Changes

It is useful to examine the relationship between the grammatical covering or production relation <math>(:>\!)</math> and the logical relation of implication <math>(\Rightarrow),</math> with one eye to what they have in common and one eye to how they differ.  The production <math>q :> W\!</math> says that the appearance of the symbol <math>q\!</math> in a sentential form implies the possibility of exchanging it for <math>W.\!</math>  Although this sounds like a ''possible implication'', to the extent that ''<math>q\!</math> implies a possible <math>W\!</math>'' or that ''<math>q\!</math> possibly implies <math>W,\!</math>'' the qualifiers ''possible'' and ''possibly'' are the critical elements in these statements, and they are crucial to the meaning of what is actually being implied.  In effect, these qualifications reverse the direction of implication, yielding <math>^{\backprime\backprime} \, q \Leftarrow W \, ^{\prime\prime}</math> as the best analogue for the sense of the production.

It is useful to examine the relationship between the grammatical covering or production relation <math>(:>\!)</math> and the logical relation of implication <math>(\Rightarrow),</math> with one eye to what they have in common and one eye to how they differ.  The production <math>q :> W\!</math> says that the appearance of the symbol <math>q\!</math> in a sentential form implies the possibility of exchanging it for <math>W.\!</math>  Although this sounds like a ''possible implication'', to the extent that ''<math>q\!</math> implies a possible <math>W\!</math>'' or that ''<math>q\!</math> possibly implies <math>W,\!</math>'' the qualifiers ''possible'' and ''possibly'' are the critical elements in these statements, and they are crucial to the meaning of what is actually being implied.  In effect, these qualifications reverse the direction of implication, yielding <math>^{\backprime\backprime} \, q \Leftarrow W \, ^{\prime\prime}</math> as the best analogue for the sense of the production.

One way to sum this up is to say that non-terminal symbols have the significance of hypotheses.  The terminal strings form the empirical matter of a language, while the non-terminal symbols mark the patterns or the types of substrings that can be noticed in the profusion of data. If one observes a portion of a terminal string that falls into the pattern of the sentential form <math>W,\!</math> then it is an admissible hypothesis, according to the theory of the language that is constituted by the formal grammar, that this piece not only fits the type <math>q\!</math> but even comes to be generated under the auspices of the non-terminal symbol <math>^{\backprime\backprime} q ^{\prime\prime}.</math>

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".

A moment's reflection on the issue of style, giving due consideration to the

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.

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

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.

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,

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.

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

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'',

the "co-product" of sets and spaces, a construction that artificially

''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.

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

One of the insights that arises out of Peirce's logical work is that the set operations of complementation, intersection, and union, along with the logical operations of negation, conjunction, and disjunction that operate in isomorphic tandem with them, are not as fundamental as they first appear.  This is because all of them can be constructed from or derived from a smaller set of operations, in fact, taking the logical side of things, from either one of two ''sole sufficient'' operators, called ''amphecks'' by Peirce, ''strokes'' by those who re-discovered them later, and known in computer science as the NAND and the NNOR operators. For this reason, that is, by virtue of their precedence in the orders of construction and derivation, these operations have to be regarded as the simplest and the most primitive in principle, even if they are scarcely recognized as lying among the more familiar elements of logic.

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.

I am throwing together a wide variety of different operations into each

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.

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

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.

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

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

creates an object tantamount to a disjoint union of sets in the resulting