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Design+ Modern interpretations of probability

What was done in this direction was too cumbersome and Infinite-valued logic for two variables whose
inconvenient. The described ideas have not found wide direct probabilities of truth are a and b is introduced by formal
application in practical activity either. At the same time, we statements: the true value of a variable is characterized by
note that the work in this direction was not in vain. They the value being on the interval of real numbers from 0 to
contributed to realizing the connection between logic and 1, negation is defined as 1-a, conjunction as min (a, b), and
probability, which led to the formation of new, practically disjunction as max (a, b).
useful scientific directions. Today, the scientific community The introduced rules made it possible to describe
perceives the degree of reasonable belief not as the actual processes and phenomena as functional dependencies
belief of the subject in the corresponding statement, but between the factors acting on the object (function
as a logical relation between statements. In this case, the arguments) and reactions of this object (function values),
measure of reasonable belief is transformed into the degree of carry out identical transformations of these dependencies,
confirmation, which is the foundation of the modern theory and form appropriate solutions. Multivalued (infinite-
of inductive logic. valued) logic with a certain degree of conventionality can
7. Multivalued logic be considered a forerunner of fuzzy logic.
Classical logic, by definition, cannot operate with concepts 8. Probability and infinite-valued logic
that are delineated concretely, ambiguously, because all At first sight, the theory of probabilities appears to be
statements (logical variables and hypotheses) in the formal analogous (or at least very close in content) to infinite-valued
systems of this logic can have only two mutually exclusive logic, in the framework of which the truth of realization of
states: “true” with truth value “1” or “false” with truth some event is characterized by the category of possibility,
value “0.” that is, it is established with some probability, which takes
The principle of ambiguity (the law of the excluded the value from 0 to 1 (zero truth means complete falsehood,
third) determines the uncompromising discreteness one truth means complete certainty). The probability of
of the representation of analyzed phenomena, leaving non-realization of some event corresponds to negation,
no room for any uncertainty, modal, and probabilistic while the probability of simultaneous realization of two
assessments of these phenomena. However, in many cases, events corresponds to conjunction.
the essence of any statement, conclusion, or hypothesis However, there is a fundamental difference between
can be perceived with more or less skepticism. After all, probability theory and infinite-valued (and multi-valued)
the statement “not full” in many cases does not mean logic. In logic, the true value of any function is completely
“empty,” just as the statement “not empty” may not fully determined by the true value of its arguments, whereas
characterize an object as “full.” Binary assessment of truth in probability theory, the probability of a complex event
in such situations imposes significant limitations on the (integrated from component events) depends not only on
realization of relevant studies and conclusions. It begins the probabilities of the component events but also on their
to hamper and restrain the activity of both the creators mutual dependence, which is determined through the
of scientific theories and the engineering community. corresponding conditional probabilities. This is manifested,
The need to reflect the uncertainty, and incompleteness in particular, in the fact that in probability theory, the
of the concepts of “truth” and “false” in the process of equivalent of the “law of the excluded third” is fulfilled. In
solving professional tasks of specialists from various fields probability theory, “some event is realized or not realized”
of knowledge led to the search for, and eventually to the is always equal to one (there is no third option), whereas,
emergence of appropriate mathematical artifacts that in infinite-valued logic, the law of the excluded third is not
significantly expanded the boundaries of bivalent logic, fulfilled. Similarly, in probability theory, the equivalent of
turning it into multivalued logic as a kind of formal logic the “law of contradiction” is fulfilled: the probability that
that allows the existence of more than two values of a “some event is simultaneously realized and not realized” is
logical variable. 18,19 always equal to zero, whereas, in infinite-valued logic, the
One of the first attempts to move beyond binary law of contradiction is not fulfilled.
logic was made in 1920 by the Polish philosopher Jan At the same time, an overly critical analysis allows us
Lukasiewicz, who introduced a three-valued logic. Within to establish the existence of some connection between the
this framework, truth values were expanded beyond the truth values of the categories of infinite-valued logic and
traditional “true” and “false” to include an intermediate the probabilities of these categories (given the classical
value: “possible.” These values corresponded to the grades interpretation of the notion of probability). This connection
“1 (true),” “0.5 (possible),” and “0 (false).” is manifested in the above provisions, which formally

Volume 2 Issue 2 (2025) 12 doi: 10.36922/dp.6387