3) SOME SEMIOTIC CONSIDERATIONS

Before we go on with the topological nature of human languages, it will be worthwhile to briefly consider the semiotic consequences of such an approach. Notice that points a)-b)-c) of chapter 2) claim epistemological adequacy  in topological linguistics because they closely describe the corresponding methods of descriptivism, generativism, and structuralism. The semiotic question has to do rather with ontological adequacy: does the above definition of topological space fit the semiotic properties of human language?
Whereas chapter 2) deals with method, the present chapter is concerned with the supposed nature of the object of study. People agree on the fact that human languages are sets of signs, but what are the properties that oppose those signs to the components of any other semiotic systems?
When trying to answer this question Ch. Hockett and S. Altmann 1968, "A note on design features", ed. T. A. Sebeok, Animal Communication, Bloomington, Indiana University Press,   61-72- compared human language to many other systems of communication found among animals (bees, apes, etc) or performed by humans (music, kinesics, etc). The conclusion they draw is that human language, unlike any other semiotic system, can be characterized by:
1) The metalinguistic capacity  of its signs; by the fact that linguistic signs describe themselves.
2) The plural articulation  of its signs; by the fact that each sign can be split at level i into units that belong to another level i+1.
3) The prevaricative capacity  of linguistic behavior; by the fact that language can be used without corresponding to its referent (in other terms: we can lie with words).
I have said above that descriptivism, generativism, and structuralism represent three alternative methods, each especially sensitive to one or another of the points a)-b)-c) which define the topological linguistic space. Regarding the nature of linguistic signs they have also emphasized one of the points 1)-2)-3)  over the others, but in this case no linguistic school has failed to point out all three of them as this would be an ontological inadequacy (to put it another way: controversies are allowed on method, but never on the object of description, and thus we find that many sciences share methodological procedures, but always differ according to the type of reality they are aiming to describe).
In descriptive schools of linguistics, points 1) and 3) have often been neglected whereas close attention has been paid to point 2). Hence, language is commonly described by means of several levels of description, each of them combining the units of the previous level according to specific sets of principles, and in order to yield a new type of units (the number of levels may vary from the pure "phonological vs. grammatical" opposition in Bloomfield's version, to the six or more that stratificational grammarians are used to recognizing). Nevertheless, 1) and 3) are also implicitly present. Point 3) derives from the descriptivist rejection of meaning, which in fact aimed to prevent metalinguistic judgements from being able to rely on the referent. Similarly, point 1) derives from Bloomfield's claim that a corpus of signs could be described without any external assistance. 
Generative grammar has typically been neither a level theory -point 2)- nor a hypothesis which is especially concerned with metalinguistic facts -point 1)-, but rather a grammar where the problem of reality and of escaping its laws is the main concern -point 3). This could seem rather surprising when one remembers that the 1957 model began by establishing a strong opposition between two strata of description, deep structure and surface structure. However, it is worth observing that these levels are not factual levels in the same sense as I.C. Analysis are: surface structure does not include a type of reality (say, phonic events) as opposed to another type of reality that deep structure would be in charge of (say, semantic events). In fact, both are grammatical structures, and twenty years of arguments among generative grammarians have only aimed to debate the gap which should exist between reality and the not immediately apparent grammatical structures, i.e. the deep structures, be this gap none (as in generative semantics), or the maximum thinkable (when later models practically merge deep and surface structures in a single level during the eighties). The lack of correspondence between grammatical deep structures and the referent, or among surface structures and the phonetic realizations create a couple of prevarication problems -point 3)-, even if each of them implies a separate analysis of semantic and phonemic events -i.e an implicit level consideration as is done in point 2)-, and also if the metalinguistic character of language -point 1)- implicitly comes up each time procedural disputations arise (compare generative semantics analysis of "kill=cause to die" with the feature proposal of, for example, the standard theory). The prevaricative nature of the problem that generative grammar is concerned with has recently been made apparent by the queries on parameters, i.e. by the relationship categories that particular languages hold to the universal tendencies they always reflect, even though they may contradict each other.

It is known that structuralism has claimed from its beginnings, an autonomous point of view in linguistics. This attitude, sustained by F. de Saussure, L. Hjelmslev, or R. Jakobson among many others, contrasts with generativist commitments to psychology, or to the behavioral assumptions held by early descriptivism, and demonstrates that structuralism conceives human language fundamentally as a self describing object -point 1)-, which is to say, as an autonomous one. Points 2) and 3) have customarily been discussed among structuralists and no agreement has been reached: thus glossematics considers that the level of phonetic facts (expression) and the level of semantic facts (content) are parallely built and that neither can be said to compose the other, whereas Praguean structuralism and particularly A. Martinet speaks of a double articulation in the sense of point 2). On the other hand, the prevarication problem has been taken on by many scholars involved in structuralist literary criticism -R. Barthes and A. J. Greimas among others-, but seldom appears in the original linguistic versions of the structuralist schools as such.

At the beginning of my own research on topological linguistics these three characteristics of human language -reflexivity (metalinguistic capacity), plural articulation, and prevarication- attracted my attention much more strongly than formal or psychological aspects of the problem. The kind of concerns that finally led to the present proposals (call it liminar grammar or topological linguistics) were originally of a semiotic nature.
Reflexivity was manifested in liminar grammar as a paradoxical statement called paradox of the boundary: whereas grammatical metalanguage is part of a natural language -i.e L contains M-, the former can only describe the latter if it is part of its alphabet -i.e. L belongs to M-, both L and M having to be kept differenciated as descriptum and descriptor. In other words: L … M, L Œ  M, L ð M.
 Plural articulation was manifested as the asymmetric sign. This concept refers to the fact that sound and meaning do not behave alike from an epistemological point of view: the first can be recognized and even split down into minimal units without bearing the second in mind, but meaning does not exist before it is given a sound-like support. In other words: people recognize foreign sounds although they have never heard the language they are listening to, and it is even possible to train some people to establish the systemic sound patterns of such an unknown language -call them "linguists". However, semantically speaking things happen to be much more difficult: no single meaning is accesible to the human brain if it is not supported by a corresponding sound as theoreticians of linguistic relativism hypotheses know well (meanings are then couples of sound and meaning). 
The asymmetric sign sound//sound+meaning: ['teibl]//['teibl] + "table"  (where a foreigner, say a Spanish speaking one, would recognize ['teibl] = [t]+[e]+[i]+[b]+[l], but he/she  would never equate ['teibl] +"table" -and not the abstract concept "table" alone-  to his/her own ['mesa] +"mesa" without help) implies a minimal plural articulation of linguistic signs, apparently a dual one. Only apparently nonetheless.
 In order to guarantee that the relationship held between sound and sound+meaning  be neither too strong nor too weak, it is necessary for both sides to allow some degree of flexibility. Notice that the meaning of words and sentences often varies polisemically and that sound patterns vary in their morphophonemic realizations, and none of them is able to support an infinite set of variations (for example, the meaning of odd  -i.e. [od] +"odd"- can be "strange" in odd question ,"left over" in to be the odd one out , "not even" in odd number , etc, but it can never mean "house" or "to smoke". On the other hand, [od] can take the suffixes -s, -ly,  and- ity,  but neither- ing,  nor -ed, etc).
This relative variability on both sides of the semiotic fraction that constitutes the sign proves that it consists of at least three main components: sound, sound+meaning, and the set of grammatical characteristics that relate both of them in what seems to be a fundamental tertiary articulation of linguistic elements: sound/grammatical features//sound+ +meaning. 
The role of the grammatical level is a central one because it is responsible for reflexivity and for prevarication at the same time. How can the metalinguistic process be explained? The linguistic environments that metalinguistically describe a linguistic item, or a series of them, operate as if they were ignorant of their own grammatical values: the environment #the...comes #, which defines the category "substantive" as potential filler of its slot (i,e the boy comes, the dog comes, etc) cannot apply its own grammatical characteristics -that is the ="determiner", comes ="verb") to the term described. Putting it in other words, we can say that metalinguistic descriptions are not equative relationships of the form A is  B  (A=B), where A and B share the same property. In this case it would be better to speak of contradiction than of tautology as B is what it is simply because it is not  A. Notice that the grammatical level remains hidden even when metalinguistic description is explicitly performed by a metalanguage, and not only in an implicit manner as above: we say that the word to see  is a verb without paying any attention to the fact that verb, on its own, is a "noun" from the grammatical point of view:                                                              [si:]/'verb'//[si:]+"see" is  [vœ:b]/'noun'//[vœ:b]+"verb". 
From now on we will conventionally represent the units of the sound level by [...], the units of the grammatical level by '...', and the units of the sound-meaning level simply by "..." (though they really are [...]+"..." as we have said).
The assumption that these levels are significant for prevarication lies in the fact that this three dimensional pattern is not a balanced one because 'grammatical' level looks much more like sound level than like meaning level. If sound units are those that contribute to meaning without having any meaning of their own, then it is clear that, for example, the features [labial] vs. [velar] which oppose [p] to [g] in pot /got  are sound-relevant factors but, for the same reason, we must also consider sound-relevant  the position of the items in the boy loves the girl/the girl loves the boy, that is to say, a theoretically 'grammatical' fact such as the opposition 'subject/object' .
The sound level will be called total sound because it has completely to do with phonetic events, whereas the grammatical level will be called partial sound for some phonetic events, but not all of them, are necessary for comprehension (compare the boy sat in the garden on a chair   with the boy sat on a chair in the garden ). Human languages can be represented, then, as sets of linguistic elements that operate in a fundamentally three dimensional space:

This space, where the three axes are intended to represent three fundamental ways of analyzing their elements (i.e metalinguistic points of view that can be adopted in relation to them), is not a balanced domain. In fact, sound clearly stands out as a Gestalt figure whereas meaning remains in the background. Therefore, all three axes are primarily interested in sound rather than in meaning: the total sound axis deals exclusively with sound, the partial sound axis does so preferentially, and the sound-meaning axis is a semantic dimension whose foundation is nonetheless constituted by sound.
Prevarication derives from this asymmetry. Simply because sound normally prevails over meaning (and hence F. de Saussure could say that language is a form, and not a substance) human languages escape reality as much as they are capable. In fact, only metalinguistic sentences suppose a violation of this natural tendency to leave meaning in the background. For this reason grammar tends to be a discourse whose signs are of the same kind as the referent and, consequently, whose aim is to deserve a universal nature (which linguistic signs could never claim). But in the normal use of language this escaping from its external referent constitutes a tendency whose extreme manifestation is precisely prevarication, that is, the total lack of correspondence with the referent.
I would like to point out that, although topological linguistics was born as a reflection on some fundamental semiotic features of human languages, the above mentioned properties result from the definition of topological space as well: 
-The paradox of the boundary is stated by definition because L contains M, L belongs to M, and L is not equal to M (the sum of metalinguistic elements being infinite).
-The plural articulation derives from the fact that the intersection of metalinguistic descriptions should be finite, which means that the entire set of metalinguistic signs cannot have any feature in common or that there are at least two different kinds of dimension (and actually three as we have seen).
-The prevaricative tendency of language use derives from a few more definitions that we have not mentioned yet. In fact, topological spaces can be defined in terms of open sets (as we have done above), or in terms of closed sets. When defining in terms of closed sets two complementary statements arise: the infinite intersection of linguistic elements (compare the finite intersection of metalinguistic ones above), and the finite sum of linguistic elements (compare the infinite sum of metalinguistic ones above) both lead to linguistic elements as well. Notice that if a finite sum of linguistic elements is just a text  (that supposedly creates its own reality), an infinite intersection of them would mean that, no matter how reality interferes with language, in the end language tends to escape from it, and only the "linguistic" features (syntagmatic and paradigmatic patterns) prevail over the metalinguistic features (universal categories) which resemble the referent.