Editing Rudolf Carnap (section)

=== Inductive logic ===
After having considered problems in semantics, i.e. the theory of the concepts of meaning and truth (''Foundations of Logic and Mathematics'', 1939; ''Introduction to Semantics'', 1942; ''Formalization of Logic'', 1943), Carnap turned his attention to the subject of probability and [[Inductive reasoning|inductive logic]]. His views on that subject are, for the most part exposed in ''Logical foundations of probability'' (1950) where Carnap aims to give a sound logical interpretation of probability. Carnap thought that, according to certain conditions, the concept of probability had to be interpreted as a purely logical concept. In this view, probability is a basic concept anchored in all inductive inferences, whereby the conclusion of every inference that holds without deductive necessity is said to be more or less likely to be the case. In fact, Carnap claims that the problem of induction is a matter of finding a precise explanation of the logical relation that holds between a hypothesis and the evidence that supports it. An inductive logic is thus based on the idea that probability is a logical relation between two types of statements: the hypothesis (conclusion) and the premises (evidence). Accordingly, a theory of induction should explain how, by pure logical analysis, we can ascertain that certain evidence establishes a degree of confirmation strong enough to confirm a given hypothesis.

Carnap was convinced that there was a logical as well as an empirical dimension in science. He believed that one had to isolate the experiential elements from the logical elements of a given body of knowledge. Hence, the empirical concept of frequency used in statistics to describe the general features of certain phenomena can be distinguished from the analytical concepts of probability logic that merely describe logical relations between sentences. For Carnap, the statistical and the logical concepts must be investigated separately. Having insisted on this distinction, Carnap defines two concepts of probability. The first one is logical and deals with the degree to which a given hypothesis is confirmed by a piece of evidence. It is the ''degree of confirmation''. The second is empirical and relates to the long-run rate of one observable feature of nature relative to another. It is the ''relative frequency.'' Statements belonging to the second concept are about reality and describe states of affairs. They are empirical and, therefore, must be based on experimental procedures and the observation of relevant facts. On the contrary, statements belonging to the first concept do not say anything about facts. Their meaning can be grasped solely with an analysis of the signs they contain. They are analytical sentences, i.e. true by virtue of their logical meaning. Even though these sentences could refer to states of affairs, their meaning is given by the symbols and relations they contain. In other words, the probability of a conclusion is given by the logical relation it has to the evidence. The evaluation of the degree of confirmation of a hypothesis is thus a problem of meaning analysis.

Clearly, the probability of a statement about relative frequency can be unknown because it depends on the observation of certain phenomena, and one may not possess the information needed to establish the value of that probability. Consequently, the value of that statement can be confirmed only if it is corroborated by facts. In contrast, the probability of a statement about the degree of confirmation could be unknown, in the sense that one may miss the correct logical method to evaluate its exact value. But, such a statement can always receive a certain logical value, given the fact that this value only depends on the meaning of its symbols.