Vladimir KUZNETSOV*1 ON TRIPLET CLASSIFICATIONS OF CONCEPTS2 Abstract: The scheme for classifications of concepts is introduced. It has founded on the triplet model of concepts. In this model a concept is depicted by means of three kinds of knowledge: a concept base, a concept representing part and the linkage between them. The idea of triplet classifications of concepts is connected with a usage of various specifications of these knowledge kinds as classification criteria. (Author). Vladimir Kuznetsov (b.1946), Dr., studied theoretical physics at the Kiev University and philosophy of science at the Institute of Philosophy in Kiev. Special interest: concept modelling, knowledge representation and organisation in the sciences and humanities; nature, composition, functions and interrelations of scientific knowledge systems. Since 1991 professor of Kiev University. Principal scientific researcher of the Kiev Institute of Philosophy. 1. Introduction Concepts have been the object of much concentrated attention in many branches of contemporary science. There are a lot of approaches to their study. In what follows concepts will be considered as elementary units of knowledge and its organisation [4]. However, concepts are not formless entities. Circumstantial evidence of this fact is that there are numerous controversial models of concepts. Practically either model has advantages and some empirical confirmation. Because of this *The draft. The final version, see: On Triplet Classification of Concepts // Knowledge Organization, 1997, 24, 3, 163-175. 1 The author thanks the Research Council of Norway for supporting this study. 2 The author recommends to select the definite concept that a reader possesses (or thinks to possess) and to try analysing it with ideas and definitions exposed in this paper. He would be very grateful for receiving the information on counter-examples to the triplet model. 2 it is reasonable to suggest that the particular model of concepts reflects several specific aspects of the "same" unit of knowledge that has been called a "concept". The hypothesis of this paper is that a concept has a rather complex and unusual structure and composition. One may think about a concept like a composite thing known only through its numerous projections. Our first aim will be to introduce such a model of concepts that has opened perspectives of the unified description of such projections. The second aim is to apply this model to classifications of concepts. 2. The Triplet Modelling of Concepts The results of current concept analysis do not permit one to be certain and final in his or her knowledge of what concepts are. There are now only reasonably detailed, formal and adequate models of a concept. These depict in various ways different properties and structures of concepts. The triplet model [6, 7, 8] has united and developed further various concept characteristics that were introduced by other models. According to the triplet model, any concept may be associated with three kinds of information. The first is the knowledge about a base of a concept. The second one is the knowledge of a representing part of a concept. The third is the knowledge about a linkage between the base and the representing part. It should be noted that there are different ways of structuring these three knowledge kinds. Let us consider the concept of some entity e. This entity may have real, ideal, or mental nature. It may be a thing, an object, a process, a state, a thought, number or an appropriate set, collection, class or group of these. Since the origin of modern science the leading strategy of investigating entities has been the selecting (monadic, dyadic, etc.) properties of entities in question, establishing and describing relationships between their properties of various orders. One may associate with this strategy the following chain (for simplicity reasons only one single entity is taken and the chain is considered as linear): an entity → first order properties of an entity → first order relations between first order properties → first order relations between entity properties and properties of other entities → ....→ n-th order properties → .... The knowledge about the concept base has been centred around such ontological structuring of the reality under study. This structuring has been expressed in so called ontological hypothesis of 3 modern culture and science. Even now we cannot be sure about accepted hypothesis on what are entities and their properties and relations. These hypotheses have been changing permanently, at least in fundamental physics and biology. However, the ontological structuring has remained without changes. One of possible ways of general depicting this ontological structuring is a usage of the constructions of abstract properties and set scale. An abstract property P is a triple (D, p, Sc) where D is a set of entities d which may possess the property in question, Sc is a scale of the property, and p is a partial function assigning the element(s) p(d) = sc ∈ Sc to an entity with the name d ∈ D [1, 3]. For example, the property of physical bodies that is usually called "velocity" may be modelled as an abstract property in the following manner. Here D is the direct product of the set of all physical bodies by the set of all physical frames of reference, Sc is the set of three dimensional vectors, and p is realised by means of some procedures of measuring and calculating the value of velocity for a given body. The set scale is built step by step through the application in definite order operations of set union, set product and constructing power-set to the basis X of the set scale S(X). The basis is a collection of sets X1, X2, ..., Xn . On each step while constructing a set scale one obtains its definite level consisting of some sets. The set scale S(X) is the union of all its levels [2]. Levels of a set scale may be used for ordering (hypothetical) knowledge about entities and their properties of various orders. The knowledge about the representing part of a concept has been structured in another way. It is organised according to the rules of human representative and communicative systems, primarily natural and artificial languages and knowledge systems. These systems have expanding and revising resources (sign, symbolic, lexical, syntactical, semantical, imagerial, modelling, operational, transformational and others). In a sense the horizon of these systems defines what we can specifically state about the general ontological structuring of reality. However, the relationship between the reality and our representative and communicative systems is not a simple one-to-one correspondence between the entities of the former and elements of the latter. The components of such relationships have been created by means of many kinds of operations and processes and are not without change. The knowledge about the linkage of a concept has centred around these operations and processes. 4 2.1. The Base of a Concept To introduce the definition of the concept base we need the following explications. Let the universe of discourse U be the set of all entities about which it is possible to think by means of concepts. The set U also contains properties of these entities, relations (dyadic properties) between entities, relations between properties, properties of relations, properties of properties (properties of the second order with respect to entities), etc. To avoid undesirable associations from here on we shall use, if necessary, capital bold symbols, letters, words, word combinations for denoting concepts. Instances of the concept denotation are C, ELEMENTARY PARTICLE, HADRON, ANIMAL, HUMAN, SOCIETY on so forth. Entities that are falling under a concept will be connoted as c, elementary particle, hadron, animal, human, society. Correspondingly, the names of a concept might be "C", "ELEMENTARY PARTICLE", "HADRON", "ANIMAL", "HUMAN", "SOCIETY." The names of the entities that are subsumed under a concept might be "c", "elementary particle", "hadron", "animal", "human", "society". It is possible to say that the name or term of a concept "is its component which conveniently summarises or synthesises and represents a concept for the purpose of designating a concept in communication" [4, p.144]. Generally as a name or term of a concept, not only lexically simple names may function, but also more complex linguistic structures like compound names, sentences, and even texts. A concept C has, as a rule, many names of the kind N(C). The same is true for the entities falling under a concept. The names in question differ in their exactness, effectiveness, simplicity and so on. There are many relationships between the names of the "same" concept as well of the names of the "same" entity falling under a concept. In this paper concepts will be considered as a complicated unit of knowledge about elements from U. It is a matter of fact that by means of a given concept C one might be informed of only about specific elements or subsets of U. Moreover, any such informing with the help of a concept C takes place in some conditions K. Aside from describing these conditions in detail, we mention only that these have been associated with individual's mental and interpretative abilities, skills and tools, available knowledge, purposes, and even psychic state. Bearing these distinctions and conventions in mind, we introduce 5 Definition 1. Under the conditions K the ground set GK(C) ⊆ U of the concept C is a set of all elements g such that 1) are denoted by the name NK(C) of the concept C and 2) are referred to by means of concept C. Under the traditional logical treatment the terms "extension" or "volume" have been frequently used for denoting the ground set of a concept. The term "category" is in use in cognitive science and psychology. Elements g ∈ GK(C) fall or subsume under the concept C. In cognitive science and psychology these elements are also called "instances" or "exemplars" of a concept. Dahlberg has used for denoting g such names as an "item of reference" and "referent" [4]. However, the association of the ground set with a concept is only a first step in its triplet modelling. Indeed, the knowing of the concept C presupposes also the possibility of indicating and describing, at least, qualitatively some properties and relations of elements from GK(C). This means that the knowledge about such properties and relations are important features of a concept. Knowledge about these is essential for a concept as a knowledge unit. Such knowledge is also a principal part of the concept use in ordinary thinking. Besides this, the usage of scientific concepts has presupposed quantitative descriptions of some properties and relations of elements from GK(C) and their values, the establishing correlation between properties under consideration, etc. As a rule, the set of some properties in question is called concept "intension" or "content". Cognitive scientists and psychologists also separate different kinds of such properties: a prototype and a core. A prototype is a set of properties that are assumed to occur in some instances. A core is a set of properties that are singly necessary and jointly sufficient for membership of an entity in the concept's category [9]. Thus, there is a need of depicting in precise terms the information, on one hand, on the concept ground set and, on the other hand, on some properties and relations of elements and subsets from the ground set. One way to do this is to use the construction of a set scale S(X) described above. In the case of triplet concept modelling, the basis X of the corresponding concept set scale necessarily includes the ground set G (G = X1 ). It is very important, that by means of selecting the appropriate basis it is possible to depict by means of the set scale properties and relations of G and its elements and subsets. For this purpose along with the ground set G, the basis should include auxiliary sets that are scales of properties and relations of elements from G. Examples of auxiliary sets are real numbers, vector spaces, truth values, etc. 6 Definition 2. The base BK(C) (in relation to the conditions K) of the concept C includes elements of GK(C) and not their properties and relations other than needed for the usage of C in conditions K. These properties and relations are modelled by means of subsets from finite number of levels of the set scale S(G*) with the basis G* = {GK, X2, ..., Xn}, where X2, ..., Xn are auxiliary sets. It may be shown that namely various structures of the concept base (under an appropriate option of the ground set, the basis of the set scale and their algebraic description) are objects of modelling that has been successfully developed by R.Wille and his collaborators [10; 11]. 2.2. The Representing Part of a Concept Apparently elements from the concept ground set and properties of these elements do not bear their names, descriptions, statements about these, etc. Such structures are human creations. Thus, any realistic concept model should take into account this fundamental, and usually neglected, fact. Without the loss of generality we may speak of only about the linguistic form of existence of these structures. Here language is understood in a very broad sense. The second triplet characteristic of a concept -its representing part -contains instances of this linguistic form. Let us assume that we use some language L with the alphabet A, the vocabulary V, the set P of word combinations, the set E of expressions (sentences) and the set T of texts. The language L may include sublanguages (sign, pictorial, natural, artificial, common, scientific, mathematical, etc.). The basis L* of set scale S(L*) of language L is {A, V, P, E, T}. In principle S(L*) contains everything expressible in the language L. Definition 3. The representing part RK(C) ⊆ S(L*) of the concept C is a set of linguistic units and structures by means of those the base BK(C) of a concept C is depicted (mapped, represented) under conditions K in some knowledge system. For example, the representing part of the physical concept ELECTRON contains the following elements: symbol e (the element of A); word "electron" (the element of V); "material carrier of elementary electric charge" (the element of P); "electron is a constituent of atom", "electrons interact by means of electromagnetic force", "electron has a rest mass of 9.1 X 10-28 gram" (the elements of E); "the electron is a fermion, a type of particle named after the Fermi-Dirac statistics that describes its behaviour. It has a half-integral spin spin constitutes the property of intrinsic angular momentum in quantum-mechanical terms" (the element of T) [5, p.435]. 7 The representing part of pre-scientific concept ATOM contains an image of small, indivisible pieces of matter. The representing part of its scientific counterpart includes quantum-mechanical wave functions, various theoretical models of atoms, schematic pictures of electron orbitals, etc. Components of the concept representing part differ in their representative and expressive capacity. Some of them only denotate the base as a whole, its selected subsets, and its individual elements. Other baptise properties and relations of elements from the ground set. The third group of components gives more or less complete and/or exact description of elements from the base or even their properties and relations. The fourth group models properties and relations in question. There are closed and non-trivial links between different kinds of elements from the representing part of scientific concepts. Moreover, these elements are intimately connected to empirical and theoretical knowledge systems and classifications available in the corresponding science. In this sense the representing part of scientific concepts is knowledge dependent. According to Dahlberg' model "a verifiable statement is the component of a concept which states an attribute of its item of reference" [4, p.144]. Such a component is a specific element of the set E that conveys verifiable knowledge about some property of elements or their combinations from the ground set of a concept. 2.3. The Linkage of a Concept The entities (from the ground set and base) have been associated with the appropriate components from the representing part by means of human activity. In this sense such associations are results of human actions. As such, these are dependent on developmental levels of civilisation, culture, language, science, person's knowledge, purposes and mental capacities. These are conditional and ephemeral, but necessary for building (forming) concepts. Thus, there is a need of more careful characterisation of links between elements and structures from the concept base and components and structures from the concept representing part. Let us point out only some aspects of links under consideration. There are many ways of their establishing: by custom, by training, by language acquisition, by convention, by analogy, by procedure, etc. From the point of view of concept functions the usage of three letters "man" for denoting MAN is accidental. Ukrainians use the set of six letters ("ëþäèíà") while Germans use another set of six letters ("Mensch"). At the same time there are universal scientific 8 procedures for finding values of such a property of macroscopic bodies as velocity for any given material body. The accuracy and exactness of these procedures may change eventually. The almost commonly accepted approach treats the links between components of concept representing part and elements from the concept base as simple naming relations. The former components play the role of names and the latter elements play the role of entities baptised by the appropriate former components. However, naming relations that assign names to entities are a specific kind of these links. For example, if the representing part of a concept contains some mathematical model of a property from the concept base, then this model not only names the property but also in principle gives the knowledge about the values of this property and even about relationships between this property and others. Without going into details, one may separate various kinds of links between the components and their sets from the representing part and the elements and subsets from the base. Among these are reference links (naming, denoting, describing, visualising, imaging), truth links, and modelling links. From what has been said it might be assumed that the knowledge on links in question is a very important part of any reasonable concept model. Definition 4. The third triplet characteristic LinK(C) of a concept C is the system of links (linkage) between the base BK(C) and the representing part RK(C). It is of fundamental importance, that for any concept this linkage is the outcome of very complex (sensual, perceptual, mental, scientific, etc.) activity. For example, for the common concept ANIMAL the linkage in question has been established by means of sensual perception. For the synonymous scientific concept the construction of such a linkage is realised in the framework of the available scientific knowledge and connected with conducting observations and measurements. It is supposed that electrons are unobservable entities. If it is true, then for different versions of scientific concept ELECTRON its linkage cannot be principally established by means of procedures of direct observation. This linkage is constructed by means of measurement and application of appropriate knowledge systems (theory of measurement, electron theory, quantum mechanics). This means that some links from the linkage of the concept ELECTRON are realised through processes of abstraction, idealisation, modelling, calculation, approximation and so forth. 9 For many scientific concepts there is a possibility of controlling the linkage between their bases and representing parts. In particular, the measurement and calculation procedures permit one to attribute quite specific linguistic and mathematical (numeric, vector, etc.) values to definite properties and relations of entities from the ground set of a concept. It should be noted that the concept linkage is transforming with the changes in scientific equipment, methods of its use and available scientific theories. 2.4. The Triplet Model of a Concept In the light of the discussion above it is apparent that the reliable concept model should take into account all three kinds of knowledge about concepts. Without any of these one may speak of only about incomplete concept modelling. Certainly, there are many successful applications of various incomplete concept models. However, the complete concept models give more profound and deep insight into concepts. From stated above one may obtain Definition 5. Under conditions K the triplet model TK(C) of the concept C is the triple (BK(C), LinK(C), RK(C)), where BK(C) is the base of C, RK(C) is the representing part of C, and LinK(C) is the linkage between BK(C) and RK(C). We would like to stress the relative nature of these and other definitions connected with concepts. The specific treatment of a concept depends not only on concept itself, but also on one's approach to it. 3. The Triplet Classifications of Concepts Let us consider briefly the idea of triplet classifications of concepts. Under it practically any working (common or scientific) concept belongs to many classes. One may take as classification criteria isolated characteristic of the base or representing part or linkage between these. The base of a concrete concept is some subset of the universe of discourse U. The classes and subclasses of concepts obtained are depicted in the tables 1-3. The sequence of dots symbolises the possibility of an extension of the type of classes mentioned above dots. 3.1. The Base Classifications 10 If one takes such characteristics of the base as its set-theoretic cardinality; the relation between the base and U; set scale composition; kinds of set-theoretical descriptions; status of entities from the ground set; the way by which the base is given to a person, etc., he or she may obtain the following (incomplete) list of concept classes. Criterion of classification Value of criterion Concepts classes and subclasses Cardinality of G The ground set contains no elements G-empty one element G-singular set of elements G-general finite set G-finite infinite set G-infinite countable set G-countable uncountable set G-uncountable Relation between G and U The ground set is: equal to U U-universal a subset of U U-non-universal a superset of U U-super-universal .......................... ......................... Ontological status of elements from G The type of elements Thing (object) G-object Event G-eventual Situation G-situational Process G-processual Action G-actional Intentions G-intentional ............................ ............................ Domain of existence of G The type of domain Physical reality G-real Psychics G-mental Communication G-communicative .......................... .......................... Set-theoretical composition of G The ground set contains only individual elements G-individual only subsets of individual elements G-collective ................................. ................................ Set scale composition of B The base contains properties {P(g ∈ G)} of individual elements g from G B{P(g ∈ G)}-attributive properties {P(G* ⊆ G)} of subsets G* of G B{(G* ⊆ G)}-attributive ............................ ............................... relations {R} between individual elements g from G G{R(g)}-relational ............................... ............................. relations between subsets G* of G G{R(G*)}-relational ................................ ............................. Set-theoretical kind of a structure Str from B 11 Standard set BStr-sharp Multiset BStr-multiset Fuzzy set BStr-fuzzy ............................... ............................... The way by which a structure Str from B is given to a person Perception BStr-perceptual Experience BStr-empirical Experiment BStr-experimental Abstraction BStr-abstracted Idealisation BStr-idealised ............................. .............................. Change of a structure Str from B No-variation of Str BStr-stable Variation of Str BStr-variative Parameter of variation of a structure Str from B Time variable BStr-temporal Space variable BStr-spatial Cause variable BStr-causal Type of cause of variation of s structure Str from B Randomness BStr-random Probability BStr-probabilistic Statistics BStr-statistical Determination BStr-deterministic .............................. Localisation of cause of a variation of a structure Str from B Inside Str BStr-internal Outside Str BStr-external Cardinality of a set of causes of a variation of a structure Str from B The set contains One cause BStr-monocausal Many causes BStr-multicausal ............................ .............................. ............................ .............................. Table 1. The base classifications of concepts 3.2. The Representing Classifications The representing classification of concepts may be constructed just as the base classification. Let L be some language. The classes followed are given relative to L. 12 Criterion of classification Value of criterion Concepts classes and subclasses Cardinality of R The representing part contains no elements from L RL-no-named one element from L RLsingle-named set of elements from L RLmulti-named finite set RLfinite-named infinite set RLinfinite-named countable set RLcountable- named uncountable set RLuncountable- named Relation between R and L L includes all needed for R elements RLexpressible L includes some needed for R elements RLpartially expressible L does not include needed for R elements RLnon-expressible R is a fuzzy subset of L RL-fuzzy-expressible ................................ ............................... Type of language L used for expression of structure Str from R The sphere of usage Us of L Common life RLUsStr-natural Science RLUsStr-scientific Units of alphabet A Pictograms RLAStr-pictogramic Signs RLAStr-sign The kind of sentence construction rules C Informal RLCStr-informal Formal RLCStr-formal The semantics Sem of sentences Assertions RLSemStr-assertoric Models RLSemStr-model Problems RLSemStr-problem Operations RLSemStr-operational Procedures RLSemStr-procedural Algorithms RLSemStr-algorithmic ............................... ................................. The kind of sentence transformation rules T Informalised RLTStr-informalised Formalised RLTStr-formalised Kind of a structure Str from R Mental images (pictures) RStr-imagerial (pictorial) Impression RStr-impressional Lexical units of L RLStr-lexical Letters RLStr-symbolic Words RLStr-lexicographic Simple words RLStr-simple- lexicographical Complex words RLStr-complex- lexicographical Word combinations RLStr-phrasal Sentences RLStr-sentential 13 Texts RLStrtextual Structure of lexical unit Un of L The unit has structure of: Scalar RLUnscalar Vector RLUnvector Spinor RLUnspinor Matrix RLUnmatrix Metric RLUn-metrical Topology RLUn-topological Fractal RLUn-fractal ....................... Kind of a set-theoretical description of a structure Str from R The theory of Standard sets RStr-sharp Multisets RStr-multiset Fuzzy sets RStr-fuzzy ............................. ...................... Access to a structure Str from R Momentary Time interval RStr-momentary RStr-temporal ....................... ......................... Psychic form of fixation of structure Str from R Consciousness RStr-conscious Unconsciousness RStr-unconscious Storage of a structure Str from R Working memory RStr-short-term Long-term memory RStr-long-term ............................... ............................. Type of knowledge system which a structure Str from R belongs to Common knowledge RStr-common General knowledge system RStr-general Special knowledge system RStr-special Science RStr-scientific Mathematics RStr-mathematical Logic RStr-logical Physics RStr-physical Social science RStr-social Psychology RStr-psychological ........................... ............................ Theology RStr-theological Philosophy RStr-philosophic ............................. ............................ Type of organisation of knowledge system which a structure Str from R belongs to Theory RStr-theoretical Formal system RStr-formal Formalised system RStr-formalised ...................... ....................... Nature of change of a structure Str from R No regularities RStr-irregular Patter-obeyed RStr-regular Kind of processing a structure 14 Str from R Ordinary thinking RStr-informal Formal thinking RStr-formal Mathematical thinking RStr-mathematical Computer processing RStr-computational ............................. .......................... ............................ .......................... Table 2. The representing classifications of concepts 3.3. The Linkage Classifications One may also construct concept classifications on the base of different characteristics of the concept linkage. Criterion of classification Value of criterion Concepts classes and subclasses Modality of a structure Str from Lin Necessity LinStr-necessary Potentiality LinStr-potential Intentionality LinStr-intentional Contingency LinStr-contingent ...................... ...................... Purposefulness of a structure Str from Lin There is a purpose LinStrpurposeful There is no purpose LinStr-non-purposeful .......................... ............................... Way of constructing of a structure S from Lin By socialisation LinStr-socialised By general education LinStr-generally educational By special education LinStr-specially educational .................................... .................................. Determination of a structure Str from Lin Unconditionality LinStr-unconditional Conditionality LinStr-conditional ........................... ....................... Character of the operation by which a structure Str from Lin is realised Without control LinStr-uncontrolled Under control LinStr-controlled Convention LinStr-conventional Ostensive indication LinStr-ostensive Operation LinStr-operational Measurement LinStr-measurable Computation LinStr-computational 15 ..................... ........................ Function of a structure Str from Lin Referring LinStr-referring Modelling LinStr-modelling Truth-bearing LinStr-adequate ..................... .................... ...................... .................... Table 3. The linkage classifications of concepts Certainly, some names of concepts appear to be very unusual. However, the triplet classifications open the way to transform such names in the terms of the future concept theory. The reader may try to find the membership of concepts (that are known to him) to classes of triplet classifications. Some memberships are rather obvious, others are in a need of special investigation and substantial knowledge. Undoubtedly, he or she will find how deep and profound is his or her knowledge associated with some concepts. 4. Further Developments It is also possible to introduce concept classifications with two or three criteria. They are the combined characters of the base and the representing part; the base and the linkage; the representing part and linkage; the base, the representing part and the linkage. The paper has realised several so called monadic classifications that have mainly based on internal structures of concepts as single monads. However, so called relational classifications are most often used. An example is the classification based on the relation of subordination between concepts. The triplet modelling of concepts permits one substantially to expand and make more precise such classifications. The triplet classifications of concepts are not without use in comparison of the maturity degrees of different concepts, in study of types and trends of concept developments, in analysis of specific of knowledge organisation at the level of concepts. References (1) Burgin, M.: Abstract theory of properties. In: V.Smirnov and A.Karpenko (eds.), Non-Classical Logics. Institute of Philosophy, Moscow, 1985, 109-118, 10 refs. (In Russian) 16 (2) Burgin, M. and Kuznetsov, V.: System analysis of scientific theories on the basis of named set theory. In: V.Sadovsky (ed.), Systems Research, Yearbook, 1985. Nauka, Moscow,1986, 136-160 (In Russian). (3) Burgin, M. and V. Kuznetsov, V.: Properties in science and their modelling. Quality and Quantity 27 (1993), 371-382, 8 refs. (4) Dahlberg, I.: A referent-oriented, analytical concept theory for INTERCONCEPT. Intern.Classificat. 5(1978) No3, 142-151, 17 refs. (5) Electron. In The New Encyclopaedia Britannica. Vol.4 (1990), 435. (6) Kuznetsov, V.: Types of fuzzy concepts. In: H.-J.Zimmermann (ed.), Second European Congress on Intelligent Techniques and Soft Computing, EUFIT'94. Verlag der Augustinus Buchhandlung, Aachen, 1994, 675-679, 9 refs. (7) Kuznetsov, V. and Kuznetsova, E.: Types of Concept Fuzziness. Fuzzy Sets and Systems (1997) (to appear),18 refs. (8) Kuznetsov, V.: A Concept and Its Structures. The Methodological Analysis. Naukova Dumka, Kiev, 1997 (In press). (9) Smith, E.E.: Concepts and thought. In: R.J.Sternberg and E.E.Smith (eds.), The Psychology of Human Thought. Cambridge, MA: Cambridge University Press, 1988, 19-49, 60 refs. (10) Vogt, F. and Wille, R.: TOSCANA A Graphical Tool for Analyzing and Exploring Data. Knowl.Org. 22 (1995) No 2, 78-81, 11 refs. (11) Wille, R.: Concept lattices and conceptual knowledge systems. Computers Math.Applic. 23 (1992) No 6-9, 493-515, 50 refs. Department of Logic and Methodology of Science Institute of Philosophy of the National Academy of the Sciences of Ukraine E-mail vladkuz@mail.itua.net. www.kuz.org.ua