Changes

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.

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 <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}).</math> It is the intention of this work to interpret this language for propositional logic, and thus to use it as a sentential calculus, an order of reasoning that forms an active ingredient and a significant component of all logical reasoning. To describe this language, the standard devices of formal grammars and formal language theory are more than adequate, but this only raises the next question:  What sorts of devices are exactly adequate, and fit the task to a "T"?  The ultimate desire is to turn the tables on the order of description, and so begins a process of eversion that evolves to the point of asking:  To what extent can the language capture the essential features and laws of its own grammar and describe the active principles of its own generation?  In other words:  How well can the language be described by using the language itself to do so?

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

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.

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,

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 <math>\mathfrak{C} (\mathfrak{P})</math> in Subsection 1.3.10.12.  Once a plausible semantics is prescribed for this language it will be possible to return to these questions and to address them in a meaningful way.

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

The prominent issue at this point is the distinct placements of formal languages and formal grammars with respect to the question of meaning. The sentences of a formal language are merely the abstract strings of abstract signs that happen to belong to a certain set.  They do not by themselves make any meaningful statements at all, not without mounting a separate effort of interpretation, but the rules of a formal grammar make meaningful statements about a formal language, to the extent that they say what strings belong to it and what strings do not.  Thus, the formal grammar, a formalism that appears to be even more skeletal than the formal language, still has bits and pieces of meaning attached to it. In a sense, the question of meaning is factored into two parts, structure and value, leaving the aspect of value reduced in complexity and subtlety to the simple question of belonging.  Whether this single bit of meaningful value is enough to encompass all of the dimensions of meaning that we require, and whether it can be compounded to cover the complexity that actually exists in the realm of meaning &mdash; these are questions for an extended future inquiry.

languages and formal grammars with respect to the question of meaning.

The sentences of a formal language are merely the abstract strings of

abstract signs that happen to belong to a certain set.  They do not by

themselves make any meaningful statements at all, not without mounting

a separate effort of interpretation, but the rules of a formal grammar

make meaningful statements about a formal language, to the extent that

they say what strings belong to it and what strings do not.  Thus, the

formal grammar, a formalism that appears to be even more skeletal than

the formal language, still has bits and pieces of meaning attached to it.

In a sense, the question of meaning is factored into two parts, structure

and value, leaving the aspect of value reduced in complexity and subtlety

to the simple question of belonging.  Whether this single bit of meaningful

value is enough to encompass all of the dimensions of meaning that we require,

and whether it can be compounded to cover the complexity that actually exists

in the realm of meaning -- these are questions for an extended future inquiry.

Perhaps I ought to comment on the differences between the present and

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.

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

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.

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

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.

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,

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.

it looks as if one sought, rather futilely, to avoid the problems of

recursion by proscribing the main program from calling itself, while

allowing any subprogram to do so.  But any trouble that is avoidable

in the part is also avoidable in the main, while any trouble that is

inevitable in the part is also inevitable in the main.  Consequently,

I am reserving the right to change my mind at a later stage, perhaps

to permit the initial symbol to characterize, to cover, to regenerate,

or to produce itself, if that turns out to be the best way in the end.

Before I leave this Subsection, I need to say a few things about

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.

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,

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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:

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 "&nbsp;" 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.

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.

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.

Already by this point in discussion the limits of the purely syntactic

approach to a language are beginning to be visible.  It is not that one

cannot go a whole lot further by this road in the analysis of a particular

language and in the study of languages in general, but when it comes to the

questions of understanding the purpose of a language, of extending its usage

in a chosen direction, or of designing a language for a particular set of uses,

what matters above all else are the "pragmatic equivalence classes" of signs that

are demanded by the application and intended by the designer, and not so much the

peculiar characters of the signs that represent these classes of practical meaning.

Any description of a language is bound to have alternative descriptions.

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.

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,

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?

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

One dimension of variation in the styles of formal grammars can be seen by treating the union of languages, and especially the disjoint union of languages, as a ''sum'', by treating the concatenation of languages as a ''product'', and then by distinguishing the styles of analysis that favor ''sums of products'' from those that favor ''products of sums'' as their canonical forms of description.  If one examines the relation between languages and grammars carefully enough to see the presence and the influence of these different styles, and when one comes to appreciate the ways that different styles of grammars can be used with different degrees of success for different purposes, then one begins to see the possibility that alternative styles of description can be based on altogether different linguistic and logical operations.

by 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

It possible to trace this divergence of styles to an even more primitive

division, one that distinguishes the "additive" or the "parallel" styles

division, one that distinguishes the "additive" or the "parallel" styles