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Design+ Modern interpretations of probability

the truth of a proposed hypothesis. Probabilistic logic establish indicators of failure-free, safety, and survivability
and logical probability are not synonyms, but essentially of such systems.
different categories and concepts for assessing the truth
of the corresponding hypotheses. In the first case, we deal 11. Fuzzy sets and fuzzy logic
with infinite-valued (multi-valued) logic; in the second, we Probability theory aims to solve problems in which the
deal with two-valued, binary logic. event under study is described by inaccurate information.
To emphasize the difference between “probabilistic In this case, for the correct application of probability
logic” and “logical probability,” the notion of deterministic theory, the frequency of realization of the event under
logic, essentially a binary logic, is introduced. With the help study ν = n/N (n is the number of realizations of the event
of deterministic logic, it is proposed to construct complex according to the results of N observations) must tend to
logical functions based on appropriate logical variables stability with increasing value A., that is, statistical stability
or the convolution (“aggregation”) of the simplest logical should be observed. If the specified stability is observed,
functions. Substitution of logical arguments by probabilities then n is a random variable that will characterize the
of their truth is further realized in the obtained logical measure of the inaccuracy of information regarding the
function by special algorithms. The possibility of applying event under study. However, if the specified stability is not
these procedures was emphasized by Andrey Nikolaevich observed, then P cannot be considered as a measure of the
Kolmogorov, the founder of axiomatic probability theory, inaccuracy of the specified information. This means that
who proved a theorem called the law of “0 and 1” (the probability theory and methods of mathematical statistics
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law of “zero”/“one”). According to this theorem, in the cannot be applied to research. At the same time, it is
case when the marginal probabilities are known and are known 23,24 that the source of the inaccuracy of information
necessarily equal to zero or one, all formulas of probability is not only its probabilistic nature but also its fuzziness. The
theory for complex logical functions f (x), whose arguments theory of fuzzy sets (fuzzy theory) was developed by Lotfi
are events independent in the aggregate, become correct Zadeh to precisely account for the inaccuracy associated
logical formulas when replacing the specified events with fuzziness. 23
with the corresponding statements. We should consider A mathematical theory of fuzzy sets allows us to
that the law of “0 and 1” assumes mutual independence describe fuzzy concepts and knowledge, operate with this
of all events of the analyzed totality. We can say that the knowledge, and make corresponding fuzzy conclusions,
variables of the analyzed population must be orthogonal. often providing sufficient information for making adequate
The orthogonalization of the space of variables is the most and effective decisions in various areas of human activity.
computationally complex procedure that accompanies the Fuzzy sets can also be formed by corresponding logical
practical application of logical probability. variables, for which in the space of fuzzy sets, it is possible
Specific values of logical probability for each synthetic to establish quite specific rules, in many respects similar to
argument depend on another judgment, which can be the rules of conventional binary logic.
interpreted as a description of a subject’s knowledge (a The main idea of introducing fuzzy sets is based on
description of the information available to this subject) realizing that real human judgments, based on appropriate
of the synthetic argument being analyzed. Given this linguistic constructions, cannot be described within
interpretation, logical probability is often referred to the framework of traditional mathematical artifacts.
as epistemological probability, that is, the probability As an alternative, fuzzy sets, classes of variables with
that depends on the available knowledge. Under certain imprecisely defined boundaries, are proposed. Such
conditions, logical probability can be treated as subjective classes are described by belonging functions, which
probability. However, the value of logical probability is characterize the degree of belonging of the corresponding
unambiguously determined by a given system of knowledge variable to a certain class (the degree of certainty that the
and available information, and hence, has an objective variable belongs to this class). At the same time, these
character. Since logical functions describe some event belonging functions can be regarded as fuzzy models
(for example, the state of a device in the corresponding of corresponding events, conclusions, and hypotheses.
technical state), logical probability can be considered a The construction of such models usually relies on expert
function that determines the corresponding characteristic evaluations, heuristics, and/or approximation methods
of such events. of generalization of experimental data. The above and
It should be noted that the concept of logical probability the development of appropriate mathematical operations
is the basis of the logical probabilistic method of analyzing on such structures provided the basis for developing a
structurally complex systems, which is currently used to new direction in modeling various objects, phenomena,

Volume 2 Issue 2 (2025) 15 doi: 10.36922/dp.6387