## CHAPTER FOUR

# LOGICAL PHYSICS

## 2. EMPIRICAL INDIVIDUALS

We shall not give the complete definition of the expression 'empirical individual' immediately. Only its partial definition, sufficient for the purposes of our presentation, will be given; it will be subject to additions as needed. The question of its complete definition will be treated again later, in connection with another problem. Partial definition is appropriate here, since we are going to consider consequences of the fact that the given objects are empirical; this consideration will be based on the available parts of the definition. Empirical individuals possess spatial dimensions. This statement represents a part of the implicit definition of the expression 'empirical individual'. Since the meaning of the linguistic expressions recording the volumes and areas of empirical individuals reduces logically to the meaning of the expression 'length', in what follows, when considering the spatial dimensions of individuals, we shall limit ourselves to considering their length. The expression 'a length a with respect to a' will have the following symbolic form: ls{a,a}. The above statement can be written as (P1. hA->(/s{a,a}>0), where A is the statement 'a is an empirical individual with respect to a'. It follows from PX that if /s{a,a} =0, in other words, if a has no spatial dimensions with respect to a, then a is not an empirical individual with respect to a. For sake of argument, it is more convenient to use a different formulation (P1. hB where B is the statement 'a is an empirical individual', and a is a variable for the methods of establishing the spatial order.

## 3. VARIATION

Assume that J,x is a state of an empirical object a at a certain time, and [y is a state of the same individual at a different time after that, i.e., J, j; supersedes J, x in the temporal order. Assume also that |— ~ (x A y\ in other words, that the states [ x and [ y exclude each other. We shall say that a transformation of the state [ x into the state j. y has taken place (or that lx has transformed into ly) or that an J,x has transformed into an I y. This will be written symbolically aslx=>ly and a[x^>a[y.The special cases of an individual's transformation from one state into another, and especially cases of variation are as follows: (1) i"] (E(a) => J, E(a) - creation of a; (2) j ~ x => j x - creation of J, x; (3) J, E(a) =>I n E(a) - annihilation of a; (4) J, x => J, ~ x - annihilation of x; (5) I ~\P(a)=>l ~\P(a) - loss of an attribute by an individual a; (6) I ~~]P(a)=>lP(a) - acquisition of an attribute by an individual a; (7) I P(x(a) =>l P/?(tf), where a > /? - decrease of a with respect to an attribute P; (8) I Pa(a) =>I P/?(a), where a < /? - increase of a with respect to an attribute P. Depending on the types of attributes with respect to which the variations of objects occur, special predicates of variation ('grew up', 'expanded', 'contracted', etc.) are introduced. This serves as a foundation for introduction of other predicates of variation, according to the rules for definitions. But their logical basis is formed by the predicate =>. Moreover, all logical properties of the derived terminology, developed for recording variations, may be reduced to it. Therefore, when we speak of variation and transformation, we imply the presence of a certain empirical object and a change of its states with time. Transformation of one object a into another b (for instance, a pupa transforms into a butterfly) usually means transformation of a state of a certain object, denoted by the term a, into another state of the same object, denoted by the term b. In order to call a certain set of variations, ordered in space and time, also a variation, it is necessary to introduce a term a by which one can denote a given cluster of empirical objects at different moments of time, regardless of the variations that occur, and to represent the results of the latter as different states of a. A special case of variation is provided by change of place in space (translation or motion). We will discuss this case separately below. Variations take place in certain spatial domains. The varying objects are characterized by spatial dimensions. But in the case of variation, one cannot introduce the spatial dimensions as such, without making certain stipulations. This is a consequence of the fact that spatial predicates are used for a different kind of empirical objects (we shall come back to this later). Variations are characterized by temporal length or duration. In the same way as one can introduce certain spatial terms and a concept of length on the basis of observation of some empirical bodies, observation of certain variations allows to present the concept of time as well as certain elementary terminology related to it. There is also a historically conditional and temporally transient quantity characteristic of the duration of a variation, necessary for the variation to be detected. The term 'the duration of variation of a with respect to a' will also be a symbolic form I'{a, a}. Every empirical variation takes place in real time, i.e., the duration of any empirical variation is greater than zero. This constitutes a part of the very definition of the expression 'empirical variation'. Suppose that A is the statement 'a is an empirical variation with respect to a'. The above statement can be written as follows: < f >2. \-A->(r{a9ot}>O). From < 1 >2 one obtains: if /'{a, a} = 0, in other words, if no time is spent on variation a, then a is not an empirical variation with respect to a. Suppose that B is the statement 'a is an empirical variation', and a is a variable for the methods of establishing the temporal order. Then the above statement can be given a different form: < f >2. hB-^(Applying the rules of the logic of relations to ($*), we obtain: {} h-B->(Va)~(/s{a,a}<), and from &2 it follows that hB-»(Va)'~(P{a,a}<), i.e., there are no empirical objects for negative lengths and durations.