² Russell, Bertrand, Human Knowledge, New York: Simon and Schuster, 1948.

³ Carnap, Rudolf, Logical Foundations of Probability, Chicago: The University of Chicago Press, 1950.

⁴ Helmer, Olaf and Oppenheim, Paul. “A Syntactical Definition of Probability and of Degree of Confirmation,” The Journal of Symbolic Logic, X, 1945, pp. 25–60.

⁵ This condition will be somewhat restricted later, see Part 2.

⁶ ‘⊢W’ expresses that W is a theorem, i.e., that it is an analytic consequence of our dependency-postulates, or simply analytic if there are no dependencies.

⁷ This CA serves to illustrate the relation between F and C. We see that F does not satisfy one of the fundamental C-principles, namely that if H ₁ follows from H ₂, then its C is at least as great as that of H ₂ with respect to any evidence. Consider the historical example: H ₁ is the bending of light-rays in a gravitational field, while H ₂ is the General Theory of Relativity, and the evidence is the evidence before 1919. Certainly F(H ₂, E) is much greater than F(H ₁, E) since H ₁ was not supported by any observation up to that time. Yet if we assigned say 80% credibility to H ₂, we would have to assign at least that much to any of its consequences. Here is a case where H ₁ has a high credibility (C) in spite of a very low degree of factual support (F)! The reasoning must be as follows: On the basis of its F-value and other considerations, simplicity of the hypothesis being the most important in this case, we assign by induction a high C-value to H ₂, and consequently H ₁ must get a C-value at least as high.

⁸ Church, Alonzo, Introduction to Mathematical Logic—Part I, Princeton: Princeton University Press, 1944, p. 91.

⁹ A theorem due to Gödel, ibid., pp. 73ff.

¹⁰ Theorems 9 and 10 hold, of course, only if the variables remain within the bounds specified in theorem 5.

¹¹ A function is homogeneous to degree k if multiplying its arguments by c multiplies the value by c^k.

¹² In order to avoid quotation marks, we will use w.f.f. as names for themselves.