7) FORMAL LAWS OF QUOTIENT TOPOLOGY AND THEIR INTERPRETATIONS WITH RESPECT TO THE HIGH LEVELS : THE FOUR MAIN STRUCTURES

Turning back to the formal laws that projective geometry is able to supply we can add four new axioms, where the theoretical notion structure happens to appear, to our well known 1)-2)-3) axioms;

4) A unit w and a node  W determine a structure W.
5) Three units wi , wj , wk also determine  a structure W.
6) A structure W and a node W also determine a unit w.
7) Three structures Wi, Wj, Wk also determine a unit w.
The axioms 4)-5)-6)-7) permit us to draw the following conclusions:
-A new level has arisen, thus we now have to deal with three levels, the first where units appear, the second where nodes appear, and the third where structures operate:
                  level n+2 : structures
                   level n+1 : nodes
                  level n     : units
-The third level, i.e. that of structures, is of a particularly surprising nature because it is not made up of two nodes, a foreground and a background node, but of two salient elements, either a unit and a node -axiom 4)-, or three units where two of them obviously compose a node and the third is added to it -axiom 5). Nevertheless, the duality principle -4) and 5)  vs. 6) and 7)- still holds.

In any case, notice that axioms 4)-5)-6)-7) can be easily interpreted within the geometrical domain, as axioms 1)-2)-3) were as well:
4) A point and a different line, i.e. a line which does not contain the point, determine a plane:

 5) Three different points also determine a plane, because two of them determine a line according to axiom 1), and this line, when confronted with the third point , allows the conditions stated in axiom 4) to be restored:

 
6) A plane and a line that is not contained in it always meet at a point:

 7) Three different planes, i.e. three planes that are neither contained in each other nor parallel to each other, always determine a point because two of them meet in a line, and this line meets the third  at a point:

 In order to make the grammatical interpretation of 4)-5)-6)-7) easier let us begin with 4). What does this axiom actually mean? That there is a level whose elements are not opposed as complementary units, but rather in a way that the first -unit w- is opposed to the second -node W-, although at the same time there could be an inclusive relationship between them because nodes are made up of units. 
This very peculiar situation can be found in the domain of subjects and predicates: in John cut the bread  the subject unit John  is opposed to the predicate cut the bread, but the latter is a node that includes at least one unit of the same kind as John , i.e. the noun phrase the bread ; for this reason we can even repeat the first unit, which yields a reflexive construction: John cut John = John cut himself.
However, "subject" and "predicate" are in fact very ambiguous notions: when considering the above sentence we can say that John  is the subject in John cut the bread, but we can also say that the bread  is the subject with respect to the bread was cut by John . 
And even more: Yesterday John came late  and John came late yesterday  only differ bcause of the fact that the pivotal notion we are speaking about (that some scholars call "topic", and others "subject") is yesterday  in the first sentence, but John  in the second.
A rather similar way of distinguishing subjects is to emphasize some words of the sentence -by means of stress, specific morphemes, etc-, and leave the rest in the background: we gave JOHN the book, we GAVE John the book, etc .
These four different schemes that any sentence can exhibit are usually recognized by grammarians and are called:
a) logical subject (vs. logical predicate): the noun phrase that is responsible for the verbal process, or that supports it (vs. the rest of the sentence).
b) grammatical subject (vs. grammatical predicate): the noun phrase that agrees with the verb (vs. the rest of the sentence).
c) topic -textual subject- (vs. comment -textual predicate-): the element previously spoken of (vs. the remaining elements).
d) focus -communicative subject- (vs. back entailment  -communicative predicate-): the element we feel is the most salient (vs. the remaining elements).

These four ways of perceiving the functional relationships in the sentence are not at the same level: in fact, a) and b) look alike on the one hand, and c) and d) do so on the other. This means that two more fundamental relationships can be established:
a)-b) The node, that is, the predicate, includes some unit(s) that could also have been the subject, and other(s) that had never been able to behave as such. For example, only nouns (noun phrases) can be either logical or grammatical subjects, but verbs or other grammatical categories can never behave as such: John cut the bread, the bread cooked,  but  *cut cooked.
c)-d) Any unit of the sentence can occur as topic -textual subject-, or as focus -communicative subject-. For example: cook the bread! / the bread cooked  ; JOHN cut the bread / John CUT the bread / John cut THE BREAD .
 It is easy to understand that axiom 4) accommodates to a)-b), while axiom 5) does to c)-d):
4) A unit and a node determine a structure, that is, a noun phrase is the subject of the sentence, and a string of formatives, where some are noun phrases which could also have been the subject, is the predicate.
5) Three units determine a structure: thus, any of them can be the subject, and the other two, that are tied together in a node, behave as a predicate.

The next step in developing our predictive method will be to distinguish a) from b), and c) from d). At any rate, it is worth noticing that the ways these axioms accommodate to human language are surprisingly accurate. The asymmetry of subjects and predicates is one of the most striking properties of syntax (as shown by P.F. Strawson, 1971, "The Asymmetry of Subjects and Predicates", Logico-Linguistic Papers, London) that can hardly be found in other domains of science, nor even of grammar. Although a specific subject can share two opposite predicates -for instance the flag is black and white-, two opposite subjects can never be related to the same predicate -John is male ; Jean is female  but *John and Jean are male, *John and Jean are female. This is due to the fact that, as we said above, predicates are built up from heterogeneous items, and thus can co-occur even if they are opposites, because only some of their components, but not all of them, will be contradictory. On the contrary, as subjects are homogeneous items (they are units), they can never co-occur if they oppose each other.
Going back to the distinction between a) and b), and between c) and d), I will guess that in each case there is a different kind of grammatical perception: first, it can happen that both, internal and external data are taken into account; second, another kind of grammatical perception would imply that only internal relationships will prove to be useful.
This claim appears to be able to stand up to empirical checking. When confronting a) and b) it becomes apparent that the fact that a noun phrase is the logical subject -a)- depends, partially at least, on the referent, but the fact that it manifests itself as grammatical subject -b)- is entirely a grammatical matter. For example, if a person whose name is John is annoying another person whose name is Mary, it would be impossible to speak of John without considering him the logical subject, but from the point of view of b), the name of either person could be the grammatical subject, either John  in John annoyed Mary, or Mary in Mary was annoyed by John .
The same distinction can be established between c) and d). The emphasized word of a sentence -d)- always depends on the assumptions shared by the speaker and the hearer in relation to the real world, but the problem of deciding which are the given words and which are the words that convey new information can only be solved by looking at the entire text where those words are included, that is, according to internal criteria. 
Suppose two people are speaking about the guests that failed to attend their party, and that they were both very interested in the attendance of John and were upset by his absence. No matter what formal device they use to emphasize John's rudeness in failing to come, if they are speaking about the above situation, they must stress the word John  or a similar group of words that denote the same referent. In JOHN was missing yesterday, The one who was absent was "precisely John", The people who were absent were Mary, Tom, and "even John" , etc. the word John  is the focus of the sentence. But in John was absent YESTERDAY, The people who were absent were John, Tom, and "even Mary", etc, John  is not the focus. However, the topic does not depend on the external world, but on the previous text: keeping in mind that the speaker and the hearer are annoyed with John, if somebody said John isn't a sociable person, then we can presumably expect John  to be the topic of the following sentence -for example, in JOHN was absent yesterday,  but if he/she were talking about what happened yesterday -for instance did you enjoy the party yesterday?, then a very probable continuation would be yesterday JOHN was absent  where John  is the focus and nonetheless the comment. 

This "external-internal perspective vs. internal perspective" duality is not only found in the high level syntactic components: in low level ones there are also two components whose units are built up from reality -phonosyntax that benefits from the phonemes provided by phonematics, and semosyntax that gets its sememes from semematics-, and one component whose units are purely internal grammatical ones -morphosyntax-. In some sense we could say that the phonic and the semic low level components arise through a double path process:
                     external-internal perspective

PHONEMATICS: phonic features combine to build up Æ
PHONOSYNTAX: phonemes combine to build up syllables
                                 internal perspective

                           external-internal perspective

SEMEMATICS: semic features combine to build up Æ
SEMOSYNTAX: sememes combine to build up clauses
                            internal perspective
                             
but the morphic component results from a single path process:
                                   internal perspective

MORPHOSYNTAX: morphemes combine to build up words
                                   internal perspective

When looking into the high level components we are faced with a formal problem because there are no specific induced topologies that provide the first step of the two path process of the outer-inner perspective like phonematics and semematics do. For this reason we need to supply the two path process in a formal way by means of the two step axioms 1) and 2) . Consequently the outer-inner perspective of axioms 3) and 4) will be completed with axioms 1) and 2) as follows:

A:3) a unit and a node determine a structure + {1) two units determine a node, 2) two nodes determine a unit}
D:4) three units determine a structure + {1) two units determine a node, 2) two nodes determine a unit}

but the internal perspective will not be completed by low level axioms, that is:

B: 3) a unit and a node determine a structure 
C: 4) three units determine a structure

What does the specific content of these enlarged axioms A, B, C, D look like? I suggest they be interpreted in the following way:
A) A logical subject and a logical predicate determine a structure according to the general principle "subject(unit) + predicate(node)". Nevertheless, by taking into account the two subordinate axioms 1) and 2) it follows that we can adopt two points of view: starting from 1) it is as if two (or more) units determined the node; starting from 2) it is as if there were two (or more) nodes that determine the unit.
 The first analysis belongs to the external type: in John cut the bread  we consider the subject John  and the object the bread  to be governed by the verb cut, that is, "cut (John, the bread) ". Notice that the dominance of the verb -its nodal character- relies on the properties of the external world as we saw above: the verb is said to be the Relational sememe and it imposes the number and the nature of the nominal arguments -Constitutive sememes- it is able to govern.
The second analysis, on the contrary, belongs to the internal type analysis: when we divide the string John cut the bread  in the form "[John] subject + [cut the bread] predicate"  we are starting from the asymmetric relationship of subjects and predicates -see above- that enables two (or more) nodes -predicates- to be said of the same unit -subject-, but never the reverse situation -i.e., John  as a [+human] subject would enable either cut the bread  or spread some butter, for example, or both (John cut the bread and spread some butter on it), but in John and Mary cut the bread  we must recognize two separate actions, although they can refer to the same piece of bread -i.e., both of them are collaborating-, or not.
The first structure predicted by the compound axiom 3/1-2 will be called the rective structure (or government structure, as it is usually known in current linguistics) because it has to do with government relationships and, since it is an external-internal perspective, we will recognize two alternative approaches to it: the outer  will be called argumental substructure, and the internal will be called predicative substructure:

                                  grammar
                                       
RECTIVE       predicative rective : logical + logical
STRUCTURE   substructure         subject   predicate
(axiom 3)            (+ axiom 2)

 RECTIVE     argumental rective : verb (subject,object1,
 STRUCTURE substructure                 object2.....objectn)     
 (axiom 3)             (+ axiom 1)
                                       
                                             world

But as we said above human languages meet reality at two points, because they not only speak about certain events, but they also serve to establish a relationship among the people who use these semiotic systems. It stands to reason that both poles of the referent be parallel, since the communicative event is simply an event undertaken by two actors that are speaking to each other. Thus we could say that a speech act is a kind of argumental rective substructure where the relationship  "verbal function (subject, object1, object2...objectn)" appears as "illocutionary function (speaker, proposition, hearer)" -for instance, I shall give you a ride  would be "to promise (speaker,   xS gives yH a ride, hearer)", while x gives y a ride  is simply of the form "give (x, a ride, y)".
Similarly we could think of the "focus vs. back-entailment" relationship in terms of "subject vs. predicate" because the asymmetric condition holds here too: a given focus subject could be raised by more than one single predicate, even by two opposite predicates, but the reverse is never true -for example if we want to charge John with a robbery, then we can say JOHN was there , but if we are trying to find an alibi for John, then we could also emphasize this very word John  in JOHN was not there, but Peter was. Nevertheless, notice that it would make no sense to put both suspects, John and Peter, in the focus, whether we want to blame John or to save him.
The above considerations allow us to read the axioms of D) as follows: 
D) "Three units determine a structure (axiom 4) + two units determine a node (axiom 1)" would mean that, although any of the main components of the communicative act -i.e. the speaker, the hearer, or the illocutionary force- have a performative nature, there is a tendency of this relationship to articulate as if the illocutionary force were a verb node that depends on the speaker and the hearer in a referential way. 
Notice that in D) both axioms, 4) and 1), must simultaneously hold, and that the "illocutionary function (speaker, hearer)" formula does not simply mirror the "verbal function (subject, object)" formula according to axiom 1), but it also agrees with axiom 4) opposed to axiom 3) -which is relevant to A). On the one hand, the speaker and the hearer are certainly some kind of Constituent unit constants -they are "pro-nouns"-, and the illocutionary force is an implicit Relational unit variable that can be made explicit as a verb. On the other hand, the performative (i.e. illocutionary) character of this structure can be found in any component, not only in the illocutionary force, but also in the speaker or in the hearer according to the "three units determine a structure" formulation. For this reason philosophers of everyday  language, who initially liked to think of performativity in terms of some specific verbs (as Austin did), usually understand them as a set of rules that tie the speaker, the hearer, and some kind of mental illocutionary force together. For instance, Searle's classical analysis for promising (J. Searle, 1969 Speech Acts. An essay in the philosophy of language, Cambridge University Press, 3.1: the Speaker promises  the Hearer to do A if "in expressing that p, S predicates a future act A of S", "H would prefer S's doing A to his/her not doing A", etc). This specific combination of axioms 4) and 1) in D) will be called enunciative dialogic substructure.

But the enunciative structure D) could be also viewed by starting with axiom 4) -that properly characterizes the structure- and with axiom 2). In this case we would introduce the asymmetry relationship shared by the focus and the back-entailment that we have just explained. Although we know that any word of the sentence, as supported by previous assumptions on the external word, could have been emphasized as a focus (i.e. "three -or more- units determine a structure"), once we have chosen the focus, it will operate as a subject-unit to which the rest of the words can be attached as asymmetric predicate nodes (that is "two nodes determine a unit"). This situation will be called the presuppositional enunciative substructure.
To sum up:  situation D), that has arisen from axioms 4) and 1) or 2), is the second external-internal structure of high level syntactic structure:

                                   grammar
                                       
ENUNCIATIVE     presuppositional: Focus+Back-entailment
STRUCTURE        enunciative  
                           substructure
(axiom 4)                   (+ axiom 2)

ENUNCIATIVE     dialogic:  illoc. force (Speaker,..,Hearer)
STRUCTURE      enunciative  
                        substructure                   
(axiom 4)              (+ axiom 1)
                                   
                                         world

Besides the external-internal structures A) and D), there are also strictly internal structures that can be approached by starting from the high level axioms 3) and 4) without further implementation. In this way we can carry out two structures:

B) "A unit and a node determine a structure" constitutes a formal statement that accommodates quite well to the structural relationship that the grammatical subject and the grammatical predicate maintain.
Notice that in the man sells cars  the grammatical subject condition of the man  is supported by the singular allomorph [..æ..] that allows it to agree with the predicate sells cars  by means of the 3d person singular allomorph [..-s] of the verb, no matter whether CAR  is singular (car) or plural (cars).
In this structure the vectorial relationship "subject-predicate" is manifested by the fact that the subject always reflects the formal variations of the predicate verb -insofar as English allomorphic paradigms allow, of course-, but the latter can remain unchanged (compare he comes / they come  with he/they must come ). In other words, we can say that the grammatical subject -a unit- stands out formally as opposed to the predicate -a node-.
However, the nodal asymmetry that we found between logical subject and logical predicate does not hold in the agreement (grammatical) domain precisely because the nodal axiom 2) has not been taken into account in this case. For example, if the boy  + dances  agree, and the girl  +  dances  also agree, then the boy  and the girl   continue with the  agreement as in the boy  and the girl  (either together or by themselves) + dance  .
We will call this structure the agreement structure:

AGREEMENT : grammatical subject = grammatical predicate
STRUCTURE 
(axiom 3)
(notice that the symbol " = " is intended to represent precisely the formal equivalence of the subject and of the predicate through agreement morphemes)

C) On the contrary, the interpretation of high level axiom 4) without further adjunction of low level axioms yields to the textual structure. I would like to point out that the statement "three units determine a structure" is a rather odd one in this high level syntactic domain because none of the components seems to be stressed. The asymmetric relationship "subject-predicate" turns out to be a very obvious one when one of the elements is strongly stressed -communicative subject or focus-, or agrees with the verb -grammatical subject-, or is even the supporter of the verbal action -logical subject-, but how can we determine which of the three units that gives rise to the C) structure will be its textual subject?
I think that such a situation accommodates to the text units string; consider the following text:
I: The men1 went to the house2. II: The door3 was locked4 . III: The key5 could not be found6 unfortunately.
As is known, scholars usually recognize two major units when dividing textual strings: the topic, i.e. the unit which carries given information and usually appears first, and the comment, i.e. the unit which carries new information and is commonly put behind the former. On the other hand, it will be noticed that the "topic" nature of a given unit is due to the fact that it has previously appeared as part of the semantic information that the precedent comment conveys. Thus, if we look at the above text we would say that 3 is the topic of sentence II (and 4 its comment) precisely because 3 is supported by comment 2 of sentence I -i.e. "a house (2) has a door (3)"- ; but if we look at sentence III, 5  becomes its topic as it is supported by the previous comment 4 of sentence II -"doors get locked (4) with keys (5)". Generally speaking, the supporting elements of the preceding sentences need not be "comments": what we are trying to make clear is that any topic relies on previous information.
Notice that the logical, the grammatical, or the communicative subject/predicate are not dependent on the surrounding sentences: for example, the door  of sentence II is the logical subject because the state expressed by the verb pivots on it, but this is independent of the fact that sentence I has a doer -the men- as its logical subject (it could also have been a static element as in the men stood at the house); the grammatical subject the door agrees in singular with the verb was, whether the previous grammatical subject agrees in singular -the man went to the house -or in plural -the men went to the house . Finally, if we emphasize one of the units of sentence II and make it the communicative subject -if we say the door WAS locked, for example-, then this will rely on assumptions from the external world (for instance, now the door is unlocked, but it was not before), but it is entirely independent of the emphatic subject of the previous sentence -as in the men went TO THE HOUSE, which could have been said if we assumed that they had gone to the shelter.
Only the textual labelling of the units of a given sentence as "topic" or "comment" is entirely dependent on environmental sentences. For this reason it makes no sense to speak of "unit" and "node" when respectively referring to the "topic" or to the "comment", because any unit(?)-topic has been node(?)-comment in the preceding sentence, and so on. The only approach we can attempt is to look at the three  elements -i.e. "preceding textual information/topic/comment"- as if they were of a similar functional nature, that is, as if they were simply "units": hence this is precisely what axiom 4) predicts.
In conclusion, the fourth structure of high level syntax will be called the textual structure, which is as follows:

TEXTUAL  : (previous comment) / topic / comment
STRUCTURE
(axiom 4)
(the "/" symbol is intended to represent the linear nature of this structure where order considerations are always at the top: notice that if sentence III had read unfortunately the key could not be found, then the topic, which now depends on some other previous information, would have been unfortunately ).

                  -------------------------

Going back to high level axioms it becomes apparent that there are still two axioms left, number 5) -"A structure and a node determine a unit"-, and number 7) -"Three structures determine a unit"-. These formulae aim to safeguard the duality principle by means of which some combinations of upper level items (in this case of structures and of nodes) are able to yield low level elements (i.e. units) again.
When looking at low level components and at the behavior of axioms 1)-2) in relation to them, it should be noticed that, before the achievement of converse axiom 2), the results of 1) are only of an abstract (i.e. code-like) nature: a root (w) and an aspectual affix (w´) certainly yield a word (farm- + -er = farmer), but it is a word of the dictionary, not a real word of the message, because real words appear in relation to other words with which they do (or do not) agree -the farmers have come, for example, where the agreement affix [-s] that results from W + W´ = w axiom (a concordant word and a concorded one determine an agreement affix) has already arisen. Similarly, unordered syllables -{V, C} = VC or CV, etc- must enter a segmental pattern with suprasegmental structure in order to convey empiric nature; sememes can fill specific functional slots in the message only after case labellings of the clause have been established.
This happens to be the case in high level syntax too: the rective structure, the agreement structure, the textual structure, and the enunciative structure display a certain type of mental reality, but only of an entirely abstract nature: in fact we would be unable to recognize them if they did not appear in the message. How do they manifest themselves in messages? In other words, what are the units that prove to be necessary in order to allow them to appear as such?
When we introduced the four main high level structures we always showed them referring to a sentence: the logical/grammatical/textual/enunciative subjects vs. predicates are the elements of a sentence that are likely to be conceived of first, precisely because sentences already exist. Consequently the sentence will be one of the high level syntax units we are looking for: when joining these structures together we will obtain, thus, a unit called sentence.
However, one could wonder how axiom 7) -three structures determine a unit- really works, as there are four structures, yet the formal statement only requires three of them. I suppose (and I am saying "suppose" because the interpretation of Wi + Wj + Wk = w is a prediction of the formal system) that this has to do with the opposition "narrated discourse vs. uttered discourse" as established by some scholars twenty years ago -E.Benveniste, 1966, "Les relations de temps dans le verbe français", Problémes de linguistique générale , Paris, Gallimard, 237-51. In fact, a text is either an actual utterance about the world or a statement that describes some facts from a non-present point of view (to a certain extent the first is performed orally, the second in writing, although this is not always true).
Keeping this in mind it is now easy to understand that a sentence must show rective relationships and agreement relationships as well, but  can be arranged either as an utterance or as a statement, or even as both. Consequently, it can only show one of the high level structures that originate in axiom 4) -i.e. the enunciative structure, or the textual structure respectively-, although both can also be found.
When people hear a knock on the door and they say come in!  their sentence shows rective and agreement relationships (you  is the logical subject that will do the action of the verb, and it is also the grammatical subject that imposes some formal pattern on it which differs from the pattern that would have imposed we, i.e. let's come, for example). In a similar way we can say that come in!  shows enunciative relationships (depending on the suprasegmental pattern and on the stress of the words of the sentence we can interpret it as an order, a request, an invitation, etc). However, in come in!  it would be hard to recognize a part that conveys given information (i.e. a topic) and another part that conveys new information (i.e. a comment), because textual antecedents are lacking.
If, on the contrary, we had extracted a sentence from a given text, for example from a novel, we would be able to discover the rective relationship, the agreement relationship, and the textual relationship, but we cannot recognize a separate enunciative information, because the external world that would support it coincides in this case with the entire text.
In many cases both structures (enunciative and textual) meet in a single sentence, but not always. For this reason we will interpret the first unit yielding axiom of high level syntax as follows:

Axiom 7)
"A sentence is a unit that consists of three structures at least - rective structure, agreement structure and {textual structure, enunciative structure}.

The fact that we have introduced the four main high level structures by means of sentential examples must not mislead the reader. We cannot take for granted that whenever a rective / agreement / textual / enunciative relationship is established that an entire sentence will be required for it to be expressed. In the car of John  there is a rective relationship that ties the "logical subject"  John  and the "logical predicate" the car of  (i.e. John has a car ). When comparing it with John's car we could moreover keep track of the previous text, and thus we can expect the car of John if we had previously spoken of cars, but not of John, and John's car everywhere else. Many languages allow adjectives to agree with the noun head -Spanish casas altas /casa alta- in noun phrases that manifest the agreement structure: in written English a similar situation can be found when we oppose the student's blood type to the students' blood types. Finally, any item of a construction can be emphasized, and hence any of them manifests the enunciative relationship -compare old GUY  with OLD guy .
What do all these constructions have in common? They are obviously phrases that manifest one of the four main syntactic structures and perhaps more than one structure alone, but that need not exhibit all of them. A phrase is then a syntactic unit where at least one syntactic structure must be present.
On the other hand, how can we recognize whether a string of formatives which belongs to a given sentence is a phrase and not something else? In the old man gave a book to the kid  the strings the old man , a book , to the kid  and gave a book to the kid  are phrases, but the strings man gave a  or book to the kid  are not. In all these cases the phrasal nature of a string of formatives emerges when one of them stands out and governs the rest, that is, when there is one formative that behaves as the nucleus while the rest behave as adjuncts. Putting it formally we could say that in a phrase there is a set of units such that one of them is of a foreground nature and it prevails over the rest which can even be lacking (compare the old man / the man came  with ∆ man is mortal ). In other words, the nucleus supports the functional behavior of the string by itself, and can thus be characterized as a node. Notice that in low level axioms of the type w + w´ = W we cannot characterize w as a potential node because many units can be w or w´ (thus [n] behaves as a consonant in no  and as vowel in didn't), but in the case of the noun phrase, for example, only the noun, and never the article or the adjective, can be the node. Therefore the association of elements belongs to a single type in low level syntax (be it the syllable, the word, or the clause), but to several (the noun phrase, the adjective phrase, the verb phrase, the prepositional phrase, etc) in high level syntax.
Now we are able to define the second axiomatic unit of high level syntax as follows:

Axiom 6)
"A phrase is a unit that consists of at least one structure -be it {rective structure, agreement structure, textual structure, enunciative structure}-, and such that one of its formatives does function as a node having nuclear power"

The first condition, i.e. the structure, is an internal patterning requirement:  notice that a string of formatives can constitute a unit of the linguistic system only if some relationships among their components are held. On the other hand, the nodal nature of one of these components, i.e. the need for one of them to become the nucleus, has to do with the external world; although the common model of syntactic structures opposes a subject and a predicate, either of them having the possibility of interchanging its functional slot with the other one (for instance John A came / Peter saw John A, he feels like dancing B / dancing B is fun), there always remains a link with the external world, where things are of an absolute nature (a stone can never be an action, etc), by means of the argumental rective substructure and its opposition "verb-node vs. arguments-units".
This simultaneously relative and absolute nature of syntactic relationships is necessary in order for language strings to show both linguistic (i.e. relative) and empirical (i.e. absolute) adequacy at the same time. But if any sentence has a verb and can be structured, among many other ways, in the argumental pattern with that verb as a node, only verbal phrases have verbs, but noun phrases, adjective phrases, etc have not. If the phrase has to be a syntactic unit capable of mirroring reality, it certainly needs a node, because it corresponds to semantic clauses that could have been made up as sentences too (compare John refused the invitation  with John's refusal of the invitation): this nodal, i.e. external-world-like governing nature of one of the formatives is predicted by axiom 6)
In conclusion we can say that, regardless of the categorization of each symbol in each specific domain (i.e. in phonosyntax, semosyntax, and morphosyntax), we have seen that high level syntax axioms are able to predict four syntactic structures (rective, agreement, textual, and enunciative), and two syntactic units (the sentence and the phrase) which, as shown above, possess empirical adequacy. 
Moreover, both high level notions, structures and units, are connected by the duality principle in the same way as the low level notions, units and nodes, precisely because subjects and predicates are either sentences or phrases:

 Axioms 3)-4): 
subjectI + predicateI = structureI [I=i,j,k..] 

Axioms 5)-6):
structurei+(structurej+structurek)={sentence, phrase}

Remember that the low level axioms w + w´ = W and the reverse W + W´ = w´´ are correlative because in phonosyntax, for example, stress (w´´) is also a phoneme capable of differenciating meanings in the same way as other phonemes, like vowels (w) or consonants (w´) do. But this is also true in the high level axiomatic approach: the units that result from structures, i.e. phrases and sentences, are at the same time the units that give rise to those structures, i.e. subjects and predicates. Sentences and phrases do function as subjects and as predicates: generally phrases are subjects (noun phrases), predicates (verb phrases), or part of any of them (prepositional phrases, adjective phrases, etc), whereas sentences behave as subjects in textual structure and as predicates in enunciative structure -they carry out previous information (topic) or provide the back entailment-, but as both in the other two structures -cfr. that he didn't arrive on time (subject) annoyed me, he said that Peter didn't arrive on time (predicate); on the other hand, both sentence concatenations would give rise to some agreement requirements having to do with the mode of the subordinate verb in many languages.

7.1) THE TOPOLOGICAL SIGNIFICANCE OF THE FOUR MAIN STRUCTURES

When we discussed the distinction between levels and dimensions, we tried to make clear that a n+1 level item is not necessarily built up from n level elements in the same sense in which we can say that a bidimensional component contains two dimensions. For example, phrases and sentences -that is, the units of high level syntax- usually consist of a single clause -that is, of a node of low level semosyntax.
In fact the concept of "level", in the code, is only intended to mean a further specification, although in the message such a successive whole-part relationship often does hold as said above (the less abstract these low level elements are, the more they are included in the higher level: thus morphosyntactic low level "words" certainly are used in joining "phrases" or "sentences", etc). A further specification can be reached if we focus more closely on a certain subject: for instance, like a zoologist that has already distinguished between vertebrate and invertebrate animals and goes a step further by classifying the first into fishes, mammals, reptiles, etc, which would become high level notions.
As said above, the fundamental low level axiom w + w´ = W and its reverse W + W´ = w´´ are located on the same topological situation: a system of closed sets  - w or W -  and a system of open sets  - w´ or W´ -  that helps to explain the former, constitute a linguistically meaningful set that behaves as a topological space. Thus, the topological significance of high level structures will be discovered by looking more accurately into the definition of topological spaces as such.
The topology of a given topological space can be described in terms of neighborhoods of elements, and consequently it must be possible to formulate a description of closed sets in terms of neighborhoods. Thus a set w contained in a is closed if its complement a-w (which is denoted by w´) is open, and hence if each element of a-w has a neighborhood that is contained in a-w. An element wi of w is said to be an adherence element of w if every neighborhood of wi contains elements of w other than wi.
Three important notions arise in relation to neighborhoods and accumulation elements. The closure of w (denoted by v) is the union of its adherence elements. An element of w is said to be interior if w is its neighborhood: the set of all interior elements of w is the interior of w  (denoted by w). Finally, the boundary of w (denoted by jw) is the set of all elements wi which are interior to neither w nor to its complement a-w.
Let us put it graphically:

 As can easily be seen, v´ (the closure of w´) is a-w, and v (the closure of w) is a-w´. Moreover, the interior w of w is open, and it is the largest open subset of w; the closure v of w is closed, and it is the smallest closed set containing w. Finally, wi is an element of the boundary if each neighborhood of wi intersects both w and w´ (= a-w): it is clear that the boundary of w is identical to the boundary of w´ (= a-w).
What is the epistemological significance of the above notions? I think that a, w, w´, v, v´, w, w´, and jw (= jw´) can be described as follows:
-There is a space a, that is a set of elements that are described by other elements (and minimally there is an element that is described by another element).
-The described set is a closed one, and hence a closure, be it v or v´.
-The descriptor set is an open one, and hence an interior, be it w or w´.
-The boundary jw (= jw´) can be attached to either w or w´, or both; it can also be considered on its own: the situation of the boundary has to do with the closures or interiors of w and w´.

We can hypothetically think up four situations :

a) w € jw € w´
The boundary is attached to neither w nor w´ : consequently both are open, i.e. w and w´.

b) v € v´
The boundary is attached to both, w and w´ : consequently both are closed, i.e. v and v´.

c) v € w´
The boundary is only attached to w, but not to w´ : hence the first is closed -i.e. v-, the second is open -i.e. w´-.

d) w € v´
The boundary is not attached to w, but to w´ : hence the first is open -i.e. w-, the second is closed -i.e. v´-.

If we now consider how subjects and predicates are related to each other in every high level structure, we will find that a)-b)-c)-d) are strictly parallel to A)-B)-C)-D) above :

a)-A) When a logical subject and a logical predicate meet together there is no formal mark in any of them that makes this fact clear: the woman  (a singular 3d person noun) can be the subject of talks, but the stone  (a singular 3d person noun) cannot -the woman talks a lot /*the stone talks a lot -. The grammarian would say that talks  requires a [+human] subject, or that the woman  allows a [+mental activity] verb, or both, but neither of these features is formally marked. Thus the subject and the predicate are open, and the boundary cluster of features -i.e. [+human, +mental activity]- that makes their association possible will remain implicit, although we can also manifest it alone -the woman talks ; -yes, she does, that is, she does  = "activity performed by a female human"-. In conclusion we can say that w € jw € w´, where w and w´ are open, predicts the rective structure.

b)-B) When two syntactic elements are linked by an agreement relationship they usually manifest the formal marks that exhibit that relationship in each of them : she cut herself , he cut himself  (in flexive languages with a rich verbal paradigm the agreement relationship between grammatical subject and grammatical predicate is always carried out by formal manifestations : Spanish el chico∆  € come∆  vs. los chicos  € comen ,"the kid eats vs. the kids eat", etc). Sometimes the formal marker of the agreement relationship is lacking in some of the related items, and even in both of them -the kid∆/s cut∆ the sausage -, but this situation is by no means associable to the one explained in a)-A) because of the obvious immanence of zero allomorphs that some changes help to make apparent (compare the kid∆ cut∆ the saussage  with the man comes  for example). Thus we can say that the grammatical subject and the grammatical predicate always exhibit the formal marks that characterize them and, as they are described by these very marks, they are closed. We can conclude that v € v´, where v and v´ are closed, predicts the agreement structure.

c)-C) Remember that when we discussed textual structure we pointed out that its subjects (topics) and predicates (comments) are neither implicit -as in A)-, nor explicit -as in B)-, but essentially momentary: the word woman is implicitly adequate to become the nucleus of the logical subject of a sentence with to sing, and as an explicit word it will support a 3d person singular agreement relationship with its grammatical predicate -for example the woman sings . On the contrary, we cannot say that woman  necessarily behaves as a topic or as a comment in a given text "Y...woman...Z  ": if we look first at woman, then woman  becomes the topic and Z  the comment, but if we had first taken Y  into account, then woman  would have become the comment and Y  the topic. In other words, in textual structure the topic nature of an element or of a string of elements is a matter of having the observer's mind begin with it, and the comment nature of the other(s) element(s) simply stems from the former as an automatic consequence. Putting it formally we could say that the topic is closed (we limit it by deciding it should be the topic), and that the comment, which lacks any limit and simply describes the former, is open. Notice that in fact the comment is the entire text that follows the topic: in some sense a tale that begins with the topic once upon a time there was a king...  is said to develop it as a comment. In conclusion we can say that v € w´, where v is closed and w´ is open, predicts the textual structure.

d)-D) At a first glance it could seem that the enunciative structure behaves in the same way as the textual structure does, that is, its subject -the focus- being put first, and its predicate -the back entailment- being discovered afterwards. Nevertheless this is the point of view of the researcher, but not that of the speaker: when people emphasize a certain element in an utterance, they do it because their previous assumptions of the world which the entire utterance refers to allowed them to do so. For example, if we know that only Jill,  of the three people Mary, John, and Jill, arrived on time to the party, and somebody has previously said that the one who made it to the party on time was Mary  (or John ), or perhaps there are some clues that favour the belief that Mary  (or John ) was the punctual person, etc, then and only then does it make sense for us to emphasize Jill  and say it was Jill who arrived on time, JILL arrived on time, etc. Thus we can say that the back-entailment is a previous describing element, i.e. an open one, and that the focus is a described element, i.e. a closed one, which allows us to draw the following conclusion: w € v´, where w is open and v´ is closed, predicts the enunciative structure. The turn-taking system of conversation (H. Sacks, E. Schegloff, G. Jefferson, 1974, "A simplest systematics for the organization of turn-taking for conversation", Language, 50, 696-735) behaves in a similar way: the first turn of an adjacent pair (as question-answer, f.ex.) is said to predict the second one (requests prefer acceptance, etc): thus the first turn -an open set- describes the second turn -a closed set- and the entire pair follows the law of enunciative structure.

 

7.2) THE GESTALT SIGNIFICANCE OF THE FOUR MAIN STRUCTURES  

When we examined the low level structure w + w´ = W and its reverse  W + W´  = w´´ we said that they accommodate to the general opposition "closed vs. open" and that their Gestalt significance reads as follows: closed sets behave as figure items that are enhanced by open sets which constitute the ground items. After having got a step further by discriminating four main topological confrontations of closed and open sets, it stands to reason that they have some Gestalt significance too.
When the first Gestalt psychologists looked into the general highlighting problem, they established four laws that collaborate in setting some of the stimuli out in front of other stimuli that remain in the background:

a´) The closure law states that stimuli tend to organize themselves in closed sets. For instance, people tend to join the four points :

together as:

rather than as:

and consequently they "see" a square, and rarely an X.

b´) The equality law states that equal stimuli tend to group together. For instance, we prefer to "see" the following picture made out of black and white points:

as if they were horizontal lines alternating black and white, i.e:

rather than as vertical mixed black-and-white lines as in:

 c´) The proximity law states that near stimuli tend to be grouped together. For instance in:

people usually see narrow rectangles separated by wide gaps as in:

rather than wide rectangles separated by narrow gaps as in:

d´) The good shape law states that stimuli tend to be grouped together following conventional guidelines which establish "good forms" accepted by the community. For instance, when looking at the picture:

people would say that it has been drawn by sketching 1+4 and 2+3, that is:

rather than in the more uncommon ways 1+2 and 3+4, or 1+3 and 2+4 as in:

These preferred groupings cannot be found in the retina, which only receives disorganized light stimuli, but are a matter of the brain: we see what we are mentally predisposed to see, and not the real world whatever it may be.
For this reason the comparison with the formal laws that govern the behavior of each high level syntactic structure prove to be accurate. Thus physical reality has nothing to do with the fact that:

a´)-A) According to rective structure rules some words require certain others: in English people "fall in love with  other people", in Spanish "they do of  them" (enamorarse de). The syntactic side of the closure law would read as follows: signs that mutually complement each other tend to group together.

b´)-B) According to the agreement structure some words agree with certain others: in English adjectives remain unchanged although the number of the noun head varies -red box/red boxes-, but possessives have to agree with the sexual gender of the head noun -the book of Mary is her book/the book of John is his book- , whereas in Spanish we find just the opposite situation caja roja/cajas rojas, but el libro de María es su libro/el libro de Juan es su libro . The syntactic side of the equality law establishes thus that signs with identical external form tend to group together.

c´)-C) As stated above the textual structure advances bits of information according to the ideas that have previously appeared in the text, without any attention paid to the referent. For instance, the newspapers can report a robbery in a store by writing the thieves broke into the store and grabbed some valuable goods from the window  (where goods means "merchandise"), or they could have written the goods of the store decreased because of the recent robbery  (in which case we tend to attribute to the word goods, that now opens the utterance, a more general meaning such as "benefits"). However, both sentences rely on the same referential state of affairs, and this very situation could not have allowed such a distinction in Spanish (because bienes, "goods", does not possess these contextual values), although other specific oppositions can also be thought up in this language (for example, los ladrones robaron una tienda  can mean either "the thieves stole a tent", or "the thieves broke into a store"; accordingly,  the utterance either took place en un camping, "in a camping ground", or en el centro , "in the downtown"). The syntactic side of the proximity law hence establishes that adjacent signs tend to group together.

d´)-D) In the case of the enunciative structure it seems hard to imagine a situation that will not depend on reality as the presuppositions shared by talking people are precisely what highlightens one specific element and makes it the focus of the utterance. Nevertheless, it is worth noticing that the set of presuppositions that constitute the back entailment always remains language specific. For example, if a young man is staying at a friend's home and is told by his friend I'm very busy, but don't worry one of my brothers will show you the town, then the young man would never answer in English If MARY comes with me, we could go dancing tonight. However, in Spanish the highlightening of María  as a focus is quite normal because hermanos  means "sisters and brothers", as an unmarked term, and hence tengo mucho trabajo, pero no te preocupes uno de mis hermanos te enseñará la ciudad  easily allows the answer si MARIA viene conmigo podríamos ir a bailar esta noche. Thus the syntactic side of the good shape law establishes that signs (not referents) which are made to co-exist by the context tend to group together.

Notwithstanding I am by no means intending to state that linguistic laws have nothing to do with reality, I am only saying that, though reality influences them (especially the outer-inner structures, i.e. enunciative and  rective), each human language behaves in its own ways when following these laws. But this can also be said of the Gestalt laws of vision. If our retina can figure out some features of reality as being more closely grouped together than others, it is probably due to the fact that reality itself behaves in this way: for example, if we see a table against the wall it is not only because the light stimuli coming from the table are more similar, closer, etc, to each other than to those coming from the wall, but also because the physical components of the table are more like each other than those of the wall. Whatever epistemological standpoint you may hold I am just pointing out that light stimuli in the retina partially rely on physical properties -i.e stimuli- of reality, whether reality actually exist or not: the same thing can be said of linguistic stimuli.
On the other hand, I would like to point out that the parallelism between Gestalt laws and high level syntactic laws extends over the behavior of both of them, and not only from a theoretical point of view. In the message the four high level structures can agree and mutually reinforce their effects, but they can also disagree. In the case of disagreement, certain structures always prevail: for instance, in the English passive construction textual structure dominates the agreement structure. Hence, starting from John gave them a book  if we want a book  to be the topic, then we will have a book was given to them (by John), but if we want they  to be the topic, then we will have they were given a book (by John), no other situations in which the grammatical subject does not coincide with the textual one would be possible -such as *a book they were given , etc-.
But the dominance of certain syntactic laws over others can be found in the Gestalt principles that govern the behavior of the retina too. For instance in:

 the equality law may tend to dominate the proximity law, and we could therefore see two wide rectangles -a black one and a white one- separated by a narrow gap, rather than colour scheme narrow rectangles separated by wide gaps.

In conclusion we find that the four main high level structures correlate with typical topological situations where a set and its complement share the boundary in four different ways, and they also correlate with the four major laws that govern the grouping together of light stimuli:

syntactic structures     topological situations     gestalt laws 
rective structure         neither w nor w´                          closure law    
                            incorporate the boundary
agreement structure    both w and w´                          quality law                             
                                      incorporate the boundary
textual structure           only w, but not w´                    proximity law
                                       incorporate the boundary
enunciative structure  only w´, but not w               good shape law
                                     incorporate the boundary

This is not surprising since this correlation could have been guessed from the first chapters of this book where linguistics, topology, and psychology were interrelated. In any case, as I will now briefly discuss, these four situations have been the focus of linguistics for the past eighty years.

7.3) THE FOUR MAIN STRUCTURES AND THE STATE OF ART

The purely introductory nature of this book excludes a detailed discussion of the many different approaches that linguists have adopted in recent years with respect to the problem of syntactic structures. Nevertheless, I would like to point out that whenever this subject has been taken up by a specific linguistic school, its advocates have always tended to propose an increasing number of structures that eventually matched the structures A)-B)-C)-D)/a)-b)-c)-d)/a´)-b´)-c´)-d´), curiously enough.
We have already mentioned the fact that traditional grammar proposed four types of subject-predicate relationships (logical, grammatical, textual, or psychological) that correlate quite well with those structures. Nonetheless, such a statement -as can be found for example in M. Sandmann, 1979, Subject and Predicate. A Contribution to the Theory of Syntax, Heidelberg, Carl Winter- was preceded by many attempts to split syntactic complexity into only two or three structures; a division which has been prevalent throughout twenty centuries of grammatical thinking. Sandmann's proposal is thus a summary of all of them.
Generative grammar and structuralism, both having developed in a rather short period of time, allow for the reaching of the four structures quite well. For instance, it is known that while earlier versions of the Praguean school distinguished between a grammatical and a textual structure -J.Firbas, 1958, "Bemerkungen über einen deutschen Beitrag zum Problem der Satzperspektive", Philologica Praguensia, 5, 2, 49-54-, in the sixties they tended to recognize three structures (the semantic level, the syntactic level, and the actual level) -F. Danes, 1966, "A three level approach to syntax", Travaux linguistiques de Prague, I,225-40-, and finally in the eighties they started speaking of semantic / syntactic / textual / pragmatic ways of analyzing an utterance -as S.C.Dik, 1980, Studies in Functional Grammar, New York, Academic Press does- which clearly mirror A)-B)-C)-D).

It would also be of interest to sketch the development of the theory of syntax in Generative grammar from the point of view we are interested in. It is a well known fact that generative grammarians opposed deep structure to surface structure, the first being mostly logical (i.e. rective -A-), the second seeming basically grammatical and language specific (i.e. it constituted the result of applying agreement and order transformational rules that obviously belong to the agreement -B- and to the textual -C- structures).  However, in the early seventies generative grammarians were forced to distinguish a third level where emphatic considerations apply (i.e. the enunciative structure -D-) as well as grammatical and logical ones: Chomsky's 1972 book -Studies on Semantics in Generative Grammar, The Hague, Mouton- constitutes the starting point of this improvement of the theory that all generative scholars, whether they agreed with Chomsky's position or not, touched on. Nevertheless, textual order considerations could no longer keep up with agreement considerations. In the late seventies and early eighties the trace theory, in which grammatical interpretation depends on the order of omitted elements, ceased to treat them simultaneously, giving rise to the four perspectives that characterize the government and binding version -N. Chomsky, 1981, Lectures on Government and Binding, Foris, Dordrecht. Starting from D-structures, which seem to be very close to non-ordered grammatical relationships, syntax generates S-structures that come near to textual ordered relationships, both having a purely internal significance; then one system of interpretive rules associates S-structures with surface representations in PF, and another system with representations in LF, the first having to do with enunciation, the second with q-roles or rective relationships, i.e. with the two external-internal components of the grammar that stand at the interface of grammatical competence and other systems such as conceptual system, pragmatic competence, speech production and so on.
Be that as it may, one wonders whether things could have happened differently. If we are forced to link linguistic signs (that is, verbal stimuli) in a way which is similar to the Gestalt habits of putting visual stimuli together, it is easy to imagine that linguists cannot avoid recognizing these very four structures (or laws) either. If they have not always done so, it was due to the fact that these structures often reinforce each other in the specific language they are taking as a model. 
Recent research in linguistic typology has shown that almost all the notions that scholars are accustomed to employing are a matter of degree -P.J. Hooper and S.A. Thompson, 1980, "Transitivity in grammar and discourse", Language, 56, 251-99-, some languages emphasizing some factors, and others stressing the remaining ones. If generative grammar had not initiated its development with the structures of the English language, or if traditional grammar had not studied the patterns of Latin, or structuralism those of Slavic and Romance languages, then they would perhaps have never limited themselves to proposing only a couple of metalinguistic perspectives. Notwithstanding, these early attempts have proven to be worthwhile, and the present four patterned state of art constitutes an indubitable test of their achievements.