It is useful to examine the relation between syntactic production 
The production 
One way to understand a production of the form
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There is a curious sort of diagnostic clue which 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 the contrasting traits, namely, the additive attitude versus the multiplicative tendency which go to make up the style in question, and even to determine the dominant inclination or point of view which establishes its perspective on the target domain.
In each style of formal grammar, the multiplicative aspect is present in the sequential concatenation of signs, in both the augmented strings and 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 to spot. It is signaled by the parallel covering of many augmented strings or sentential forms by the same non‑terminal symbol. In active terms, it calls for the independent rewriting of that non‑terminal symbol by a number of different successors, as in the following scheme.
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It is possible to trace the divergence of formal grammar styles to an even more primitive division, distinguishing between the additive or parallel styles and the multiplicative or 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 things 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 the broad division of styles, the scope and sweep of their dimensions of variation can be delineated in the following way.
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. The analogous style in propositional logic leads to the disjunctive normal form (DNF).
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. The analogous style in propositional logic leads to the conjunctive normal form (CNF).cc: Academia.edu • BlueSky • Laws of Form • Mathstodon • Research Gate
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Along with the distinctions we see evolving among different styles of grammar and the preferences different observers display toward them, there naturally arises the question: What is the root of that evolution?
One dimension of variation in formal grammar style can be seen by treating a union of languages, and especially a disjoint union of languages, as a sum 
If one examines the relationship between grammars and languages closely enough to detect the influence of the above two styles and comes to appreciate how different grammar styles may be used with different degrees of success for different purposes then one begins to see the possibility that alternative styles of description might be based on altogether different linguistic and logical operations.
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The questions about boundary conditions we keep encountering betray a more general issue. Already by this point in the discussion the limits of a purely syntactic approach to language are becoming visible. It is not that one cannot go a long way by that road in the analysis of a particular language and the study of languages in general but when it comes to understanding the purpose of a language, extending its use in a chosen direction, or designing a language for a particular set of uses, what matters above all are the pragmatic equivalence classes of signs demanded by the application and intended by the designer and not so much the peculiar characters of signs representing the classes of practical meaning.
Any description of a language is bound to have alternative descriptions. In particular, a formally circumscribed description of a formal language, as any effectively finite description is bound to be, is certain to suggest the equally likely existence and possible utility of other descriptions. A single formal grammar describes but a single formal language, but any formal language is described by many formal grammars, not all of which afford the same grasp of its structure, provide equivalent comprehensions of its character, or yield interchangeable views of its aspects. Even with respect to the same formal language, different formal grammars are typically better for different purposes.
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Having broached the distinction between objective propositions and syntactic sentences, its analogy to the distinction between numbers and numerals becomes clear. What are the implications of that distinction for the realm of reasoning about propositions and its representation in sentential logic?
If the purpose of a sentence is precisely to denote a proposition then the proposition is simply the object of whatever sign is taken for the sentence. The computational manifestation of a piece of reasoning about propositions thus amounts to a process taking place entirely within a language of sentences, being a procedure which can rationalize its account by referring to the denominations of sentences among propositions.
As far as it bears on our current context of problems, the upshot is this: Do not worry too much about what roles the empty string 
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The pragmatic theory of sign relations is called for in settings where everything that can be named has any number of other names, that is to say, the usual case. Of course we’d like to replace the multiplicity of signs with an organized system of canonical signs, one for each object that needs to be named, but reducing the redundancy too far, beyond what is necessary to eliminate the factor of “noise” in the language and thus clear up its effectively useless distractions, can destroy the utility of natural languages and bespoke formal systems, which are evolved to provide a ready means for expressing present situations, clear or not, and to describe ongoing conditions of experience in just the way they present themselves. Within a fully 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’s a well‑known fact but an often forgotten truth that no one computes with numbers, but solely with numerals in respect of numbers, and numerals themselves are symbols. Among other things, that renders all discussion of numeric versus symbolic computation a bit beside the point, since it’s only a question of what types of symbols are required for one’s immediate application or for one’s selection of ongoing objectives. The numerals everyone knows best are just the canonical symbols, the standard signs or the normal forms for numbers, and the process of computation is a matter of getting from the obscure signs a situation impresses on us in the form of data to the indications of the situation’s character which can be rendered clear enough to motivate action.
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Expanding our perspective on the options for formal grammar style brings us to questions 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 that, if not for any better reason than to aid in clarifying the kinds of languages people are accustomed to use, to assist in writing computer programs capable of parsing real sentences, and to serve in designing programming languages people would like to become accustomed to use.
As it happens, the only time formal language theory becomes too picky, or a bit too myopic in its focus, is when it leads one to think one is dealing with the thing itself and not just the sign of it, in other words, when the people who use the tools of formal language theory forget they are dealing with the mere signs of more interesting objects and not the objects of ultimate interest in and of themselves.
There are then a number of deleterious effects at risk of arising from the extreme pickiness of formal language theory, arising, as often the case, when theorists forget the practical context of theorization. The exacting task of defining the membership of a formal language leads one to think that object and that object alone is the justifiable end of the whole exercise. The distractions of the mediate objective render one liable to forget one’s penultimate interest lies always with various equivalence classes of signs, not entirely or exclusively with their more meticulous representatives.
When that happens, one typically goes on working oblivious to the circumstance 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 fact 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 to the final constitution of those equivalence classes of signs which answer for the genuine objects of the whole enterprise of language.
Whenever a form of theory, against its initial and its best intentions, succumbs to an absence of mind no longer benign in its main effects, a counterbalancing form of theory is needed to restore the presence of mind all forms of theory are meant to support.
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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 the changes are minor, in the sense they are not intended to alter the classes of languages able to be generated but only to clear up the ambiguities and obscurities affecting their conception.
Perhaps most importantly, the conventional scope of non‑terminal symbols is expanded to include the sentence symbol, mainly on account of all the contexts where initial and 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 is introduced in the fashion of a distinction between the initial and the intermediate symbols, and that serves its purpose in all the contexts where the two kinds of symbols need to be treated separately.
At present I remain a bit worried about the motivations and the justifications for introducing that distinction 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 question has to do with how it all ends. The excuses of efficiency and expediency I offered as 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 cover themselves, not to mention being allowed to cover the empty string, and so the very types of traps one exerts oneself to avoid at the outset are always there to afflict the process at all the intervening times.
If one reflects on the form of grammar 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 avoidable in the part is also avoidable in the main, while any trouble inescapable in the part is also inescapable 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, cover, produce, or regenerate itself, if that turns out to be the best way in the end.
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Before the capacity of a language to describe itself can be evaluated, the missing link to meaning must be supplied for each of its expressions. That means opening a dimension of semantics to be navigated by means of interpretation, topics to be taken up in the case of 
The pressing 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 strings of signs which 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 they say what strings belong to it and what strings do not.
A formal grammar, then, a formalism appearing even more skeletal than a 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 to the simple question of belonging. Whether that single bit of meaningful value is enough to encompass all the dimensions of meaning we require, and whether it can be compounded to cover the complexity which actually exists in the realm of meaning — those are questions for an extended future inquiry.
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