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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:

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:

: <math>X \Rightarrow Y\ \operatorname{and}\ Y \Rightarrow Z.</math>

| <math>X \Rightarrow Y\ \operatorname{and}\ Y \Rightarrow Z.</math>

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:

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:

: <math>X\ \le\ Y\ \le\ Z.</math>

| <math>X\ \le\ Y\ \le\ Z.</math>

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:

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:

: <math>Z \Leftarrow Y\ \operatorname{and}\ Y \Leftarrow X.</math>

| <math>Z \Leftarrow Y\ \operatorname{and}\ Y \Leftarrow X.</math>

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.

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.

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 "&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.

The application of these considerations in the immediate setting is this:  Do not worry too much about what roles the empty string <math>\varepsilon \, = \, ^{\backprime\backprime\prime\prime}</math> and the blank symbol <math>m_1 \, = \, ^{\backprime\backprime} \operatorname{~} ^{\prime\prime}</math> are supposed to play in a given species of formal languages.  As it happens, it is far less important to wonder whether these types of formal tokens actually constitute genuine sentences than it is to decide what equivalence classes it makes sense to form over all of the sentences in the resulting language, and only then to bother about what equivalence classes these limiting cases of sentences are most conveniently taken to represent.

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.

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.

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.

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

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

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:

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

from the "multiplicative" or the "serial" styles.  The issue is somewhat

confused by the fact that an "additive" analysis is typically expressed

in the form of a "series", in other words, a disjoint union of sets or a

linear sum of their independent effects.  But it is easy enough to sort

this out if one observes the more telling connection between "parallel"

and "independent".  Another way to keep the right associations straight

is to employ the term "sequential" in preference to the more misleading

term "serial".  Whatever one calls this broad division of styles, the

scope and sweep of their dimensions of variation can be delineated in

the following way:

1.  The "additive" or "parallel" styles favor "sums of products" as

# The ''additive'' or ''parallel'' styles favor ''sums of products'' <math>(\textstyle\sum\prod)</math> as canonical forms of expression, pulling sums, unions, co-products, and logical disjunctions to the outermost layers of analysis and synthesis, while pushing products, intersections, concatenations, and logical conjunctions to the innermost levels of articulation and generation.  In propositional logic, this style leads to the ''disjunctive normal form'' (DNF).

    canonical forms of expression, pulling sums, unions, co-products,

# The ''multiplicative'' or ''serial'' styles favor ''products of sums'' <math>(\textstyle\prod\sum)</math> as canonical forms of expression, pulling products, intersections, concatenations, and logical conjunctions to the outermost layers of analysis and synthesis, while pushing sums, unions, co-products, and logical disjunctions to the innermost levels of articulation and generation.  In propositional logic, this style leads to the ''conjunctive normal form'' (CNF).

    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,

There is a curious sort of diagnostic clue, a veritable shibboleth,

that often serves to reveal the dominance of one mode or the other

that often serves to reveal the dominance of one mode or the other