5) LEVELS, DIMENSIONS, AND COMPONENTS

The dimensions of a given space are the parameters we ought to consider when attempting to describe it. For example, it would make no sense to ask for a sheet of paper that is 40 inches long, 20 inches wide, and 3 inches deep: a sheet of paper is a two dimensional object whose length and width, but never its depth, can be considered. Note that the dimension problem has no ontological significance: of course a sheet of paper has a certain depth, but in everyday life it is of no importance.
When we think about the problem of dimensions in linguistics we discover a very similar situation: the question of how many aspects of language are worth describing -whether a description takes into consideration a greater or smaller number of dimensions, it is considered to lack descriptive adequacy- produces more than one answer among linguists.
Another methodological concept which is usually associated with "dimensions" and "components", but that deserves specific attention is the notion of level. It is important not to confuse them as scholars sometimes  do. Two different dimensions means two entities that exist as such and can combine therefore with each other in order to provide a complex dimension, that is, a component. On the other hand, levels follow segmentally and hence the units of each one consist of the units of the subordinate level and become part of the units of the upper ordinate level. No combination of levels can be conceived of  unless the n-mlevel units are functionally equivalent to the nlevel units and therefore belong to the latter.
Let´s look at an example: length and width are dimensions that, when combined, yield a two dimensional component -the area-, but inches and feet are levels of measurement that apply to the same dimension (be it length, width, or depth) and that cannot be combined in order to yield a new component. They can only be added -when we say that something is three feet and two inches long we mean that its length is three feet and part of one foot. 
The confusion between dimensions and levels in linguistics is probably due to the fact that human languages are multi-articulated systems, as we know. Phrases and sentences are levels, not dimensions, and in fact both can apply to the same syntactic element. Therefore we can add phrases to phrases in order to obtain sentences, or even add phrases to sentences, but phrases never combine with sentences. On the contrary, total-sound and sound-meaning are dimensions whose combination yields the morphological component, i.e., the component where units are provided with characterization of sound and meaning at the same time. 
Nevertheless we can break down sentences into phrases or phrases into words, but also words into morphemes or morphemes into phonemes: by doing so, succesive levels apparently yield new dimensions and components, that is syntax > morphology > phonology. However, this conclusion does not hold: in fact any component turns out to be operative at any level. Thus there are certain phonological units, such as intonational patterns that extend over the entire sentence, or morphological phenomena, such as agreement morphemes, that link the subject noun to the predicate verb, as elements of the sentence. We will conclude by saying that dimensions and components are points of view, whereas levels are units of measurement. Linguists are concerned with both, no matter what kind of assumptions they may make on the nature of human language, and regardless of the names they give to them (for example, in generative grammar the so-called phonological vs. syntactical "levels" are obviously "components", i.e "complex dimensions").
Returning to the problem of dimensions, you will notice that it is related to the question of how many components need describing. For example, when considering a block it would only be relevant to talk about the entire volume or about any of its three sides, as no other domain has any real significance, but this restriction relies on the fact that the three dimensions of space of a block -l:length, w:width,d:depth- can combine either lwd, or dw, or lw, or ld. 

We said earlier that language together with its metalanguages constitutes a topological space. Consequently the dimension problem applies to language as it does to any spatial object, and we must therefore first solve the problem of determining how many linguistic dimensions there are, their nature, and how they can be combined to produce methodological perspectives, call them "components" of the grammar.
Topological procedures will help us here too: it is said that the number of dimensions of a space is always one unit higher than the number of dimensions of its boundaries. For example, blocks are limited by planes and, knowing that planes are two dimensional, we could infer that blocks must have three dimensions; likewise two dimensional planes are limited by one dimensional lines, and one dimensional lines by zero dimensional points. If we did not intuitively know this, geometrical dimensions could be computed by following the boundary process to its end as shown below:

volume   >>>   plane   >>>   line >>>   point
3 dim.  <<<<<   2 dim.<<<<<  1 dim.<<<<< 0 dim. 

When we consider the space of language it is by no means obvious how many dimensions it should have, and hence we must proceed as above. Notice that in linguistics the limitation or definition process is conveyed by metalinguistic descriptions: when we say that a certain sentence has such and such structural characteristics we are able to distinguish it from other sentences of the corpus in the same way that a segment belonging to a line is said not to mix up with other segments on the same line because it is limited by two points. But this metalinguistic limitation process can be carried out exactly three times without needing to repeat it to infinity:

            1                             2                     3
red  is an adjective >> adjective  is a noun >> noun  is a noun
3 dim........(2 dim) <<<< 2 dim.........(1 dim) << 1 dim....(0 dim) 

Consequently we can say that the linguistic topological space is three dimensional: linguistic signs have three dimensions, but specific metalinguistic elements or linguistic elements metalinguistically employed  use only two of them when applied to the former, and even one when applied to themselves (this is only conceivable with the label "noun", of course).
What are the dimensions we are trying to discover? If you keep track of the types of elements we distinguished in the topological linguistic space before quotient topology was introduced, you could easily deduce that sound-meaning, partial-sound, and total-sound are precisely the dimensions we are looking for. Nevertheless, this conclusion cannot be drawn without arguments to support it, because it could be said, for instance, that the social reaction brought about by the sign, the grade of emotion it carries, and the physical alterations of the air stream it is responsible for, are the three dimensions of linguistic space, and this would prove to be numerically adequate too (this does not mean that linguistics is not interested in the exterior links that signs maintain with other systems, but they must be established through a product topology as we said above, and not by means of a quotient topology). 
We can discover what these dimensions are by looking into the process of dimensional loss outlined above:
i) The step "red  is an adjective" cannot be accomplished without putting the grammatical value or partial-sound dimension of the sign "adjective" in brackets. When we say "red  is an adjective", we use the total-sound /ædjektiv/ with its grammatical sound-meaning "adjective", but we ignore the fact that its grammatical value is not 'adjective' but 'noun' as any other metalinguistic label. Consequently we can say that the partial-sound is precisely the first dimension that metalinguistic discourse ignores, and hence one of the three dimensions we are trying to discover. 
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ii) The step "adjective  is a noun", on the contrary, leaves out the sound-meaning dimension "adjective" of the sign adjective, because we agree that it is grammatically speaking a noun, even though its referential meaning is "adjective" (notice that surprisingly enough this referential meaning is found in the grammar and is, thus, grammatical too). The second dimension of linguistic space we are interested in is sound-meaning.
iii) The step "noun  is a noun" is obviously a tautological one where only the identity relation A=A appears. Notice that such a formal property can only be said of intensional classes (whether of partial-sound as a grammatical class, or of sound-meaning as a conceptual class in our example), but never of individuals, because empirical appearances of stimuli are always different from each other. Consequently we can say that the total-sound, the unique physical dimension of linguistic signs, is lost in this step and constitutes the third factor we were trying to find.
These three dimensions of linguistic signs are not the same: whereas total-sound and sound-meaning each have a linguistic aspect, but also a non-linguistic support (the articulatory and acoustic properties of sounds, and the physical or psychological referent, respectively), the partial-sound leans on the former two, and thus is only accessible through a previous examination of both. The partial-sound dimension cannot be considered a fact, but only a relationship; it is not real, but virtual. For this reason we have represented it by means of a dotted line, and the other two, that properly give rise to it, by continuous lines:

At this point we can recognize three main components of the grammar, each consisting of the two axes that determine its plane as opposed to the remaining axis :  

                                                                                                                                               

The above scheme will be read as follows:
I) phono-syntax is a two dimensional component that deals with the functions or relationships -i.e., partial-sound aspects- of total-sounds according to the sound-meaning effects they are able to manifest. Notice that when we oppose the Agent meaning of Paul  in Paul saw Peter  to the Patient meaning of this very Paul  in Peter saw Paul  we are making use of two differential positions in the chain -i.e.,the total-sound patterns- that also deliver a meaningful opposition precisely because they have a functional significance: total-sound differences without partial-sound effects do not permit the deduction of any meaning effects (compare the above situation with Latin Paulus Petrum vidit  @ Petrum Paulus vidit , etc). Nevertheless the extent to which phonological considerations apply is not fixed: we oppose the total-sound /kon'tent/ with its partial-sound 'adjective' to the total-sound ['kontent] with its partial-sound 'noun' in order to distinguish the meanings "content1" (@ "happy") and "content2" (@ "conceptual value" ). Similarly we say that [t] and [p] are two phonemes of the English language, that is two total-sound + partial-sound (functional) units, because by substituting one for the other we can obtain two different sound chains -[pot] vs. [pop]-, each with its corresponding functional grammatical value -i.e., pot :'countable noun', pop : 'not countable noun'- that show different sound-meaning.
The entire methodology of structural phonology is outlined in I) statement, but it is worth noticing that, contrary to common opinion, grammar (i.e.,. partial-sound) must always be present: two signs that share the same grammatical properties and whose respective total-sounds differ in some respect cannot give rise to any meaning differences but only to dialectal ones -compare A.E. and B.E. pronunciations of the word better  for instance.

II) semo-syntax is a two dimensional component that deals with the functions or relationships (i.e. about the partial-sounds) of sound-meanings according to the total-sound patterns that determine them. In the domain of human languages no meaning differences can be found that lack a total-sound patterning, but at the same time these meaning differences are of a grammatical nature: if not, they only show dialectological significance -compare the connotations of the word boy  in A.E. and in B.E. for example-. As above, notice that functional sound-meanings can be identified by referring to large signs -for instance the verbal phrase scheme- or to smaller ones -to a single word-, and both syntagmatically or paradigmatically (a large paradigmatic sound-meaning of English would be "adjective", a smaller one "kinship term").

III) morpho-syntax is a two dimensional component that deals with total-sound and sound-meaning associations according to their grammatical values. This component is the most language specific of the three. Hence human languages always differ in morphosyntactic aspects, despite sharing many total-sound or sound-meaning characteristics. What we call a [p] or a "kinship term" in English is not entirely equivalent to Spanish [p] and "pariente", but they coincide in some respects: nonetheless English 'indicative' has nothing to do with Spanish 'indicativo' .
 Since morpho-syntactic categories are purely grammatical, they only apply to the entire domain of the sentence or phrase, that is, to the grammatical patterns which result from adding minor lexical units together. Semo-syntactic categories, on the contrary, are tied to the referent and apply to single words. The reader may perhaps be surprised by the fact that we consider "adjective" a semo-syntactic notion instead of a morpho-syntactic one, despite the fact that we called it a partial-sound feature some pages back. Nevertheless, I would like to pay close attention to the difference that exists between dimensions and components: all three components (phono-syntax, morpho-syntax, and semo-syntax) of the grammar we have just introduced are of a functional grammatical nature  and, consequently, manifest the partial-sound dimension. As dimensions total-sound and sound-meaning must relate to reality, whether acoustic or referential, and belong to paradigmatic rather than syntagmatic components: we will discuss them when we deal with the question of coordination later on. For the time being, I would only like to point out that each of the real axes (total-sound and sound-meaning) gives rise by itself  to a paradigmatic component where certain data taken from the real world are combined in order to create the minimal units of description, whereas the virtual axis has to take its empirical pieces of information from the patterns previously generated by grammar. This fact explains how "adjective", although it is a semo-syntactic category, appears as the partial-sound 'adjective' in signs.
The paradigmatic components we have alluded to are as follows:
IV) phonematics is a one dimensional component that relies on total-sound axis and whose units -the phonemes- are originated as a result of combining data which come from acoustic and/or articulatory reality.
V) semematics is a one dimensional component that relies on sound-meaning axis and whose units -the sememes- are originated as a result of combining data that come from referential or mental reality.
VI) morphematics is a one dimensional component that relies on the partial-sound axis and whose units -the morphemes- are provided by I)-II)-III), that is, by the two dimensional components of the grammar.

The disciplinary division outlined above represents only a first step in the metalinguistic process of describing a language. Remember that, as we said earlier, the virtual axis (partial-sound) is the result of the relationship established between total-sound axis and sound-meaning axis, but it has no existence on its own. If we now take into account the sound-meaning axis, then it is easy to understand that the confrontation of sound and meaning inside of it will yield a relationship that is a secondary virtual axis too. This secondary partial-sound axis is responsible for a secondary subcomponent division which only appears in the components that are produced by this sound-meaning axis, that is, in the morphosyntax and in the semosyntax :

1.- Phono-syntax
2.- Morpho-syntax
2.1.- Morphophonemics
2.2.- Derivate Morphology
2.3.- Inflectional Morphology
3.- Semo-syntax
3.1.- Formal Semo-syntax
3.2.- Actantial Semo-syntax
3.3.- Constructional Semo-syntax

These subcomponents, that allow us to go a step further in the description of morphosyntactic or semosyntactic facts, do not exclude the previous II) or III) types, but are added to them. They would look as follows :
II') formal semosyntax aims to describe the semosyntactic functions that are dependent on purely formal facts, for example the subject and object functions that respectively precede and follow the verb, etc. Notice that this component is limited by two internal semosyntactic axes and by the partial-sound axis.
II'') actantial semosyntax deals with the aspects of semosyntactic behavior that deserve semantical consideration, for example the Agent, Instrument, Experiencer, etc. roles of noun phrases. It is limited by two internal semosyntactic axes and by the sound-meaning axis.
II''') constructional semosyntax is limited by purely internal semosyntactic axes, and hence it will describe semosyntactic facts that depend neither on the referent, nor on the formal pattern of the sentence.  For example, the active/passive opposition has nothing to do with the real world and, on the other hand, is carried out by morphological distinctions, but not by the positional patterns of the sentence.
III') morphophonemics or phonological morphosyntax aims to describe the phonological patterns of alternance of morphemes, that is to say, the aspects of morphemic behavior that have phonological significance because it is limited by two internal morphsyntactic axes and the total-sound axis.
III'') derivative morphology or semantic morphosyntax aims to describe the semantic values of word formation procedures (affixes, etc), that is, the aspects of morphological behavior that have to do with semantics because they are limited by two internal morphosyntactic axes and the sound-meaning axis.
III''') inflectional morphology or morphological morphosyntax, because it is only limited by internal morphosyntactic axes, deals with the aspects of morphological behavior that are unrelated to other types of axes, that is, the grammatically conditioned allomorphs of morphemes.

In order to understand what I am really proposing with this disciplinary division let me give you an example. The grammatical description of the well known (almost) Sapirean sentence the farmer killed the duck  could not be achieved without pointing out the following facts :
i) Phonosyntax. The phonemes defined by the Phonematic component do not all behave in the same way: some of them can occupy the prenuclear and postnuclear syllabic position (thus [f-] in farmer  and [-f] in laugh ), but others cannot ( [d-] in the  in front of *[-d]); there are major pauses before the farmer  and after the duck, and a minor pause between farmer  and killed, etc.
ii) Semosyntax. Killed  is a verb, duck  is a noun, the  is a determiner, etc. Notice that semosyntactic labels only apply to minor constituents as we said above.
iii) Morphosyntax. Killed  is a past form. On the contrary, although linked with some specific forms, these labels apply to all of the elements of the sentence: not only to kill, but also the farmer  and the duck  refer to entities in the past.
iv) Phonematics. The phoneme [k] consists of a [+stop] feature, a [+velar] feature, and a [+voiceless] feature, etc. Notice that by defining a phoneme by means of a set of features, we are really listing its coordinates, in the same sense as we do when we say that a point is the [+3,-2] one in a Cartesian space.
v) Semematics. Similarly we could define kill  by the semic coordinates [+action], [+physical], [+directed towards a being], etc.
vi) Morphematics. The list of morphemes and allomorphs that result from the grammatical analyses, that is the, kill-,-ed, farm-,-er, etc.

ii') Formal semosyntax. The "subject" function of the farmer  and the "object" function of the duck  are both manifested by the respective positions of these words.
ii'') Actantial semosyntax. The "Agent" value of the farmer  and the "Experiencer" value of the duck  are labels that are referentially tied.
ii''') Constructional semosyntax. The "transitive" and "active" nature of the entire sentence. 
Notice that ii'), ii''), and ii''') represent a subspecification of the analysis undertaken by ii) because, according to ii), farmer  is a "noun", and can be the "subject" -ii')-, or "Agent" -ii'')- and, because of the general meaning of the "verb" -ii)-  to kill, the above sentence can be "transitive" -ii''')- etc.
iii') Morphophonemics. The [+voiced} performance of the past morpheme in killed  -i.e., ['kild]- as opposed to the [+voiceless] performance of the same morpheme in looked  -i.e.,. ['lu:kt]- for example.
iii'') Derivative morphology. The [+doer] value of the -er  suffix in farmer.
iii''') Inflectional morphology. The fact that the verbal form of the English simple past is unmarked -I/you/he/she/it/we/you/they killed  - as opposed to the present continuous, for example, which uses a form of the verb "to be" in agreement with the subject pronoun.
As above, it is valuable to realize that only after iii) has been established can we establish iii'), iii''), and iii'''): it is because killed  is a past form -iii)- that -ed  constitutes an allomorph -iii''')- whose phonetic performance is [-d] -iii')- (compare single-bed  where -ed  is not an allomorph and its performance is not [-d]); it is because of the "noun phrase" nature of the farmer  -iii)- that the root farm  can take the suffix -er  (compare with John farms potatoes ).
 Earlier I repeatedly pointed out that any topological approach to language cannot help but have a Gestalt significance. Take heed of the fact that this is also the case in the disciplinary division I have just proposed. When painters paint a landscape they never reproduce reality as such: each painter selects certain features of the referent he/she is perceiving and leaves the rest in the background; hence their paintings are always different although they talk about  the same things. When describing a corpus linguists do it in a similar way: some of them emphasize the total-sound + partial-sound dimensions and leave the sound-meaning axis as a background that helps to make the other two stand out; a procedure known as phonology. If, on the contrary, they had emphasized the partial-sound + sound-meaning dimensions and had only outlined the total-sound dimension, then they would be practising another procedure called semosyntax, and so on. 
One could think that this comparison is not a very solid one as painters are artists and their pictures always have an individual value, whereas linguists aim to be scientists that describe regularities found in human languages. Nevertheless, paintings are also made according to general techniques that select certain features of reality over others -impresionism, expresionism, etc-, even if each painter uses them in a strictly individual manner. In the same way linguists must learn general techniques (the so-called components of the grammar) that will determine the general point of view of their descriptions (whether they are talking about total-sound or sound-meaning, etc), although each linguist is always free to carry out his/her description in a way that allows him/her to perform individually. As in painting or other arts, linguistics lies halfway between natural sciences and humanities, and thus shares the properties of both: in short, linguistics is an empirical and a hermeneutic discipline at the same time. 
A further question that could be raised is whether the above mentioned components constitute all the points of view we can adopt with regard to the sentence the farmer killed the duck , or if there are still other problems we would have to talk about. Someone might notice, for example, that by saying the farmer killed the duck  we are presupposing the proposition "the duck was alive" was true before, or that this sentence can be followed by another utterance, such as and even two ducks .
 Other readers would possibly say that, if the above statement were expressed the day before Thanksgiving in the U.S.A., it could easily be echoed by but the typical dish is turkey, etc.
There are several suggestions that could be added to our previous analysis of the farmer killed the duck. However, the observations made in the first period above, and those made in the second, are of an entirely different nature. The latter have to do with socio-cultural contexts of use of this sentence, and would therefore belong to sociolinguistics. The former, on the other hand, are independent of the specific context where the sentence is used and can be derived from general laws: it is a "presupposition" of the verb to kill  that the killed being was previously alive; one of the characteristics of "conversational implicatures", as stated by H. P. Grice -1975, "Logic and Conversation", in Cole and Morgan, Syntax and Semantics 3: Speech Acts Academic Press, 41-58-, is that they can be cancelled, and for that reason the "quantity principle" gives rise to the removal of the number of ducks from the sentence.
In the first case the descriptum-descriptor relationship has been reversed: in sociolinguistics we cannot say that sociological facts explain linguistic ones, but we can say that linguistic phenomena are used as pointers that underline specific sociological events. In these cases linguistic utterances, as a whole, only deserve metalinguistic significance. Although they are of obvious interest, sociolinguistics, psycholinguistics, and natural logic are interdisciplinary sciences in which linguistics has only a small role to play.
The second case, that of presuppositions, conversational implicatures, or performative verbs, is a very different one. Although these notions always have to do with the context, they are of a general nature and can be stated as rules or laws of language and language items, and they need not rely on other parts of reality.
We will call pragmatics the general component of grammar that studies the relation of language to its context. Despite the fact that grammar has many components of purely internal significance, we will distinguish only three main branches of pragmatics, only those in which language  relates directly to context, whether through its articulatory/acoustic aspect, or through its referential aspect, or, finally, through its entire syntactical system:

 Pragmatics, as any other component of grammar, is interested in the description of language as a  figure, that is, as a closed set. But in this case it does not take advantage of part of the linguistic set as a base -i.e., as an open set-, but benefits instead from the data of the context in order to play a descriptive role. External linguistics (natural logic, sociolinguistics, psycholinguistics, etc) operate in the opposite way by constituting the context as the object of study and language simply as its descriptor.
The three subcomponents of pragmatics would read as follows:
VII) phonic pragmatics is acoustic phonetics, that is, a physical science interested in the physical properties of linguistic sounds that phonematics makes use of.
VIII) lexical pragmatics is a psychological and anthropological science interested in how the referents become signs of a given language. As can easily be seen, this branch of pragmatics has a strong epistemological significance.
IX) communicative pragmatics relates language as a system (or as the component of grammar in which minimal elements are joined in the sentence, that is, syntax) to its context of use. It is this part of linguistics that scholars refer to as pragmatics.