1. As a linguist, who thinks his discipline an empirical science, I will be not so much concerned with either language philosophy, formal logics or mathematics, but mainly with the study of meaning as it is constituted in spoken or written texts used in the process of communication. Rather than focussing on the fiction of an 'ideal speaker' or the formal rules of an abstract and mere theoretical language usage, my linguistic point of view implies that I am much more interested in the analysis and description of natural language regularities that real speakers/hearers follow and/or establish when they interact verbally by means of texts in order to communicate.
For any description of natural language meaning, however, we are in need of a formally adequate meta-language to depict semantic phenomena, and for any analysis of natural language meaning we need methods and procedures which are empirically adequate. Both, the postulates of formal and empirical adequacy will have to be met by a communicative theory of semantics that is comprehensive and satisfactory. Such a theory - that should be stressed here and kept in mind throughout the following - does not exist and no one has yet presented even the outlines of it - and I shall not either. But I think that the concept of fuzzy sets may prove to serve as an at least formally satisfactory and numerically flexible link or joint to connect the two main, seemingly divergent lines of research in modern semantics: namely, the more theoretically oriented models of what formal semanticists feel an 'ideal' speaker should, or would do when he produces meaningful sentences a n d the more empirically oriented methods and procedures of experimental semanticists that try to find out what real speakers actually do when they produce texts for communicative purposes.
In general, most linguists will probably agree that - whatever else has to be dealt with - natural language meaning presents two major problems:
firstly, what is known as the connotational or structural aspect of how the words and sentences of a language are related to one another;
and secondly, what is known as the denotational or referential aspect of how the words or sentences of a language are related to the objects and/or processes they refer to.
To start with the latter, referential semantic theory has developed along the line of Frege, Russel, early Wittgenstein and Carnap. Their relevance to linguistics and to linguistic semantics in particular has been recognized during recent years only. In the meantime, the increasing interest in formal semantics among linguists has produced quite a number of different models which share the fiction though, that natural language sentences ought to be either 'true' or 'false' or at worst have a third value like 'undetermined'. Like the truth-conditions for predicates, those for natural language sentences are analogously introduced in terms of classical set theory. Accordingly, the meaning of a word is basically identified with a set of points of reference in the universe of discourse, allowing a truth-value to be assigued to any (declarative) natural language sentence. These truth-value models now tend to exhibit all the formalisms and abstractions mathematical rigor calls for. They do so, however, at the price of a rather limited coverage of basic and very obvious characteristics of natural language meaning, one of which had to be excluded totally: that is vagueness.
Unlike referential theory, structural semantics has considered this very notion of vagueness to be fundamental to natural language meaning. Structuralists have therefore been concerned with the question of how the lexical meanings of words - rather than being related to extra-lingual sets of objects - are intra-lingually related to one another, constituting relational systems which people obviously make use of when communicating. According to structural theory, the meaning ('sense') of each term is to some extent depending on the position it occupies in that system. It is argued, that, although the terms may referentially be vague, the position of each term in the system relative to each other, will nevertheless be defined with precision.
This fiction of 'structural preciseness' as opposed to 'referential impreciseness' has inspired linguists since Saussure, Trier and Weisgerber up to Coseriu, Greimas and Lyons and even scholars from nonlinguistic disciplines like Osgood, Goodenough or Wallace - to mention only these few. Their models and methods have undoubtedly been fertile and influencial for some time and/or discipline. But as they were based mainly upon intuitive introspection and probands questioning, they do not seem to have achieved either the theoretical consistency or the methodological objectivity that empirical theory calls for. Thus, apart from the ethno-sciences or experimental psychology, structuralistic ideas seem to be of decreasing influence in modern linguistics and its recent semantic theories.
If, however, it is agreed that on the one hand natural language semantics should be an empirical science and as such be an integral part of modern linguistics, one obviously cannot be content to rely on traditional structural methods and related procedures of people looking into their minds, each into his own and some into others'. If on the other hand one is just as malcontent with the highly theoretical and most abstract concepts formal semantics can offer to cope with very real and concrete problems of natural language meaning, one is apt to think of the afore-mentioned comprehensive semantic theory which is both, empirically and formally adequate.
These issues became involved, when the concept of fuzzy sets was introduced into linguistic semantics. Basic [1] to the notion of fuzzy sets is - other than in classical set theory - that the elements of fuzzy sets show gradual rather than abrupt transition from non- to full-membership. Fuzzy sets are defined by characteristic- or membership-functions which associate with each element a real, nonnegative number between 0 and 1 with 0 equaling 'non-membership' and 1 equaling 'full-membership' in the classical sets-theoretical sense. Let A be a subset of X, then A can be defined by a membership-function
In his 1971 paper on 'Quantitative Fuzzy Semantics' Zadeh adopted a strictly reference-theoretical model, into which he successfully incorporated the notion of fuzziness. He was able to show that the meaning of a word or term may well be vague in the sense that it refers to a set of reference-points whose boundary is not sharply defined, thus constituting a fuzzy set in the universe of discourse.
"In fact it may be argued that in the case of natural languages, most of the words occurring in a sentence are names of fuzzy rather than non-fuzzy sets, with the sentence as a whole constituting a composite name for a fuzzy subset of the universe of discourse"The second aspect raised by Zadeh in the same paper, whether
"fuzziness of meaning can be treated quantitatively, at least in principle"has however been dealt with only formally. The empirical side of it, concerning questions of how the meaning of a term described as a fuzzy set may be detected, or how the membership-grades may be ascertained and associated with the elements of a descriptor set in a particular case, these questions have not even been touched upon. We are informed instead that membership-functions
"can be defined in a variety of ways: in particular (a) by a formula, (b) by a table, (c) by an algorithm (recursively), and (d) in terms of other membership functions (as in a dictionary)"From the empirical linguist's point of view this is rather unsatisfactory. As clearly as he does recognize the relevance of fuzzy sets theory for the description of natural language meaning, he also will find that - what its analysis is concerned - fuzzy sets theory does not offer any new method. It seems that it merely allows a somehow quantified notation of more or less subjective, more or less acceptable results, which traditional methods of linguistic introspection may yield anyway.
I therefore would like to propose fuzzy sets theory to be combined with methods of statistical text-analysis in order to arrive at a generative model of structural semantics for which the notion of vagueness is constitutive.
2. It is assumed that the structural meaning of any lexical item (word, lexeme, stem, etc.) depends on its pragmatics and hence may be detected from sets of natural language texts according to the use the speakers/writers make of an item when they produce utterances in order to communicate. Such utterances are called 'pragmatically homogeneous' if they were written or spoken by real communicants in sufficiently similar situations of actually performed or at least intended verbal interaction.
It has been shown elsewhere [5] that in a sufficiently large sample of pragmatically homogeneous texts, called corpus, only a restricted vocabulary, i.e. a limited number of lexical items will be used by the communicants however comprehensive their personal vocabularies in general might be. Consequently, the lexical items employed in these texts will be distributed according to their communicative properties, constituting semantic regularities which may be detected empirically [6]. For this purpose a modified correlation coefficient has experimentally been used. It allows to compute the relational interdependence of any two lexical items from their textual frequencies. Those items which co-occur frequently in a number of texts will positively be correlated and hence called 'affined', those of which only one (and not the other) frequently occurs in a number of texts will negatively be correlated and hence called 'repugnant'. Different degrees of word-repugnancy and word-affinity - indicated by numeric values ranging from -1 to +1 - may thus be ascertained without recurring to an investigator's or his probands' knowledge of the language (competence), but solely from the regularities observed in a corpus of texts spoken or written by real speakers/writers in actual communication (performance).
Let T be such a corpus that consists of a number of texts t satisfying the conditions of pragmatic homogeneity. For illustrative purposes, we will consider a simplified case where the vocabulary V employed in these texts shall be restricted to only three word-types, say, i, j, and k which have a certain overall token-frequency. The correlation-coefficient a will measure the regularities of usage by the 'affinities' and 'repugnancies' that may hold between any one lexical item and all the others used in the texts. That will yield for any item an n-tuple of correlation-values, in this case for the lexical item i with n = 3 the tripel of values ii, ji, and ik. These correlation-values are now interpreted as being coordinates, that will define for each lexical item i, j or k one point ai, aj or ak in a three-dimensional space.
This is illustrated in Fig. 2. There we have three axes representing the three word-types i, j and k which cross in front of the three planes cutting the axes at their +1 values. The point ai is defined by the correlation-values ii = +1, ij = -.25 and ik = -.75; it is therefore situated in the i-plane with the interrupted lines (parallel to the j- and k-axis) representing the ii- and ik-values. The other points aj and ak are defined analogously. The position of ai, in this space now obviously depends on the regularities the lexical item i has been used with in the texts of the corpus. ai therefore is called corpus-point of i in the a- or corpus-space.
Two a-points in this space will consequently be the more adjacent to each other, the less their regularities of usage differ. This difference may be calculated now by a distance measure d between any two a-points, illustrated in this figure by the dotted lines.
These distance-values, which are real, non-negative numbers, do represent a new characteristic which may be interpreted in two ways:
firstly: die dotted distances between any one a-point and all the others are interpreted as new coordinates: then these coordinates will again define a point in a new n-dimensional space, called semantic-space. The position of such a meaning-point in the semantic space will depend on all the differences (d- or distance-values) in all the regularities of usage (a- or correlation-values) any lexical item shows in the texts analyzed;
secondly: the dotted distances between any one a-point and all the others are interpreted as membership-grades: then - after these d-values have been transformed appropriately into m-values, ranging from 0 to +1 - the differences of a lexical item's usage-regularities may well be represented by a fuzzy set with the vocabulary serving as its descriptor set.
Both these interpretations of d-values, as coordinates of points in the semantic space or as membership-grades of fuzzy subsets in the vocabulary, are equivalent: they will equally map the 'meaning' of a word as a function of all its differences in all its regularities onto the vocabulary, according to the usage a lexical item is made of by the speakers/writers in a corpus of pragmatically homogeneous texts.
Apart from that, the fuzzy-sets-theoretical interpretation allows an considerable extension of this analytical model of structural meaning. Some basic definitions and formal operations may now be introduced which will allow an empirically based and formally satisfactory explication of linguistic sense-relations and - even more important than that - the formal generation of (at least in principle) infinitely many n e w meanings from the finite number of those lexical meanings, which prior to that have been analyzed empirically from the text-corpus.
Assuming that the definition - as proposed by
Zadeh [7] - are well known, I will confine myself
to show their semantic correspondences in this
linguistic model of structural lexical meanings.
Synonymy of meanings may be explicated as equality of two fuzzy sets;
Partial synonymy of meanings may be defined in terms of a similarity-formula, introducing a threshold-value s;
Hyponymy of a meaning relative to another may be explicated as containment of fuzzy sets.
What the operations of negation, adjunction and conjunction are concerned, there has been quite a bit of critical discussion lately, particularly from the empiricists' point of view. For the generation of new meaning-points in the semantic space, I have so far gone back on those definitions proposed by Zadeh. Modified definitions of adjunction and conjunction proposed are, however, experimented with at the moment.
3. Coming to the end, I would like to give you some examples from the computer-analysis of a corpus of 19th and 20th Century German students poetry, the first part of which covering the early 19th Century comprises some 500 texts and a vocabulary of 315 lemmatized word-types/21000 tokens.
As there are serious difficulties in visualizing a 315-dimensional semantic space on the one hand, and, as there is, on the other, but little illustrative use in reproducing an n-tupel of 315 d-values, defining a meaning-point or fuzzy set respectively of, say, the lexical item baum/tree (Fig. 3), I have thought of some other means to give an impression of the lexical structure.
To illustrate the position of a meaning-point, I have tabulated those points which are nearest to it in the semantic space, constituting something like a meaning-point's topological environment As I have shown elsewhere [8], these environments prove to be very similar to what linguists have called paradigmatic or semantic fields.
When you let your eyes pass along the meanings-points listed in Fig. 4 and Fig. 5, showing the environments of baum/tree and friedhof/graveyard, you will get an idea of the semantic fields of these words as used in the German poems of the early 19th Century. What the paradigmatic relations are concerned, I think they are rather self-evident to a native speaker of German, or, to say the least, they are not contra-intuitive [9].
As I have only started with the testing [] of the operations defined to generate new meanings, I have chosen two lexical items, namely baum/tree and blüte/blossom which paradigmatically are closely related. The idea was, that the new meaning-points ` baum/tree Ù blüte/blossom' (Fig. 6) and ` baum/tree Ú blüte/ blossom' (Fig. 7) resulting from conjunction and adjunction of these two items, should be positioned somewhere in the same region of the semantic space, which in fact they are.
As you might have noticed, my approach to the analysis and description of natural language meaning is still very tentative and far away from a consistent theory of semantics; but it is hoped, that this approach will arrive at a model which in its abstract (algebraic) parts may linguistically be interpreted as a corpus-independent theory of semantic competence ('langue'), whereas its empirical (quantitative) parts will represent the performative data ('parole') which are corpus-dependent and hence will vary according to the texts analyzed.
1Published in: Trappl, R./Hanika, P./Pichler, F.R. (Eds.): Progress in Cybernetics and Systems Research (Vol. V), Washington/New York/London (Wiley & Sons) 1979, pp. 495-503.