Logical Probability and Confirmation

Carnap distinguishes sharply between what he calls Probability₁ (the degree of confirmation of a hypothesis given evidence — logical probability, the subject of inductive logic) and Probability₂ (the relative frequency of an event type in a long sequence — the statistical concept used in the natural sciences). These are not two imprecise uses of the same concept but two entirely different concepts that happen to share a name. Inductive logic is concerned exclusively with Probability₁ — with the rational degree of belief that a hypothesis merits in the light of given evidence — and it is this concept that Carnap attempts to define precisely.

Carnap defines a family of confirmation functions c(h, e) that assign to each pair of a hypothesis h and an evidence statement e a real number between 0 and 1, representing the rational degree of belief in h given e. These functions are constrained by axioms of probability and by the requirement that confirmation be invariant under the permutation of individual constants in the language (the requirement of symmetry or exchangeability). Different confirmation functions correspond to different values of the parameter λ, which governs the weight given to prior information versus observed frequencies. The special function c* is Carnap's preferred choice for most purposes.

The project of inductive logic was criticised from several directions. Popper argued that there is no such thing as confirmation — science proceeds by falsification, not by the accumulation of positive evidence. Goodman's new riddle of induction showed that any confirmation function that licenses inductive extrapolation from observed cases also licenses extrapolation using "grue" and other gerrymandered predicates that generate absurd predictions. Quine argued that the very notion of logical probability presupposes analyticity, which he had rejected. Carnap spent his later decades refining and defending the project; it has had more influence in Bayesian statistics and formal epistemology than in philosophy of science, where the turn to naturalism reduced interest in a priori confirmation theory.

Logical Foundations of Probability was published in 1950, with a second enlarged edition in 1962. It was followed by The Continuum of Inductive Methods (1952), which elaborated the λ-parameter family of confirmation functions. Carnap was working on a further systematic presentation of inductive logic at the time of his death in 1970; the manuscript was edited and published posthumously as Studies in Inductive Logic and Probability (1971).