THE FORMALIZATION OF THE SYSTEM'S DEFINITIONS
It is known that various authors give various definitions of the system's concept. Part of them cannot distinguish "system" from "not system". These definitions are "too wide." e.g., "systems are sets of objects with some relations between them" (Kleene, 1952).
We can express such a definition with the aid of the following TDL formula:
[(ιA )S]=def [a(ιA)] (4.1)
Here the symbol S denotes "a system". That definition is too wide because any object has some relations.
Another kind of definition is connected with specification of the type of relation which must have systems. According to Bertalanffy (1956) systems are "complexes of interacting elements." Here, instead of a simple relation, we have a relation of a specific kind – an interaction. Its properties must be definite and given in advance as t. Thus, we can have the definition of the next type:
(ι A)S = def [(a)t](tA) (4.2)
Ackoff criticized the definition by Bertalanffy as too narrow. It doesn't embrace conceptual systems, elements of which are inter connected, but they do not interact with each other (Ackoff, 1964).
The definition of systems as "complexes of interconnected elements" has the same scheme (4.2). It can be criticized as "too narrow" as well. There can be no interconnection between historical events in various parts of the Earth, but if all of them are arranged in time, we have a system. In this case, instead of interconnection, we have to take an order. The scheme of our new definition should be the same as (4.2), but the interpretation of t should be different. Now t is the combination of the three properties: antireflexivity, antisymmetricity, transitivity. But the definition of systems with the aid of those properties is
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too narrow as well. It doesn't embrace some physical systems, e.g., electronic gas, in which it is impossible to introduce any order.
How can we find a way out of such a situation? It is useless to try to find an appropriate t. Every concrete t has its defects. From our point of view the solution of the problem is in the changing of the interpretation of t in (4.2). We should change a concrete t by t in general. Let t be any definite property given beforehand. In this case we can read (4.2) not as a scheme of definitions but as a definition:
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"A system is an arbitrary thing in which a relation having a definite property is realized".
We can rewrite (4.2) in a more compact form:
(ι A)S =def{[a(* ι A)])t (4.3)
The majority of definitions given in literature can be considered as particular cases of our definition. Nevertheless there are some exceptions. At times the role of definite t plays a relation. Some properties correspond to that relation. A system is an object which is having those properties (Rapoport, 1966). In such cases instead of (4.2–4.3) we should have:
(ιA)S=def (ιA)[t(a)]
(ι A)S=def t([(ι A*)a}) (4.4)
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."
References
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Aristotle, Topics. 152 b 25-30.
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Boulding, K. (1956) "General systems theory – the skeleton of science," General Systems, 1,11-17.
Churchman, C.W. (1964) "An approach to general systems theory." Views on General Systems Theory (John Wiley and Sons, Inc, New York, London, Sydney).
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Cornacchio, J.V. (1972) "Topological concepts in the mathematical theory of general systems." In: Trends in General Systems Theory, G.J. Klir (Ed.) (John Wiley, New York).
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Grzegorczyk, A. (1974) An Outline of Mathematical Logic. (D. Reidel Publishing Company, Dordrecht, Boston) (Polish Scientific Publishers, Warszawa).
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Screider, Y.A. and Scarov, A.A. (1982) The Systems and Models (Radio and Syjaz, Moscow), p. 8 (In Russian).
Tachtajan, A.L. (1972) "Tectology: history and problems." Systems Research (Yearbook 1971, Moscow), pp. 200-277 (In Russian).
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Uyemov, A.I. (1983) "Self-applicability as the property of the formal language of the Parametrical General Systems Theory." In: VII Int. Congress for Logic, Methodology and Philos. of Science (Abstracts), Sections 6, 8-13 (Zalzburg, Moscow), pp. 68-70.
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Avenir I. Uyemov was born in the village near Ivanovo (Russia) in 1928. He graduated from Moscow University in 1949. The degree of candidate of science (Logic) was received in 1952. In 1964 he received the degree of doctor of science (Logic). He was the head of the department of Philosophy in Odessa State University from 1964 to 1973. Since 1973 till 1996 he was the Head of the Management Theory and Systems Analysis Department of the Institute for Market Problems and Economic-Ecological Researches of Ukrainian National Academy of Sciences. Since 1996 he has been a professor of Odessa State University. His main research interest is Logic and Methodology of Science. Since the 1960s he has worked on the problems of construction of Parametric General Systems Theory and its Formal apparatus.
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