ATTRIBUTIVE SYSTEM’S PARAMETERS.

AS A FORMALISM FOR THE PARAMETRIC

GENERAL SYSTEMS THEORY : PART II

AVENIR I. UYEMOV

Odessa University, Odessa 65000, Ukraine

This part is a continuation of the first part of my article that was published in International Journal of General Systems, vol. 28 (4-5), pp.351-366. In the Part II we deal with the development of the conceptual content of the Ternary Description Language and formalization in it the most important concepts of the Parametric General Systems Theory such as system descriptors and system parameters. Formal models of the 40 values of the binary attributive system’s parameters are given.

Keywords: System’s descriptor; system’s parameter; attributive, relational, mereological, neutral implications; truth; contrary falsity; contradictory falsity; systems: conceptual point, structural point, rigid, totalitarian, minimal, immanent, centric, homeomery, elementary, unique, automodel, internal, homogeneous etc.

THE BRIEF EXPOSITION OF THE PART I

This section serves as an introduction to the Part II. It is a brief outline of what I have argued previously. First, I have showed the insufficiency of the existing mathematical and logical apparatus to express main concepts of General Systems Theories (GST). The formal apparatus appropriate for this purpose has to borrow some fundamental features from natural languages. One of them is intensionality, and the other is incrementality. (A formalism is called 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.) Furthermore, our desirable formal language must be self-applicable just as natural one. (It was argued that this feature does not lead necessarily to paradoxes if special methods of language construction are used). Finally, it is desirable to broaden the sphere of the logical conclusions within the language by including not only deductions from judgments, but also deductions from terms.

Below follows the account of the essentials of logical formalism called Ternary Description Language (TDL) that pretends to be a language of Parametric GST and possesses features mentioned above. Firstly, it differs from traditional (Aristotelian) logic and also from Predicate logic in its categorial framework, which includes three categories: Things, Properties and Relations. (This accounts for the name TDL of our formalism). The peculiarities of our philosophical approach to these categories were described in the previous paper. Here it can be said that we do not reduce the difference between

DOI: 10.1080/03081070290017886

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Property and Relation to the difference of one-placed and many-placed predicates. The essential feature of our conceptual framework is the contextual character of distinction between the categories. It means that a thing in one context can be a property or relation in another context. For instance, in the sentence “Love is a good affection” the word “Love” expresses a thing (= object = entity). In the sentence “That affection is love” the word “love” expresses a property. In the sentence “John loves Margaret” the word “loves” denotes a relation.

Things, properties and relations can be definite, indefinite and arbitrary. We denote the definite object by the symbol t, an indefinite object by the symbol a, an arbitrary object by the symbol A. Formulae

I. t, a, A

are elementary well-formed formulae (WFF) of our formalism – TDL. The other types of WFF are formed in the following manner:

II. (A) A

– Arbitrary thing (= object = entity) has an arbitrary property. Here we can substitute A for any WFF, and the result of such a substitution is considered as WFF too, e.g., (A)a, (t)a, (a)t, ((A) a) a are WFF. The same is valid for formulae given below.

III. A(A)

– Arbitrary thing has an arbitrary relation. a(A), a(t), t(a), a(a(A)) are special cases of that type of WFF.

IV. (A*) A

– This type of WFF differs from (II) in the direction of the predicate relation. The formula means that an arbitrary property belongs to an arbitrary thing, e.g.: (a*) A, (a*) t, (a*) a, (a*) (a) t.

V. A(*A)

– An arbitrary relation realizes on an arbitrary thing, e.g.: A(*a), t(*a), a(*a), t(a)(*a).

The formulae of the types (II)-(III) may be called direct ones, and the formulae of the types (IV)-(V) – inverse formulae.

VI. [A]

– A formula of this type means the what may be called the conceptual closure of the formula A. If A expresses a proposition, then [A] denotes the concept corresponding to that proposition. The conceptual closure of (A) A gives us the formula [(A) A] that is interpreted as “an arbitrary thing possessing an arbitrary property.” Similarly [(A*)A] denotes “an arbitrary property inherent in the arbitrary thing”, etc.

The formulae of the type (II)-(V) are open, while the formulae with square outermost brackets are closed.

VII. {A}

– Curly brackets have an ancillary character. They are used in the case when the inclusion of one formula into another as a subformula leads to ambiguity. E.g. (A) a(A) may be understood as “A possesses the property a(A)”, and also as “A possesses the relation (A) a”. The first interpretation is expresses as (A){ a(A)}, the second as {(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 they do not suppose any relation between their components. Note that the order of formulae in a list is ignored. The combinations of symbols t, a and a, t are regarded as one and the same combination.

Nevertheless, the order of symbols is very essential in the other types of WFF. We have seen it in the examples of direct and inverse formulae. The importance of that order gets its manifestation in the role symbol’s place in a formula plays in its interpretation. The concrete meanings of A and a objects (but not t object) depend on their environment in a formula.

Let us explain this dependence. We distinguish the first – initial, and the second – final parts in every two-membered formula listed above. In direct formulae initial parts are included in parentheses. They denote things. In inverse formulae the initial parts are placed outside of parentheses. They denote properties and relations. In the open formulae, {(A)A}, {A( A)}, {(A*)A}, {A(*A)}, both A are completely arbitrary objects. In the closed formulae, A denotes the completely arbitrary object only if it is placed on their final parts. The arbitrariness of A on the initial part of a formula is restricted, e.g. in the [(A) t] the symbol A denotes an

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arbitrary object that is restricted by the condition: “having the property t”. An indefinite thing a, when it is placed on the initial part of an open formula, has an unlimited range of indefiniteness. However, when a appears in the final part of a formula, it has a restricted indefiniteness. In {(a) a} the second a is an indefinite object, but is also a property of the first a. Correspondingly in {a(a)} the first (that is final) a is a relation of the second (initial) a. In {(a*) a} the final a 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 non-elementary subformulae, we can also (recursively) define the initial part, i.e. the beginning of the formula. The indefiniteness placed on the beginning of an open formula will be called initial, while the indefiniteness restricted by the sentence context – contextual. In the closed formulae the indefiniteness can be contextual even on the initial place, e.g. [(a) t].

If there are two or more occurrences of the symbols a or A in the same formula, this does not mean, that they necessarily denote the same object. On the other hand, different subformulae can denote one and the same thing (just like in natural languages).

In those cases, when it is known that various occurrences of the same or of different subformulae denote the same object, this fact should be expressed with the aid of additional symbols in front of these subformulae. There is no need to include these symbols to the list of WFF’s types since formulae with identifying symbols can be formally defined through WFF listed above, as it was shown in our previous paper. We have constructed our formal definition of identity based on well known principle that was formulated by Aristotle and is usually called Leibniz’s principle: “That what is said about one thing should be said about the other”.

Speaking about identity, we should take the direction of the identification into account. It is particularly important for us, because without the distinguishing of directions of identification we could not distinguish the operations of synthesis and analysis.

We use the small Latin letter j (italic jay) to denote an object with which the identification is being carried out: jA. Jay bold-faced letter in front of the formula denotes an object being identified: jA. The assertion of the identity of any object to any object will have the form: jA jA. In particular: { ja ja }, { jA ja }.

Formulae with jay operators are analogous to the identity relation that is represented by “=” symbol, e.g. in algebra: (a+b)2 = a2+2ab+b2 . But in algebra we can observe other type of identifications, which are related to separate terms in formula. E.g., both occurrences of a in a2+2ab+b2 denote identical numbers. Algebra does not require a special symbol for that kind of identity because of the assumption that it is expressed by the identity of the forms of symbols. But in our case the same symbol a (or A) can denote different objects in different occurrences. Therefore if those denoted objects are factually identical, it is necessary to use special marks for the corresponding occurrences of terms.

In this paper we shall restrict ourselves to the case of the undirected identity of terms. We denote it by the Greek letter i (iota) in front of formulae representing identified objects. It can be shown that iota operators can be formally defined through jay operators. Examples of iota operators usage: (i A)i A, (i A*)i A. Not one, but many different identifications can occur in the same formula. In this case, several types of iota operators are used. In order to obtain the necessary variety of these operators the letter iota can have various subscripts, be doubled, tripled, etc. E.g.: ([(i A)ii A])[(i A)ii A], ([(i8A)i13A])[(i8A)i13A].

Let us now turn to problems of GST. Various authors give different definitions of the system’s concept. Some are too “broad”, e.g. “Systems are sets of objects with some relations between them”. We can express such a definition with the aid of the following TDL formula:

( i A )S =def a(i A) (1.1)

Another kind of definition is connected to the specification of the “system-making” relation. Systems are defined as “Complexes of interacting elements”, “Complexes of interconnected

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elements”, or “Ordered sets of elements”, etc. Such definitions can be expressed with the help of TDL formula:

( i A )S =def [(a) t ] (i A) (1.2)

All definitions of the form (1.2), where t is a concrete property, are too “narrow”, because an attempt to find t that is appropriate for any system research fails. Every concrete t has its defects. From our point of view the solution of the problem is in the permission to change the interpretation of t. Instead of concrete t we will refer to t in general. In this case we can take the above formula not as a scheme of definitions but as a pure definition: “A system is an arbitrary thing in which a relation having a definite property is realized”.

We can rewrite formula (1.2) in equivalent but more compact form:

( i A )S =def ( [ a(*i A) ] ) t (1.3)

The majority of definitions given in the literature can be considered as particular cases of our definition (1.3). Nevertheless there are some exceptions. At times the role of definite t in the system definition moves to a relation (see Rapoport, 1966). In such cases we should have:

( i A )S =def (i A) [ t (a) ] (1.4)

( i A )S =def t ( [ (i A*) a] ) (1.5)

In words: “A system is an arbitrary thing in which properties having a definite relation between them are realized”. This definition is dual to the previous one in respect to transformation “property – relation.”

In the following sections of the part II the concepts of system’s descriptors and attributive system’s parameters are analyzed. Several types of implications are introduced. With the help of implications the concept of Truth and (contrary and contradictory) Falsity are defined. In the final section the explications of values of several binary attributive parameters are given.

SYSTEM’S DESCRIPTORS AND

ATTRIBUTIVE SYSTEM’S PARAMETERS.

Formulae (1.3), (1.5) of the previous section give evidence concerning that the system presentation of an object (i.e. construction of its system model) presupposes the separation of three aspects, which may be named system descriptors. First and foremost, t must be defined. Because t is immediately connected with the initial conception of the system, the term concept is prescribed to it. If the concept is a property as in the formula (1.3), then it is an attributive one. If the concept is a relation, as in the formula (1.5), then it is a relational one.

From this point on we shall restrict our consideration to the systems models with an attributive concept. Indicate the attributive concept with the symbol P (from Latin proprietas – property). A relation having the attributive concept as a property is called a structure (relational) and indicated with the symbol R (from Latin relatio – relation). A thing, on which a structure is realized, forms a substratum of systems. We indicate it with the letter m (from Latin materia – matter, substance, material). The concept, structure and substratum are the system’s descriptors of the first order. They form the integral elements of a system model of any objects.

Note that, for any object m, there is always a concept P and such a structure R, which satisfies this concept, that will be realized by that object. Therefore in relation to those P and R a

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given object m will be a system. On the other hand, there are always such P and R, that a given object m will not be a system in relation to these P and R. This gives evidence on the relativity of the system’s concept and the fact that systems are not a species of objects, but their models, which always can be constructed.

The aim of the system model construction consists in receiving a specific system information about an object. To reveal the weight or color of an object, it is not necessary to present this object as a system. It is another matter, if questions are posed about homogeneity or heterogeneity, or about the presence of a center, or about the stability or simplicity-complexity et cetera. For answers to these questions, it is necessary to present the object under investigation as a system.

The system information is determined by that class to which a system under consideration belongs. Thus we encounter the problem of the systems classification. In what way can that classification be constructed? The model of system, which was proposed above, gives the possibility to formalize the answer to that question.

Fundamenta divisionis in the systems classification are systems descriptors characteristics. Among those descriptors, which were defined above, only the concept has meaning in its own right. Its properties can be given without any relation to a structure and substratum. As for properties of the structure and substratum, they are determined only in relation to another system descriptors.

Relations between descriptors of the first order, i.e. the concept, structure and substratum, are descriptors of the second order. Their properties are essential as fundamenta divisionis of division of the general concept of system into classes. Indicate such descriptors. It is natural to express various modes of realization of the concept on a structure with the symbol P/R. The contrary relation – the structure to the concept is expressed as R/P. Relation of the structure to a substratum – R/m. It can be called the structure organization. Similarly – the relation of the substratum to the structure – substratum organization – m/R. On the basis of combinatory considerations it is possible to relate the concept P to the substratum m and vice versa – the substratum m to the concept P. However, such a relation has no practical interest, because the relation between the concept and the substratum always goes through the intermediary of the structure.

The order of the system’s descriptors may be enhanced if we relate a descriptor of the second order to a descriptor of the first or the second order. In such a manner a system’s descriptor of the third order would be obtained. However in practice such a descriptor and descriptors of higher order have seldom been used. In what follows we shall restrict ourselves to system’s descriptors of the first and on the second order. Descriptors of a higher order will be taken into account only as a theoretical possibility.

The next scheme of the systems classes’ definition may be concluded from the above:

(m)System of a definite class =def ([R(*m)])P · (2.1)

· {(P)a Ú (P/R)a Ú (R/P)a Ú (R/m)a Ú (m/R)a}

The meaning of the brackets here is the same as in TDL. a – the symbol of a property. The point denotes the sign of related list, joint acceptance, which is analogous to the conjunction. Ú – the sign of disjunction, which is not included into the set of primitive symbols of TDL, but may be formally defined in its framework. The symbols of system descriptors are not formulae of TDL. Hence the whole formula (I) is not a formula of TDL. This is only a graphic demonstration, the purpose of which is an explication of the mechanism of construction of systems classes. A class of systems is determined when an indefinite property a receives its formal expression in a definite formula of TDL. It is evident that for all sets of descriptors it is possible to obtain an unlimited number (counting infinity) of TDL

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formulae, which express the properties of those descriptors. Therefore for each set of systems descriptors we shall receive an unlimited number of system classes in the same way.

Concrete formulae expressing peculiarities of system’s descriptors determine those which can be named values of attributive system’s parameters. Each of those values determines the corresponding class of systems.

The supplementary value of an attributive system’s parameter determines the supplementary class of systems. All systems can be divided into two classes that are supplementary to one another, e.g. classes for having a center and not having a center, or homogeneous and heterogeneous systems, et cetera.

We call such attributive system’s parameters that have only two supplementary to one another values binary. In some cases, determination of the two supplementary values of the attributive system parameters is inadequate, e.g. making evaluation of complexity or integrity of a system. Here it is necessary to determine the multitude of values, which are ordered in a linear way. We call such parameters linear ones.

The preceding means that an unlimited quantity of attributive system parameters may be correlated to every system’s descriptor.

Any information received about a system can be expressed through characterizing it with values of attributive system’s parameters. The purpose of a general systems theory is not only a description of systems in the language of this theory, but also to receive consequences of that description. M.Mesarovic and Y.Takahara are right when they note that “Study of logical consequences from the fact that systems have definite properties must be the basic content of any general systems theory” (Mesarovic, et.al., 1975).

There are two methods to obtain such results. The first is based on statistical relationships, which are established between values of attributive system’s parameters. The parametric general systems theory has been developed in this way during the first stages of its existence. (Problems of the Formal Analysis of Systems, 1968; Portnov, et.al., 1972).

Combinations of values of attributive system’s parameters on concrete objects were investigated. In this manner 31 general systems laws of statistical character were determined. E.g.: “There is no centric system that is not stationary” (Uyemov, 1978). On the basis of this law if we are confronted with a centric system, then we can suppose with high probability that it is stationary.

The statistical character of interconnections between values of system’s parameters can give us only a probability of the obtained conclusion. Therefore in cases when a completely reliable conclusion is required, statistical methods are unsuitable. Here it is possible to use another, i.e. the analytical method of obtaining conclusions.

It may be used even if there is no axiomatic construction of the general systems theory. The difficulty of finding a general systems theory axioms was mentioned above. However such axioms turn out to be unnecessary if there is a formal apparatus with the help of which on the basis of formal models of some system’s parameters values it is possible to determine formal models of another system’s parameters values. TDL gives such a possibility. But in order to be able to express this, it is necessary to enrich its formalism so that it is possible to construct formal models of attributive system parameters values within its framework.

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