Overview
The Two Concepts of Probability is not a monograph but a single sustained essay—an article-length philosophical paper rather than a book—in which Rudolf Carnap sets out to disentangle a confusion that he believed had distorted a generation of debate about probability and induction. Its scope is exactly that argument: it does not narrate a life, survey the history of probability, or develop a full formal calculus in the manner of his later, much longer works. What it does is state a distinction, defend it against both logical and empiricist objections, and sketch what a theory built on that distinction would have to look like.
The central move is terminological and conceptual at once. Carnap observes that the word “probability” is used in everyday speech, among scientists, and in the technical literature in at least two quite different senses, which he labels probability₁ and probability₂. Probability₁ is a logical concept: a relation between two sentences or propositions, namely the degree to which a hypothesis h is supported by evidence e. Carnap identifies this with his concept of degree of confirmation, written c(h, e), a phrase quoted here from the essay itself: probability₁ “is a logical concept, a certain logical relation between two sentences”; “it is the same as the concept of degree of confirmation.” Probability₂, by contrast, is an empirical concept: the relative frequency of a property within a class, taken as a limit in the long run, of the kind used in statistical physics and in the statistical testing of hypotheses.
Having drawn the distinction, Carnap argues that the long-running quarrel between the “logical” conception of probability, represented by Keynes and Jeffreys, and the “frequency” conception, represented by von Mises and Reichenbach, is largely futile. These theories, he maintains, are not rivals at all: they are theories of two different concepts, both of which are indispensable to science, and they therefore supplement rather than contradict one another. The error he diagnoses is the assumption that only one of the two can be the “real” meaning of probability, so that the other must be reduced to it or dismissed as metaphysical.
A substantial portion of the essay is devoted to clarifying what kind of concept probability₁ is, and to answering the charge that it smuggles a priori knowledge into science. Carnap’s reply is that statements of the form c(h, e) = r are analytic or logical in character: they assert a relation of partial entailment between two statements, not a fact about the world. Logical probability, on this account, no more violates empiricism than deductive logic does. The empirical content of any inductive conclusion comes entirely from e, the evidence, which is a factual report; the logical machinery merely makes explicit what follows from it. He is explicit that inductively derived statements of this kind cannot by themselves yield substantive factual conclusions; empiricism is violated only if one asserts a factual sentence without adequate empirical backing, or if one adopts an apriorist metaphysics.
Connected with this is Carnap’s account of inductive logic as a discipline parallel to deductive logic. Deductive logic studies the relation of logical consequence; inductive logic studies the relation of degree of confirmation, and both belong to the analysis of meanings rather than to empirical inquiry. Carnap distinguishes three kinds of confirmational concept: a classificatory one (a hypothesis is or is not confirmed by given evidence), a comparative one (a hypothesis is more strongly confirmed by one body of evidence than another), and a quantitative or metrical one (the degree of confirmation assigned a numerical value). The essay’s interest is chiefly in the third, and it outlines how such a value might be defined: within a suitably formalised language, one specifies a measure over the state descriptions of that language, and the degree of confirmation of h on e is then the measure of the h-and-e cases divided by the measure of the e cases. This is presented as the general pattern that any systematic inductive logic must follow, together with the requirements of symmetry or invariance that the underlying measure should satisfy, and with explicit acknowledgement that more than one such system can be constructed—so that the choice among admissible systems is itself a substantive problem.
The essay is candid about the limits of what had been achieved. Carnap concedes that the logical concept of probability is not yet applicable to any but the simplest problems of statistical inference, and that statistical practice must for the time being continue to work with probability₂, the frequency concept; results reached in this way he regards as unsatisfactory from a philosophical point of view, and the elaboration of the logical concept is presented as work still to be done. Related difficulties that recur in the discussion include the confirmation of universal generalisations—which, on finite evidence, threaten to receive vanishing or zero confirmation under simple measures—and the correct treatment of analogy and of the predictive inference from observed to unobserved individuals. Carnap also stresses the requirement, sometimes called the principle of total evidence, that a degree of confirmation be assessed relative to all the evidence available rather than to a convenient selection of it, and he draws out the bearing of the analysis on the use of Bayes’ theorem in inductive reasoning.
The final strand of the essay is polemical. Carnap examines and rejects a prominent line of criticism, associated with Reichenbach, which holds that a genuinely empiricist account must collapse logical probability into statistical probability, since otherwise the logical concept is a disguised a priori element. Carnap’s response is that this objection rests precisely on the conflation the essay was written to expose: it assumes that both concepts answer the same question. Once probability₁ is seen as a semantical relation between statements and probability₂ as an empirical frequency, the alleged conflict disappears, and the demand that one be reduced to the other loses its force. The essay closes with the conviction that the clarification of the two concepts removes a major obstacle to the development of inductive logic as a rigorous discipline, while leaving that development, in its technical detail, as a task for the future.
The logical concept is further characterised as a generalisation of entailment: complete entailment is the limiting case in which the evidence makes the hypothesis certain, while the degree of confirmation measures partial entailment. This is why the essay treats inductive logic as a branch of logic rather than as a descriptive study of belief; the confirmation relation is objective and depends on the statements serving as evidence, not on the psychological state of a particular inquirer. The frequency concept, by contrast, is reserved for empirical statements about sequences or populations, so that the two concepts operate at different levels of discourse. The essay’s significance lies in making that division explicit and in proposing a formal setting within which degrees of confirmation can be defined and compared.
