Object Definition in Semantic Space

Assuming that we have description matrices   of every informant's interactions ( - an informant's index in the selected mentality segment) with reality in different situations (or with a certain class of phenomena , ) in the property space . In fact, it is similar to performing a  independent series of measurements (an informant here is equivalent to a meter) of phenomena (situations) by indices.

; .

First and foremost, the matrices assigned in a property space, are recounted into  , defined in the semantic space.

Let us designate the estimation matrix of the -th situation  presented by all informants in the semantic space by .

; .

It is evident that vectors  of situation must make up considerably narrow "clusters" of correlated "opinions" in the context of a specific mentality, in the -dimensional semantic space,   - unit vectors possibly correlating with them (see Fig.2).

Fig.2  Definition of object states  by evaluaion results .

(Comments in the text)

Let us define independent object states or a set of stable interpretations for each in the context of the studied mentality. Since all the are unit vectors, every line of matrix represents decomposition of   into all , (). (Here and further the upper index   denotes transposition).

As far as the vector scalar product equals the sum of their coordinate bundle or the product of their lengths and cosine of the angle between them, therefore:

.

In the last expression   is an angle between the two unit property-vectors (-th и -th). It should be noted that the cosine of this angle absolutely equals the projection onto and  covariance of two random vectors. Therefore:

Evidently enough, matrix is symmetric to .  

We can further calculate major directions, alongside which the estimation vectors form clusters (have a maximal sum of projection squares), as well as correlate unit vectors to them along these directions (). It is evident that they characterize the most expected estimates of the given object by this mentality. In fact, represent orthogonal basis of the vector space defined by matrix , whereas coincides with the rank of this matrix (the number of linearly independent vectors). The search for reduces to the solution of a standard characteristic equation , all -eigenvectors and -characterisitc values/eigenvalues of matrix complying with it (7). In the case of symmetric matrix, all eigenvalues are positive. The matrix of eigenvectors prescribes major directions, along which the estimation vectors have the maximum sum of projection squares.

Let us standardize the eigenvectors so that , then and (the square sum of all unit vectors-observations, represented in the basis of eigenvectors ). To find the semantic representation of al the object states, let us employ the correlation: (projection of the -th observation of the -th object onto the -th component). Thus, for each state of each object on the basis of estimates of respondents one can form an equation system:

Since normally , the system is overdetermined and is solved with the use of residual vector. It should be noted that the relation represents a part of the whole sampling estimates of the -th object, over the -th state or probability of the -th interpretation , defined by state . Therefore, is equivalent to the number of interpretations , defined by state . It is usually supposed that . There is a theoretical possibility of the -solution to the characteristic equation, yet it makes no sense to consider the states with the corresponding eigenvalues lower than measurement errors. Consequently, . You can find the expanded form of this equation in  Appendix 1.

Let us determine that .  In the case of the general population (infinite experience), with eigenvectos in Hilbert space pass to the eigenfunction of operator. Different dimensions of a physical object are normally indexed with the help of . Therefore, and would determine the probability of finding object in the -state.

The average state can be defined as a sum of all estimate projections onto -state, referred to the part of these interpertations :. The average for all states equals:

.

Here:.

Object rigidity reflects the value and stability of all its estimates and, therefore, it is proportional to the average estimation vector and inversely proportional to their dispersion. Let

, therefore:

Since the vector length does not depend on the basis, then

 , и

 .

In a classical case, object is defined through a single state with probability . In case there are several states, especially when their observation conditions are mutually exclusive (for example, observation conditions for the states with descriptors "leadership" and "submission" in psychology, and wave properties in quantum mechanics), vector as a superposition of states , could be correlated with the  -th object.  Thus, .

Therefore, rigidity of the -th object - , rigidity of the object state equals , while its probability would equal .

Let us juxtapose a unit vector to each object state . While performing semantic analysis of sociological and psychological data, the study of average values as formal descriptors of a situation is not as necessary as considering real points of view (states)  with the indication of their stability (rigidity) and prevalence (probability) in a given mentality.

Let:

here is the probability of accepting estimate , while is the probability an opposite.

Let us juxtapose vector . to each object. It is evident that:.

A state rigidity can be decomposed in the same way:

here is rigidity of the positive state meaning, while is rigidity of the negative; is probability of the positive polarity, while   is the probability of the negative polarity within a given state.

If we introduce an operator-matrix :

, then and, therefore, the average module for all states equals:

.

In fact, we have just demonstrated how objects and their states "spring up" on the basis of the subject-environment interaction matrix as a method of outlining the results of that interaction with outside reality on the subject's mental map. Description of mutually disjoint states of one object defined by the mutually exclusive observation conditions makes the object turn from its classical (real) state to quantum (virtual) state. This is the cause of most paradoxes of the quantum mechanics. These paradoxes must definitely be part of the language as well.

The semantic description of objects that we obtained is virtually redundant. This is related to the fact that a specific descriptor set to define a certain semantic subspace is applicable only to a strictly specific class of objects.  These objects can basically be viewed as outside means of satisfying a certain requirement. If one runs out of inner means of requirement realization (for example, biological mechanisms of temperature-control), he makes use of the outside means that make up such object classes as: "clothing", "dwelling", etc. as for the food requirement, food products would be the means of its satisfaction, for example, "sausage". It should be noted that objects are the means of satisfying a need (here it is hunger), and not its subjects (it is not the sausage that is the subject of a food requirement). But that means that a certain class of objects that is related to a certain goal is united by one common function. Object properties within this class are not random or arbitrary. In any case, when elements are selected to form a class, there always functions a non-arbitrary factor that correlates decriptors of this class, itself being the reason for this kind of correlation. It does not mean that a semantic space has lost its orthogonality. It is just that objects within a certain class possess a certain type of similarity related to the commonness of function or another selection factor. For example, such qualities as "green" and "unripe" in the class of "berries" can be independent for one type of territory, yet for our latitude they are correlated. By analogy: a negative correlation between the number of professors and farm animals can only be traced with the progress of urbanization. This very fact makes semantic object description through properties really redundant.

One of the ways to overcome redundancy of such object representation is the transition from the primary characteristics to orthogonal factors that unite the descriptions synonymous in a given class of objects. That considerably shrinks the size of a semantic space. Essentially, there works an operation similar to the state selection operation, only concerning the properties. It is natural that both the lengths and angular ratios of semantic vectors must remain intact. Since every object class is associated with a definite goal (requirement), certain conditions and means of realization, the factor actually reflects a psychological and social perception and estimation attitude [8] toward objects of this class in a certain mentality. It is clear that in a different class perception, the interpretation attitudes of objects and phenomena also change. Since every factor is determined by a regression equation that links properties by a specific correlation in this class of objects, we automatically obtain specific laws of property correlation therein. Factor calculation is presented in  Appendix 2.

Thus, we obtain the image of object states i m-dimensional space of independent factors (attitudes):

.

VARIMAX-rotation of the factors by orthogonal transformation (9), simplifies their lexical expression through primary properties and completes the transition from the factor semantic space to the categorial semantic space :

.

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