Changes

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.

In the usual way of proceeding on formal grounds, meaning is added by giving each grammatical sentence, or each syntactically distinguished string, an interpretation as a logically meaningful sentence, in effect, equipping or providing each abstractly well-formed sentence with a logical proposition for it to denote.  A semantic interpretation of the cactus language is carried out in Subsection 1.3.10.12.

==The Cactus Language : Syntax==

===The Cactus Language : Syntax===

{| align="center" cellpadding="0" cellspacing="0" width="90%"

{| align="center" cellpadding="0" cellspacing="0" width="90%"

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

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

===Grammar 1===

====Grammar 1====

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

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

# 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.

# A grammar rule that invokes a notion of decatenation, deletion, erasure, or any other sort of retrograde production, is frequently considered to be lacking in elegance, and a there is a style of critique for grammars that holds it preferable to avoid these types of operations if it is at all possible to do so.  Accordingly, contingent on the prescriptions of the informal rule in question, and pursuing the stylistic dictates that are writ in the realm of its aesthetic regime, it becomes necessary for us to backtrack a little bit, to temporarily withdraw the suggestion of employing these elliptical types of operations, but without, of course, eliding the record of doing so.

===Grammar 2===

====Grammar 2====

One way to analyze the surcatenation of any number of sentences is to introduce an auxiliary type of string, not in general a sentence, but a proper component of any sentence that is formed by surcatenation.  Doing this brings one to the following definition:

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:

Grammar&nbsp;2 achieves a portion of its success through a higher degree of intermediate organization.  Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math> but it is clearly not explained by this simple circumstance alone, since it is taken for granted that the intermediate symbols serve a purpose, a purpose that is easily recognizable but that may not be so easy to pin down and to specify exactly.  Nevertheless, it is worth the trouble of exploring this aspect of organization and this direction of development a little further.

Grammar&nbsp;2 achieves a portion of its success through a higher degree of intermediate organization.  Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math> but it is clearly not explained by this simple circumstance alone, since it is taken for granted that the intermediate symbols serve a purpose, a purpose that is easily recognizable but that may not be so easy to pin down and to specify exactly.  Nevertheless, it is worth the trouble of exploring this aspect of organization and this direction of development a little further.

===Grammar 3===

====Grammar 3====

Although it is not strictly necessary to do so, it is possible to organize the materials of our developing grammar in a slightly better fashion by recognizing two recurrent types of strings that appear in the typical cactus expression. In doing this, one arrives at the following two definitions:

Although it is not strictly necessary to do so, it is possible to organize the materials of our developing grammar in a slightly better fashion by recognizing two recurrent types of strings that appear in the typical cactus expression. In doing this, one arrives at the following two definitions:

With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})</math> that is afforded by Grammar&nbsp;2, and, taking that as a point of departure, explore other avenues of possible improvement in the comprehension of these expressions.  In order to observe the effects of this alteration as clearly as possible, in isolation from any other potential factors, it is useful to strip away the higher levels intermediate organization that are present in Grammar&nbsp;3, and start again with a single intermediate symbol, as used in Grammar&nbsp;2.  One way of carrying out this strategy leads on to a grammar of the variety that will be articulated next.

With the distinction between empty and significant expressions in mind, I return to the grasp of the cactus language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) = \operatorname{PARCE} (\mathfrak{P})</math> that is afforded by Grammar&nbsp;2, and, taking that as a point of departure, explore other avenues of possible improvement in the comprehension of these expressions.  In order to observe the effects of this alteration as clearly as possible, in isolation from any other potential factors, it is useful to strip away the higher levels intermediate organization that are present in Grammar&nbsp;3, and start again with a single intermediate symbol, as used in Grammar&nbsp;2.  One way of carrying out this strategy leads on to a grammar of the variety that will be articulated next.

===Grammar 4===

====Grammar 4====

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

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

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.

Achieving a grammar that respects this convention typically requires a more detailed account of the initial setting of a type, both with regard to the type of context that incites its appearance and also with respect to the minimal strings that arise under the type in question.  In order to find covering productions that satisfy the intermediate significance condition, one must be prepared to consider a wider variety of calling contexts or inciting situations that can be noted to surround each recognized type, and also to enumerate a larger number of the smallest cases that can be observed to fall under each significant type.

===Grammar 5===

====Grammar 5====

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

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

Finally, it is worth trying to bring together the advantages of these diverse styles of grammar, to whatever extent that they are compatible.  To do this, a prospective grammar must be capable of maintaining a high level of intermediate organization, like that arrived at in Grammar&nbsp;2, while respecting the principle of intermediate significance, and thus accumulating all the benefits of the context-free format in Grammar&nbsp;5.  A plausible synthesis of most of these features is given in Grammar&nbsp;6.

Finally, it is worth trying to bring together the advantages of these diverse styles of grammar, to whatever extent that they are compatible.  To do this, a prospective grammar must be capable of maintaining a high level of intermediate organization, like that arrived at in Grammar&nbsp;2, while respecting the principle of intermediate significance, and thus accumulating all the benefits of the context-free format in Grammar&nbsp;5.  A plausible synthesis of most of these features is given in Grammar&nbsp;6.

===Grammar 6===

====Grammar 6====

Grammar&nbsp;6 has the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with the production set <math>\mathfrak{K}</math> as listed in the next display.

Grammar&nbsp;6 has the intermediate alphabet <math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} S' \, ^{\prime\prime}, \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \},</math> with the production set <math>\mathfrak{K}</math> as listed in the next display.

|}

|}

==Generalities About Formal Grammars==

===Generalities About Formal Grammars===

It is fitting to wrap up the foregoing developments by summarizing the notion of a formal grammar that appeared to evolve in the present case.  For the sake of future reference and the chance of a wider application, it is also useful to try to extract the scheme of a formalization that potentially holds for any formal language.  The following presentation of the notion of a formal grammar is adapted, with minor modifications, from the treatment in (DDQ, 60&ndash;61).

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&ndash;61).

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

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

==The Cactus Language : Stylistics==

===The Cactus Language : Stylistics===

{| align="center" cellpadding="0" cellspacing="0" width="90%"

{| align="center" cellpadding="0" cellspacing="0" width="90%"

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.

There is a measure of ambiguity that remains in this formulation, but it is the best that I can do in the present informal context.

==The Cactus Language : Mechanics==

===The Cactus Language : Mechanics===

{| align="center" cellpadding="0" cellspacing="0" width="90%"

{| align="center" cellpadding="0" cellspacing="0" width="90%"

</ol>

</ol>

==The Cactus Language : Semantics==

===The Cactus Language : Semantics===

{| align="center" cellpadding="0" cellspacing="0" width="90%"

{| align="center" cellpadding="0" cellspacing="0" width="90%"

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 <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question:  What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.

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 <math>(\rightarrow)</math> and <math>(\leftarrow),</math> and (2) the assertions called ''implications'' and symbolized by the signs <math>(\Rightarrow)</math> and <math>(\Leftarrow)</math>, and, in a related question:  What has happened to the distinction that is equally insistently made between (3) the connective called the ''biconditional'' and signified by the sign <math>(\leftrightarrow)</math> and (4) the assertion that is called an ''equivalence'' and signified by the sign <math>(\Leftrightarrow)</math>?  My answer is this:  For my part, I am deliberately avoiding making these distinctions at the level of syntax, preferring to treat them instead as distinctions in the use of boolean functions, turning on whether the function is mentioned directly and used to compute values on arguments, or whether its inverse is being invoked to indicate the fibers of truth or untruth under the propositional function in question.

==Stretching Exercises==

===Stretching Exercises===

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

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