Inductive logic and confirmation theory: is probability logically definable as objective-Bayesian c-functions (Logical Foundations of Probability 1950, Continuum 1952, the late Basic System), against subjectivist Bayesianism (Ramsey, de Finetti, the Lewis-Jeffrey tradition)?

Rudolf Carnap, answered from the texts and cited to the page.

The position in brief

Probability is logically definable. The c-functions of Logical Foundations of Probability (1950) and the late Basic System of Inductive Logic give the apparatus: probability assignments to sentences are determined by the logical-structural properties of the sentences relative to the constructed language framework, not by either subjective degrees of belief or by observed frequency ratios alone.¹²³

Inductive logic is therefore an objective discipline parallel to deductive logic, with degrees of confirmation (c(h,e)) standing to inductive arguments as deductive validity stands to deductive arguments. Against the subjectivist Bayesian tradition (Ramsey, de Finetti, Jeffrey, Lewis) that takes probability to be subjective degrees of belief constrained only by the coherence requirement: probability has objective logical structure that can be made precise in a constructed framework.

The project was developed across the entire late period from LFP through the unfinished Basic System and remains defensible in the contemporary objective-Bayesian revival.⁴⁵

Sources

1. Who is right, the man of common sense or the critical philosopher? We see here, as so often, that both are partially right. Hume's criticism of the customary forms of induction was correct. But still the basic idea of common sense thinking is vindicated: induction, if properly reformulated, can be shown to be valid by rational criteria.

   Studies in Inductive Logic and Probability Vol I, p. 41

2. Studies in Inductive Logic and Probability VOLUME I Studies in Inductive Logic and Probability VOLUME I Rudolf Carnap and Richard C.

   Studies in Inductive Logic and Probability Vol I, pp. 7–8

3. My Basic Conceptions of Probability and Induction I shall here briefly summarize my basic conceptions of probability and induction for convenience of reference in the subsequent discussions of the contributions by Kemeny, Burks, Putnam, Nagel and Popper.

   Replies and Systematic Expositions, p. 998

4. Let us now consider rational Cred-functions. We shall not assume that there can be only one such Cred-function, but rather leave open the possibility that two reasonable persons, or one reasonable person during different periods of his life, may have different Cred-functions.

   Replies and Systematic Expositions, pp. 1002–1003

5. For these considerations a principle of indifference would be necessary. Yet I recognized that, in contrast to the very strong principle in its classical form which had been correctly criticized, we had to take a much weaker, essentially restricted version of the principle. Somewhat later I studied Harold Jeffreys' Theory of Probability (1939).

   Intellectual Autobiography, pp. 84–85

Where the argument is made

• Logical Foundations of Probability (1950) — the canonical statement of the logical-probability conception
• The Continuum of Inductive Methods (1952)
• the late "Basic System of Inductive Logic" in Studies in Inductive Logic and Probability Vol I (Carnap + Jeffrey eds., 1971, Part 1) and Vol II (Jeffrey ed., 1980 posthumous, Part 2)
• "Inductive Logic and Rational Decisions" (1971, opening chapter of Vol I)
• the relevant sections of Philosophical Foundations of Physics (1966) on inductive logic in late form
• the Replies and Systematic Expositions (Schilpp ed., 1963) on the foundations of probability

How the argument was received

settled-with-late-extension-still-being-worked-out