6) FORMAL LAWS OF QUOTIENT TOPOLOGY AND THEIR INTERPRETATIONS WITH RESPECT TO THE LOWER LEVELS

One of the main advantages of hypothetical deductive methods is that the formal laws of mathematics permit the predicton of many new situations not previously collected, and they also permit such predictions in a variety of referential domains. 
Allow me to illustrate this with an example. Consider the simple mathematical relationship a•b=c. When physicists began to make use of it in the sixteenth century they first applied it to the description of situations of motion in the form "time•speed=space". The most interesting aspect of this law lies in the fact that it can account not only for previously given facts, but also for never registered ones, or even for those that would be impossible to register at all with current measuring devices, for example, the space that a rocket would cover in three million years. In other words, the above law -that I have expressed in a very simplified manner in order to simplify what I am trying to say- has predictive capacity. But at the same time the relationship a.b=c extends to other domains and gives rise to other types of categories too. When applied to the description of dynamic situations, it would appear as "acceleration•mass=force", when applied to electric domains as "voltage•intensity=resistance", in thermodynamics, it would be volume.pressure=temperature, and so on.
The formal laws we will borrow from topology are required to fulfill the following two conditions:
1) They should describe general classes that can be applied to many specific situations. In other words, topological symbols and rules will predict categories, and not particular items.
2)  The same set of topological symbols and rules should be applicable to many domains of the grammar, that is to say, they will be given an interpretation in several grammatical components.
Since the formal statements differ according to the topology of the space they are aiming to describe, we ought to successively consider one by one the symbols and laws that belong to quotient topologies, to induced topologies, and to product topologies: as the latter has scarcely been investigated by researchers of topological linguistics, I will omit it in this introduction.
A set of formal relationships that can be derived from quotient topologies are the following:
1) Two opposite units -call them w and w´-, that belong to a certain level "n", determine a complex unit or node -call it W- that belongs to the upper level "n+1".
2) Two opposite nodes -for example W and W´-,  that belong to an "n+1" level, determine a unit of the subordinate "n" level when they are confronted.
3) There are four alternatively opposed units of the "n" level: two units of type "w";  wi and wj,  and two units of type "w´";  wi´ and wj´, such that only three of them can be confronted in order to determine the same node. In other words, only wi+wj+wi/j´ or wi´+wj´+wi/j are allowed.
These three axioms could be formally represented as follows:
1) w + w´= W
2) W + W´= w´´ (either w or w´, or another unit of the "w" type)
3){wi+wj+wi/j´, wi´+wj´+wi/j}= W
where w,w´,w´´ stand for "n" level units, W,W´ stand for "n+1" level units (nodes), and + means "linked with" or "related to".
As odd as these axioms may appear, it is worth noticing that they  fulfill the laws of projective geometry quite well. Suppose that w is a point in a space, and that w´ is another different point; then the association of these points yields a unit of the next level (a node), that is a line W. As we can see, the behavior of points (level 1) and lines (level 2) is precisely explained by our axiom 1) when its variables are conveniently substituted by geometrical notions:
1) Two different points determine a line, that is, between two points we can draw only one line:

But the converse relationship also applies to points where two different lines intersect according to axiom 2):
2) Two different lines (not parallel ones) meet at only one point:

Finally, in order to prevent these notions from belonging to only a one dimensional space where axiom 1) would also be true, the formal system states that :
3) Four different points exist in such a way that three of them do not belong to the same line, but three of them do:

Since axioms 1)-2)-3) are of a very simple nature it would be hard to conceive of a scientific domain other than geometry where they would be applicable. The reason lies in the fact that these axioms follow the principle of duality, which means that 1) and 2) are the same principle when we substitute w symbols (w,w´) by W symbols (W,W´) and vice versa. In chemistry single elements are joined together to produce molecules, but the combining of molecules does not produce single elements. In biology the association of cells is called tissue, but the association of tissues does not give rise to cells again. In anthropology several families compose a tribe, but several tribes do not compose a family. 
When generation is involved it would appear that 1)-2) requirements are fulfilled: for example, in biology a male individual and a female of the same species create a new individual that, if coupled with another individual of the opposite sex, can produce individuals again, etc. However, this process is not properly of the type 1)-2) outlined above: the individuals of the second generation do not belong to a different level "n+1" when we compare them with their first generation ancestors, that is, they are not nodes.
You can now see that the assumption that language is a space is much more than a useful metaphor because, surprisingly enough,1)-2)-3) hold true in the three main components of linguistic space stated in chapter 5) as well.

6.1) LOW LEVEL PHONOSYNTAX

As we said above, the minimal units of the phonosyntactic component are the phonemes provided by the phonematic component through a coordination -i.e. feature adding- process. Given the set of phonemes of a human language, axiom 1) establishes:
That there is a fundamental difference between "w" units and "w´" units, that is, the phonemes of a language cannot be arranged as a continuum: there is a threshold that clearly opposes phonemes of one type and phonemes of the other type. In other words, although human sounds constitute a continuum from a physical point of view, phonosyntactically there are two main types, call them vowels -w- and consonants -w´-. No matter how difficult the labelling of a concrete phone may be -for example in the case of glides-, or even despite the fact that what in a certain language A is considered a consonant -[r] in English- in another language B can work as a vowel -in Czech f.ex.-,  phonosyntactically the phones of a human language must nevertheless be classified in one of them.
That the association of a vowel -w- and a consonant -w´- gives rise to a unit of the upper level, that is, it determines a node -W-: this node will obviously be the syllable.
Notice the fundamental nature of all the categories predicted by axiom 1). There are many languages that lack some types of phonemes -French has no [q], English has no [ü], etc-, but there is no human language without vowels, without consonants, or without syllables, i.e., a compound of "one vowel + one consonant" (as CV, as VC, or both: the axiom does not predict the order of their manifestation).
But CV or VC are not the only phonosyntactic combination that languages allow. As axiom 3) clearly establishes:
One of the phonemic types can be repeated in the syllabic structure, but never the other  -we can have either wiwj... wkw´ or wi´wj´...wk´w, but not both. It is easy to understand that human languages repeat the consonantal type because they prefer the wi´wj´...wk´w pattern: VC/CV (at, be), CVC/CCV/VCC (tip, tree, aft ), CVCC/CCVC (tilt, trip ), etc.
As a result of what we are saying, diphthongs will not be considered as an association of vowels, that is, as syllables of the form *VVC , *CCVV, etc., but rather as special units occurring in the syllabic patterns CDC (ripe ), CCD (try ), DC (use ), etc. This conclusion suits English quite well as the phonetic components of its diphthongs do not exist on their own, but it does not adequately suit other languages, such as Spanish whose diphthongs can at times be pronounced as if they were two different syllables -that is viaje ,"trip", as [bi-'a-xe] or as ['bia-xe]- but this very fact demonstrates that, although Spanish diphthongs are not minimal units provided by vocalic coordination as English ones are (see later), they cannot be considered a sum of vowels either. When we compare English and Spanish consonantal clusters it becomes apparent that Spanish diphthongs are a matter of discourse, rather than a matter of linguistic system: we could split never the English conclude  -i.e [kon-'klu:d]- in this way *[konk-'lu:d], or the Spanish estructura  -pronounced [es-truk-'tu-ra]- in this way *[est-rukt-'u-ra]. Whatever explanation our formal system provides for this peculiar alternative splitting of diphthongs in Spanish, it certainly has nothing to do with a hypothetical interpretation of  axiom 3) as *ViVj...VkC.  Axiom 3) can only be read as CiCj...CkV (whatever the order of its symbols may be), or as ∆V as in I -['ai], due to the fact that the pause is the absolute stop of air stream (that is, the strongest consonant). 
However, the most interesting fact about these axioms is their reversible nature. As stated by axiom 2), it also turns out to be true that:
Two different syllables, when they are put together, give rise to a phoneme again. How can such a statement be interpreted?
First of all notice that "different" means here "functionally opposite" and that it cannot be understood in a material sense: from the point of view of axiom 1) -w + w´ = W- the phonemes [t] and [p] are not different from each other, nor are the phonemes [æ] and [i], but we can only say that any consonant, as it is the syllabic border, differs from any vowel, as it is the syllabic nucleus. In a similar way the syllable [ri] of receptive -[ri'septiv]- does not differ from syllable [tiv] because both of them belong to the same stress pattern, that is, they are unstressed, in opposition to syllable ['sep]. 
Now we are able to suggest a suitable reading for axiom 2) -W + W´ = w´´- that makes sense. When a stressed syllable is added to an unstressed one or, as axiom 3) enables us to generalize, when any number of unstressed syllables are added to a stressed one -cfr. discourage, enthusiastic -, it causes a physical property of sound to stand out that will permit us to differentiate meanings in the same way that phonemes do, that is, a stress toneme which opposes ['kontent]noun to [kon'tent]adjective for example will come into consideration.
It would be advisable to point out the relative character of these notions once more. First, the physical property involved in W and W´ need not necessarily be stress -in many languages it is pitch, as in Chinese, or even duration, as in Latin. Secondly, from a physical point of view, all syllables in an utterance always possess some stress, thus W will only mean the most noticeably stressed syllable as opposed to W´. This relative character of stress reminds us of the relative nature of vocalic sounds, because vowels are not completely unstopped streams of air, but have a smaller stop when compared to consonants. Similarly ∆ stress syllables -i.e unaudible ones- can be added to a stress syllable in order to determine an acceptable stress pattern as in go  (to put it simply: stressed syllables are never alone).
In conclusion: by providing a suitable reading for axioms 1)-2)-3) we obtain the following low level phonosyntactic categories and relationships :
1) There are two kinds of phonemes, vowels and consonants, whose association gives rise to syllables, that is, nodes, which belong to the following level. 
2) There are two kinds of syllables, tonemiced syllables and untonemiced syllables (for example stressed and unstressed), whose association gives rise to tonemes that make up another kind of phonemes which belong to the preceding level.
3.1) Syllables can consist of more than one consonant, or even of none, whereas the vocalic element is obligatory and can stand alone. 
3.2) Tonemes can be determined by patterns that consist of more than one unstressed syllable, or even of none, whereas the stressed segment is obligatory and can stand alone.
Before continuing with our interpretation of axioms 1)-2)-3), I would like to underline that the notions we have been speaking of, simple as they may seem, are by no means obvious. We cannot take for granted that any sound of any human language must be either nuclear (call it vowel), or marginal (call it consonant), nor that their association must have a vowel as its nucleus, nor that such an association must yield elements of the upper level (call them syllables), nor that these nodes must recognize an outstanding type and a muffled one, nor that their confrontation must give rise to a unit of the subordinate level (call it toneme) again, etc.
All of these statements are predictions of empirical facts whose corroboration is enabled by events, but that had never been automatically produced by them. For this reason it took inductive linguistics a long time to discover them -remember the vowel / consonant distinction for example-, and they still hesitate when dealing with some of the less observable events, such as suprasegmental facts. I think that, at least as a heuristic procedure, topological axioms have proven to be a fundamental aid.

6.2) LOW LEVEL MORPHOSYNTAX

A suitable proposal concerning the predictions that axioms 1)-2)-3) make with respect to the morphological domain could be the following  (units of the italized examples are spelled in capital letters, nodes in small letters):
1) Two opposite morphemes, that is, a radical and an aspectual affix, determine a word (i.e clear-  + -ance  = clearance )..
2) Two opposite words, that is, a concordant word and a concorded word, determine an agreement affix (i.e. the boy-∆  come-S ).
3.1) Words can be made up of a radical and more than one aspectual affix, or even of no aspectual affix at all (i.e. pre-  + -determine-  + -ation  = predetermination  ; sting-  + -∆  =  sting ).
3.2) Agreement chains, where agreement morphemes appear, can consist of one concordant word and more than one concorded word, or even of no concorded word at all (i.e they (3d person plural) will get there by  them-(3d person plural)-selves (plural) ; I have (many) books).
Notice that the basis of morphosyntax is the duality principle once again. Morphemes are the minimal units of syntactic description that have sound and a meaning that does not directly refer to external things as sememes do. But the radicals measure-, astonish-, run-, etc., are morphemes just like the aspectual affixes -er, -ment  are (which associated with these roots give rise to runner, measurement, astonishment, etc.), or like the agreement affix -s  (in measurements, runners ). The only difference between the former and the latter is that radicals and aspectual affixes appear in the dictionary, but agreement affixes have to belong to a previously established concordance relationship of words, that is, the first are units of level n, whereas the second is a unit that only appears after the confrontation of nodes (of units of the n+1 level) has been performed.
I would also like to point out that axioms 1)-2)-3) do not necessarily have a material linear interpretation. Hence, maternal  and maternity  consist of the aspectual affixes -al  (as in medicinal ) and -ity  (as in possibility) together with a stem matern-  that exists not only in this form, but as mother  as well. Similarly, mice  consists of mouse-  + -(plural agreement affix), and to limp  of the functional equivalents of lame-  and -en  (as to deafen, etc.), that is, it merges a root morpheme and an aspectual affix morpheme .
When providing a suitable interpretation for axiom 3) we are faced with a problem that reminds us of that of diphthongs: it is easy to understand that the number of affixes that make up a word can be greater than one, but how can we justify that the radicals can never be repeated in a single word when considering compound words like lunchtime, bedroom, etc.? If we compare English with Romance languages, for example, it would become apparent that these formations are quite common in the first, and also that the second translates them either by noun phrases -Sp. hora de comer-, or by radical+affix chains -Sp. dormitorio.  However, it is important to realize that English compounds do not consist of several radicals and only one aspectual affix as the reverse reading of axiom 2) -i.e. wiwj...wkw´- would require: nappyliner  consists of two words, nappy  + linen (better than liner) , the first being analyzed as radical nap-  + affix -y  , the second as radical line-  + affix -en.  English compounds behave rather like Spanish diphthongs, in that they appear to be associations of words in the discourse which follow a specific law that the axiomatic system has to justify (and, in fact, will do so later). In the same way that a Spanish diphthong can be pronounced diphonematically, an English compound sometimes doubts between being considered a single word, or a couple of them -for example, the sergeant put off licenses and any kind of permits/the company opened off-licences all over the country.

6.3) LOW LEVEL SEMOSYNTAX

When trying to apply axioms 1)-2)-3) to the domain of semosyntax some specific problems arise. First of all it is by no means clear what the units of this component look like. Whatever difficulties linguists have to solve when establishing the list of phonemes and syllables of a human language, it is obvious that speakers generally recognize them as such, and for this reason spelling systems always tend to adopt either an alphabetic -i.e. phonemically shaped- form, or a syllabic one. In the morphosyntactic domain a similar situation occurs: linguists have often discussed how words can be defined, what their limits are, and even whether or not a certain label should be given scientific significance. Nevertheless, speakers are aware of the existence of words, they use them, and they continuously create new ones by combining roots and aspectual affixes that they know by heart.
Successful predictions of the semosyntactic component seem even harder to. Let's take a look at the problems concerning axiom 1) in an interrogative manner: what are the minimal units -sememes- of the meaningful component? They are terms like horse, dog, sweet  or to come  of course. But in this case, what do they really look like? Since they cannot be words -which are morphosyntactic nodes-, are sememes abstract logical concepts, or mental psychological representations? And once we have made up our mind, what are the primarily opposite units w and w´ that allow axiom 1) to work?
If we look at the phonosyntactic or at the morphosyntactic components, it would be apparent that we must not begin by posing the formulation question. We should never commit ourselves to the problem of deciding what a phoneme or a morpheme is (that is, what their ontological or epistemological nature should be), but we do need to ask how they work. Phonemes work either as syllabic nucleus -as vowels-, or as syllabic margins -as consonants-. For this reason, the entire set of phonemes of a given language can and must be divided into two separate subsets, that of vowels and that of consonants, no matter how the continuous stream of sounds acquired discreteness, nor how it became apparent to consciousness. Similar reflections can be made with respect to morphemes.
Going back to the semosyntactic component I would like to propose that the fundamental question be this one: what are the two opposite roles that any sememe has necessarily to play? Notice that, as fundamental as the distinction between noun and verb could seem, there are sememes that work neither as nouns, nor as verbs, for example on, briefly, etc. This means that, although grammarians usually distinguish two separate sets of semosyntactic rules to account for the distinction between nouns and verbs -cfr. Chomsky's 1965 Aspects of the Theory of Syntax, § 2.3.3  for example-, in fact these systems of features underlie a much more fundamental opposition that enlarged Xbar-syntax versions -as R.S. Jackendoff's 1977, Xbar-Syntax: A Study of Phrase Structure, Ling. Inq. Mon .2- tried to capture. However, it is necessary to distinguish syntax from semosyntax: syntactically there are more than two categories as Xbar syntax clearly established, but semantically, meaning behaves in only two fundamental ways which Chomsky's rules intended to reflect.
Where do sememes join together?: in the clause. And, what is the formal property that any sememe cannot avoid exhibiting when included in a clause?: the fact that it either governs another/other sememe(s), or is governed by it/them.
Let us compare this again with phonosyntax or with morphosyntax: [r] will phonemically be a vowel in the languages where it can occupy a syllabic nucleus, that is, when it need not be helped by other phonemes in order to exist syllabically; on the contrary, it will be a consonant when it needs help, that is, when it is a dominated or governed phoneme.
When considering the government quality of sememes it turns out that any sememe can be classified in one of these two classes:
Relational Units (RU) are sememes that cannot be conceived of without another/other sememe(s) that they dominate. This concept is not necessarily tied to the verbal category, but it has a much greater field of application. For example, the sememe wish always relies on two or three sememes, be it manifested as a verb -John1wishes me2 well3-  or as a noun -my1 best3 wishes on your birthday2- ; the sememe in  is needed to support another two sememes, though it can appear as a preposition -people1 in society2-  or as an adjective -the inner tube1 of the tyre2 -.
 Constitutive Units (CU) are sememes that can be conceived of without governing other sememes: they can appear as nouns -lip-, as adjectives -white-, and even as adverbs -cfr.bye in bye-bye; as a preposition (by)  it is an RU that links two sememes together.
Thus the fundamental distinction that holds among sememes is  whether they are dominating/dominated: there are sememes that cannot be thought of if we do not think about some potential sememe(s) that they are dominating at the same time, and there are sememes that do not behave in this way. But both relational and constitutive sememes are abstract notions: grammatical categories are a step further in their manifestation, and phonological shaping as well. Notice that we cannot speak of one light sememe, but of two different ones: light1 that is constitutive and appears as a noun, an adjective, or an adverb, and light2 that is relational and appears only as a verb -compare the light on the table, a light colour, to touch lightly  with John lighted (lit) the cigarette.
The association of an RU and a CU is also of an entirely abstract nature and we will call it a clause. For instance, the relational unit discover and the constitutive units Volta and battery, when joined together, constitute the clause discover (Volta, battery) that can be manifested either as a sentence -i.e.Volta discovered the electric battery-, or as a phrase -Volta's discovery of the electric battery. This situation is very similar to what happens in the phonosyntactic component when the abstract syllable {vowel [a:] + consonant [f]} can be manifested as either [a:f] -in after -, or [fa:] -in far.
Notice that the above clause has been obtained by combining axiom 1) with axiom 3) which permits us to enunciate a general statement concerning clause structure: a clause can consist of only one RU and of one or more CU, or even of no CU at all -cfr. it is raining  whose relational unit rain only exhibits an empty pronoun support-, that is, only type wi´ wj´ wk´...w will be allowed.
Thus clauses and sentences show radically different behaviors, and are never synonymous notions: a sentence can contain as many clauses as RUs it exhibits -John came1 yesterday  has one clause, John wanted1 me to go2 home  or John's disapproval1  annoyed2  me  have two clauses- while, on the contrary, a clause can consist of more than one single sentence -cfr. John said(sentence 1): bye! (sentence 2)-.
  In short, axioms 1) and 3) when applied to the semosyntactic component will predict the following relationships:
1) The association of a relational sememe and of a constitutive sememe, i.e., the semosyntactic units of level n, yields a clause, i.e, a node which is the semosyntactic unit of level n+1.
3.1) The clause may consist of more than one CU, and even of no CU at all, but RU can neither be repeated nor lacking.

One should now wonder whether the duality principle can be maintained in the semosyntactic component: how does the association of two opposite clauses yield a sememe again?
First of all, consider how the similar confrontation of nodes W and W´ starts in the phonosyntactic or in the morphosyntactic component: a stressed syllable is a more stressed syllable than an unstressed one, a concordant word is a word whose agreement morphemes are more its own than the agreement morphemes of a concorded word. The property that opposes node W to node W´ is then a property that in each case characterizes the entire node and that manifests a different degree in each of them.
Taking these circumstances into account, remember that a clause is a set of relationships that is held by an RU and by several CUs: consequently we will take for granted that the W node types of clause will be the clauses that exhibit all their CUs, and the W´ node types of clause will be the clauses that are only able to manifest some of them. We will call the former ones the normal clause type, and the second ones the derived clause type: for instance, John opened the door with a key  is a normal clause, the key opened the door   or the door opened , are derived ones.
By confronting John opened the door with a key  -W- with the key opened the door  -W´- it seems that the only thing that this comparison makes clear is the absence of the CU John  in the second clause. Nevertheless, the word John  is not absent, but remains as an implicit term (compare with Peter came  where John  is really absent): the key opened the door because John used it. In other words, the confrontation of the normal clause with the derived clause not only helps to make clear that John  is implicit in the second, but also that he is implicitly the doer or agent of the action expressed by the verb. On the other hand, when confronting the door opened  with the normal clause John opened the door with a key  through the intermediate step the key opened the door, it is evident that the key  is an implicit instrument of the clause the door opened, and that the remaining noun phrase the door  would be called the neutral one.
These notions -agent, instrument, neutral object, etc- have been called cases in current literature and have been well known to grammarians since at least the 13th century, but it is worth pointing out two facts about them. First, cases can be conceived of neither by looking at the real world, nor by proposing some kind of logical -i.e. universal- definition for each one, but only by taking into account their behavior in a specific language, that is, by confronting all the clauses -the normal and the derived ones- where they appear as I have suggested above. The word key  need not always play an instrumental role (although keys are instruments in the external world), as it can also fill locative slots as in Mary sat on the key, or objective slots as in Susan lost the key, etc. On the other hand, it is a well known fact that case grammarians have been unable to find a universal set of cases because each language uses some of them and is not aware of the rest (or, as I would like to express it, each language splits the casual continuum in a specific manner).
The second fact that is worth noticing about cases is that they happen to belong to the same inventory as RUs and CUs, all three of which are units of the semosyntactic component -for this reason I propose to call cases definitional units or DUs. Some people may not agree with me and say that RUs (like to come ) and CUs (like dog ) are linguistic terms, but DUs (like agent  or instrument ) are metalinguistic elements. However, I want to point out that specific metalinguistic labels (i.e., the words of grammar) are never encountered except in grammatical discourse -the terms syllable, morpheme, or adjective  do not appear in everyday conversation-  but cases are by way of a set of specific linguistic items that are incorporated in every human language. For example, somebody does something somewhere  or who does what to whom?  are linguistic sentences whose CUs have no meaning except: in the first example, "animate being" -somebody -, "inanimate being" -something -,  "place" -somewhere -;  in the second, "animate active" -who -, "inanimate" -what -,  "animate passive" -whom , and the RU that links them means simply "action". Hence the clauses they make up when put together are 'action [Agent, Object, Location]'  or 'action [Agent, Object, Experiencer]'. Cases are the result of joining the most general sememes and therefore they are semosyntactic entities that can constitute a homogeneous class together with RUs and CUs.
The semosyntactic interpretation of axioms 1) and 3.1) that we have discussed above will be completed now with the following proposed interpretations for axioms 2) and 3.2) in semosyntax: 
2) When a normal clause and a derived clause are confronted, they specify which definitional units (cases) are implicitly or explicitly present.
3.2) The manifestation of casual notions may require that more than one derived clause, or even no derived clause at all, be confronted with the normal clause, although the latter must obviously never be repeated.

        _____________________________________________

When comparing the units and nodes of the three low level components of syntax with each other, they turn out to be adequate not only in the code, but also in the message:

A) Phonosyntax: Phonemes: Vowel+Consonants=SyllablesÆ
                                                                  (that make upÆ)
B) Morphosyntax:  Morphemes: Root+Aspectual affixes=WordsÆ        
                                                    (that make upÆ)
C) Semosyntax: Sememes: UR+UCs=Clauses

If the nodes of A)  always fit the units of B), and the nodes of B)  always fit the units of C) -that is, if any morpheme consisted of a syllable, and any sememe consisted of a word-, then our three components would be levels of description at the same time. Human languages certainly have a tendency to behave in this way so as to make the adjustments between the code and the messages it produces easier, but it is only a tendency. A syllable can manifest more than one individual morpheme; -[tid] in prevented  can take either past or past participle, and any grammatical person of the past tense. On the other hand, some morphemes can require more than one syllable in order to be manifested in discourse -the present continuous morpheme of any verb requires a discontinuous syllabic manifestation form of to be ...-ing. Similarly, words can exhibit more than one sememe -camp  means "a kind of place" and "effeminate"-, and the reverse -"an emotional pain" is a grief  or a distress , etc. 
There are certain linguistic methodologies that have paid close attention to this kind of splitting of the utterances of a language, because they are primarily interested in messages (see distributionalism as characterized above). However, as a matter of fact, levels must not be confused with domains in the code, although they obviously tend to mix with the message in order to facilitate an economical communication process (in a similar way, anatomy and physiology are two separate branches of science, but they collaborate when we move our legs).

6.4) THEIR GESTALT BACKGROUND

As I said in chapter 1), topological linguistics is a Gestalt linguistics from a psychological point of view. The question we will now attempt to answer is how this psychological background manifests itself with respect to axioms 1)-2)-3) .
When looking into the formal nature of w+w´=W and of W+W´=w´´, we discover that the common layout of both axioms consists of one more outstanding unit/node that, when confronted with a less outstanding unit/node, gives rise to a new node/unit. 
Vowels are not sounds that are empirically opposed to consonants, but rather they are sounds that stand out in the syllable, and they do so because the surrounding consonants do not stand out (compare did not, i.e. [did]+[not], with didn't, sometimes pronounced [di]+[dnt] with [n] alternatively occurring as a consonant or as a vowel). Morphemes need not necessarily be a radical stem or an aspectual affix, but can sometimes occur in both functional positions (compare outcome, with out  as a prefix, with outer,  where out  is the root). Sememes can also vary in their manifestation, thus some RUs (for instance to work  in John works a lot ) can appear as CUs (as in she made up her mind to work ).
However, to a large extent the w and the W elements remain w and W ones, the w´ and W´ continue to be  w´ and  W´. This is due to the fact that in many cases more natural performances of each item exist, although they may always theoretically vary.
Now the Gestalt condition of w+w´=W and W+W´=w´´ laws should be apparent: the w or W item is always a foreground element that stands out, but it can do so only because the w´ or W´ item is a background element that is weakend; when we invert the figure/ground relationship that both maintain a previously w´ or W´ item can at times become w or W, although such changes are not frequent as they always imply a radical alteration of the natural condition of the items. 
On the other hand, axiom 3) states the well known fact that in some cases the background of what we are perceiving can be very slight (as when we see a square whose only contrasting background is provided by the four lines that limit its surface), or, on the contrary, it can be very strong, being composed of several w´ or W´ items that practically substitute the highlightened nature of the w or W foreground (as when a white square is surrounded by a black background). For this reason, in "give (John, the book, to Mary)" the three CUs almost replace the empty RU "give", as opposed to "run (John)"  where, for example, the only CU "John" does not predict the RU "run" at all.

        w+w´                   w+∆w´             w+w´i w´j w´k...
       W+W´                    W+∆W´                 W+W´iW´jW´k...

Notice by the way that w+w´ and W+W´ manifest a fundamental topological relationship, w or W as foreground items being the closed sets, and w´ or W´ as background items being the open sets: consequently, both w+w´ and W+W´ must be conceived of as topological spaces.