THE VALUES OF BINARY ATTRIBUTIVE SYSTEM’S
PARAMETERS AND THEIR FORMALIZATION
IN THE TDL FRAMEWORKS.
Let’s return to the scheme (2.1). Consider the first member of our disjunction (P)a which determines those characteristics of the concept which can be used to single out the value of a binary attributive system’s parameter. The concept P is a property, a – some property of P. Therefore the classification of properties is supposed.
In the framework of such a classification let us use the distinction between the simple definite object t and the limited definite object Lt from the previous section. The property that is associated with the limited definite object may be called a point property. It doesn’t allow for any variations. Aristotle gave us an example of such a property – “triangular” (Categories). This can be distinguished from the property “white”, which permits some variations. This is an example of “a non-point property.”
All the systems can be divided in two big classes according to these properties. The first includes those systems, which have point properties as concepts. The second includes those systems, which have non-point concepts. We call the first – conceptual-point systems, the second – conceptual-non-point ones.
As an example of the conceptual-point system one may take the natural series of numbers, the concept of which cannot be more or less. The relations of the type: “more clever”, “more beautiful” do not satisfy the concept of the full or strict ordering. Here only partial ordering may exist, which permits some deviations from the strict order.
The value of the system’s parameter “conceptual-point” may be formalized in such a way:
( i A )Conceptual-point system =def (i A){([ a(*i A)])Lt} (4.1)
The complementary class of systems is defined so:
( i A )Conceptual-non-point system =def
(i A){ {([ a(*i A)])t }·( t ÞLt )F} (4.2)
These definitions belong to the class of explications. Here definiendums are not formulae of TDL. They include some extralogical expressions, e.g. “system”, “conceptual-point”. Definitions do not make them formulae of TDL. They give only formal equivalents of those expressions. As distinct from explications, other kinds of definitions represent an introduction of denotations, with the help of which a simple symbol in definiendum is equated to more complicated one in definiens. Definitions (3.39-3.48) are definitions of such a type. Definiendums of those definitions became formulae of TDL with meanings taken from their definienses.
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In both cases of definitions definiendums and definienses may be as nucleary, i.e. non-valent, as valent formulae. Non-valent formulae can be made valent ones by putting of the mark T after the definiendum and definiens simultaneously. According to the convention c) from the previous section, the mark T may be omitted after the definiendum and definiens simultaneously.
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There exist not only conceptual-point systems, but also structural-point systems. A structural-point system is a system, the structure of which is uniquely determined, without any variations. To put it otherwise, the system’s structure must be a limited object. At the same time the structure need not to be a definite object.
The definition of a limited indefinite object is possible to obtain by substitution of A for a in the definition scheme (3.48). We will have:
L i a =def [(i a ){ { { i a ·ii a } Þ i a } É { i a É ii a } }] (4.3)
Changing iota-operators, we will obtain the next definition:
( i A )Structural-point system =def (i A){( [ L ii a (*i A)] )t } (4.4)
In definiens here you see a single iota operator ii a. However, there are paired iota operators in the definiens of L ii a definition.
The complementary class of structural-non-point systems is defined below:
( i A )Structural-non-point system =def
(i A){ {( [ii a (*i A)] )t} ·(ii a Þ L ii a )F } (4.5)
In definitions (4.4-5) we have used the convention c) about omitting the mark T after the definiendum and definiens of definition. We shall continue to use this convention.
While the division of systems according to the distinction between “conceptual-point system – conceptual-non-point system” pertains to the concept (taken as a system’s descriptor); the system’s parameter with values “structural-point/non-point system” supposes another system’s descriptor – the relation of the structure to the concept – R/P. Another parameter which pertains to that descriptor is a fundamentum divisionis of systems into structurally open and structurally closed ones. In the first case, the structure permits its complication without going out of the frameworks of the concept t, i.e. changing into another system. The other situation would take place in a structurally closed system. Here any complication of the structure demolishes the system. Plato supposed that a master and his slaves constitutes a structurally closed system, which is founded solely on the relation of domination – submission. Another view was Aristotle’s who admitted a friendly relation between a master and his slaves. Therefore Aristotle regarded the system “Master – slaves” as structurally open (Aristotle, Politics).
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The next formal definitions might be assumed:
( i A )Structurally open system =def
(i A){ {( [ii a (*i A)] )t} · {( [ii aD(*i A)] )t}} (4.6)
( i A )Structurally closed system =def
(i A){{( [ii a (*i A)] )t}·{( ( [ii aD(*i A)] )t )F}} (4.7)
The next parameter that reflects other relation of a structure to a concept is “structural variability / non-variability”. In structurally non-variable systems there are no relations that are different from the system-forming ones. As in the examples of the natural series of numbers or the abstract notion of a triangle. A structurally variable system has some relations that are different from the structure of a system. So, in a student group there are relations – of friendship, hostility etc., – that are different from group-forming ones. Let us give the formal definitions:
( i A )Structurally non-variable system =def
(i A){{( [ii a (*i A)] )t}·{ [A(*i A)] Þ ii a} } (4.8)
( i A )Structurally variable system =def
(i A){{([ii a(*i A)] )t}·{ ([A(*i A)] Þ ii a)n} } (4.9)
Note that in this definition the contradictory negation is used. It negates the truth of an attributive implication. By doing so, it asserts that for some relations in the system this
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implication is false, i.e. there is ([a(*i A)]Þii a)F. This is not contradict the case where for some other relations that implication is true, i.e. ([a(*i A)]Þii a)T also takes place.
Let us now change the direction of system descriptor R/P. We receive P/R, i.e. the relation of the concept to a structure. According to this descriptor it is possible to single out a new parameter. Let us consider a system, where the structure is uniquely determined by the concept. Such systems should be called rigid. For example of a rigid system, consider a problem which has only one feasible solution. Formally, it can be expressed as:
( i A )Rigid system =def (i A){{( [ii a (*i A)] )t}·{ t ® ii a}} (4.10)
In the other cases the structure is not uniquely determined by the concept. The concept can be realized by various structures. This occurs in such problems that can be solved by various methods. Such systems should be called non-rigid. It is possible to give them the next formal definition:
( i A )Non-rigid system =def
(i A){{( [ii a (*i A)] )t}·{( t ® ii a )F}} (4.11)
Another parameter, which is distinguished according to the descriptor P/R, has as its first value the kind of systems, which was the ideal of every totalitarian state, e.g. “State” by Plato, “1984” by Orwell or the non perfect realization of such ideas in the empire under Stalin, the third Reich under Hitler, China under Mao, North Korea, Kampuchea etc. This ideal holds that any relation in a state must be determined by its goal – the concept of a system. Such type of systems should be called totalitarian ones. Its formal definition is:
( i A )Totalitarian system =def
(i A){{( [ a(*i A)] )t} ·{ t ® [A(*i A)]}} ( 4.12)
Systems of that type do not exist in society only. A good example of such a system is the natural series of numbers. All relations in this system are determined by its concept – the full order.
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The complementary class – “non-totalitarian systems” may be obtained by the negation of the second set of braces in the definiens of the “totalitarian systems” definition. In negating the second set of braces we do not negate the possibility that some relation in the substratum could be the consequence of the concept. This means that the negation must have a contradictory, non-contrary character. Therefore we have the next formal definition:
( i A )Non-totalitarian system =def
(i A){{( [ a(*i A)] )t} ·{( t ® [A(*i A)])n}} (4.13)
Many system’s parameters correspond to the system’s descriptor R/m, i.e. they determine the type of relations of the structure to the substratum. Let’s begin the consideration of those parameters with the division of systems between non-minimal and minimal.
Non-minimal systems admit a removal of some of their elements without a destruction of the system as a whole. E.g. troops at war can suffer losses without ceasing to function as a system. Formally it can be defined as:
( i A )Non-minimal system =def
(i A){{( [ii a(*i A)] )t} ·{( [ii a(*i AÈ)] )t}} (4.14)
There are also systems, which do not admit the removal of elements without the destruction of the system as a whole. E.g. a quadrangle without an angle is not a quadrangle at all. Such systems we call minimal ones. Give the next definition:
( i A )Minimal system =def
(i A){{( [ii a(*i A)] )t}·{(( [ii a(*i AÈ)] )t)F}} (4.15)
Here we use an usual or contrary negation because there is no arbitrary object in the negated formula.
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Another system’s parameter is connected with the question: does a system-forming relation embrace only the system substratum elements or it embraces some elements outside this substratum? The movement of our troops forms a system, and it may corresponds strongly with the movement of hostile troops. However the movement of hostile troops is not included in the substratum of our system. The movement of troops on the parade is a different matter. Here a system-forming relation embraces only the substratum of a system. Naturally, in this case it is also necessary to take the conditions of the surroundings into account. This account has a different character from the account of a movement of hostile troops. Unsuitable conditions can spoil a parade, but they do not determine its structure.
We will call immanent a system, where the structure is not beyond its substrata. The complementary class is non-immanent systems. The formal definitions:
( i A )Non-immanent system =def (i A){( [a(*i A ·i A° )] )t} (4.16)
( i A )Immanent system =def
(i A){{([ii a(*i A)] )t } ·{( [ii a(*i A · [(A)i A°] )] )F}} (4.17)
Bear in mind that symbol i A° denotes disparate of the substratum i A. The contrary falsity (F) means that the ascription of the structure ii a to the synthesis of the substratum i A with any disparate [(A)i A°] is false.
Note that the change of a value of the considered system’s parameter may lead to fundamental discoveries. So – from Newton’s point of view – time flows “sine relatione ad externum quodvis”, i.e. represents an immanent system. As for Einstein, he represented time as non-immanent system relating time flow to movement of matter.
The next system’s parameter is centricity. There may be an element among all the elements of a system, such that relations between any of the elements of the system can be established only through the relations to this (central) element. Those systems can be named internal-centric. As an example take a “man-machine” system, in the case when interactions between the elements of the machine can be realized only with the help of a man, i.e. the machine is not an automaton.
If a machine is considered separately from a man who controls it, it forms a system of external-centric type because the center is outside the system.
The notion that generalizes internal and external centricity is simply centricity. The complementary (to centricity) value of the parameter is non-centricity. Give the next formal definition:
( i A )Internal-centric system =def
(i A){{([a(*i A)])t} ·{ii {i AÈ}·{[A(*i A)] Þ [a(*ii {i AÈ})]}}} (4.18)
The second set of braces means that an arbitrary relation of the substratum is some relation which is realized on a subobject of the substratum – the center – ii {i AÈ} .
( i A )External-centric system =def
(i A){{([a(*i A·i A°)])t} ·{ii{i A°}·{[A(*i A)] Þ [a(*ii{i A°})]}}} (4.19)
Here the first set of braces means that system-forming relation embraces not only the substratum i A but its disparate i A° too. The second set of braces indicates that an arbitrary relation in the substratum is a relation to some of its disparate – the center – ii {i A°} .
( i A )Centric system =def
(i A){{([a(*i A)])t} · {ii a ·{[A(*i A)] Þ [a(*ii a)]}}} (4.20)
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Here an internal center ii {i AÈ} or external center ii {i A°} is replaced by the simple center – ii a.
( i A )Non-centric system =def
(i A){{([a(*i A)])t } ·{(ii A ·{[A(*i A)] Þ [a(*ii A)]})F}} (4.21)
Here in the second set of braces there is a contrary negation which means the impossibility to be a center for an arbitrary object.
The parameter homeomerity was examined in ancient times. The term “homeomeria” (ta homoimere) is defined by Aristotle. By this term he meant things, in which the structure of the parts is the same as the structure of the whole. Examples are: copper, gold, silver, tin, iron, stone, etc., and also in an animal and plant: flesh, bones, tendons, skin, etc. (Aristotle, Meteorology).
We can say with reasonable confidence that Aristotle denoted the value of the system parameter, which we call homeomerity. The definition is:
( i A )Homeomery system =def
(i A){ {( [ii a(*i A)] )t} · {( [ii a(*[(A)i AÈ] )t} } (4.22)
The second set of braces in the definiens means that the structure immediately relates to any subsystem (part) of the defined system.
The negation of homeomerity may be defined so:
( i A )Non-homeomery system =def
(i A){ {( [ii a(*i A)] )t } · {(( [ii a(*[(A)i AÈ] )t)T)n} } (4.23)
The contradictory character of the negation represents the fact that in a non-homeomery system there may be some parts which are similar to the whole.
Further, all systems may be divided into elementary and non-elementary ones. Any element of a system may be considered individually as a system. There may be two cases. In the first, none of the elements of the system is a system in the same sense as the whole. In other words, the individual elements are not systems with the same concept as the initial system. Such systems we call elementary. E.g. a family is an elementary system because none of its elements is a system in the same sense in which the family is a system.
In a non-elementary system there are elements which are systems with the same concept that the whole system has. E.g. the Solar system is a system of that kind because there are such subsystems in it – Jupiter, Saturn with their satellites – which themselves are similar to the Solar system. Formally the distinction between the non-elementary and elementary types of systems can be expressed with the help of the next definitions:
( i A )Non-elementary system =def
(i A){{( [ a(*i A)] )t}·{( [ a(*i AÈ)] )t }} (4.24)
Note the absence of iota operators applied to the structure in the definition. That indicates the possibility of various structures of the initial system and those, which exist in substratum elements. Here only the identity of concepts is essential.
( i A )Elementary system =def
(i A){{( [a(*i A)] )t }·{((( [a(*i AÈ)] )t )T)n}} (4.25)
Using the definition of the contradictory negation, the definiens of that definition can be expressed in the next form:
(i A){{(( [a(*i A)] )t)T} · {(( [ A(*i AÈ)] )t )F}} (4.26)
Further, systems may be unique and non-unique. In unique systems a system-forming relation can’t be realized on any object different from the substratum. Works of famous
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artists are unique systems. Any reconstruction of their structures on other substrata, no matter how precise, is considered as a forgery. On the other hand, polygraphic copies of those works are non-unique systems, because they are realized on various substrata. As for other examples, the problem of uniqueness of the human personality is very interesting.
( i A )Non-unique system =def
(i A){{( [ii a(*i A)] )t} · {( [ii a(*i A¢ )] )t }} (4.27)
( i A )Unique system =def
(i A){{( [ii a(*i A)] )t }·{(( [ii a(*i A¢ )] )t )F}} (4.28)
Now let us analyze substratum-open and -closed systems. Structurally open and closed systems were considered above. They belong to the system’s descriptor R/P, i.e. the relation of the structure to the concept. Substratum openness and closeness belong to the system’s descriptor R/m, i.e. the relation of the structure to the substratum.
Substratum-open systems permit the addition of new elements to substratum without changing the system’s character. E.g. an edition of a book may be updated, new students may come during the lecture, etc. However adding a new angle to a quadrangle destroys that quadrangle: it is a substratum-closed system.
( i A ) Substratum-open system =def
(i A){{( [ii a(*i A)] )t} · {( [ii a(*i AD)] )t }} (4.29)
( i A )Substratum-closed system =def
(i A){{( [ii a(*i A)] )t }·{(( [ii a(*i AD)] )t )F}} (4.30)
Here we have finished the consideration of system’s parameters which are connected with the system’s descriptor R/m. Now we shall analyze some parameters, which are connected with the inverse meaning of that descriptor, i.e. with the relation of the substratum to the structure – m/R.
Consider the parameter automodelity. Automodel systems are such systems, in which every element has properties of the system as a whole. Therefore the system models itself in everyone of its elements. Let us take “Material thing” as a system. Material thing consists of material things, i.e. elements of a material thing are material things. Any properties of the system as a whole refer to anyone of its elements.
Note the difference between automodel and homeomery systems. The price of a pound of gold is much more then the price of a gram. Hence homeomery system “a pound of gold” is not automodel.
Give the formal definition:
( i A ) Automodel system =def (i A) {3 {1 ( [ii a(*i A)] )t 1} ·
·{2( [(A)i AÈ]*)[([(i A) {1[ii a(*i A)] )t1} ]*)A] 2} 3} (4.31)
( i A ) Non-automodel system =def (i A) {3 {1 ( [ii a(*i A)] )t 1} ·
·{2((( [(A)i AÈ]*)[([(i A) {1[ii a(*i A)] )t1} ]*)A] 2} )T)n3} (4.32)
Internal and external systems. If there are numbers, e.g. 4 and 5, then there is the relation between them: 4<5 and it is impossible to be otherwise. We call the relation “<” an internal relation between numbers. But if there are John and Peter, then we do not know who is a superior. “To be a superior” is an external relation between people.
The question about existence of external relations was called “the Great Question of Philosophy” (W.James). For us, as well as for James, there is no doubt about the positive solution of that question. We use it in the division of all systems into internal, the structure of which are internal relations, and external, the structure of which includes external relations. For the formalization of this distinction let us use the
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relational implication f that was defined in (3.4):
( i A ) Internal system =def (i A){{( [ii a(*i A)] )t} ·{i A f ii a}} (4.33)
( i A )External system =def (i A){{( [ii a(*i A)] )t}·{(i A f ii a)F}} (4.34)
As distinct from simply internal systems there are elementary internal ones. If in the first case the structure is relationally implicated by the substratum as a whole and nothing is said about that implication for each one of its elements, then in second case it is supposed that the structure is implied by each element of the system. E.g. a collection of tractor’s details is an internal system. It is known that three points determines a circle. Therefore any three points of a circle implicates the circle as a whole. Thus a system of three points of the circle is elementary internal. On the contrary, a system, elements of which consists of a single or two points of a circle, is not an elementary internal system.
Formally it can be expressed so:
( i A ) Elementary internal system =def
(i A){{( [ii a(*i A)] )t} ·{[(A)i AÈ]f ii a}} (4.35)
( i A ) Non-elementary internal system =def
(i A){{( [ii a(*i A)] )t} ·{([(A)i AÈ]f ii a)n}} (4.36)
We finish the consideration of system parameters with the substratum homogeneity-heterogeneity. The first type includes systems that consist of homogeneous elements. As an example we can take all copies of a draft of a geometrical theorem proof. Any property of one of them is a property of the others.
Formally:
( i A ) Substratum homogeneous system =def
(i A){{( [a(*i A)] )t} ·{([(A)i AÈ]*)[([(a)i AÈ]*)A]} } (4.37)
Substratum heterogeneous system supposes that there are elements in it, which have various properties. E.g., a watch, a human body. Formally it can be expressed as:
( i A ) Substratum heterogeneous system =def
(i A){{( [a(*i A)] )t} ·{[(i AÈ)ii A] ·[(i AÈ)ii A¢]}} (4.38)
Note that one and the same object may be a heterogeneous and homogeneous system in accordance with their concepts and structures. So a sand heap can be considered as building material and is a homogeneous system. A different situation arises when sand is subjected to physical or chemical analysis. Such an analysis may discover the essential diversity of properties of separate grains of sand.
References.
Aristotle, Categories, 10b25-11a15
Aristotle, Politics, 1255b13-b16
Aristotle, Meteorology, 388a13-a20
Carnap, R. (1946) Introduction to Semantics. (Cambridge, p.26).
Luschei, E. (1962) The Logical Systems of Lesniewski (North-Holland Publ. Co., Amsterdam)
Mesarovic, M. and Takahara, Y. (1975) General Systems Theory: Mathematical Foundation. (Academic Press, New York).
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Portnov, G.G. and Uyemov, A.I. (1972) Investigation of dependencies between systems parameters with the help of computers. (Systems Researches. Yearbook 1971. “Nauka” Publ. House, Moscow, p.103-127) –in Russian
Problems of the Formal Analysis of Systems. (1968). Eds. V.Sadovskiy and A.Uyemov. (“Vyschaja schkola” Publ. House, Moscow) –in Russian
Rapoport, A. (1966) Mathematical aspects of general systems analysis (General Systems, XI, p.3-11)
Ujomov, A.I. (1965) Dinge, Eigenschaften und Relationen (Akademie Verlag, Berlin, p.94-101)
Uyomov, A.I. (1978) System Approach and the General Systems Theory (“Nauka” Publ. House, Moscow, p.183) –in Russian
Uyomov, A.I. (1984) A Formalism for Parametric General Systems Theory (Systems Researches. Yearbook. “Nauka” Publ. House, Moscow, p.152-180) –in Russian
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