**FROM THE LOGICAL TO THE STATISTIC
PROBABILITY**

**IN THE SCIENTIFIC RESEARCHES**

Nikolay
Ivanov Petrov

* **Technical** University – *

** Abstract: **The
article features the ideas of an aviation engineer, specialist in reliability
and management systems. These ideas have been formed on the basis of quotations
of world known scientists in the fields of philosophy, mathematics and the theory
of management.

The following thesis
is supported: “The measurement of the probability degree for making a
hypothetical conclusion is defined by the information possibility for
generalization found in its contents.”

** Key words:** logical probability; statistic probability; information
possi-bility for generalization

** Introduction:** The idea of probability appeared in the remote
past. Then it was the characteristic of human knowledge. Besides, the existence
of probable knowledge was recognized in contrast with the true (real) knowledge
and the false (delusion) knowledge. As Bertrand Russel points out two
scientists – skeptics, Charneades and Chlitomachus, “revolted against the
belief in gods, magic and astrology, which had been gaining popularity. They
developed a con-structive doctrine which defined the probability degree as a
sense of trust.”

The role of the probability theory in the
structure of contemporary mathematics is essential. “Every specialist in the
probability theory knows well that mathematics is actually part of the
probability theory.” - these words were
the beginning of the lecture delivered by the world famous American scientist
George Doob in front of the Moscow Mathematics Society in 1960.^{3}
Of course, his words were
accepted as a joke but everybody knows the old saying that “in every joke there
is some truth”. It is a fact that in modern mathematics we can bear witness to an
intensive development of probability theory.

Isaac Yaglom thinks that “the relative weight
of the probability theory has dramatically increased in comparison with the other
mathematic sciences.”^{3}

The famous Bulgarian
scientists Academic Ivan Popchev, Prof. Yordan Zapryanov and Prof. Stoyan
Markov say that: “The formal theory can, not only, provide for the building of large system models, but
also, because of its specific structure, strict logic of conclusion and
simplicity of terms is a tool of knowledge, allowing the achievement of new
results which cannot be possible within the content description. Mathematic models as a class of sign models
are actually an illustrative, concrete and simplified image of the original. If
there is compliance between the model and the original, the model can be used for experimental
research, its properties can be studied and transferred to the original, and
thus a theory can be build. A model is a mathematical description of the
information and logical side of the formal system structure.”^{4}

* *

** Expose: **The use of probability and its distributions is extremely wide – from
the every day language to all scientific researches. Together with that, the
interpretation of the probability is well-founded and it shows the large scale
of interaction with the fundamental models of being and knowledge.

The development of the probability idea had led
to the fact that in the philosophy’s methodological literature of the 30s of 20^{th}
century were differentiated two types of probability – logical (inductive) and
statistic (frequency) probability. The scientist Rudolph Carnap points out:
“There exist two fundamentally different types of probability and I call the
first one “statistic probability” and the other one - “logical probability”.
Unfortunately, the word “probability” is used with two completely different
meanings. This interpretation of the different meanings of the term
“probability” is the reason for dramatic contradictions in the books of philosophic
sciences. It is also the reason for philosophic discussions due to the
different speculations of the scientists.”^{5}

As he mentioned his interest in probability
interpretation, Karl Popper says that together with Rudolph Carnap they
decided, in the 30s, “to differ sharply probability as a term used in the physics
hypotheses (especially in the quantum theory) on the one hand, and on the other
hand – the so called “probability hypothesis or
a degree of its proof (a degree of confirmation).”^{6} The idea of inductive (logical)
probability is older than the idea of statistic (frequency) probability. In the
remote past the term “degree of probability” was used as a compliant
characteristic of some or other hypothetical statements. A similar approach to
probability is quite popular nowadays, too, especially in every day life.

In the spoken language the phrase “quite
probably”, “probably”, “unlikely, improbably” and “impossible” are frequently
used. They are used in describing the perspective of the development of some
events – from the simple life is truth to the scientific and political
prognoses. For example, it is said: “It is very probable NASA to build a space
base on the Moon in ^{st}
century.”, “It is absolutely improbable that the idea of trajectory has played
an important role in the development of the elementary part theory”, etc. In
the above mentioned cases probability is shown as a measurement for subjective
confidence, focusing on different outcomes of the events. It makes it possible
to express an attitude towards the reliability of particular single
statements.

Such an attitude towards probability, as a
characteristic of logical relations, is practically independent from the direct
experience appeal. This problem is typical for the period of recognition of the
probability theory and its classical stage of development. The language of such
theory is formed on the basis of analogy with the separate single statements
and the analyzing of the different degree of their confirmation. „For the scientists of the classical
period”, Rudolph Carnap says, “ probability is a degree of reliability of
future events or the trust in them. That is a logical probability, not a
probability in the statistic sense.”^{5} Correspondently, an event with a degree of probability
1(one) is considered reliable, while an event with zero probability –
improbable and an event with a probability degree different from zero – as a
probable one with a certain degree of reliability. It is interesting that the
term possibility is divided into two types. Max
Bunge states: “There exist two radically different types of possibilities-
conceptual and real.”^{7} The real possibilities refer to the formal
interpretation of probability, while conceptual possibilities – to the logical
probability. By considering the two approaches to probability interpretation,
Yan Haking discusses its dual nature.^{8}

Inductive probability expresses the logical
relation between two opinions (statements). The mutual subordination between
the opinions (statements) has different character and they are mostly divided
into deductive and inductive.

Deduction is a term which means that data, taken out of a batch of conclusions, are as true as the batch itself.

Induction is different from deduction as with it the state of the problem is completely different. The reliability of the inductive conclusion is always undefined, i.e. it is not a result reached by logical conclusion based on the batch, which makes it unreliable.

Here comes the question: Can the degree of reliability of the inductive hypothetical conclusion be evaluated in quantity? In the course of probability theory development such questions were frequently discussed each time the possibility to talk about a probability degree of hypothetical conclusion appeared in the study of nature objects. In all such cases, the value of probability cannot be strictly measured or given quantity expression. Here we have to consider that the logical, i.e. inductive probability does not allow the development of the mathematic apparatus of the probability theory.

The mathematical development of the probability
theory is connected with the other type of probability – the statistic
(frequency) probability in the study of phenomena. The achievements of the
probability science are strongly connected with the fact that they allow the
introduction of mathematical image (notion) of thinking in the study of the
processes. That image is based on the development of the mathematical apparatus
during the centuries. The development of the probability notion expresses
mainly the strength and power of mathematics and its modern apparatus.
Namely, the mathematical nature of probability defines its success. Without the
probability idea of the mathematical basis of the model in use, the development
of science is impossible.

The frequency approach to the interpretation of
the term “probability” is defined in the process of its use and mathematical
modeling. Its defining was done in the
20s of 20^{th} century and it is connected with the names of Robert
Myses and Georg Reichenbach and nowadays with the scientistst Andrey
Kolmogorov, Norbert Winer and Andrey Hinchin.

The establishment of the frequency approach
corresponds in time with the transformation of the probability theory to the
part of classic mathematics. In its
early years the probability theory was studied not as strict mathematics
science but as a separate nature science. It was often compared to astrology
and alchemy.

For the proving that the probability theory is
a strict mathematics science, the scientific works of Andrey Kolmogorov had an
important contribution. He stated the following: “If in 1920 we could say the
probability theory was not part of mathematics, in 1936 it is not possible to
say that any more.”^{9}

The success of the mathematical formal expressions of knowledge is due to the fact that mathematics is closely connected with the nature of theoretical knowledge. It is now accepted that the probability theory has an independent value of its own. Its meaning is global and is defined by the analysis of its predicting function. The nature of general knowledge is also important as well as the essence of its relations. The “independence” of the probability theory is expressed by the fact that mathematics exists and continues its development. Besides, it shows itself as a tool of knowledge. The development of the researches provokes the formation of new ideas, based on mathematics and its terms and notions.

*As Freeman Dyson points out: „**Mathematics is the main source of
notions and principles in science. It serves as means for the development of
new nature theories.” ^{10} *

Mathematics is a science with an abstract structure. The laws of its functioning and development are actually a science of operations on a great number of terminable and interminable nature objects. In the study of the mathematical development of nature theories, the fact that mathematics is the main form for expressing the corresponding regularities becomes obvious. With its help, the basic equations forming the core of the scientific theories are defined.

The main advantage of mathematics is that its abstract objects and relations express the inner organization of our knowledge and the corresponding nature processes. The historical character of the mathematic science is defined by its interaction with the other fields of knowledge. That means that in the process of science development, a change of the mathematics disciplines which interact most closely with natural science occurs.

It is characteristic for classical mechanics to use the common classical analysis (differential and integral calculations); for classical electrodynamics of Maxwell – the vector analysis; for the theory of relativity – the tensor analysis; for quantum mechanics – the theory of Hilbert spaces; for the theory of elementary parts – the theory of the groups and the generalized functions. In correspondence with that, the theory of probability is used in natural sciences and that theory is in constant progress, especially with the development of information and technology.

In order to understand the peculiarities of the use of mathematics in real knowledge, the problem of the sources of its development should be studied. As an example of the „mathematization” of physics is the fact that the development of mathematics is faster than that of physics. The mathematics’ disciplines, characteristic for most of the modern physics parts, have been “basically out-lined” independently from the corresponding physic theories.

Nikola Burbaki wrote the following: “Since the
revolutionary development of modern physics started, a lot of efforts have been
put in mathematics
to give birth the convincing** **experimental
truths (proofs).”^{11} He claims that in the development of quantum physics, the macroscopic intuition of
reality was demonstrated, hidden by “microscopic” phenomena of a completely
different nature. There is a need of such divisions of mathematics for the
study of those phenomena which probably have not been developed to be used in
the experimental study. On the other
hand, the axiom method shows that “the truths” which must be the bases of the
conclusions in mathematics are in fact a particular aspect of the general
concept of science development.

In order to develop new mathematical disciplines for working with creative imagination, particular mathematical models need to be attracted and applied. New mathematical models as well as entire new disciplines can appear in the process of analyzing factual material which has not been initially considered worthy. A typical example of that is the initial project of the theory of probability. It has to be pointed out that the basic language of studying this theory is the research of the ‘games of chance’. Of course, such a choice of elementary models does not contain any complicating factors and it serves as a basis for the development of new notions. It is typical not only for mathematics.

In that sense a famous example is the great “contribution” of genetics to the development of medical science and its further improvement – genetic engineering and the science of organ transplantation.

The general consideration of the nature of mathematics and its application has to be considered when the reasons for the success of the probability ideas are analyzed. The basic notions of the theory of probability are similar to all basic mathematical notions. They carry an abstract character and, in their essence, they are away of the concrete nature of the real phenomena. The development and application of the probability theory is based on the research of more general forms and nature systems.

Including the ideas and methods of the the
theory of probability in the development of the quantum theory was
the reason for the establishment of the physics bases of the self-organization phenomena and then the synergetic science. That provokes the improvement
of the respective concept by using probability. The abstract nature and the
development of the particular methods are both reasons for the later success of
the theory in question and knowledge of nature phenomena.^{12}

The mathematical bases of the theory of subordination and the systems of high-level of organization are considered as another contribution to the theory of probability. That is taken as a base for the demonstration of the different degree of variability and versatility of the separate levels of the complex theoretical and technical systems. Having in mind that the stable levels are illustrated by generalized characteristics while the more changeable and versatile ones – by the language of the basic characteristics.

The possibility to synthesize logically
different notions in a wholesome secure** **system is granted due to the fact that the corresponding rules are formed by existing probability
distributions and their cognitive application.

**Conclusions of the authors’ thoughts**

*The measure of the probability degree of a
hypothetical conclusion is based on the information possibility found in its
contents of notions and knowledge, world perspective and heuristic
possibilities of mind and soul. *

*In the history of contemporary physics,
mathematics and philosophy there has not been even a single concurrence between
mathematics and the modern experimental physics. *

*Due to that particular “overtaking”, the
strength of mathematical pro-bability theory is demonstrated and that defines
its independent development. *

**Bibliography**

1. *Russel, B.* History of Western Philosophy. M., 1959, p. 76, p. 257.

2. *Russel,
B*. History of Western Philosophy.

3. *Doob**,**G,** **I.Yaglom.
*About some problems of
modern mathematics and Cybernetics.

4. *Popchev**,**Iv.**,**Y.Zaprianov**, **St**. **Markov**.** *Hierarchical
decentralized mana- gement
systems. Technika Publishing House, S., 1985, p.
7.

5. *Carnap**, **R**.* Philosophical
Basis of Physics. Nauka Publishing House,

6. *Popper**,**K**.* World’s
predisposition.
Two new points of view on causality. Man and Philosophy. Part II, Nauka PH,

7. *Bunge, M.** *Possibility and probability. Foundations of Probability The- ory, Statistical Inference and Statistical Theories of Science. Vol.
III, Dordrecht-Holland, 1976, p. 31.

8. *Hascing,
I.* The
Emergence of Probability. Cambridge, 1978, p.11-17.

9. *Kolmogorov**,**A.N. *Main terms of the
probability theory. 2 P.H., Moscow-Leningrad, 1936, p. 180.

10.*Dyson**,**F**.** *Mathematics and Physics. UFN Publishing House,

11.*Burbaki**, **N**.** *Articles on the history of Mathematics. Nauka Publishing House.

12.*Petrov**,**N**.* Probability,

**Nikolay Ivanov Petrov **received M. Sc. Degree from
the

In 2007 he awarded Professor by specialty,
Atomized Systems for Treatment of Information and Control. He has more than 300
scientific works, publications and developments, 50 of which – abroad. He
published 33 scientific books and textbooks, 10 of which – monographs.

**Nikolay Iv. Petrov,
Prof., Dr.Sc.**

1784 Sofia, Poligona Str., ¹ 8, Fl. 70

Phone: +359-2-974-46-91

E-mail: nikipetrov_1953@abv.bg