THE CATEGORIAL FRAMEWORK AND THE WELL-FORMED FORMULAE OF THE TERNARY DESCRIPTION LANGUAGE
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International Journal of General Systems, 1999, vol. 28 (4-5), p. 351-366
Int. J. General Systems, Vol. 28(4-5), pp. 351-366
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THE TERNARY DESCRIPTION
LANGUAGE AS A FORMALISM FOR
THE PARAMETRIC GENERAL
SYSTEMS THEORY: PART 1
AVENIR I. UYEMOV
Odessa University, Odessa 270000, Ukraine
(Received 15 June 1994; In final form 18 March 1997)
In this paper we discuss the problem of appropriate formal apparatus for General systems theories. Drawbacks of actual mathematical and logical formalisms are treated. A new type of formalism – the Ternary Description Language (TDL) – is proposed. It is a kind of non-classical, deviant logic and essentially differs from Predicate Calculus. The concepts of set and number are not used in TDL as initial ones. That formalism is based on two triplets of categories: Thing (= Object, Entity), Property, Relation and Definite, Indefinite, Arbitrary. The types of well-formed formulae of TDL are' listed and explained. Those formulae are applied to generalization and formalization of various definitions of the concept "the system". Two definitions of that concept which are dual, one to another, are received.
Keywords: Thing; property; relation; definite; indefinite; arbitrary; system; general systems theory; formalism
WHAT KIND OF FORMALISM IS THE MOST APPROPRIATE TO THE CREATING OF GENERAL SYSTEMS THEORIES?
One of the most poignant questions in the development of General Systems Theories (GST) has been the question of their formal apparatus. Why is such an apparatus necessary for GST? The first variant of GST, "Tectology" by Bogdanov, had no formal apparatus at all (Bogdanov, 1989). This was not an accident. In the opinion of
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Bogdanov, mathematics is the science of neutral complexes only, where 1 + 1=2 and 0 + 0 = 0. In more complicated cases we can have 1 + 1= 0 and 0 + 0 = 1. It depends upon a character of organization.
The point of view, according to which GST can be excavated only in the frameworks of native languages, has proponents even now (Tachtajan, 1972). But the majority of investigators understand the importance of a formal apparatus for GST.
There are two kinds of such a formal apparatus. The first is a purely mathematical one. That is a special mathematical language such as the language of geometrical drawings, or differential equations, or algebraic operations, or set-theoretical relations.
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Mathematical expressions are much more precise, clear-cut, uniquely determined then corresponding utterances of natural languages. It is possible to derive one mathematical expression from another with the help of pure formal means. But the point is, what mathematical language is adequate for GST? Every pair of natural languages, e.g. English and Russian, is mutually translatable. It is an entirely different situation with artificial languages such as mathematical ones. Each of them was constructed for solving special tasks, which were put well before the origin of the first variants of GST.
For the reason given there is no foundation for a priori assurance that a mathematical language is appropriate to GST. In the course of development of GST various mathematical languages were used. Bertalanffy, Rapoport and their followers have tried to express system properties in terms of differential equations. But many systems, such as a family, a syllogism, a text of a book and so on, do not allow such a description.
More recently, investigators in the field of GST have used the Group theory (Urmantzev, 1978), and other parts of algebra (Kalman et ai, 1969; Gissin and Tsalenko, 1984; Cornacchio, 1972). All of these mathematical means have the same drawbacks as differential equations. Mesarovic and Takahara were correct when they wrote that for very complicated events, the specific language of classical theories, which are based on concrete mathematical structures (such as differential equations, arithmetic or abstract algebras), do not allow the adequate description of reality (Mesarovic and Takahara, 1975).
Mesarovic and Takahara themselves tackled the problem by choosing a minimal mathematical structure and adding to it new
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mathematical structures, which are necessary for explorations of various system properties. From our point of view this method has two flaws. The first is the lack of integrity of the mathematical apparatus. The second is even more important. The minimal mathematical structure according to Mesarovic and Takahara is a set-theoretical one. The system's concept itself is defined as a type of set.
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However, there is a well justified opinion that set-theoretical and system-theoretical approaches are opposite to one another in their character (Screider and Scarov, 1982; Kantorovic and Plisko, 1983). The main opposition is the opposition of priority of elements in set theory to priority of the whole in systems theory. Let us take a simple example. If we have the row: 1,3,4,5, it is possible to ask, what numeral is omitted? But this question is correct only from the system-theoretical point of view. The set-theoretical approach makes it senseless.
We have considered the difficulties which are connected with a purely mathematical formal apparatus of GST. The other type of for mal construction is a logical one. In this case there is no necessity in identification of system characters with special mathematical models. The logical formulae can be immediately related to utterances of a natural language. Therefore a logical language can express more than a concrete mathematical language.
If we state the fundamental presumption of a theory with the help of logical formulae, we can deduce from them other assertions on the basis of corresponding logical calculi. But there are difficulties in this approach. The attempts of axiomatic constructions of traditional sciences such as Physics, Biology, etc. were not successful.
There are two main causes of this. First, the necessary axioms are hard to formulate, and second, there are essential drawbacks in the actual logical calculi themselves. In spite of the great diversity of modern logical systems, most of them are based on statement calculus and predicate calculus.
Statement calculus is a very good deductive system. But its expressivity is weak. It is impossible to express the structure of simple sentence in terms of this calculus. Predicate calculus has advantages in that respect. Here simple sentences are divided in things (individuals) and predicates (properties and relations). But broadly speaking the expressivity of predicate calculus as well as statement calculus is
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too low for being an appropriate formal logical apparatus of modern sciences, especially of GST. Predicate calculus is very good for solving special mathematical problems, i.e., as a part of metamathematics (Kleene, 1952). According to Grzegorczyk "predicate calculus – the most important achievement of contemporary logic – did not emerge through successive improvement of ancient logic. It grew out of the specifically modern mathematical investigations connected with infinite operations and the concepts of continuity and limit, which are fundamental for mathematical analysis and the whole of contemporary mathematics" (Grzegorczyk, 1974, p. 567).
For the reason given predicate calculus is suitable for the expression of those utterances of natural language which are used in mathematical investigations. But it is not adapted to the logical analysis of a natural language as a whole.
There are essential differences between predicate calculus and any natural language. The main difference is a set-theoretical approach to the interpretation of the properties and relations in predicate calculus. According to it the properties and relations are identified with corresponding sets of objects. Therefore, from that point of view, properties and relations are the same if they correspond to the same sets. For instance "to be an angel", "to be a devil" and "to be a round square" are the same property because of their corresponding to the same (the zero) set.
As distinct from it any natural language is intentional, i.e., here properties and relations corresponding to the same set of object can be entirely diverse. There is a strong criticism of predicate calculus in point of insufficient expressivity (Materna et al, 1989). The authors are opposed to this calculus so-called "Transparent Intentional Logic" (TIL) (Tichy, 1988). The logical background of this system is a combination of the types' theory and A-calculus. Both of them are valuable tools for analysis of mathematical texts, but their application to the simple text in natural languages extra mathematics seems rather cum bersome. Any natural language is self-applicable, i.e., it is possible to speak about a language in terms of the same language. It is forbidden in the types' theory. Therefore the phrase "This statement is written in English" is inexpressible in that theory.
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Now we may anticipate that which is most appropriate to the construction of GST formalism has not been taken as one among existing
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formal systems. The best way is to design it specially for solving problems of GST. For the reason given it must suit some requirements. Inasmuch as assertions of GST in general can be made in frameworks of a natural language, our formalism does not have to be a purely mathematical one. Logical formalism has a great advantage because of its applicability directly to natural language utterances. But in contradiction to usual logical systems our "ideal formalism" should not necessitate an axiomatic arrangement of GST. As was mentioned above, a formulation of adequate axioms for GST is very difficult – even impossible. Axioms referred to any system must be so general that they become banal. It is not surprising then that some authors have insisted on triviality of GST as a theory which describes objects (Boulding, 1956; Sadovskii, 1974). From their viewpoint GST has the sense of a method of special system theories' construction, i.e., as a metatheory. Of course, there are attempts to create axiomatics for GST (Churchman, 1964; Urmantzev, 1978). But there is no way to deduce the real content of GST from those axioms.
Notice that the use of mathematical formalisms, such as arithmetical, algebraical, geometrical and differential equations, is not connected directly with the process of the logical inference. The latter refers to propositions. Logicians argue that propositions are the only thoughts which have valent characteristics (true, false or something between). But what kind of valent characteristics has, for example, the mathematical expression 3 + 3 + 4? Mathematicians are not interested in it. But they can transform that expression into 5 + 5, into 20 – 10, and so forth. Such expressions are called terms (Kleene, 1952). It is possible to apply terms to a description of an object directly without use of a special axiomatics of that object.
Our logical formalism would meet the case of GST much better if it had the features described above in common with mathematics. In other words, it has to be applied to the operations over terms as well as to the deductive relations between judgments.
Another requirement for our formalism is connected with generality of GST. Usual theories such as physical, chemical, biological, etc., apply to a concrete part of the world. The specific features of that part of the world are mapped onto sets of concrete predicates, for example, velocity, mass, energy, and charge. The relations between such predicates are formulated in axioms of applied predicate calculus. These
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axioms are added to the axioms of "pure" predicate calculus. There is no special part of the world for GST. There is the specific-systemic point of view only. That viewpoint can be expressed not with the help of concrete predicates, but through the structures of desirable formulae.
In this case there is no need to divide the calculus into two parts: pure and applied. Because of this there is no need to construct special axioms for GST. Formulae of our language should be sufficient for expression of GST concepts, and judgments, as well as axioms and rules, should be sufficient for the deductive process.
Furthermore, our formalism should have an intentional character. As was pointed out above, the extentional languages, which are based on set-theoretical concepts, have a deficit of expressibility. In this respect our formal language should have something in common with "Transparent Intentional Logic". But as distinct from it the desirable formalism must have another essential characteristic. We may call it Incrementality . A formalism is incremental in relation to a nonformal language if it is possible to increment the expressivity of the formalism with the help of that formalism itself until it copes with a challenge.
And the last. Our desirable language must be self-applicable just as a natural one. It is particularly important for the parametric GST (Uyemov, 1983). Is it necessary that self-application of a language leads to paradoxes within it? The affirmative answer to that question after the works by Russell and Tarsky has been commonly accepted. But it was possible to show that even in natural languages the paradox of the Liar could be avoided (Uyemov, 1976). As was noted by Gupta, "There are many sentences that involve self-reference that are not in any obvious way problematic." (Gupta, 1982). He tried to find criteria which distinguish trouble-free sentences from the troublesome ones in the framework of the classical first order quantificational language.
THE CATEGORIAL FRAMEWORK AND THE WELL-FORMED FORMULAE OF THE TERNARY DESCRIPTION LANGUAGE
The first and the most important step in the construction of the formal apparatus, which fills the requirement listed above, is the choice of
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appropriate categorial framework. According to Stephan Korner, "To indicate a thinker's categorial framework is to make explicit: (i) his categorization of object, (ii) the constitutive and individuating principles associated with the maximal kinds of his categorizations, (iii) the logic underlying his thinking" (Korner, 1970).
Aristotelian logic was based on the categorial opposition: thing (subject)-property (predicate). The relational logic (De Morgan, Povarnin, Serruce) presupposed the oppositions: things-relation. Instead of the traditional form of judgment: "S est P" in that logic, the scheme aRb was accepted.
The classical predicate logic (Frege, Russell, Pierce) has separated the world into two categories: individuals and predicates. Here the distinction between properties and relations has been reduced to a pure quantitative one. A one-place predicate expresses a property, a two-place, a three-place, etc. predicate expresses a relation.
In our categorial framework there are three basic categories: Things, Properties, Relations. This accounts for the name of our formalism – the Ternary Description Language.
The constitutive and individuating principles associated with those categories are the following. Everything which can be named or described is a thing (= object = entity). For example – "love", "the struggle", "the identity". Everything which distinguishes one thing from another is a property, e.g., "red", "old". Everything which constructs one thing from other is a relation, e.g., "read", "marry." If "John reads a book," we have a pair which consists of John and a book. If "John gets married to Margaret," we have a new object – a family. Of course, old John is a new object in relation to John. But old John is John. Nevertheless the family is not John, as well as the family is not Margaret. That is the most essential difference between properties and relations. The numerical difference is not essential. John loves Margaret. And John loves himself. In both cases "loves" is a relation in spite of John who loves and John who is loved, is the same object.
Certainly, the definitions listed above are not rigorous. They presuppose the knowledge of such things as "named," "described," "distinguishes," "constructs." If we had defined them, we should use the terms "thing, property, relation". Therefore, we may be accused of the mistake called "circulus vitiosus". But that mistake is inevitable in the definitions of categories. For the reason given "Thing," "Property,"
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and "Relation" must be considered as primary, undefinable notions. Our "definitions" are, more specifically, elucidations which can be useful for understanding our conception of those categories.
Another essential feature of our conceptual framework is the con textual character of distinction between the categories. It means that a thing in one context can be a property or a relation in another context. For instance, in the sentence "Love is a good affection," the word "Love" expresses a thing (=an object, an entity). In the sentence, "That affection is love," "love" is a property. In the sentence, "John loves Margaret," the word "loves" denotes a relation.
Things, properties and relations can be, in their turn, definite, indefinite and arbitrary. With the help of these categories we reflect extra logical reality in an essentially different manner from that done by the categories of set, element and quantifier. Let us take an example. "Some men are clever." Set-theoretical analysis presupposes that the sets "men" and "clever person" must be specified. It is not easy. We have to answer such questions: Is the Neanderthal man a man? Is it possible to include into that set future persons, e.g., in CLX century? Will mankind exist in that time? In practice we don't know such things. Actually in spite of the logicians' drill we don't separate the set "men" as a subject from "some" as a quantifier. We think that the subject of our thought is "some men" or indefinite man – a man. From our point of view "a man", "the man" and "any man" are three kinds of "man". All of them can be a subject of the corresponding judgments.
In general we can denote the definite object (thing, entity), i.e., the object by the symbol t, an indefinite object, i.e., an object by the symbol a, and an arbitrary object – any object by the symbol A. Formulae t, a, A are elementary well-formed formulae (WFF) of our formalism – Ternary Description Language.
The other types of WFF are formed in following manner:
II {A)A – Arbitrary thing (= object = entity) has arbitrary property. Certainly, that judgment is wrong. But it is a WFF. We can substitute instead of A, any WFF. All the results of such substitutions must be WFF. e.g., (A)a, (t)a, (a)t are the results of the substitutions of A by elementary formulae. In the case of ((A)a)a, we have a substitution non-elementary formula (A)a instead of A (thing) and elementary formula a instead of A (property).
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III A(A) – Arbitrary thing has arbitrary relation. WFF: a(A), a(t), t{a), a(a(A))are special cases of the WFF of that type.
IV (A*)A. This type of WFF differs from (II) in the direction of the predicate relation. That formula means that an arbitrary property belongs to an arbitrary thing. Of course it isn't true. But some special cases of (IV) are true: (a*)A, (a*)t, (a*)a, (a*)(a)t.
V A(*A). This formula is analogous to the preceding one. It means that an arbitrary relation can be realized on an arbitrary thing. It is wrong. The next formulae of that type are true: A(*a), t(*a), a(*a), a(t)(*a).
The formulae of the (II) – (III) types may be called direct ones and the formulae of the (IV)-(V) types – inverse ones.
VI [A]. A formula of this type means that which may be called the conceptual closure of the formula A. If A is of a propositional nature, i.e., express a proposition, then [A]denotes the concept corresponding to the statement A.
The conceptual closure of (A)A gives us the formula [(A)A] which must be interpreted as an "arbitrary thing possessing arbitrary property", in particular [(a)a] – a thing possessing a property. Correspondingly [(A*)A] – arbitrary property inherent in an arbitrary thing. [{a*)a] – a property inherent in a thing. By the same token, mutatis mutandis, we can interpret the formulae [A(A)], [A(*A)].
We may say that the formulae of type (II)-(V) are open, while the formulae with square outermost brackets are closed. If the formula A is closed, its closure doesn't change that formula. Therefore [[A]]means the same as [A].
VII {A}.Curly brackets have an ancillary character. They are used in the case when an inclusion of one formula into another as a subformula leads to ambiguity. For example, the formula (A)a(A) is well-formed. But it is possible to understand it in two ways: as "A possesses the property a(A)" and as "A possesses the relation (A)a." It is quite possible to adopt both interpretations simultaneously. But when only one of them should be adopted, the corresponding subformula is enclosed in curly brackets: (A){a(A)} or {(A)a}(A).
VIII A,A This type of WFF is a simple list of WFF. We shall call the formulae of such a type free lists because it doesn't suppose any relation between its components.
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Note that the order of the formulae in a list is not simply inessential; it is ignored, just as the differences between typographical marks for one and the same symbol are ignored. For the reason given, the combination of symbols t , a and a, t are regarded as one and the same combination of symbols.
Nevertheless the order of symbols is very essential in the other types of WFF. We saw it above on the examples of direct and inverse formulae. The importance of that order has its manifestation in the role of a symbol's place in a formula in the interpretation of that symbol. It doesn't hold true for the symbol of definite object – t, because t must be defined in advance without regard to a formula in which t is included. But the concrete meaning of an arbitrary object A and especially – indefinite object – a are function of their surroundings in a formula.
Let us distinguish the first – initial, and the second – final, parts in every formulae listed above. In direct formulae initial parts are included in parentheses. They denote things. Those are true both for the open and closed formulae. If the initial parts of the formulae are underlined we receive: {(A)A}, {A(A)},[(A)A], [A(A)]. In inversed formulae the initial parts denote properties and relations. Accordingly we have: {{A*)A}, {A(*A)}, [{A*)A], [A(*A)].
Note that a marked out formula is not a formula of a new kind. Marks are only means for the better understanding of a formula. In the open formulae: {(A)A), {A(A)}, {(A*)A}, {A(*A)}both A are completely arbitrary objects. Nevertheless in the closed formulae: [(A)A], [A(A)], [(A*)A)] and[A(*A)]the symbols of the completely arbitrary objects are placed only on the final parts. The arbitrariness of A on the initial parts of the formula are restricted. In [(A) A] the symbol A on the initial part of the formula denotes the arbitrary thing, which is having an arbitrary property. Of course, we cannot find such object in our world. But [{A)t] – an arbitrary thing having the property t, is a real object, if t is real. Correspondingly [t(A)], [(t*)A], [A(*t)]are examples of other kinds of restricted arbitrary things.
An indefinite thing, the symbol of which is placed on the initial part of an open formula, has an unlimited range of indefiniteness. e.g., it is so in the formulae {(a)a}, {a(a)}, {(a*)a}, {a(*a)}. But the second a – in the final parts of those formulae – has a restricted indefiniteness. In (a)a it is such an a which is a property of the first a. Correspondingly in {a(a)}the second a is a relation of the first a. In {(a*)a} the final a
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is a thing to which an initial a is prescribed as a property, and in {a(*a)} – as a relation.
In the case of complicated formulae which consist of many sub-formulae of various types we can find the initial part, i.e., the beginning of the formula step by step, e.g., in the formula ([((A*)A)A])A we can determine the initial part – [((A*)A)A].The next step is finding the initial part of that initial part. It is (A*)A. And finally we received the beginning formula at whole – (A*)A. The indefiniteness placed on the beginning of an open formula will be called initial, while the indefiniteness restricted by the sentence's context – contextual.
In the closed formulae the indefiniteness can be contextual even on the initial place, e.g. {{a)t]. Butin the case of [(a)a], there is no actual limitation, because a thing always has a property.
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