20th WCP: From Lullus to Cognitive Semantics: The Evolution of a Theory of Semantic Fields

Introduction

Although field-semantics was only created at the beginning of the 20th century, some of its major features have precursors dating back to antiquity. Two disciplines have contributed to it: logic on the one hand and models of the world / cosmology on the other hand. My specific concern will be the rise of a space-orientated concept of a semantic field because, as the word "field" indicates, the ideas of dimensionality (one two- or multi-dimensionality) lies at the heart of the image-schema "field" in its theoretical use.

1 The circular fields of Llull

The first systematic spatial organization of lexical items (their concepts) was put forward by Raymundus Lullus (Ramón Llull: 1232-1314). All conceptual systems of his Ars Magna are arranged in a linear order with (normally) nine segments. Since the extremes of this 'belt' are joined, we have a circular field. Every concept has two neighbours, and by adding specific figures (triangles, squares, etc.) one can join three, four, etc. concepts to create a sub-network. The concepts of an area of knowledge may be organized into a set of such nine-tuple 'fields'. On top of all the more specific conceptual fields (arrays of nine concepts), stands a universal field, which contains those qualities of God that are at the origin of all further entities and their concepts. The semantic system has an ontological and metaphysical foundation in the tradition of Aristotelian and medieval logic; the circular organization is shown in Figure 1.

Figure 1: The Lullian circles (Lullus's first figure)

The idea that concepts/words form linear arrays, that the extremes may be glued together, and that a hierarchy of such arrays exists, is a first realization of 'field-semantics'. But Lullus did not stop at the static idea of a (circular) field of concepts: he proposed a combinatory mechanism which may have been motivated by the 'machinery' of medieval syllogistics, but which contained a new mathematical impulse which allowed the later development of computing machines by Leibniz, Pascal, and others (cf. Wildgen 1998: 84f).

I will not go further with analysis of the Lullian system. It should be clear that a hierarchy of linear (and circular) fields, and a combinatorial dynamics on it, already constitute a powerful theoretical instrument for organizing the universe of concepts.

2 The houses (atria) of memory introduced by Giordano Bruno

In the late sixteenth century, Giordano Bruno (1548-1600) began to elaborate the Lullian system, devising a new system of conceptual organization based on the analogy between the macrocosm (the universe) and the microcosm (man, his mind). He replaced Lullus's closed linear field with a regular, bidimensional pattern extending to infinity. I shall comment only on the consequences of this new geometry of 'fields'.

Mathematically, Lullus's field is a circular segment divided into nine sub-segments. If instead of linear segments the basic unit is a regular surface, we may consider either the filling of an (infinite) area by circular surfaces (spheres), or its filling by regular surfaces (polygons) or bodies (polyhedra). The corresponding mathematical problem is that of an (optimal) package of circles/spheres or polygons/polyhedra. The 'practical geometry' of the sixteenth century (as elaborated by Bovillus) had already considered this problem.

Bruno decided, that the filling of a surface by squares is most adequate, therefore the semantic universe is constructed on the basis of a square grid. Bruno went further. Referring to the tradition of artificial memory dating back to antiquity, he conceived parks with square houses, the atria (shaped after the classical roman house). The atrium consists basically of 3 x 3 = 9 square rooms; if the central room is open, we have one centre and eight rooms in its periphery. Figure 2 shows this arrangement. Note that the central atrium has eight neighbours, four of which have a line as their borders, and four have only a point of contact.

Figure 2: The atrium

As a consequence of the regular pattern, all fields show self-similarity; on every scale we find the same type of pattern; the universe is homogeneous (isotropic); and there is no global centre. I will give an example for the filling of such a geometrical structure with concepts in Table 1.

Table 1: The atrium altaris (cf. Bruno 1591 (De imaginum ... compositione): 37 and in the translation of 1991: 55)

But this is only the static aspect of the system; there are two types of dynamic elaborated by Bruno:

The generation of phrases and sentences. The atrium is embedded in a square. The four corners of the square may be assigned different syntactic functions, e.g., subject, absolute predicate, relational predicate, and circumstances.

A second dynamic directly concerns the fillers of the memory pattern. Every word in this pattern may be replaced by its metaphor or its metonym. Thus, a text which was first generated along the lines of basic meanings associated with the terms filled into the system can now produce a totally different text or text interpretation by using a set of metaphorical and metonymical processes.

Historically, we may say that the consequence of Bruno's parallel work on cosmology and artificial memory is a new model of semantic fields which was so radical in its time that the first modern followers (although ignorant of this tradition) are the Von-Neumann automata and the neural net systems of the 1980s (cf. Wildgen 1998: 39, 237f).

3 Leibniz' "caracteristica" and semantic fields

The tradition of Lullus and Bruno was still alive when Leibniz designed his "De Synthesi et Analysi universali seu Arte inveniendi et judicandi". Leibniz refers to the work by Athanius Kircher "Ars Magna Lucis et Umbrae, 1645", which was only published when he had already found his own solution of the problem in his "early youth". (1) Leibniz' solution is an arithmetic one and can be interpreted as a precursor of feature-semantics today. It associates cognitively primitive (i.e. non-definable) concepts with prime numbers. All definable concepts correspond to non-prime numbers which can be decomposed into prime-numbers. Leibniz eliminates as Lullus did the basic distinction between subject and predicate, and practically considers only two levels: primitive and (by definition) composite concepts.

In a second system called "geometrical characteristic" (in a letter to Huyghens, 1679) Leibniz sketches a constructive device which generalizes the methods of Euclid and applies them to conceptual systems. The transition from the arithmetical to the geometrical characteristic corresponds to the transition between possible (conceivable) worlds, i.e., pure intention, to the real world, to the spatialization and temporization of intentional concepts. Central notions are geometrical congruence (i.e., the identity of geometrical objects under spatial transposition) and the intersection of geometrical figures. (2) Leibniz demonstrates only how the simplest notions like (universal) space, point, line, plane, circle, position in space, etc., are constructed. This type of conceptual characteristic has the merit that all entities defined can be constructed and Leibniz imagines how his system if elaborated can be used to describe plants and animals and to invent machines. The geometrical characteristic would allow man to do this with symbolic techniques in his imagination without the help of concrete figures and models.

Leibniz' critique of image-like models can be generalized to all too specific and ad hoc pictorial descriptions. This strategy already condammed by Leibniz 300 years ago is systematically tried by cognitive semantics which work with ad hoc figures and with pictures which have no theoretical status. Semantics of this type will soon accumulate a chaotic universe of ad hoc figures and will loose the capacity to find general and stable regularities which is a central aim of any scientific enterprise. Thus Leibniz geometrical characteristic is a kind of deconstruction of cognitive semantics in the style of Lakoff and Langacker.

4 Peirce's existential graph

Charles Sanders Peirce (1839-1914) developed his 'existential graph' as an intermediate formalism between mathematical logic, which was about to assume the form that it has today, and topology, which was a new mathematical discipline which went beyond geometry (cf. the works of Felix Klein, Möbius, and others in the nineteenth century).

The basic ideas of Peirce's system is that every utterance may be inscribed on an abstract utterance sheet. The absolute terms are represented by dots on the sheet. If two dots are referentially identical they are joined by a line. The negation of a dot is represented by a closure which contains the dot. Insofar as no general geometrical pattern is presupposed, a dot can be placed anywhere. Relations between concepts (schemata) are represented by relational graphs with different valences. Figure 3 shows the representation of valences from 0 to 3.

Figure 3: The graphical representation of valences

What is new in Peirce's existential graphs?

Lullus's relational concept is elaborated by the concept of valence. Fillmore would later call these schemata for the organization of concepts into a unitary macro-concept: frames (cf. Fillmore, 1977).

Negation, and in a similar manner modality (described by a permeable closure), is included.

The whole system can be translated into the sequential language of modern predicate calculus (it is formally equivalent to it).

The first two innovations are of immediate relevance to our purposes. The relevance of a formal logical model of meaning for field-semantics is open to discussion (many field-semantics may be translated into logical models). The first innovation is perhaps the most important one: it has to do with the consolidation of chemistry in the nineteenth century (the term 'valence' is a clear indicator of this origin). It would be crucial for any later field-semantics to cope with the phenomenon of 'valence'.

5 Gestalt theory ( Kurt Lewin) and semantic fields

At the origin of Gestalt theory stands philosophy and psychology, which were not yet institutionally separated in Germany. Thus Christian von Ehrenfels' 1890 article 'Über Gestaltqualitäten' follows the tradition of Goethe and Humboldt and considers mainly perceptual 'Gestalts'. In the various schools of Gestalt psychology (Berlin, Graz, Leipzig) different aspects were foregrounded: psychophysiological aspects in Berlin (e.g., Wertheimer, Koffka, Köhler, Lewin), intellectual forces as Gestalt-foundation in Graz (e.g., Meinong, Benussi), emotional and symbolic aspects in Leipzig (e.g., Cornelius, Bühler).

As early as 1912 Kurt Lewin foresaw that a scientific psychology would have to make use of 'topology' and of the dynamics which could be conceived in a topological structure (cf. Lewin 1969: 9). Lewin's central idea was that of a 'psychological life-space' (psychologischer Lebensraum). Life-space is constituted by the individual and a situation relevant for the individual at a given moment. The life-space of an individual has two aspects:

Every partial domain of an individual's life-space corresponds to a 'psycho-domain', containing the person, structures of the life-space specifically relevant for the person (individual situations) and structures of the life-space which are constituted independently from the person (standard situations).

Psychic 'locomotion', i.e., paths in the life-space with preferred routes, barriers and obstacles.

The development of a child or an adult may be described as a change in life-space. The life-space of a child alters as soon as it learns to grasp, to control, to walk, to speak, and so on. A prisoner has a dramatically reduced life-space and some situations may contain attractors (cf. emotional attractors, sympathy, love, etc.) or repellers (situations of frustration, anger), which provoke reactions of escape.

Thus far we have considered the person to be an integral component of a life-space. But persons may themselves be regarded as a topological field with an inner area (intrapersonal domain), a periphery of this domain, and a sensor-motorical domain, which lies between the person and his/her context (the situation). In Lewin (1969: 185) the following diagram is used to depict this structure.

Figure 4: Lewin's structure of the person

Language has two major functions in this framework:

It transfers internal states of the person to his/her environment. This corresponds roughly to Bühler's expressive function (Ausdrucksfunktion).

Language enables a type of indirect locomotion in psychic space: for example, surrogate locomotion as in the speech act of ordering or in social coordination via language, change of social and personal relations and cognitive influence, which creates new possibilities of psychic locomotion via learning. The topological psychology of Lewin was later elaborated by Fritz Heider and his 'attributional psychology'. Heider strengthened the relation between 'life span' categories and semantic categories, for example, perceptual, experimental, affecting, causing, evaluation, part-whole relations (possessive), can, trying, wanting, etc. (cf. Heider, 1958).

Thus, the psychic and the symbolic world have the structure of a field, and topological and dynamic (vectorial) notions from contemporary mathematics are used to specify these fields.

6 The link between Gestalt psychology and cognitive semantics

In 1968 Charles Fillmore published his famous article "A Case for Case". Later he generalised the notion of (abstract) case to frames (Fillmore 1977), and to constructions (Fillmore 1988). In the context of my historical introduction I shall only mention the fact that the concept of 'frame' introduced in 1977 by Fillmore was anticipated by use of the term in Minsky (1975). It cannot be historically proved, but my impression is that Gestalt psychology was finally accepted by major trends in computational vision research, and via vision research it reached the centres of cognition research, one of which was founded in Berkeley in 1968, and of which Fillmore and his colleagues in Berkeley were members. Thus, after an odyssey comprising persecution (after 1938) in Germany, emigration, and then difficult integration in the USA where it conflicted with the dominant behaviourist trends in psychology, and after the rather mechanistic first stages of computational psychology, this strand of ideas entered mainstream theorizing in 1975, contributing to the development of cognitive linguistics. In 1977 Lakoff gave a paper on linguistic Gestalts at the Summer School on Mathematical and Computational Linguistics in Pisa. In the same period Leonard Talmy wrote his articles "Rubber Sheet Cognition in Language" and "Figure and Ground in Complex Sentences", and in 1979 Langacker published the first article - entitled "Grammar as Image" - on what would become 'space grammar' and later 'cognitive grammar'. Thus, between 1976 and 1979 the new 'wave' of topological and dynamic semantics finally reached California and soon thereafter began to spread through Italy, Germany, and France - cultural areas in which, half a century earlier, the major trends in Gestalt theory and corresponding applications to linguistics had been created. But this 'comeback' was something more: it had inherited the missionary attitude of American linguistics after Chomsky.

A philosophically and formally less isolated type of cognitive semantics has been proposed by René Thom since 1968 (cf. the translation of his early articles in Thom, 1983), but was ignored by theoretical linguistics in the Chomskian tradition. It has been elaborated in further research since 1976 and major results are summarized in Wildgen 1994 (in English) and 1999 (in French).

Notes

(1) Cf. G.W. Leibniz, 1966, vol. I: 41.

(2) Cf. Leibniz, ibid.: 80.

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