# Polysemy

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1. Problem and history

The simplest problem connected with polysemy is its frequency distribution in the dictionary. One can easily see that in a monolingual dictionary lexical items have different number of explications. Sometimes the explications are numbered and indicate meaning difference or nuance. This is the only source of approaching the problem. The number of meanings of a word is sometimes called semantic size or semantic volume (cf. Tuldava 1998: 119). Some reserchers stated that the mean semantic volume of different word classes differs significantly (cf. Višnjakova 1976; Tuldava 1979; Ufimceva 1968: 89) Historically, G.K. Zipf (1935, 1945a, 1949) was probably the first who has shown that meaning is associated with frequency, with length and other properties analyzed in synergetic linguistics (see Vol. 2). Nevertheless, the distribution of the semantic volume can be studied separately (see Introduction, Chapter 2), since it represents a loop in the system. Another question is the frequency of polysemic words in text. The first trials to find a theoretical distribution can be found in Krylov, Jakubovskaja (1977), Krylov (1982) who derived the geometric distribution, Tuldava (1979) who used on empirical grounds a kind of exponential function and Levickij, Drebet, Kiiko (1999) who used rather the mixed geometric distribution, assuming that the words in the dictionary are of a mixed character. A good survey of problems can be found in Hoffmann (2001).

2. Hypothesis

The number of words ($f_x$) with x meanings is a function of x.

3. Derivation

3.1. Waring distribution

Wimmer and Altmann (1999a) considered the building of polysemy as a simple Poissonian birth-and-death process and set up the equations

(1)$\lambda_0 P_0 = \mu_1 P_1\quad$

$(\lambda _x + \mu_x)P_x = \lambda_{x-1}P_{x-1} + \mu_{x+1}P_{x+1}, \quad x= 1, 2,...$

Inserting $\lambda_x = a+x, \mu_x = a+b+x$ and displacing the distribution one step to the right – because zero-semic words are not taken into account - they obtained the Waring distribution

(2)$P_x = \frac{ba^{(x-1)}}{(a+b)(a+b+1)^{(x-1)}},\quad x=1, 2, ...$

3.2. Bissinger-geometric distribution

Since the Waring distribution can be presented by a backward recurrence formula, Wimmer and Altmann (1999a) consider also the possibility of forward dependence using a parent distribution {Px* }x≥0 and building partial sums in the form

(3)$P_0= f_0 (P*_1 + P*_2 + P*_3 + P*_4 + ...)$

$P_1 = f_1 (P*_2 + P*_3 + P*_4 +...)$

$P_2 = f_2 (P*_3 + P*_4 +...)$

From the great number of possibilities (cf. Wimmer, Altmann 2000) they chose the simple form

(4)$f_1 = (P*_{i+1} + P*_{i+2} + ...) = c \sum_{j=i+1}^\infty\frac{P*_1}{j}, \quad i=0, 1, ...$

yielding $c = \frac{1}{1-P*_0}$ as a normalizing constant.. Using the geometric distribution as parent distribution and displacing the result one step to the right they obtain

(5)$P_x = \frac{p}{q}\sum_{j=x}^\infty\frac{q^j}{j}, \quad x= 1, 2, 3, ...$

known as Bissinger-geometric distribution (cf. Wimmer, Altmann 1999).

Example: Polysemy in Maori (New Zealand)

Using the data ascertained by V. Krupa, Wimmer and Altmann (1999a) fitted the Waring and the Bissinger-geometric distributions to Maori dictionary data as shown in Table 1.

3.3. Tuldava´s version $y_x = a \exp (-bx^c), \quad y_x$ = number of words with x meanings

3.4. Krylov-Jakubovskaja´s version (

$y_x = e^{-bx}$]]]

4. Authors: U. Strauss, G. Altmann

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