar !G!.
In order to facilitate this active employment of the grammar, it is
conventional to write the characterization (S_1, S_2) in either one
of the next two forms, where the more generic form is followed by
the more specific form:
| S_1 :> S_2
|
| Q_1 · q · Q_2 :> Q_1 · W · Q_2
In this usage, the characterization S_1 :> S_2 is tantamount to a grammatical
license to transform a string of the form Q_1 · q · Q_2 into a string of the
form Q1 · W · Q2, in effect, replacing the non-terminal symbol q with the
non-initial string W in any selected, preserved, and closely adjoining
context of the form Q1 · ... · Q2. Accordingly, in this application
the notation "S_1 :> S_2" can be read as "S_1 produces S_2" or as
"S_1 transforms into S_2".
An "immediate derivation" in !G! is an ordered pair (W, W')
of sentential forms in !G! such that:
| W = Q_1 · X · Q_2,
|
| W' = Q_1 · Y · Q_2,
|
| and (X, Y) in !K!,
|
| i.e. X :> Y in !G!.
This relation is indicated by saying that W "immediately derives" W',
that W' is "immediately derived" from W in !G!, and also by writing:
W ::> W'.
A "derivation" in !G! is a finite sequence (W_1, ..., W_k)
of sentential forms over !G! such that each adjacent pair
(W_j, W_(j+1)) of sentential forms in the sequence is an
immediate derivation in !G!, in other words, such that:
W_j ::> W_(j+1), for all j = 1 to k-1.
If there exists a derivation (W_1, ..., W_k) in !G!,
one says that W_1 "derives" W_k in !G!, conversely,
that W_k is "derivable" from W_1 in !G!, and one
typically summarizes the derivation by writing:
W_1 :*:> W_k.
The language !L! = !L!(!G!) = <!G!> that is "generated"
by the formal grammar !G! = ("S", !Q!, !A!, !K!) is the
set of strings over the terminal alphabet !A! that are
derivable from the initial symbol "S" by way of the
intermediate symbols in !Q! according to the
characterizations in K. In sum:
!L!(!G!) = <!G!> = {W in !A!* : "S" :*:> W}.
Finally, a string W is called a "word", a "sentence", or so on,
of the language generated by !G! if and only if W is in !L!(!G!).
Reference
| Denning, P.J., Dennis, J.B., Qualitz, J.E.,
|'Machines, Languages, and Computation',
| Prentice-Hall, Englewood Cliffs, NJ, 1978.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.10 The Cactus Language: Stylistics
| As a result, we can hardly conceive of how many possibilities there are
| for what we call objective reality. Our sharp quills of knowledge are so
| narrow and so concentrated in particular directions that with science there
| are myriads of totally different real worlds, each one accessible from the
| next simply by slight alterations -- shifts of gaze -- of every particular
| discipline and subspecialty.
|
| Herbert J. Bernstein, "Idols", page 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing: The Sciences, Society, & Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.
This Subsection highlights an issue of "style" that arises in describing
a formal language. In broad terms, I use the word "style" to refer to a
loosely specified class of formal systems, typically ones that have a set
of distinctive features in common. For instance, a style of proof system
usually dictates one or more rules of inference that are acknowledged as
conforming to that style. In the present context, the word "style" is a
natural choice to characterize the varieties of formal grammars, or any
other sorts of formal systems that can be contemplated for deriving the
sentences of a formal language.
In looking at what seems like an incidental issue, the discussion arrives
at a critical point. The question is: What decides the issue of style?
Taking a given language as the object of discussion, what factors enter
into and determine the choice of a style for its presentation, that is,
a particular way of arranging and selecting the materials that come to
be involved in a description, a grammar, or a theory of the language?
To what degree is the determination accidental, empirical, pragmatic,
rhetorical, or stylistic, and to what extent is the choice essential,
logical, and necessary? For that matter, what determines the order
of signs in a word, a sentence, a text, or a discussion? All of
the corresponding parallel questions about the character of this
choice can be posed with regard to the constituent part as well
as with regard to the main constitution of the formal language.
In order to answer this sort of question, at any level of articulation,
one has to inquire into the type of distinction that it invokes, between
arrangements and orders that are essential, logical, and necessary and
orders and arrangements that are accidental, rhetorical, and stylistic.
As a rough guide to its comprehension, a "logical order", if it resides
in the subject at all, can be approached by considering all of the ways
of saying the same things, in all of the languages that are capable of
saying roughly the same things about that subject. Of course, the "all"
that appears in this rule of thumb has to be interpreted as a fittingly
qualified sort of universal. For all practical purposes, it simply means
"all of the ways that a person can think of" and "all of the languages
that a person can conceive of", with all things being relative to the
particular moment of investigation. For all of these reasons, the rule
must stand as little more than a rough idea of how to approach its object.
If it is demonstrated that a given formal language can be presented in
any one of several styles of formal grammar, then the choice of a format
is accidental, optional, and stylistic to the very extent that it is free.
But if it can be shown that a particular language cannot be successfully
presented in a particular style of grammar, then the issue of style is
no longer free and rhetorical, but becomes to that very degree essential,
necessary, and obligatory, in other words, a question of the objective
logical order that can be found to reside in the object language.
As a rough illustration of the difference between logical and rhetorical
orders, consider the kinds of order that are expressed and exhibited in
the following conjunction of implications:
"X => Y and Y => Z".
Here, there is a happy conformity between the logical content and the
rhetorical form, indeed, to such a degree that one hardly notices the
difference between them. The rhetorical form is given by the order
of sentences in the two implications and the order of implications
in the conjunction. The logical content is given by the order of
propositions in the extended implicational sequence:
X =< Y =< Z.
To see the difference between form and content, or manner and matter,
it is enough to observe a few of the ways that the expression can be
varied without changing its meaning, for example:
"Z <= Y and Y <= X".
Any style of declarative programming, also called "logic programming",
depends on a capacity, as embodied in a programming language or other
formal system, to describe the relation between problems and solutions
in logical terms. A recurring problem in building this capacity is in
bridging the gap between ostensibly non-logical orders and the logical
orders that are used to describe and to represent them. For instance,
to mention just a couple of the most pressing cases, and the ones that
are currently proving to be the most resistant to a complete analysis,
one has the orders of dynamic evolution and rhetorical transition that
manifest themselves in the process of inquiry and in the communication
of its results.
This patch of the ongoing discussion is concerned with describing a
particular variety of formal languages, whose typical representative
is the painted cactus language !L! = !C!(!P!). It is the intention of
this work to interpret this language for propositional logic, and thus
to use it as a sentential calculus, an order of reasoning that forms an
active ingredient and a significant component of all logical reasoning.
To describe this language, the standard devices of formal grammars and
formal language theory are more than adequate, but this only raises the
next question: What sorts of devices are exactly adequate, and fit the
task to a "T"? The ultimate desire is to turn the tables on the order
of description, and so begins a process of eversion that evolves to the
point of asking: To what extent can the language capture the essential
features and laws of its own grammar and describe the active principles
of its own generation? In other words: How well can the language be
described by using the language itself to do so?
In order to speak to these questions, I have to express what a grammar says
about a language in terms of what a language can say on its own. In effect,
it is necessary to analyze the kinds of meaningful statements that grammars
are capable of making about languages in general and to relate them to the
kinds of meaningful statements that the syntactic "sentences" of the cactus
language might be interpreted as making about the very same topics. So far
in the present discussion, the sentences of the cactus language do not make
any meaningful statements at all, much less any meaningful statements about
languages and their constitutions. As of yet, these sentences subsist in the
form of purely abstract, formal, and uninterpreted combinatorial constructions.
Before the capacity of a language to describe itself can be evaluated,
the missing link to meaning has to be supplied for each of its strings.
This calls for a dimension of semantics and a notion of interpretation,
topics that are taken up for the case of the cactus language !C!(!P!)
in Subsection 1.3.10.12. Once a plausible semantics is prescribed for
this language it will be possible to return to these questions and to
address them in a meaningful way.
The prominent issue at this point is the distinct placements of formal
languages and formal grammars with respect to the question of meaning.
The sentences of a formal language are merely the abstract strings of
abstract signs that happen to belong to a certain set. They do not by
themselves make any meaningful statements at all, not without mounting
a separate effort of interpretation, but the rules of a formal grammar
make meaningful statements about a formal language, to the extent that
they say what strings belong to it and what strings do not. Thus, the
formal grammar, a formalism that appears to be even more skeletal than
the formal language, still has bits and pieces of meaning attached to it.
In a sense, the question of meaning is factored into two parts, structure
and value, leaving the aspect of value reduced in complexity and subtlety
to the simple question of belonging. Whether this single bit of meaningful
value is enough to encompass all of the dimensions of meaning that we require,
and whether it can be compounded to cover the complexity that actually exists
in the realm of meaning -- these are questions for an extended future inquiry.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.10 The Cactus Language: Stylistics (cont.)
Perhaps I ought to comment on the differences between the present and
the standard definition of a formal grammar, since I am attempting to
strike a compromise with several alternative conventions of usage, and
thus to leave certain options open for future exploration. All of the
changes are minor, in the sense that they are not intended to alter the
classes of languages that are able to be generated, but only to clear up
various ambiguities and sundry obscurities that affect their conception.
Primarily, the conventional scope of non-terminal symbols was expanded
to encompass the sentence symbol, mainly on account of all the contexts
where the initial and the intermediate symbols are naturally invoked in
the same breath. By way of compensating for the usual exclusion of the
sentence symbol from the non-terminal class, an equivalent distinction
was introduced in the fashion of a distinction between the initial and
the intermediate symbols, and this serves its purpose in all of those
contexts where the two kind of symbols need to be treated separately.
At the present point, I remain a bit worried about the motivations
and the justifications for introducing this distinction, under any
name, in the first place. It is purportedly designed to guarantee
that the process of derivation at least gets started in a definite
direction, while the real questions have to do with how it all ends.
The excuses of efficiency and expediency that I offered as plausible
and sufficient reasons for distinguishing between empty and significant
sentences are likely to be ephemeral, if not entirely illusory, since
intermediate symbols are still permitted to characterize or to cover
themselves, not to mention being allowed to cover the empty string,
and so the very types of traps that one exerts oneself to avoid at
the outset are always there to afflict the process at all of the
intervening times.
If one reflects on the form of grammar that is being prescribed here,
it looks as if one sought, rather futilely, to avoid the problems of
recursion by proscribing the main program from calling itself, while
allowing any subprogram to do so. But any trouble that is avoidable
in the part is also avoidable in the main, while any trouble that is
inevitable in the part is also inevitable in the main. Consequently,
I am reserving the right to change my mind at a later stage, perhaps
to permit the initial symbol to characterize, to cover, to regenerate,
or to produce itself, if that turns out to be the best way in the end.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.10 The Cactus Language: Stylistics (cont.)
Before I leave this Subsection, I need to say a few things about
the manner in which the abstract theory of formal languages and
the pragmatic theory of sign relations interact with each other.
Formal language theory can seem like an awfully picky subject at times,
treating every symbol as a thing in itself the way it does, sorting out
the nominal types of symbols as objects in themselves, and singling out
the passing tokens of symbols as distinct entities in their own rights.
It has to continue doing this, if not for any better reason than to aid
in clarifying the kinds of languages that people are accustomed to use,
to assist in writing computer programs that are capable of parsing real
sentences, and to serve in designing programming languages that people
would like to become accustomed to use. As a matter of fact, the only
time that formal language theory becomes too picky, or a bit too myopic
in its focus, is when it leads one to think that one is dealing with the
thing itself and not just with the sign of it, in other words, when the
people who use the tools of formal language theory forget that they are
dealing with the mere signs of more interesting objects and not with the
objects of ultimate interest in and of themselves.
As a result, there a number of deleterious effects that can arise from
the extreme pickiness of formal language theory, arising, as is often the
case, when formal theorists forget the practical context of theorization.
It frequently happens that the exacting task of defining the membership
of a formal language leads one to think that this object and this object
alone is the justifiable end of the whole exercise. The distractions of
this mediate objective render one liable to forget that one's penultimate
interest lies always with various kinds of equivalence classes of signs,
not entirely or exclusively with their more meticulous representatives.
When this happens, one typically goes on working oblivious to the fact
that many details about what transpires in the meantime do not matter
at all in the end, and one is likely to remain in blissful ignorance
of the circumstance that many special details of language membership
are bound, destined, and pre-determined to be glossed over with some
measure of indifference, especially when it comes down to the final
constitution of those equivalence classes of signs that are able to
answer for the genuine objects of the whole enterprise of language.
When any form of theory, against its initial and its best intentions,
leads to this kind of absence of mind that is no longer beneficial in
all of its main effects, the situation calls for an antidotal form of
theory, one that can restore the presence of mind that all forms of
theory are meant to augment.
The pragmatic theory of sign relations is called for in settings where
everything that can be named has many other names, that is to say, in
the usual case. Of course, one would like to replace this superfluous
multiplicity of signs with an organized system of canonical signs, one
for each object that needs to be denoted, but reducing the redundancy
too far, beyond what is necessary to eliminate the factor of "noise" in
the language, that is, to clear up its effectively useless distractions,
can destroy the very utility of a typical language, which is intended to
provide a ready means to express a present situation, clear or not, and
to describe an ongoing condition of experience in just the way that it
seems to present itself. Within this fleshed out framework of language,
moreover, the process of transforming the manifestations of a sign from
its ordinary appearance to its canonical aspect is the whole problem of
computation in a nutshell.
It is a well-known truth, but an often forgotten fact, that nobody
computes with numbers, but solely with numerals in respect of numbers,
and numerals themselves are symbols. Among other things, this renders
all discussion of numeric versus symbolic computation a bit beside the
point, since it is only a question of what kinds of symbols are best for
one's immediate application or for one's selection of ongoing objectives.
The numerals that everybody knows best are just the canonical symbols,
the standard signs or the normal terms for numbers, and the process of
computation is a matter of getting from the arbitrarily obscure signs
that the data of a situation are capable of throwing one's way to the
indications of its character that are clear enough to motivate action.
Having broached the distinction between propositions and sentences, one
can see its similarity to the distinction between numbers and numerals.
What are the implications of the foregoing considerations for reasoning
about propositions and for the realm of reckonings in sentential logic?
If the purpose of a sentence is just to denote a proposition, then the
proposition is just the object of whatever sign is taken for a sentence.
This means that the computational manifestation of a piece of reasoning
about propositions amounts to a process that takes place entirely within
a language of sentences, a procedure that can rationalize its account by
referring to the denominations of these sentences among propositions.
The application of these considerations in the immediate setting is this:
Do not worry too much about what roles the empty string "" and the blank
symbol " " are supposed to play in a given species of formal languages.
As it happens, it is far less important to wonder whether these types
of formal tokens actually constitute genuine sentences than it is to
decide what equivalence classes it makes sense to form over all of
the sentences in the resulting language, and only then to bother
about what equivalence classes these limiting cases of sentences
are most conveniently taken to represent.
These concerns about boundary conditions betray a more general issue.
Already by this point in discussion the limits of the purely syntactic
approach to a language are beginning to be visible. It is not that one
cannot go a whole lot further by this road in the analysis of a particular
language and in the study of languages in general, but when it comes to the
questions of understanding the purpose of a language, of extending its usage
in a chosen direction, or of designing a language for a particular set of uses,
what matters above all else are the "pragmatic equivalence classes" of signs that
are demanded by the application and intended by the designer, and not so much the
peculiar characters of the signs that represent these classes of practical meaning.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.10 The Cactus Language: Stylistics (cont.)
Any description of a language is bound to have alternative descriptions.
More precisely, a circumscribed description of a formal language, as any
effectively finite description is bound to be, is certain to suggest the
equally likely existence and the possible utility of other descriptions.
A single formal grammar describes but a single formal language, but any
formal language is described by many different formal grammars, not all
of which afford the same grasp of its structure, provide an equivalent
comprehension of its character, or yield an interchangeable view of its
aspects. Consequently, even with respect to the same formal language,
different formal grammars are typically better for different purposes.
With the distinctions that evolve among the different styles of grammar,
and with the preferences that different observers display toward them,
there naturally comes the question: What is the root of this evolution?
One dimension of variation in the styles of formal grammars can be seen
by treating the union of languages, and especially the disjoint union of
languages, as a "sum", by treating the concatenation of languages as a
"product", and then by distinguishing the styles of analysis that favor
"sums of products" from those that favor "products of sums" as their
canonical forms of description. If one examines the relation between
languages and grammars carefully enough to see the presence and the
influence of these different styles, and when one comes to appreciate
the ways that different styles of grammars can be used with different
degrees of success for different purposes, then one begins to see the
possibility that alternative styles of description can be based on
altogether different linguistic and logical operations.
It possible to trace this divergence of styles to an even more primitive
division, one that distinguishes the "additive" or the "parallel" styles
from the "multiplicative" or the "serial" styles. The issue is somewhat
confused by the fact that an "additive" analysis is typically expressed
in the form of a "series", in other words, a disjoint union of sets or a
linear sum of their independent effects. But it is easy enough to sort
this out if one observes the more telling connection between "parallel"
and "independent". Another way to keep the right associations straight
is to employ the term "sequential" in preference to the more misleading
term "serial". Whatever one calls this broad division of styles, the
scope and sweep of their dimensions of variation can be delineated in
the following way:
1. The "additive" or "parallel" styles favor "sums of products" as
canonical forms of expression, pulling sums, unions, co-products,
and logical disjunctions to the outermost layers of analysis and
synthesis, while pushing products, intersections, concatenations,
and logical conjunctions to the innermost levels of articulation
and generation. In propositional logic, this style leads to the
"disjunctive normal form" (DNF).
2. The "multiplicative" or "serial" styles favor "products of sums"
as canonical forms of expression, pulling products, intersections,
concatenations, and logical conjunctions to the outermost layers of
analysis and synthesis, while pushing sums, unions, co-products,
and logical disjunctions to the innermost levels of articulation
and generation. In propositional logic, this style leads to the
"conjunctive normal form" (CNF).
There is a curious sort of diagnostic clue, a veritable shibboleth,
that often serves to reveal the dominance of one mode or the other
within an individual thinker's cognitive style. Examined on the
question of what constitutes the "natural numbers", an "additive"
thinker tends to start the sequence at 0, while a "multiplicative"
thinker tends to regard it as beginning at 1.
In any style of description, grammar, or theory of a language, it is
usually possible to tease out the influence of these contrasting traits,
namely, the "additive" attitude versus the "mutiplicative" tendency that
go to make up the particular style in question, and even to determine the
dominant inclination or point of view that establishes its perspective on
the target domain.
In each style of formal grammar, the "multiplicative" aspect is present
in the sequential concatenation of signs, both in the augmented strings
and in the terminal strings. In settings where the non-terminal symbols
classify types of strings, the concatenation of the non-terminal symbols
signifies the cartesian product over the corresponding sets of strings.
In the context-free style of formal grammar, the "additive" aspect is
easy enough to spot. It is signaled by the parallel covering of many
augmented strings or sentential forms by the same non-terminal symbol.
Expressed in active terms, this calls for the independent rewriting
of that non-terminal symbol by a number of different successors,
as in the following scheme:
| q :> W_1.
|
| ... ... ...
|
| q :> W_k.
It is useful to examine the relationship between the grammatical covering
or production relation ":>" and the logical relation of implication "=>",
with one eye to what they have in common and one eye to how they differ.
The production "q :> W" says that the appearance of the symbol "q" in
a sentential form implies the possibility of exchanging it for "W".
Although this sounds like a "possible implication", to the extent
that "q implies a possible W" or that "q possibly implies W", the
qualifiers "possible" and "possibly" are the critical elements in
these statements, and they are crucial to the meaning of what is
actually being implied. In effect, these qualifications reverse
the direction of implication, yielding "q <= W" as the best
analogue for the sense of the production.
One way to sum this up is to say that non-terminal symbols have the
significance of hypotheses. The terminal strings form the empirical
matter of a language, while the non-terminal symbols mark the patterns
or the types of substrings that can be noticed in the profusion of data.
If one observes a portion of a terminal string that falls into the pattern
of the sentential form W, then it is an admissable hypothesis, according to
the theory of the language that is constituted by the formal grammar, that
this piece not only fits the type q but even comes to be generated under
the auspices of the non-terminal symbol "q".
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.10 The Cactus Language: Stylistics (cont.)
A moment's reflection on the issue of style, giving due consideration to the
received array of stylistic choices, ought to inspire at least the question:
"Are these the only choices there are?" In the present setting, there are
abundant indications that other options, more differentiated varieties of
description and more integrated ways of approaching individual languages,
are likely to be conceivable, feasible, and even more ultimately viable.
If a suitably generic style, one that incorporates the full scope of
logical combinations and operations, is broadly available, then it
would no longer be necessary, or even apt, to argue in universal
terms about "which style is best", but more useful to investigate
how we might adapt the local styles to the local requirements.
The medium of a generic style would yield a viable compromise
between "additive" and "multiplicative" canons, and render the
choice between "parallel" and "serial" a false alternative,
at least, when expressed in the globally exclusive terms
that are currently most commonly adopted to pose it.
One set of indications comes from the study of machines, languages, and
computation, especially the theories of their structures and relations.
The forms of composition and decomposition that are generally known as
"parallel" and "serial" are merely the extreme special cases, in variant
directions of specialization, of a more generic form, usually called the
"cascade" form of combination. This is a well-known fact in the theories
that deal with automata and their associated formal languages, but its
implications do not seem to be widely appreciated outside these fields.
In particular, it dispells the need to choose one extreme or the other,
since most of the natural cases are likely to exist somewhere in between.
Another set of indications appears in algebra and category theory,
where forms of composition and decomposition related to the cascade
combination, namely, the "semi-direct product" and its special case,
the "wreath product", are encountered at higher levels of generality
than the cartesian products of sets or the direct products of spaces.
In these domains of operation, one finds it necessary to consider also
the "co-product" of sets and spaces, a construction that artificially
creates a disjoint union of sets, that is, a union of spaces that are
being treated as independent. It does this, in effect, by "indexing",
"coloring", or "preparing" the otherwise possibly overlapping domains
that are being combined. What renders this a "chimera" or a "hybrid"
form of combination is the fact that this indexing is tantamount to a
cartesian product of a singleton set, namely, the conventional "index",
"color", or "affix" in question, with the individual domain that is
entering as a factor, a term, or a participant in the final result.
One of the insights that arises out of Peirce's logical work is that
the set operations of complementation, intersection, and union, along
with the logical operations of negation, conjunction, and disjunction
that operate in isomorphic tandem with them, are not as fundamental as
they first appear. This is because all of them can be constructed from
or derived from a smaller set of operations, in fact, taking the logical
side of things, from either one of two "solely sufficient" operators,
called "amphecks" by Peirce, "strokes" by those who re-discovered them
later, and known in computer science as the NAND and the NNOR operators.
For this reason, that is, by virtue of their precedence in the orders
of construction and derivation, these operations have to be regarded
as the simplest and the most primitive in principle, even if they are
scarcely recognized as lying among the more familiar elements of logic.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.10 The Cactus Language: Stylistics (cont.)
I am throwing together a wide variety of different operations into each
of the bins labeled "additive" and "multiplicative", but it is easy to
observe a natural organization and even some relations approaching
isomorphisms among and between the members of each class.
The relation between logical disjunction and set-theoretic union and the
relation between logical conjunction and set-theoretic intersection ought
to be clear enough for the purposes of the immediately present context.
In any case, all of these relations are scheduled to receive a thorough
examination in a subsequent discussion (Subsection 1.3.10.13). But the
relation of a set-theoretic union to a category-theoretic co-product and
the relation of a set-theoretic intersection to a syntactic concatenation
deserve a closer look at this point.
The effect of a co-product as a "disjointed union", in other words, that
creates an object tantamount to a disjoint union of sets in the resulting
co-product even if some of these sets intersect non-trivially and even if
some of them are identical "in reality", can be achieved in several ways.
The most usual conception is that of making a "separate copy", for each
part of the intended co-product, of the set that is intended to go there.
Often one thinks of the set that is assigned to a particular part of the
co-product as being distinguished by a particular "color", in other words,
by the attachment of a distinct "index", "label", or "tag", being a marker
that is inherited by and passed on to every element of the set in that part.
A concrete image of this construction can be achieved by imagining that each
set and each element of each set is placed in an ordered pair with the sign
of its color, index, label, or tag. One describes this as the "injection"
of each set into the corresponding "part" of the co-product.
For example, given the sets P and Q, overlapping or not, one can define
the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the
copy of P into the first part of the co-product and the copy of Q into the
second part of the co-product, in the following manner:
P_[1] = <P, 1> = {<x, 1> : x in P},
Q_[2] = <Q, 2> = {<x, 2> : x in Q}.
Using the sign "]_[" for this construction, the "sum", the "co-product",
or the "disjointed union" of P and Q in that order can be represented as
the ordinary disjoint union of P_[1] and Q_[2].
P ]_[ Q = P_[1] |_| Q_[2].
The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is
just the cartesian product of sets L_1 x L_2 without the extra x's,
but the relation of cartesian products to set-theoretic intersections
and thus to logical conjunctions is far from being clear. One way of
seeing a type of relation is to focus on the information that is needed
to specify each construction, and thus to reflect on the signs that are
used to carry this information. As a first approach to the topic of
information, according to a strategy that seeks to be as elementary
and as informal as possible, I introduce the following set of ideas,
intended to be taken in a very provisional way.
A "stricture" is a specification of a certain set in a certain place,
relative to a number of other sets, yet to be specified. It is assumed
that one knows enough to tell if two strictures are equivalent as pieces
of information, but any more determinate indications, like names for the
places that are mentioned in the stricture, or bounds on the number of
places that are involved, are regarded as being extraneous impositions,
outside the proper concern of the definition, no matter how convenient
they are found to be for a particular discussion. As a schematic form
of illustration, a stricture can be pictured in the following shape:
"... x X x Q x X x ..."
A "strait" is the object that is specified by a stricture, in effect,
a certain set in a certain place of an otherwise yet to be specified
relation. Somewhat sketchily, the strait that corresponds to the
stricture just given can be pictured in the following shape:
... x X x Q x X x ...
In this picture, Q is a certain set, and X is the universe of discourse
that is relevant to a given discussion. Since a stricture does not, by
itself, contain a sufficient amount of information to specify the number
of sets that it intends to set in place, or even to specify the absolute
location of the set that its does set in place, it appears to place an
unspecified number of unspecified sets in a vague and uncertain strait.
Taken out of its interpretive context, the residual information that a
stricture can convey makes all of the following potentially equivalent
as strictures:
"Q", "XxQxX", "XxXxQxXxX", ...
With respect to what these strictures specify, this
leaves all of the following equivalent as straits:
Q = XxQxX = XxXxQxXxX = ...
Within the framework of a particular discussion, it is customary to
set a bound on the number of places and to limit the variety of sets
that are regarded as being under active consideration, and it is also
convenient to index the places of the indicated relations, and of their
encompassing cartesian products, in some fixed way. But the whole idea
of a stricture is to specify a strait that is capable of extending through
and beyond any fixed frame of discussion. In other words, a stricture is
conceived to constrain a strait at a certain point, and then to leave it
literally embedded, if tacitly expressed, in a yet to be fully specified
relation, one that involves an unspecified number of unspecified domains.
A quantity of information is a measure of constraint. In this respect,
a set of comparable strictures is ordered on account of the information
that each one conveys, and a system of comparable straits is ordered in
accord with the amount of information that it takes to pin each one of
them down. Strictures that are more constraining and straits that are
more constrained are placed at higher levels of information than those
that are less so, and entities that involve more information are said
to have a greater "complexity" in comparison with those entities that
involve less information, that are said to have a greater "simplicity".
In order to create a concrete example, let me now institute a frame of
discussion where the number of places in a relation is bounded at two,
and where the variety of sets under active consideration is limited to
the typical subsets P and Q of a universe X. Under these conditions,
one can use the following sorts of expression as schematic strictures:
| "X" , "P" , "Q" ,
|
| "XxX", "XxP", "XxQ",
|
| "PxX", "PxP", "PxQ",
|
| "QxX", "QxP", "QxQ".
These strictures and their corresponding straits are stratified according
to their amounts of information, or their levels of constraint, as follows:
| High: "PxP", "PxQ", "QxP", "QxQ".
|
| Med: "P" , "XxP", "PxX".
|
| Med: "Q" , "XxQ", "QxX".
|
| Low: "X" , "XxX".
Within this framework, the more complex strait PxQ can be expressed
in terms of the simpler straits, PxX and XxQ. More specifically,
it lends itself to being analyzed as their intersection, in the
following way:
PxQ = PxX |^| XxQ.
>From here it is easy to see the relation of concatenation, by virtue of
these types of intersection, to the logical conjunction of propositions.
The cartesian product PxQ is described by a conjunction of propositions,
namely, "P_<1> and Q_<2>", subject to the following interpretation:
1. "P_<1>" asserts that there is an element from
the set P in the first place of the product.
2. "Q_<2>" asserts that there is an element from
the set Q in the second place of the product.
The integration of these two pieces of information can be taken
in that measure to specify a yet to be fully determined relation.
In a corresponding fashion at the level of the elements,
the ordered pair <p, q> is described by a conjunction
of propositions, namely, "p_<1> and q_<2>", subject
to the following interpretation:
1. "p_<1>" says that p is in the first place
of the product element under construction.
2. "q_<2>" says that q is in the second place
of the product element under construction.
Notice that, in construing the cartesian product of the sets P and Q or the
concatenation of the languages L_1 and L_2 in this way, one shifts the level
of the active construction from the tupling of the elements in P and Q or the
concatenation of the strings that are internal to the languages L_1 and L_2 to
the concatenation of the external signs that it takes to indicate these sets or
these languages, in other words, passing to a conjunction of indexed propositions,
"P_<1> and Q_<2>", or to a conjunction of assertions, "L_1_<1> and L_2_<2>", that
marks the sets or the languages in question for insertion in the indicated places
of a product set or a product language, respectively. In effect, the subscripting
by the indices "<1>" and "<2>" can be recognized as a special case of concatenation,
albeit through the posting of editorial remarks from an external "mark-up" language.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.10 The Cactus Language: Stylistics (cont.)
In order to systematize the relations that strictures and straits placed
at higher levels of complexity, constraint, information, and organization
have with those that are placed at the associated lower levels, I introduce
the following pair of definitions:
The j^th "excerpt" of a stricture of the form "S_1 x ... x S_k", regarded
within a frame of discussion where the number of places is limited to k,
is the stricture of the form "X x ... x S_j x ... x X". In the proper
context, this can be written more succinctly as the stricture "S_j_<j>",
an assertion that places the j^th set in the j^th place of the product.
The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained
to a frame of discussion where the number of places is restricted to k,
is the strait of the form X x ... x S_j x ... x X. In the appropriate
context, this can be denoted more succinctly by the stricture "S_j_<j>",
an assertion that places the j^th set in the j^th place of the product.
In these terms, a stricture of the form "S_1 x ... x S_k"
can be expressed in terms of simpler strictures, to wit,
as a conjunction of its k excerpts:
"S_1 x ... x S_k" = "S_1_<1>" & ... & "S_k_<k>".
In a similar vein, a strait of the form S_1 x ... x S_k
can be expressed in terms of simpler straits, namely,
as an intersection of its k extracts:
S_1 x ... x S_k = S_1_<1> |^| ... |^| S_k_<k>.
There is a measure of ambiguity that remains in this formulation,
but it is the best that I can do in the present informal context.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.11 The Cactus Language: Mechanics
| We are only now beginning to see how this works. Clearly one of the
| mechanisms for picking a reality is the sociohistorical sense of what
| is important -- which research program, with all its particularity of
| knowledge, seems most fundamental, most productive, most penetrating.
| The very judgments which make us push narrowly forward simultaneously
| make us forget how little we know. And when we look back at history,
| where the lesson is plain to find, we often fail to imagine ourselves
| in a parallel situation. We ascribe the differences in world view
| to error, rather than to unexamined but consistent and internally
| justified choice.
|
| Herbert J. Bernstein, "Idols", page 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science & The Reconstruction of Knowledge", pages 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing: The Sciences, Society, & Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.
In this Subsection, I discuss the "mechanics" of parsing the
cactus language into the corresponding class of computational
data structures. This provides each sentence of the language
with a translation into a computational form that articulates
its syntactic structure and prepares it for automated modes of
processing and evaluation. For this purpose, it is necessary
to describe the target data structures at a fairly high level
of abstraction only, ignoring the details of address pointers
and record structures and leaving the more operational aspects
of implementation to the imagination of prospective programmers.
In this way, I can put off to another stage of elaboration and
refinement the description of the program that constructs these
pointers and operates on these graph-theoretic data structures.
The structure of a "painted cactus", insofar as it presents itself
to the visual imagination, can be described as follows. The overall
structure, as given by its underlying graph, falls within the species
of graph that is commonly known as a "rooted cactus", and the only novel
feature that it adds to this is that each of its nodes can be "painted"
with a finite sequence of "paints", chosen from a "palette" that is given
by the parametric set {" "} |_| !P! = {m_1} |_| {p_1, ..., p_k}.
It is conceivable, from a purely graph-theoretical point of view, to have
a class of cacti that are painted but not rooted, and so it is frequently
necessary, for the sake of precision, to more exactly pinpoint the target
species of graphical structure as a "painted and rooted cactus" (PARC).
A painted cactus, as a rooted graph, has a distinguished "node" that is
called its "root". By starting from the root and working recursively,
the rest of its structure can be described in the following fashion.
Each "node" of a PARC consists of a graphical "point" or "vertex" plus
a finite sequence of "attachments", described in relative terms as the
attachments "at" or "to" that node. An empty sequence of attachments
defines the "empty node". Otherwise, each attachment is one of three
kinds: a blank, a paint, or a type of PARC that is called a "lobe".
Each "lobe" of a PARC consists of a directed graphical "cycle" plus a
finite sequence of "accoutrements", described in relative terms as the
accoutrements "of" or "on" that lobe. Recalling the circumstance that
every lobe that comes under consideration comes already attached to a
particular node, exactly one vertex of the corresponding cycle is the
vertex that comes from that very node. The remaining vertices of the
cycle have their definitions filled out according to the accoutrements
of the lobe in question. An empty sequence of accoutrements is taken
to be tantamount to a sequence that contains a single empty node as its
unique accoutrement, and either one of these ways of approaching it can
be regarded as defining a graphical structure that is called a "needle"
or a "terminal edge". Otherwise, each accoutrement of a lobe is itself
an arbitrary PARC.
Although this definition of a lobe in terms of its intrinsic structural
components is logically sufficient, it is also useful to characterize the
structure of a lobe in comparative terms, that is, to view the structure
that typifies a lobe in relation to the structures of other PARC's and to
mark the inclusion of this special type within the general run of PARC's.
This approach to the question of types results in a form of description
that appears to be a bit more analytic, at least, in mnemonic or prima
facie terms, if not ultimately more revealing. Working in this vein,
a "lobe" can be characterized as a special type of PARC that is called
an "unpainted root plant" (UR-plant).
An "UR-plant" is a PARC of a simpler sort, at least, with respect to the
recursive ordering of structures that is being followed here. As a type,
it is defined by the presence of two properties, that of being "planted"
and that of having an "unpainted root". These are defined as follows:
1. A PARC is "planted" if its list of attachments has just one PARC.
2. A PARC is "UR" if its list of attachments has no blanks or paints.
In short, an UR-planted PARC has a single PARC as its only attachment,
and since this attachment is prevented from being a blank or a paint,
the single attachment at its root has to be another sort of structure,
that which we call a "lobe".
To express the description of a PARC in terms of its nodes, each node
can be specified in the fashion of a functional expression, letting a
citation of the generic function name "Node" be followed by a list of
arguments that enumerates the attachments of the node in question, and
letting a citation of the generic function name "Lobe" be followed by a
list of arguments that details the accoutrements of the lobe in question.
Thus, one can write expressions of the following forms:
1. Node^0 = Node()
= a node with no attachments.
Node^k_j C_j = Node(C_1, ..., C_k)
= a node with the attachments C_1, ..., C_k.
2. Lobe^0 = Lobe()
= a lobe with no accoutrements.
Lobe^k_j C_j = Lobe(C_1, ..., C_k)
= a lobe with the accoutrements C_1, ..., C_k.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.11 The Cactus Language: Mechanics (cont.)
Working from a structural description of the cactus language,
or any suitable formal grammar for !C!(!P!), it is possible to
give a recursive definition of the function called "Parse" that
maps each sentence in PARCE(!P!) to the corresponding graph in
PARC(!P!). One way to do this proceeds as follows:
1. The parse of the concatenation Conc^k of the k sentences S_j,
for j = 1 to k, is defined recursively as follows:
a. Parse(Conc^0) = Node^0.
b. For k > 0,
Parse(Conc^k_j S_j) = Node^k_j Parse(S_j).
2. The parse of the surcatenation Surc^k of the k sentences S_j,
for j = 1 to k, is defined recursively as follows:
a. Parse(Surc^0) = Lobe^0.
b. For k > 0,
Parse(Surc^k_j S_j) = Lobe^k_j Parse(S_j).
For ease of reference, Table 12 summarizes the mechanics of these parsing rules.
Table 12. Algorithmic Translation Rules
o------------------------o---------o------------------------o
| | Parse | |
| Sentence in PARCE | --> | Graph in PARC |
o------------------------o---------o------------------------o
| | | |
| Conc^0 | --> | Node^0 |
| | | |
| Conc^k_j S_j | --> | Node^k_j Parse(S_j) |
| | | |
| Surc^0 | --> | Lobe^0 |
| | | |
| Surc^k_j S_j | --> | Lobe^k_j Parse(S_j) |
| | | |
o------------------------o---------o------------------------o
A "substructure" of a PARC is defined recursively as follows. Starting
at the root node of the cactus C, any attachment is a substructure of C.
If a substructure is a blank or a paint, then it constitutes a minimal
substructure, meaning that no further substructures of C arise from it.
If a substructure is a lobe, then each one of its accoutrements is also
a substructure of C, and has to be examined for further substructures.
The concept of substructure can be used to define varieties of deletion
and erasure operations that respect the structure of the abstract graph.
For the purposes of this depiction, a blank symbol " " is treated as
a "primer", in other words, as a "clear paint", a "neutral tint", or
a "white wash". In effect, one is letting m_1 = p_0. In this frame
of discussion, it is useful to make the following distinction:
1. To "delete" a substructure is to replace it with an empty node,
in effect, to reduce the whole structure to a trivial point.
2. To "erase" a substructure is to replace it with a blank symbol,
in effect, to paint it out of the picture or to overwrite it.
A "bare" PARC, loosely referred to as a "bare cactus", is a PARC on the
empty palette !P! = {}. In other veins, a bare cactus can be described
in several different ways, depending on how the form arises in practice.
1. Leaning on the definition of a bare PARCE, a bare PARC can be
described as the kind of a parse graph that results from parsing
a bare cactus expression, in other words, as the kind of a graph
that issues from the requirements of processing a sentence of
the bare cactus language !C!^0 = PARCE^0.
2. To express it more in its own terms, a bare PARC can be defined
by tracing the recursive definition of a generic PARC, but then
by detaching an independent form of description from the source
of that analogy. The method is sufficiently sketched as follows:
a. A "bare PARC" is a PARC whose attachments
are limited to blanks and "bare lobes".
b. A "bare lobe" is a lobe whose accoutrements
are limited to bare PARC's.
3. In practice, a bare cactus is usually encountered in the process
of analyzing or handling an arbitrary PARC, the circumstances of
which frequently call for deleting or erasing all of its paints.
In particular, this generally makes it easier to observe the
various properties of its underlying graphical structure.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.12 The Cactus Language: Semantics
| Alas, and yet what 'are' you, my written and painted thoughts!
| It is not long ago that you were still so many-coloured,
| young and malicious, so full of thorns and hidden
| spices you made me sneeze and laugh -- and now?
| You have already taken off your novelty and
| some of you, I fear, are on the point of
| becoming truths: they already look so
| immortal, so pathetically righteous,
| so boring!
|
| Friedrich Nietzsche, 'Beyond Good and Evil', Paragraph 296.
|
| Friedrich Nietzsche,
|'Beyond Good and Evil: Prelude to a Philosophy of the Future',
| trans. by R.J. Hollingdale, intro. by Michael Tanner,
| Penguin Books, London, UK, 1973, 1990.
In this Subsection, I describe a particular semantics for the
painted cactus language, telling what meanings I aim to attach
to its bare syntactic forms. This supplies an "interpretation"
for this parametric family of formal languages, but it is good
to remember that it forms just one of many such interpretations
that are conceivable and even viable. In deed, the distinction
between the object domain and the sign domain can be observed in
the fact that many languages can be deployed to depict the same
set of objects and that any language worth its salt is bound to
to give rise to many different forms of interpretive saliency.
In formal settings, it is common to speak of "interpretation" as if it
created a direct connection between the signs of a formal language and
the objects of the intended domain, in other words, as if it determined
the denotative component of a sign relation. But a closer attention to
what goes on reveals that the process of interpretation is more indirect,
that what it does is to provide each sign of a prospectively meaningful
source language with a translation into an already established target
language, where "already established" means that its relationship to
pragmatic objects is taken for granted at the moment in question.
With this in mind, it is clear that interpretation is an affair of signs
that at best respects the objects of all of the signs that enter into it,
and so it is the connotative aspect of semiotics that is at stake here.
There is nothing wrong with my saying that I interpret a sentence of a
formal language as a sign that refers to a function or to a proposition,
so long as you understand that this reference is likely to be achieved
by way of more familiar and perhaps less formal signs that you already
take to denote those objects.
On entering a context where a logical interpretation is intended for the
sentences of a formal language there are a few conventions that make it
easier to make the translation from abstract syntactic forms to their
intended semantic senses. Although these conventions are expressed in
unnecessarily colorful terms, from a purely abstract point of view, they
do provide a useful array of connotations that help to negotiate what is
otherwise a difficult transition. This terminology is introduced as the
need for it arises in the process of interpreting the cactus language.
The task of this Subsection is to specify a "semantic function" for
the sentences of the cactus language !L! = !C!(!P!), in other words,
to define a mapping that "interprets" each sentence of !C!(!P!) as
a sentence that says something, as a sentence that bears a meaning,
in short, as a sentence that denotes a proposition, and thus as a
sign of an indicator function. When the syntactic sentences of a
formal language are given a referent significance in logical terms,
for example, as denoting propositions or indicator functions, then
each form of syntactic combination takes on a corresponding form
of logical significance.
By way of providing a logical interpretation for the cactus language,
I introduce a family of operators on indicator functions that are
called "propositional connectives", and I distinguish these from
the associated family of syntactic combinations that are called
"sentential connectives", where the relationship between these
two realms of connection is exactly that between objects and
their signs. A propositional connective, as an entity of a
well-defined functional and operational type, can be treated
in every way as a logical or a mathematical object, and thus
as the type of object that can be denoted by the corresponding
form of syntactic entity, namely, the sentential connective that
is appropriate to the case in question.
There are two basic types of connectives, called the "blank connectives"
and the "bound connectives", respectively, with one connective of each
type for each natural number k = 0, 1, 2, 3, ... .
1. The "blank connective" of k places is signified by the
concatenation of the k sentences that fill those places.
For the special case of k = 0, the "blank connective" is taken to
be an empty string or a blank symbol -- it does not matter which,
since both are assigned the same denotation among propositions.
For the generic case of k > 0, the "blank connective" takes
the form "S_1 · ... · S_k". In the type of data that is
called a "text", the raised dots "·" are usually omitted,
supplanted by whatever number of spaces and line breaks
serve to improve the readability of the resulting text.
2. The "bound connective" of k places is signified by the
surcatenation of the k sentences that fill those places.
For the special case of k = 0, the "bound connective" is taken to
be an expression of the form "-()-", "-( )-", "-( )-", and so on,
with any number of blank symbols between the parentheses, all of
which are assigned the same logical denotation among propositions.
For the generic case of k > 0, the "bound connective" takes the
form "-(S_1, ..., S_k)-".
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.12 The Cactus Language: Semantics (cont.)
At this point, there are actually two different "dialects", "scripts",
or "modes" of presentation for the cactus language that need to be
interpreted, in other words, that need to have a semantic function
defined on their domains.
a. There is the literal formal language of strings in PARCE(!P!),
the "painted and rooted cactus expressions" that constitute
the langauge !L! = !C!(!P!) c !A!* = (!M! |_| !P!)*.
b. There is the figurative formal language of graphs in PARC(!P!),
the "painted and rooted cacti" themselves, a parametric family
of graphs or a species of computational data structures that
is graphically analogous to the language of literal strings.
Of course, these two modalities of formal language, like written and
spoken natural languages, are meant to have compatible interpretations,
and so it is usually sufficient to give just the meanings of either one.
All that remains is to provide a "codomain" or a "target space" for the
intended semantic function, in other words, to supply a suitable range
of logical meanings for the memberships of these languages to map into.
Out of the many interpretations that are formally possible to arrange,
one way of doing this proceeds by making the following definitions:
1. The "conjunction" Conj^J_j Q_j of a set of propositions, {Q_j : j in J},
is a proposition that is true if and only if each one of the Q_j is true.
Conj^J_j Q_j is true <=> Q_j is true for every j in J.
2. The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J},
is a proposition that is true if and only if just one of the Q_j is untrue.
Surj^J_j Q_j is true <=> Q_j is untrue for unique j in J.
If the number of propositions that are being joined together is finite,
then the conjunction and the surjunction can be represented by means of
sentential connectives, incorporating the sentences that represent these
propositions into finite strings of symbols.
If J is finite, for instance, if J constitutes the interval j = 1 to k,
and if each proposition Q_j is represented by a sentence S_j, then the
following strategies of expression are open:
1. The conjunction Conj^J_j Q_j can be represented by a sentence that
is constructed by concatenating the S_j in the following fashion:
Conj^J_j Q_j <-< S_1 S_2 ... S_k.
2. The surjunction Surj^J_j Q_j can be represented by a sentence that
is constructed by surcatenating the S_j in the following fashion:
Surj^J_j Q_j <-< -(S_1, S_2, ..., S_k)-.
If one opts for a mode of interpretation that moves more directly from
the parse graph of a sentence to the potential logical meaning of both
the PARC and the PARCE, then the following specifications are in order:
A cactus rooted at a particular node is taken to represent what that
node denotes, its logical denotation or its logical interpretation.
1. The logical denotation of a node is the logical conjunction of that node's
"arguments", which are defined as the logical denotations of that node's
attachments. The logical denotation of either a blank symbol or an empty
node is the boolean value %1% = "true". The logical denotation of the
paint p_j is the proposition P_j, a proposition that is regarded as
"primitive", at least, with respect to the level of analysis that
is represented in the current instance of !C!(!P!).
2. The logical denotation of a lobe is the logical surjunction of that lobe's
"arguments", which are defined as the logical denotations of that lobe's
accoutrements. As a corollary, the logical denotation of the parse graph
of "-()-", otherwise called a "needle", is the boolean value %0% = "false".
If one takes the point of view that PARC's and PARCE's amount to a
pair of intertranslatable languages for the same domain of objects,
then the "spiny bracket" notation, as in "-[C_j]-" or "-[S_j]-",
can be used on either domain of signs to indicate the logical
denotation of a cactus C_j or the logical denotation of
a sentence S_j, respectively.
Tables 13.1 and 13.2 summarize the relations that serve to connect the
formal language of sentences with the logical language of propositions.
Between these two realms of expression there is a family of graphical
data structures that arise in parsing the sentences and that serve to
facilitate the performance of computations on the indicator functions.
The graphical language supplies an intermediate form of representation
between the formal sentences and the indicator functions, and the form
of mediation that it provides is very useful in rendering the possible
connections between the other two languages conceivable in fact, not to
mention in carrying out the necessary translations on a practical basis.
These Tables include this intermediate domain in their Central Columns.
Between their First and Middle Columns they illustrate the mechanics of
parsing the abstract sentences of the cactus language into the graphical
data structures of the corresponding species. Between their Middle and
Final Columns they summarize the semantics of interpreting the graphical
forms of representation for the purposes of reasoning with propositions.
Table 13.1 Semantic Translations: Functional Form
o-------------------o-----o-------------------o-----o-------------------o
| | Par | | Den | |
| Sentence | --> | Graph | --> | Proposition |
o-------------------o-----o-------------------o-----o-------------------o
| | | | | |
| S_j | --> | C_j | --> | Q_j |
| | | | | |
o-------------------o-----o-------------------o-----o-------------------o
| | | | | |
| Conc^0 | --> | Node^0 | --> | %1% |
| | | | | |
| Conc^k_j S_j | --> | Node^k_j C_j | --> | Conj^k_j Q_j |
| | | | | |
o-------------------o-----o-------------------o-----o-------------------o
| | | | | |
| Surc^0 | --> | Lobe^0 | --> | %0% |
| | | | | |
| Surc^k_j S_j | --> | Lobe^k_j C_j | --> | Surj^k_j Q_j |
| | | | | |
o-------------------o-----o-------------------o-----o-------------------o
Table 13.2 Semantic Translations: Equational Form
o-------------------o-----o-------------------o-----o-------------------o
| | Par | | Den | |
| -[Sentence]- | = | -[Graph]- | = | Proposition |
o-------------------o-----o-------------------o-----o-------------------o
| | | | | |
| -[S_j]- | = | -[C_j]- | = | Q_j |
| | | | | |
o-------------------o-----o-------------------o-----o-------------------o
| | | | | |
| -[Conc^0]- | = | -[Node^0]- | = | %1% |
| | | | | |
| -[Conc^k_j S_j]- | = | -[Node^k_j C_j]- | = | Conj^k_j Q_j |
| | | | | |
o-------------------o-----o-------------------o-----o-------------------o
| | | | | |
| -[Surc^0]- | = | -[Lobe^0]- | = | %0% |
| | | | | |
| -[Surc^k_j S_j]- | = | -[Lobe^k_j C_j]- | = | Surj^k_j Q_j |
| | | | | |
o-------------------o-----o-------------------o-----o-------------------o
Aside from their common topic, the two Tables present slightly different
ways of conceptualizing the operations that go to establish their maps.
Table 13.1 records the functional associations that connect each domain
with the next, taking the triplings of a sentence S_j, a cactus C_j, and
a proposition Q_j as basic data, and fixing the rest by recursion on these.
Table 13.2 records these associations in the form of equations, treating
sentences and graphs as alternative kinds of signs, and generalizing the
spiny bracket operator to indicate the proposition that either denotes.
It should be clear at this point that either scheme of translation puts
the sentences, the graphs, and the propositions that it associates with
each other roughly in the roles of the signs, the interpretants, and the
objects, respectively, whose triples define an appropriate sign relation.
Indeed, the "roughly" can be made "exactly" as soon as the domains of
a suitable sign relation are specified precisely.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.12 The Cactus Language: Semantics (cont.)
A good way to illustrate the action of the conjunction and surjunction
operators is to demonstate how they can be used to construct all of the
boolean functions on k variables, just now, let us say, for k = 0, 1, 2.
A boolean function on 0 variables is just a boolean constant F^0 in the
boolean domain %B% = {%0%, %1%}. Table 14 shows several different ways
of referring to these elements, just for the sake of consistency using
the same format that will be used in subsequent Tables, no matter how
degenerate it tends to appears in the immediate case:
Column 1 lists each boolean element or boolean function under its
ordinary constant name or under a succinct nickname, respectively.
Column 2 lists each boolean function in a style of function name "F^i_j"
that is constructed as follows: The superscript "i" gives the dimension
of the functional domain, that is, the number of its functional variables,
and the subscript "j" is a binary string that recapitulates the functional
values, using the obvious translation of boolean values into binary values.
Column 3 lists the functional values for each boolean function, or possibly
a boolean element appearing in the guise of a function, for each combination
of its domain values.
Column 4 shows the usual expressions of these elements in the cactus language,
conforming to the practice of omitting the strike-throughs in display formats.
Here I illustrate also the useful convention of sending the expression "(())"
as a visible stand-in for the expression of a constantly "true" truth value,
one that would otherwise be represented by a blank expression, and tend to
elude our giving it much notice in the context of more demonstrative texts.
Table 14. Boolean Functions on Zero Variables
o----------o----------o-------------------------------------------o----------o
| Constant | Function | F() | Function |
o----------o----------o-------------------------------------------o----------o
| | | | |
| %0% | F^0_0 | %0% | () |
| | | | |
| %1% | F^0_1 | %1% | (()) |
| | | | |
o----------o----------o-------------------------------------------o----------o
Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%,
of which there are precisely four. Here, Column 1 codes the contents of
Column 2 in a more concise form, compressing the lists of boolean values,
recorded as bits in the subscript string, into their decimal equivalents.
Naturally, the boolean constants reprise themselves in this new setting
as constant functions on one variable. Thus, one has the synonymous
expressions for constant functions that are expressed in the next
two chains of equations:
| F^1_0 = F^1_00 = %0% : %B% -> %B%
|
| F^1_3 = F^1_11 = %1% : %B% -> %B%
As for the rest, the other two functions are easily recognized as corresponding
to the one-place logical connectives, or the monadic operators on %B%. Thus,
the function F^1_1 = F^1_01 is recognizable as the negation operation, and
the function F^1_2 = F^1_10 is obviously the identity operation.
Table 15. Boolean Functions on One Variable
o----------o----------o-------------------------------------------o----------o
| Function | Function | F(x) | Function |
o----------o----------o---------------------o---------------------o----------o
| | | F(%0%) | F(%1%) | |
o----------o----------o---------------------o---------------------o----------o
| | | | | |
| F^1_0 | F^1_00 | %0% | %0% | ( ) |
| | | | | |
| F^1_1 | F^1_01 | %0% | %1% | (x) |
| | | | | |
| F^1_2 | F^1_10 | %1% | %0% | x |
| | | | | |
| F^1_3 | F^1_11 | %1% | %1% | (( )) |
| | | | | |
o----------o----------o---------------------o---------------------o----------o
Table 16 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%,
of which there are precisely sixteen in number. As before, all of the boolean
functions of fewer variables are subsumed in this Table, though under a set of
alternative names and possibly different interpretations. Just to acknowledge
a few of the more notable pseudonyms:
The constant function %0% : %B%^2 -> %B% appears under the name of F^2_00.
The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15.
The negation and identity of the first variable are F^2_03 and F^2_12, resp.
The negation and identity of the other variable are F^2_05 and F^2_10, resp.
The logical conjunction is given by the function F^2_08 (x, y) = x · y.
The logical disjunction is given by the function F^2_14 (x, y) = ((x)(y)).
Functions expressing the "conditionals", "implications",
or "if-then" statements are given in the following ways:
[x => y] = F^2_11 (x, y) = (x (y)) = [not x without y].
[x <= y] = F^2_13 (x, y) = ((x) y) = [not y without x].
The function that corresponds to the "biconditional",
the "equivalence", or the "if and only" statement is
exhibited in the following fashion:
[x <=> y] = [x = y] = F^2_09 (x, y) = ((x , y)).
Finally, there is a boolean function that is logically associated with
the "exclusive disjunction", "inequivalence", or "not equals" statement,
algebraically associated with the "binary sum" or "bitsum" operation,
and geometrically associated with the "symmetric difference" of sets.
This function is given by:
[x =/= y] = [x + y] = F^2_06 (x, y) = (x , y).
Table 16. Boolean Functions on Two Variables
o----------o----------o-------------------------------------------o----------o
| Function | Function | F(x, y) | Function |
o----------o----------o----------o----------o----------o----------o----------o
| | | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% | |
o----------o----------o----------o----------o----------o----------o----------o
| | | | | | | |
| F^2_00 | F^2_0000 | %0% | %0% | %0% | %0% | () |
| | | | | | | |
| F^2_01 | F^2_0001 | %0% | %0% | %0% | %1% | (x)(y) |
| | | | | | | |
| F^2_02 | F^2_0010 | %0% | %0% | %1% | %0% | (x) y |
| | | | | | | |
| F^2_03 | F^2_0011 | %0% | %0% | %1% | %1% | (x) |
| | | | | | | |
| F^2_04 | F^2_0100 | %0% | %1% | %0% | %0% | x (y) |
| | | | | | | |
| F^2_05 | F^2_0101 | %0% | %1% | %0% | %1% | (y) |
| | | | | | | |
| F^2_06 | F^2_0110 | %0% | %1% | %1% | %0% | (x, y) |
| | | | | | | |
| F^2_07 | F^2_0111 | %0% | %1% | %1% | %1% | (x y) |
| | | | | | | |
| F^2_08 | F^2_1000 | %1% | %0% | %0% | %0% | x y |
| | | | | | | |
| F^2_09 | F^2_1001 | %1% | %0% | %0% | %1% | ((x, y)) |
| | | | | | | |
| F^2_10 | F^2_1010 | %1% | %0% | %1% | %0% | y |
| | | | | | | |
| F^2_11 | F^2_1011 | %1% | %0% | %1% | %1% | (x (y)) |
| | | | | | | |
| F^2_12 | F^2_1100 | %1% | %1% | %0% | %0% | x |
| | | | | | | |
| F^2_13 | F^2_1101 | %1% | %1% | %0% | %1% | ((x) y) |
| | | | | | | |
| F^2_14 | F^2_1110 | %1% | %1% | %1% | %0% | ((x)(y)) |
| | | | | | | |
| F^2_15 | F^2_1111 | %1% | %1% | %1% | %1% | (()) |
| | | | | | | |
o----------o----------o----------o----------o----------o----------o----------o
Let me now address one last question that may have occurred to some.
What has happened, in this suggested scheme of functional reasoning,
to the distinction that is quite pointedly made by careful logicians
between (1) the connectives called "conditionals" and symbolized by
the signs "->" and "<-", and (2) the assertions called "implications"
and symbolized by the signs "=>" and "<=", and, in a related question:
What has happened to the distinction that is equally insistently made
between (3) the connective called the "biconditional" and signified by
the sign "<->" and (4) the assertion that is called an "equivalence"
and signified by the sign "<=>"? My answer is this: For my part,
I am deliberately avoiding making these distinctions at the level
of syntax, preferring to treat them instead as distinctions in
the use of boolean functions, turning on whether the function
is mentioned directly and used to compute values on arguments,
or whether its inverse is being invoked to indicate the fibers
of truth or untruth under the propositional function in question.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
In this Subsection, I finally bring together many of what may
have appeared to be wholly independent threads of development,
in the hope of paying off a percentage of my promissory notes,
even if a goodly number my creditors have no doubt long since
forgotten, if not exactly forgiven the debentures in question.
For ease of reference, I repeat here a couple of the
definitions that are needed again in this discussion.
| A "boolean connection" of degree k, also known as a "boolean function"
| on k variables, is a map of the form F : %B%^k -> %B%. In other words,
| a boolean connection of degree k is a proposition about things in the
| universe of discourse X = %B%^k.
|
| An "imagination" of degree k on X is a k-tuple of propositions
| about things in the universe X. By way of displaying the kinds
| of notation that are used to express this idea, the imagination
| #f# = <f_1, ..., f_k> is can be given as a sequence of indicator
| functions f_j : X -> %B%, for j = 1 to k. All of these features
| of the typical imagination #f# can be summed up in either one of
| two ways: either in the form of a membership statement, stating
| words to the effect that #f# belongs to the space (X -> %B%)^k,
| or in the form of the type declaration that #f# : (X -> %B%)^k,
| though perhaps the latter specification is slightly more precise
| than the former.
The definition of the "stretch" operation and the uses of the
various brands of denotational operators can be reviewed here:
055. http://suo.ieee.org/email/msg07466.html
057. http://suo.ieee.org/email/msg07469.html
070. http://suo.ieee.org/ontology/msg03473.html
071. http://suo.ieee.org/ontology/msg03479.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
1.3.10.13 Stretching Exercises
Taking up the preceding arrays of particular connections, namely,
the boolean functions on two or less variables, it possible to
illustrate the use of the stretch operation in a variety of
concrete cases.
For example, suppose that F is a connection of the form F : %B%^2 -> %B%,
that is, any one of the sixteen possibilities in Table 16, while p and q
are propositions of the form p, q : X -> %B%, that is, propositions about
things in the universe X, or else the indicators of sets contained in X.
Then one has the imagination #f# = <f_1, f_2> = <p, q> : (X -> %B%)^2,
and the stretch of the connection F to #f# on X amounts to a proposition
F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as
the "stretch of F to p and q". If one is concerned with many different
propositions about things in X, or if one is abstractly indifferent to
the particular choices for p and q, then one can detach the operator
F^$ : (X -> %B%)^2 -> (X -> %B%), called the "stretch of F over X",
and consider it in isolation from any concrete application.
When the "cactus notation" is used to represent boolean functions,
a single "$" sign at the end of the expression is enough to remind
a reader that the connections are meant to be stretched to several
propositions on a universe X.
For instance, take the connection F : %B%^2 -> %B% such that:
F(x, y) = F^2_06 (x, y) = -(x, y)-.
This connection is the boolean function on a couple of variables x, y
that yields a value of %1% if and only if just one of x, y is not %1%,
that is, if and only if just one of x, y is %1%. There is clearly an
isomorphism between this connection, viewed as an operation on the
boolean domain %B% = {%0%, %1%}, and the dyadic operation on binary
values x, y in !B! = GF(2) that is otherwise known as "x + y".
The same connection F : %B%^2 -> %B% can also be read as a proposition
about things in the universe X = %B%^2. If S is a sentence that denotes
the proposition F, then the corresponding assertion says exactly what one
otherwise states by uttering "x is not equal to y". In such a case, one
has -[S]- = F, and all of the following expressions are ordinarily taken
as equivalent descriptions of the same set:
[| -[S]- |] = [| F |]
= F^(-1)(%1%)
= {<x, y> in %B%^2 : S}
= {<x, y> in %B%^2 : F(x, y) = %1%}
= {<x, y> in %B%^2 : F(x, y)}
= {<x, y> in %B%^2 : -(x, y)- = %1%}
= {<x, y> in %B%^2 : -(x, y)- }
= {<x, y> in %B%^2 : x exclusive-or y}
= {<x, y> in %B%^2 : just one true of x, y}
= {<x, y> in %B%^2 : x not equal to y}
= {<x, y> in %B%^2 : x <=/=> y}
= {<x, y> in %B%^2 : x =/= y}
= {<x, y> in %B%^2 : x + y}.
Notice the slight distinction, that I continue to maintain at this point,
between the logical values {false, true} and the algebraic values {0, 1}.
This makes it legitimate to write a sentence directly into the right side
of the set-builder expression, for instance, weaving the sentence S or the
sentence "x is not equal to y" into the context "{<x, y> in %B%^2 : ... }",
thereby obtaining the corresponding expressions listed above, while the
proposition F(x, y) can also be asserted more directly without equating
it to %1%, since it already has a value in {false, true}, and thus can
be taken as tantamount to an actual sentence.
If the appropriate safeguards can be kept in mind, avoiding all danger of
confusing propositions with sentences and sentences with assertions, then
the marks of these distinctions need not be forced to clutter the account
of the more substantive indications, that is, the ones that really matter.
If this level of understanding can be achieved, then it may be possible
to relax these restrictions, along with the absolute dichotomy between
algebraic and logical values, which tends to inhibit the flexibility
of interpretation.
This covers the properties of the connection F(x, y) = -(x, y)-,
treated as a proposition about things in the universe X = %B%^2.
Staying with this same connection, it is time to demonstrate how
it can be "stretched" into an operator on arbitrary propositions.
To continue the exercise, let p and q be arbitrary propositions about
things in the universe X, that is, maps of the form p, q : X -> %B%,
and suppose that p, q are indicator functions of the sets P, Q c X,
respectively. In other words, one has the following set of data:
| p = -{P}- : X -> %B%
|
| q = -{Q}- : X -> %B%
|
| <p, q> = < -{P}- , -{Q}- > : (X -> %B%)^2
Then one has an operator F^$, the stretch of the connection F over X,
and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with
the following properties:
| F^$ = -( , )-^$ : (X -> %B%)^2 -> (X -> %B%)
|
| F^$ (p, q) = -(p, q)-^$ : X -> %B%
As a result, the application of the proposition F^$ (p, q) to each x in X
yields a logical value in %B%, all in accord with the following equations:
| F^$ (p, q)(x) = -(p, q)-^$ (x) in %B%
|
| ^ ^
| | |
| = =
| | |
| v v
|
| F(p(x), q(x)) = -(p(x), q(x))- in %B%
For each choice of propositions p and q about things in X, the stretch of
F to p and q on X is just another proposition about things in X, a simple
proposition in its own right, no matter how complex its current expression
or its present construction as F^$ (p, q) = -(p, q)^$ makes it appear in
relation to p and q. Like any other proposition about things in X, it
indicates a subset of X, namely, the fiber that is variously described
in the following ways:
[| F^$ (p, q) |] = [| -(p, q)-^$ |]
= (F^$ (p, q))^(-1)(%1%)
= {x in X : F^$ (p, q)(x)}
= {x in X : -(p, q)-^$ (x)}
= {x in X : -(p(x), q(x))- }
= {x in X : p(x) ± q(x)}
= {x in X : p(x) =/= q(x)}
= {x in X : -{P}- (x) =/= -{Q}- (x)}
= {x in X : x in P <=/=> x in Q}
= {x in X : x in P-Q or x in Q-P}
= {x in X : x in P-Q |_| Q-P}
= {x in X : x in P ± Q}
= P ± Q c X
= [|p|] ± [|q|] c X.
Which was to be shown.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
