7 formal laws of quotient topology and their interpretations with respect to the high levels : the four main structures



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Topological linguistics

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
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