Diversification

1. Problem and history

Diversification is a process of enlarging the number of forms or meanings of any linguistic entity. It can be paradigmatic, e.g. the rise of cases, numbers, tenses, etc., phono-morphemic, e.g. the rise of allophones, allomorphs etc., geographical, e.g. the increase in the number of different expressions of a concept, social, e.g. the rise of different words or meanings of a word or different pronunciations, idiolectal within a community, semantic, e.g. the increase in synonymy and polysemy, contextual, e.g. the increase in the usage of a unit in different contexts. Diversification comprises a number of phenomena dispersed in this volume.

For the sake of illustration let us show some concrete examples:

1) The word can enlarge its class membership without any change, e.g. through conversion: “the hand”, “to hand”.

2) The stem enlarges its class membership through derivation, e.g. German "Bild", "bilden", "bildhaft", or vocalization in Semitic languages, etc.

3) The stem can enlarge its applicability within one class through derivation e.g. German "Blut", "Blutung", "Bluter", or through vocalization, etc.

4) The stem can enlarge its applicability within one class through compounding e.g. "Blut", "Blutdruck", "Blutdurst", etc.

5) If a language abandons isolating morphology, then morphemes diversify into several morphs because of agglutination or inflection (sequential or syntactic dependence).

6) The word can enlarge its applicability in the sentence by acquiring several functions, i.e. it enlarges its dispositional properties, which are different from the constant grammatical properties, e.g. practically every word can become the subject of a sentence.

7) Verbs can enlarge their valence, i.e. their combinability with different cases.

8) The word can enlarge its cotextuality (cf. Köhler 1986), i.e. its ability to occur in several contexts (where "context" can be defined in several ways). The reverse of this kind of diversification process is part of style formation, where a "position" diversifies, i.e., a position in a given context can be filled with different units (words, sentences, etc.).

9) A concept can be expressed by different forms, giving rise to dialects, sociolects, idiolects, or to synonymy.

10) A word can acquire different meaning (polysemy).

11) Every word can acquire different associations (connotations).

Diversified entities abide by a ranking law, i.e. if the members of the diversified entity are ordered according to their frequency, then the frequencies are “lawfully” connected.

The factors of diversification can be as follows (Altmann 2005):

a) Random fluctuation which is omnipresent in any language phenomena.

b) Environmentally conditioned variation forcing an element to acquire different forms or meaning nuances in different environments.

c) Conscious change through conscious creation, borrowing, emotionality etc.

d) Self-organisatory triggering of a process to a limit, causing changes on other levels, too.

e) System modification joined with local or global modifications in a subsystem,

f) Köhlerian requirements (Köhler 1986, 1987, 1989, 1990, 1991) forcing to take into account collateral pressures form different sides. They are as follows: (i) The trend for minimal coding and deciding effort, (ii) sufficient redundancy, (iii) the coding requirement in general, (iv) context economy vs. context specificity, and (v) invariance vs. flexibility of relation between expression and meaning.

The concepts of diversification and unification go back to G.K. Zipf (1935, 1949). Together they are called “Zipfian processes”. The scope of the phenomena is enormous. Semantic phenomena have been examined by Beöthy and Altmann (1984a,b, 1991), Altmann (1985a), Altmann, Best, Kind (1987); grammatical phenomena were analyzed in the omnibus volume edited by Rothe (1991), where also a study on spelling errors in English can be found, and dialectal diversification was studied by Altmann (1985b).

The laws hold usually for ranked nominal classes of limited size.

2. Hypothesis

Every linguistic entity diversifies, i.e. it generates variants and secondary forms and acquires membership in different classes. The ranked frequencies of individual entities abide by a rank-frequency distribution (or a rank-frequency series).

A “rank-frequency distribution” (series) is a function expressing the decrease of frequencies ranked according to their magnitude. There are, eo ipso, no bell-shaped rank-frequency distributions.

“Variants” are all free or conditional “non-standard” forms of the entity, e.g. allophones, allomorphs, dialectal or sociolectal expressions of a concept, etc.

“Secondary forms” are in some way derived from the primary form, e.g. secondary meanings (polysemy), cases, times, moods, aspects, etc.

“Classes” are built by a class-building criterion, e.g. derivates, compounds, declination classes, word classes (Wortarten), even semantic classes, etc.

Corollary: If the above hypothesis holds, then the frequencies of elements of a linguistic class are not distributed uniformly.

In a “uniform distribution” all frequencies are equal. The corollary is rather a well corroborated inductive generalization. Some theoretical rank-frequency distributions can result in the discrete uniform distribution for special values of parameters but they are not actual in linguistics.

3. Derivation

3.1. Altmann´s approach A (1991).

Because the entities are ranked and because of the corollary, it is true that for the probabilities of classes it holds that

$P_x\le P_{x-1}$

Since $P_x$ and $P_{x-1}$ ($x$ = 2,3,…) are joined in a law-like manner, we can write

(1) $P_x=g(x)P_{x-1}\quad$, where $g(x)\le 1\quad$.

Furthermore, $g(x)$ can be written as

$g(x)=\frac{f(x)}{h(x)}$,

where $f(x)$ is a function composed of a language constant $a$ and the diversifying effect of the speaker $bx$, i.e. $f(x) = a+bx$, while $h(x)$ contains the controlling, regulating effect of the hearer (community) $cx$, i.e.

$g(x)=\frac{a+bx}{cx} \le 1$, (a, b, and c are assumed positive),

so that

(2) $P_x=\frac{a+bx}{cx}P_{x-1}$.

In order to obtain a known distribution, one can reparametrize (2) by writing $a/b = k-1$ and $b/c = q$, and solving (2) for $P_x$. One obtains

(3) $P_x=\begin{pmatrix}k&+&x&-&1\\&&x\end{pmatrix}\frac{p^kq^x}{1-p^k}, \quad x=1,2,3,...$

yielding the zero-truncated (positive) negative binomial distribution. The condition $g(x)\le 1$ is fulfilled if $kq\le 1$. Using (1) Altmann (1991) showed a number of other possibilities of obtaining a diversification distribution.

3.2. Alternative derivation (Altmann 1985b)

For the purposes of dialectal variation captured in terms of numbers of lexeme variants on maps of a dialect atlas, Altmann (1985) used the birth-and-death process based on the following assumptions:

(a) In a time interval Δ$t$ the birth of a new variant is proportional to the length of the interval, i.e. $a$Δ$t$.

(b) The assertion of a variant against $x$ rivals is propotional to the number of rivals and the length of the interval, i.e. $bx$Δ$t$.

(c) The death of a variant is proportional to the number of variants and the length of the interval, i.e. $cx$Δ$t$.

(d) No change (birth, death or assertion) in Δ$t$ is given as the complement to the above changes: 1 – [$a+(b+c)x$]Δ$t$, ignoring intervals smaller then Δ$t$.

(e) The events are independent and the probability of more then one event in the interval is zero.

Thus the probability that there are $x-1$ variants and a new variant arises or asserts itself against $x-1$ rivals is

$a\triangle tP_{x-1}(t) + b(x-1)\triangle tP_{x-1}(t)$;

the probability that there are $x+1$ variants and one dies is

$c(x+1)\triangle tP_{x+1}(t)$;

the probability that nothing happens in Δ$t$ is

${{1-[a+(b+c)x]\triangle t}}P_x(t)$.

Putting these probabilities together we obtain the probability that in the interval ($t$, $t$+Δ$t$) there will be exactly $x$ variants as

$P_x(t+\triangle t) = [a+b(x-1)]\triangle tP_{x-1}(t) + c(x+1)\triangle tP_{x+1}(t) + {1-[a+(b+c)x]\triangle t}P_x(t)$.

Substracting $P_x$ from both sides and dividing them by Δ$t$, we obtain

$\frac{P_x(t+\triangle t)-P_x(t)}{\triangle t}= [a+b(x-1)P_{x-1}(t)+c(x+1)P_{x+1}(t)-[a+(b+c)x]P_x(t)]$.

Letting $\triangle t\rightarrow 0$ we finally get

$\frac{dP_x(t)}{dt}=[a+b(x-1)P_{x-1}(t)+c(x+1)P_{x+1}(t)-[a+(b+c)x]P_x(t)]$

$\frac{dP_0(t)}{dt}=cP_1(t)-aP_0(t)$

Solving the balancing equations holding for the steady state

$-aP_0+cP_1=0,\quad$

$-[a+(b+c)x]P_x+[a+b(x-1)]P_{x-1}+c(x+1)P_{x+1}=0, \quad x\ge 1,$

and setting b/c = q and a/b = k results again in the negative binomial distribution

(4) $P_x=\begin{pmatrix}k&+&x&-&1\\&&x\end{pmatrix}p^kq^x, \quad x=0,1,2,...$

For dialect maps, (4) is to be understood as the probability that the basic lexeme has $x$ variants, i.e. if on a map there is only one unique form, then $x$ = 0.

Example: Goebl´s law (dialectal diversification)

Goebl (1984) studied the dialect maps of North West France and Italy and brought the distribution of the numbers of variants in the atlases. Since dialectal variants of a concept arise by a birth-and-death process, the number of maps containing $x$ variants must follow the negative binomial distribution. One of these distributions is shown in Table 1 (Fig. 1).

Fig. 1.Fitting the negative binomial distribution to Goebl´s data

Example: Beöthy´s law (semantic diversification)

According to this law the ranked frequencies of the elements of a semantic class are distributed according to (3) or (5) (see below). Rothe (1991c) brings a survey of semantic classes abiding by these laws. Testing has been perfomed for meanings of different Hungarian verbal prefixes (Beöthy, Altmann 1984a,b, 1991), Slovak verbal prefixes (Nemcová 1991), the Japanese postposition ni (Roos 1991), German compounds (Raether, Rothe 1991), the German particle von (Best 1991), the German preposition auf (Fuchs 1991), the English preposition in (Hennern 1991), the Polish preposition w (Hammerl, Sambor 1991), Russian conjunctions a and no (Kuße 1991), the French conjunction et (Rothe 1986), the German genitive (Rothe 1991b), word class distribution in Latin, German and Chinese (Schweers, Zhu 1991), in German (Best 1994, 1997b, 2000b, 2001e; Hammerl 1989; Judt 1995), in Arabic (Altmann 1991a), in Portuguese (Ziegler 1998, 2001), in French (Judt 1995), spelling errors by Japanese English-users (Rothe 1991c), word building patterns in Early High German (Best 1990). In the example (Table 2, Fig. 2) one finds the ranked distribution of German neologisms of the type “Noun + Noun” categorized in 13 groups from Raether, Rothe (1991).

The result shows that nominal classifications of language entities abide by this type of diversification law.

Fig. 2. Fitting the positive negative binomial distribution (3) to Raether-Rothe data

3.3. Hřebíček ´s approach (1996)

Hřebíček used two assumptions:

(i) The logarithm of the ratio of the probabilities $P_1$ and $P_x$ is proportional to the logarithm of the classe size, i.e

$\ln(P_1/P_x)\propto\ln x\quad$

(ii) the proportionality function is given by the logarithm of Menzerath´s law ($\rightarrow$ Hierarchy), i.e.

$\ln(P_1/P_x)=\ln(AX^b)\ln x\quad$,

yielding the solution

(5) $P_x=P_1x^{-(a+b\ln x)}, \quad x=1,2,3,...$.

If (5) is considered a probability distribution, then $P_1$ is the norming constant, otherwise it is estimated as the size of the first class ($x$ = 1). Since the frequency of the first class $x$ = 1 is decisive for the form of the distribution, one usually ascribes it a special value α, modifying (5) as

(6) $P_x=\begin{cases}\alpha, & x=1\\\frac{(1-a)x^{(a+b\ln x)}}{T}, & x=2,3,...,(n)\end{cases}$

where $T=\sum_{j=2}^nj^{-(a+b\ln j)}$, 0 < α < 1, $a,b\in\mathfrak{R}$ so that $P_x$ converges for $n\rightarrow\infty$. This version corroborates again the relevance of Menzerath´s law ($\rightarrow$). Distributions (5) or (6) are called Zipf-Alekseev distributions. If $n$ is finite, (6) is called modified right truncated Zipf-Alekseev distribution (see Wimmer, Altmann 1999). Even though (3) and (5) are quite different, it can be shown that they are special cases of the Siromoney-Dirichlet distribution

(7) $P_x=\frac{a_xe^{-\theta b_x}}{f(\theta)}, \quad x=1,2,3,...$ $f(\theta)=\sum_{j=1}^\infty a_je^{-\theta b_j}<\infty$

(i) If $a_x = k^{(x)}/x!, b_x = x, e^{-\theta} = q\quad$, we obtain the positive negative binomial distribution with parameters ($k$,$p$) ($q = 1-p$);

(ii) if $\theta = 1, a_x = 1, b_x = (a+b \quad\ln \quad x)\ln x$, we obtain the Zipf-Alekseev distribution ($a,b$);

(iii) the 1-displaced negative binomial distribution, which would be obtained with the conventional displacement of (4), would result if $a_x = k^{(x-1)}/(x-1)!, b_x = x-1, e^{-\theta} = q\quad$.

Formula (7) admits to the development of further theoretical approaches (see Wimmer, Altmann 1999).

Example: Association law

The connotations of a word diversify because everybody can have different associations. Nevertheless, within a community of speakers, they are distributed in a very regular way suggesting a background mechanism which can be captured as a law. In the dictionaries of word associations (see e.g. Palermo, Jenkins 1964), the responses to a stimulus word are ordered according to the number of test persons that gave the same response, i.e. they are ranked according to their frequency of occurrence. The persons tested are usually classified according to age, sex, education, occupation, social status etc. Quantitative modelling began most probably in Horvath (1963) and continued in Haight (1966), Haight, Jones (1974), Lánský, Radil-Weiss (1980) who used the logarithmic, the Yule, the Borel and the Haight-zeta distributions, none of which gave satisfactory results. Dolinskij (1988, 1994) proposed the Zipf-Alekseev distribution, Altmann (1992) added the 1-displaced negative binomial and modified the Zipf-Alekseev distributions. In Table 3 (Figure 3) one finds the fitting of the Zipf-Alekseev distribution to the rank-frequency of associations of the word “high” (4th grade, male) as given by Palermo, Jenkins (1964).

Table 3

Fitting model (5) to the associations of the word “high” (4th grade, male)

given by Palermo, Jenkins (1964)

The result represents a perfect fit that has been found in all cases of associations.

Fig. 3. Fitting the Zipf-Alekseev distribution (5) to the word associations of “high”

4. Author: U. Strauss, G. Altmann

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