There are similarities and differences
between the virtual distributions which are supported by set {R}, or
by its intervals, briefly presented above, and the *realizable*
*distributions* (which are also called in this paper as
*systemic*). The similarity is that the systemic distributions
also consist in a set of assignment relations between the value of
the dependent variable and the values of the independent variable,
and the domain of potential values of the independent variable
makes-up the distribution’s *support*. The basic
difference between the virtual and the systemic distributions is
represented by the fact that the elements of the virtual
distributions (according to their definitions) are mostly *virtual
*objects (non-realizable as the class instances, because the
support singular values which belong to set {R} contain an infinite
quantitative information amount), while the systemic distributions
are *realizable* objects, either only abstract, or abstract and
material.

Comment
2.3.1: The following chapters will allow a better understanding of
the concept regarding the *realizability* of an object. For the
time being, it is enough to mention that an object is *abstract
realizable* if its related information is finite, therefore, the
information amount may be included in a finite Information Support
System (ISS). For example, an usual numerical value is contained in a
finite number of digits. An abstract object is virtual if it is
associated with an infinite information amount; this kind of objects
cannot be actually realized (as instances of the objects class). An
abstract realizable object whose properties may belong to a material
object shall be a *material* *realizable object*.

As we have already noticed from the
previous subchapter, the primary distribution element is represented
by a singular value *y*_{k} of the distributed
attribute, assigned to a singular value *x*_{k}
of the support attribute by means of a relation *f*_{k}.
The point of view on the singular numerical values and on the support
domain amount is the essential difference between the virtual and the
realizable distributions. If the virtual (mathematic) distributions
are supported by the infinite domains and AAV (see Dirac distribution
described in the previous chapter), the realizable distributions
allows only finite interval supports, limited by the two boundaries,
with values which were estimated in relation to the inner reference,
and these intervals consist of __a finite number__ of singular
values.

Therefore, taking into account the
support domain amount, a realizable object shall always have its
properties distributed on a non-null and finite support domain. The
most simple domain is the *pointwise domain*, any other domain
could be made-up by means of composition (concatenation, joining) of
a finite set consisting of such identical elements. As for this basic
element of support domain, the objectual philosophy proposes a
special denomination: *domain point* or *dimensional point
*(DP) which is the geometrical representation of *a normal
singular value* (for details, see annex X.3).

**Definition
2.3.1:** An absolute accurate singular value *x* which is
associated with a non-determination interval^{1}1
with the amount
,
becomes a **normal singular value.**

Comment
2.3.2: The non-determination interval **
has the role to substitute the infinite number of digits required for
the display of an AAV, which should follow after a reasonable series
of digits representing the normal singular value, by providing in
this way the finiteness of the information amount required by the
abstract realizability. For example, the following numerical value:

If
it would be written according to its value from {R}, we should also
have an infinite series of 3 figures after the decimal point.
Practically, if it would be written as x = 0.333, this means that we
have drop out the infinite interval as a number of figures included
in brackets, that part becoming a non-determination interval. The
association of this non-determination interval was done since the
ancient times up to nowadays, the people always operating with normal
singular values, but without considering the discrepancy between the
elements of set {R} (according to its definition) and these normal
numerical values. Further on in this paper, when we shall be dealing
with singular numerical values, these will be normal values, if not
otherwise mentioned. According to the fashionable language, the
dimensional point (DP) may be considered as a quantum for a variable
domain, but unlike the fundamental quanta *h* from the current
physics field, the quantum DP does not have an universal size, but it
depends on the Information Processing System (IPS) type which uses
it. If it is operated with a single-digit after the decimal point,
then,
,
and if the calculus is made with numbers with 6 decimals, then,

In order to comply with the definitions
which were above-mentioned in this chapter, we must warn the reader
that if we have defined DP as __a domain__ (interval), even if it
is pointwise, this fact means that its boundaries must be settled
according to definition 2.2.4. Because DP is __informationally
__equivalent with a single-determined numerical value (normal
singular value), another known value is needed, namely the inner
reference of other adjacent DP. In this way, the amount of the
non-determination range related to DP is the difference between two
normal singular, successive values of the support domain.

Comment
2.3.3: In this way, the continuity and coherence of the structure of
the elements belonging to the objectual philosophy is provided, as we
are about to minutely see in the following chapters, after we shall
enlighten ourselves by using the *distribution* concept about
what does the *object* mean and what does the *inner reference*
of an object mean. We shall also see that the relations between two
objects are actually relations between their inner references, in
case of two DP-type of objects, the reference are the two RAV which
have been associated with the non-determination intervals (see annex
X.3).

On such a support
domain element (DP type), which, once again, __informationally
speaking, is equivalent with only a single known value__ (a
singular value), the result is, according to the definition 2.2.1,
that the value distributed by means of relation *f* shall also
be a single one (but which have also an associated non-determination
interval), therefore, the basic element of *realizable* primary
distribution is *a normal singular value, distributed on a DP
support domain, *or otherwise speaking, a normal value *y*_{k}
of the distributed attribute, determined through the relation *f*_{k}
by a normal support value *x*_{k}.

**Attention! **Do
not confound a __non-determination interval__ of a variable value
with an __interval of deterministic variation__ of the same
variable. If the inner differential quantitative information is null
within the non-determination interval (the equivalent of an uniform
distribution, equiprobable), this information is non-null within the
deterministic variation (e.g. linear) interval. This aspect is
approached in annex X.3, when we shall be dealing with the difference
(in case of axis X, for example) between the domain quantum **_{x}
and the elementary domain *dx.*

As for the realizable distributions, based on the above-presented issues, we may notice that the support of the primary achievable distributions is no longer continuous (as it was in case of the virtual distributions), but discrete, any of its finite interval being made-up from a finite number of normal singular values.

11 As we are about to see later on in this paper, non-determination means the lack (absence, nonexistence) of information.

Copyright © 2006-2011 Aurel Rusu. All rights reserved.