## CHAPTER TWO

# THE SPECIAL THEORY OF INFERENCE AND TERMS

## 1. THE LOGICAL EXPLICATION OF TERMS

The task of logic is not just to study the general properties of terms and expressions, associated exclusively with the fact that they represent terms and expressions, and with the logical operators and the combinations of logical operators, terms and statements contained in them. It also studies various terms and expressions through establishing their meaning, or by realizing their explication by the methods of the general theory of inference and terms. Examples of such expressions are 'individual', 'class', 'cluster', 'exists', 'possible', etc. From this point of view, logic represents the general semantics of science.

## 2. EMPIRICAL AND ABSTRACT OBJECTS

It is only in the final chapter that the distinction between empirical and abstract objects will be of significant importance. But since many of the definitions accepted in this chapter do not agree with the usual definitions of the corresponding expressions that have been formed precisely in connection with abstract objects and that are not applicable to empirical objects, we shall, in order to give an explanation, have to take into account the difference between abstract and empirical objects. Therefore, in any case, we shall offer a few remarks here concerning this issue. The objects that are perceived, sensed, observed, etc. by the investigator, in other words, the objects that act on the scientist's sensory apparatus, are empirical objects. They exist at a certain time and in certain areas of space; they appear, vary and become destroyed, influence each other, etc. To put it briefly, they include everything that may be said in general concerning the research activities of people The investigator may make a decision not to take into account certain characteristics of the objects under study, or to take into consideration only some of their characteristics. This decision may be carried out by selection of the appropriate research situation or through an artificial creation of such situation. In this case, however, the objects under study remain empirical objects, but considered under special conditions. The situation is quite different if one adopts a decision to neglect those characteristics of object without which the empirical objects cannot exist. For instance, the investigator may decide to ignore the dimensions and shape of physical bodies when considering their motion, under the assumption that these bodies have no spatial dimensions (they represent 'material points'). Such a decision may be realized by the assumption of the existence of special objects that are called abstract, or ideal. This kind of object does not exist empirically, by the very nature of the underlying assumption. And study of such objects will not represent a process of observation. Instead it will be a process of determination of the appropriate definitions and axioms (postulates) and of deducing necessary conclusions from them. The terms for abstract objects are introduced, first of all, through definitions, which may be presented explicitly and schematically as follows: (1) we shall call by a term a those objects that have the attributes P1,..., PM (n > 1) but do not have the attributes Q1,..., Qm (m > 1); the attributes Q1,...,Qm are such that if the empirical objects have the attributes P1,..., PM, they also have the attributes Q1,..., Qm; (2) we shall call by a term a those objects that have the attributes P1,...,P", and if from the fact that they have these attributes, and from other assertions accepted in a given science, it does not follow logically that Q(a), then we have HQ(tf); the attribute Q may be such that if an empirical object has attributes P1,. •>?", it also has Q. Then the terms for abstract objects are introduced according to the general rules for construction of terminology with the terms, introduced by Definitions (1) and (2), used as the initial base. Examples of terms for abstract objects: the terms 'material point', 'ideal gas', 'incompressible fluid', etc. In the case of a material point it is assumed that a physical body has mass but no spatial dimensions. The objects denoted by terms for abstract objects do not exist empirically according to the very conditions for construction of these terms. It does not follow from this that these terms do not denote anything. Thus, the term 'material point' denotes physical bodies possessing mass but not possessing spatial dimensions. As far as the relation of these terms to empirical objects is concerned, it is governed by the following rule: a decision is taken to consider certain empirical objects a such that a-^b, where b is a term for abstract object. For instance, a decision is taken to consider the planets of the Solar system and the Sun as material points.

It is an extralogical question, how the objects a are selected. In this case it is important that the relation a-^b is accepted according to the investigator's choice. Then from a-^b and from certain statements containing b, one can obtain, as logical conclusions, the statements containing a. The objects a are selected in such a way that these latter statements could be accepted as true. Deviations from the desirable result presented here do not change the essence of the operation. In practice the situation is as follows: there are known empirical objects a, having the attributes P1,..., P"; it is known that their attributes Q* may be neglected; and the term for the abstract object b is introduced in such a way that a decision is taken beforehand to accept a-^b; but the set of objects that may be regarded as objects b is not limited to the objects a; in principle it is not bounded, and any objects, similar to objects a (from the point of view of the above attributes), may be regarded as objects b. The selection of the empirical object a such that a-*b represents the empirical interpretation of the abstract object. If the terms for abstract objects are complex, and there are derived predicates introduced for them, then the empirical interpretation of abstract objects proves to be a rather subtle and not always possible process. And a whole number of methodological difficulties results from violation of the rules for such interpretation. If b is a term for abstract object, and P is a predicate, introduced for the terms of abstract objects and appropriate with respect to b, then in the empirical interpretation of b the problem of the possibility of expanding the sphere of application of the predicate P over a should be solved in a specific way in each case. It is not excluded that this interpretation of b will represent the interpretation of P(fc), for which one selects for the object a a certain predicate Q such that J,Q(a)—\|,P(6).

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