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Design+ Modern interpretations of probability

of a random event from some statistical set of events is of hypothesis H cannot be deduced based on logical
assigned a certain probability, then the same probability conclusions. For the same reason, the probability P = 0
can be correlated with a judgment characterizing this does not mean that hypothesis H is false with respect to
event, that is, an unambiguous correspondence between the data E.
events and judgments concerning these events can be The validity of the numerical determination of the
established. Probabilistic logic, however, relies on a probability of some statements based on other statements
logical interpretation of probability, according to which is currently a subject of discussion. It is solved differently by
this probability is considered as a relation between the representatives of different branches of probabilistic logic.
premises and conclusions of induction. The first systems of However, the calculation of the probability of complex
probabilistic logic arose precisely in the framework of the hypotheses (complex logical functions), which provided
logical interpretation of probability, so logical probability that the probabilities of the individual components of
is often defined as an inductive probability. such hypotheses are known, in all systems of probabilistic
The substantive essence of probabilistic logic consists logic is carried out according to the rules of mathematical
in the fundamental impossibility of constructing certain calculus of probabilities, which today is based on the
logical structures to avoid, and completely exclude from axiomatic system. Such a system defines the properties
consideration, the category of incomplete reliability of abstract categories for which probabilistic reasoning is
(relative truthfulness) of premises and conclusions, applicable, as well as the corresponding rules for obtaining
which is inherent in any knowledge based on inductive the probabilities of certain categories (taking into account
conclusions, that is, any inductive knowledge. As a logical the formation of their corresponding structure) based on
system, probabilistic logic is a type of infinite-valued logic. the probabilities of others. The probabilistic-logical concept
The multivaluedness (or rather infinite-valuedness) of the fits into the field of logic and acquires its essence when
probabilistic evaluation of a hypothesis does not deny the analyzing (based on logical conclusions) the connections
fact that the hypothesis itself can have only one of two between individual hypotheses.
truth values (“true” or “false”). Probability remains binary,
because the value of probability characterizes the relation 10. Logical probability (probability logic)
of a given hypothesis to reality not directly (the direct Logical probability is a logical relation between two
relation of a hypothesis to reality remains two-valued), but hypotheses (events, variables, and statements), the
indirectly, through other hypotheses based on the available characteristic feature of which is that this relation
information (knowledge) concerning the events being establishes the degree (quantitative value) of confirmation
analyzed, that is, on the information that is available at the of some hypothesis H by evidence E. The subject of
present moment. Hence, the concept of probabilistic logic probability logic is the calculation of the probability of the
treats probability as a characteristic of a logical variable. truth of random events that take exclusively two values: 1
Rudolf Carnap first outlined the theory of probabilistic or 0. 21,22
logic in its most complete form in his work Logical To provide a mathematical interpretation of the above
Foundations of Probability. In Carnap’s interpretation, the definition of logical probability, consider some logical
concept of probability is considered a certain category of function f (x), which can take a true value on a set of data x. It
inductive logic. In this case, probability characterizes the is known that the acceptance of a true value by the function
logical connection between judgments and the degree of f (x) can occur only with a certain probability R, which is
confirmation of hypothesis H by the data E; in other words, the corresponding logical probability. Mathematically, it is
relative to the data E, hypothesis H has a probability P. In written as in Equation II.
his work, Carnap argues that such probabilistic-logical
structures should be considered analytical because they say P {f (x) = 1} = R (II)
nothing about the environment, and are independent of the The above should be interpreted as follows. There
empirical truth of both E and H. However, these categories is relevant information regarding the truth of a set of
can be determined predominantly by empirical data. variables x. Based on this information, we should establish

Note that within the framework of this concept, the the probability of the logical function f (x) taking a true
concept of probability has nothing in common with the value. Of course, the more information there is about the
concept of truth. Attributing to hypothesis H, the degree truth of variable x, the higher the value of the specified
of probability P = 1 with respect to certain data E does probability.
not mean the truth of this hypothesis, because the data E Regarding the statistical interpretation of probabilities,
itself may be false, and under such conditions, the truth conditional probability is the function for confirming

Volume 2 Issue 2 (2025) 14 doi: 10.36922/dp.6387