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Design+ Modern interpretations of probability

and processes, especially when numerical and linguistic adapted to handle large data sets from which conclusions
variables describe their properties. are drawn. The theory of fuzzy sets is adapted to “work”
The place of fuzzy theory among other mathematical with a significantly smaller amount of data, allowing their
theories, as well as the objective need for practice in this incompleteness, inaccuracy, and non-numerical nature
theory, is best reflected by its author Zadeh: (and simultaneously, within the study of one object). This
distinction leads to greater stability of the results obtained
The term fuzzy logic is currently used in two different based on fuzzy set theory when there are little data to assess
senses. In a narrow sense, fuzzy logic is a logical system what is happening. In many cases, this becomes the main
that aims at a formalization of approximate reasoning… factor in choosing fuzzy algorithms as a tool for solving
In a broad sense, fuzzy logic is almost synonymous with practical problems.
fuzzy set theory. Fuzzy set theory, as its name suggests, is
basically a theory of classes with unsharp boundaries. Fuzzy Based on the introduced concept of fuzzy sets, it was
set theory is much broader than fuzzy logic in its narrow reasonable to generalize classical binary logic based on
sense and contains the latter as one of its branches. 25(p.78) consideration of an infinite set of truth values, that is, to
pass to fuzzy logic. In the proposed version of fuzzy logic,
As noted above, a characteristic feature of axiomatic the set of truth values of statements is generalized to the
probability theory is that the objects analyzed in the interval [0, 1], that is, it includes binary logic and finite
framework of this theory form a set S of unit measure P multi-valued logic (including three-valued Lukasiewicz
(S) = 1. Such a measure in mathematics, as is known, is logic) as special cases. Thus, it becomes possible to
called a probability measure. However, other sets of objects consider proposed hypotheses (logical variables) with
and measures correspond to these sets that have remained different truth values, to rely on a small amount of
unattended by probability theory. In particular, there are available information, to give experts some freedom in
known sets of objects for which the measure can be greater assessing the situation, and to form conclusions with
or less than one. Such sets constitute the subject of fuzzy set some, but for most applications with quite acceptable
theory. Usually, these measures are considered as a measure uncertainty, which, moreover, quickly decreases in the
of possibilities (P (S) > 1) and a measure of necessities (N (S) process of receipt of additional information regarding the
< 1). It is quite clear that by simultaneously considering all proposed hypotheses.
three measures (P, P, and N), we go beyond the limitations
of probability theory, cover the whole space of objects, 12. Probability, logic, and fuzziness as
information about which, under certain conditions, can engineering tools
be provided in incomplete, inaccurate, and non-numerical
form. According to the classical interpretation, probability is
the ratio of the number of selected realizations of some
As it has been repeatedly mentioned above, the experiment to the number of all its possible realizations.
existence of statistical stability of the results of observations At the same time, all of these realizations are considered
related to establishing the frequency of realization of the equally possible. Criticism of the shortcomings of this
corresponding event among a set of events of a certain class definition led to the emergence of the concept of frequency
can rarely be guaranteed and practically cannot be verified. probability (based on which mathematical statistics
Nevertheless, despite these methodological difficulties, emerged), and later to the axiomatic theory of probability.
today, specialists mainly use well-developed and widely Today, probability theory and mathematical statistics are
known methods of probability theory to interpret the well-developed and comprehensively understood. Despite
results of such observations. In many respects, this situation certain difficulties in substantiating the corresponding
is promoted by the current system of training specialists of algorithms and interpreting the obtained results, both of
corresponding (mainly engineering) specialties, in which these tools (both probability theory and mathematical
too little attention is paid (and often not at all) to the theory statistics) are widely used in the processes of realizing
of fuzzy sets, but a detailed study of probabilistic methods the life cycle of technical objects, in social research, in
and practical calculations based on these methods is the construction of artificial intelligence systems, and in
assumed, often without a thorough analysis of the limits of general, all situations where the available information is
their application. sufficient to obtain acceptable adequacy of conclusions.
There is a significant difference between probability The measure of adequacy becomes the values of confidence
theory and fuzzy set theory concerning the requirements intervals and calculations. Among the tasks that are
related to the accuracy of the raw data and the conclusions successfully solved based on the methods of probability
drawn from the data. By definition, probability theory is theory and mathematical statistics, the following are

Volume 2 Issue 2 (2025) 16 doi: 10.36922/dp.6387