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Degree of Factual Support

Published online by Cambridge University Press:  14 March 2022

John G. Kemeny  and
Paul Oppenheim
John G. Kemeny
Affiliation:
Princeton, New Jersey
Paul Oppenheim
Affiliation:
Princeton, New Jersey
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Extract

We wish to give a precise formulation of the intuitive concept: The degree to which the known facts (the evidence) support a given hypothesis.

Type
Research Article
Information
Philosophy of Science , Volume 19 , Issue 4 , October 1952 , pp. 307 - 324
Copyright
Copyright © 1952, The Williams & Wilkins Company

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Footnotes

1

In the main the contribution of Paul Oppenheim was limited to propounding some of the fundamental ideas.

References

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 ck.

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