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Foundations of Probability in Mathematical Logic

Published online by Cambridge University Press:  14 March 2022

Theodore Hailperin
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Extract

It is the purpose of this paper to present a theory of probability derived from two-valued logic—the logic of which an aspect is given in Part I, Section A, of Principia Mathematica. The symbolic system of Mr. Keynes, given in his Treatise on Probability, will be shown to be a part of our system. We have, however, little if anything in common with his philosophical analysis; a definition of Keynes’ fundamental probability relation, free from psychological or material reference, will be given, enabling us to offset some of the objections to his theory.

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Type
Research Article
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Philosophy of Science , Volume 4 , Issue 1 , January 1937, pp. 125 - 150
Copyright
Copyright © Philosophy of Science Association 1937

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References

1 I wish to express my thanks here to Dr. Ernest Nagel of Columbia University for his generous counsel and assistance in the development of this theory.

2 Rice Inst. Pamph. Vol. XVI, no. 4. In this he is following the Hilbert school.

3 This does not say that for any propositions p and q the probability of “p or q“ is greater than that of “p and q“; see 2.32 below. It will be recalled that A and B represent p v q and p ⋅ q only when p and q are special kinds of propositions.

4 As a rule, only for less evident propositions will proofs be given. The propositions preceded by numbers are to be understood as valid for any propositions.

5 Also of Jeffries and Wrinch (Jeffries, Scientific Inference, p. 15n).

6 It is always the case that there is a function f such that fq ≡ p, for if p and q are any propositions in general we always have

It is to be remembered that p/q is not a truth-function of p and q and that before we can apply the “matrix method” to propositions containing p/q it must be changed to ft.

7 In the rest of this section we shall not trouble to state the hypothesis of 3.92; besides, it is approximately satisfied in ordinary cases.

8 More accurately, … then P(pk i, ⊃ q) will be near zero and consequently P(p ⊃ q) will be near zero, for P(p) = o means that p is a contradiction.

9 Monist, Vol. 42, Oct. 1932. The reader is recommended to this paper for a very capable and more detailed exposition of the subject of many-valued logics.

10 The rules for multiplication and addition are of course modified. When computing with a scale of radix r we carry or borrow r instead of the usual 10. Thus using the radix 9

For further details see any text such as Hawkes, Advanced Algebra.