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On carnap and popper probability functions

Published online by Cambridge University Press:  12 March 2014

Hugues Leblanc  and
Bas C. van Fraassen
Hugues Leblanc
Affiliation:
Temple University, Philadelphia, PA 19122 University of Toronto, Toronto, CanadaM5S 1A1 University of Southern California, Los Angeles, CA 90007
Bas C. van Fraassen
Affiliation:
Temple University, Philadelphia, PA 19122 University of Toronto, Toronto, CanadaM5S 1A1 University of Southern California, Los Angeles, CA 90007
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Extract

With PC understood to be the propositional calculus of [3], call a binary function Pr from the wffs of PC to the reals a Carnap (probability) function if it meets requirements A1—A5 in Table I (with ‘⊢…’ short in A3—A4 for ‘… is a tautology’), and call the function a Popper (probability) function if it meets requirements Bl—B6 there:

Leblanc established in [3] that every Carnap function is a Popper one, and he tendered proof of the converse. As reported by Stalnaker in [5], the proof unfortunately was incomplete, a mishap due to Leblanc's abbreviating ‘Pr(A, ∼ A) = 1’ as ‘⊢ A’ when Pr is a Popper function. Borrowing from [4], Leblanc did show, as Harper notes in [2, footnote 17], that

(1) If Pr is a Popper function and A is a tautology, then Pr(A, ∼ A) = 1. He did not, however, show that

(2) If Pr is a Popper function and Pr(A, ∼ A) = 1, then A is a tautology.

Nor could he have done so: (2) is false, as the simplest of counterexamples shows. Denied (2), Leblanc had no hope of proving that every Popper function is a Carnap one: of a Popper function Pr it is easily ascertained that Pr meets requirement A4 if and only if Pr(A, ∼ A) = 1 just in case A is a tautology.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 3 , September 1979 , pp. 369 - 373
Copyright
Copyright © Association for Symbolic Logic 1979

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References

REFERENCES

[1]Carnap, R., Logical foundations of probability, University of Chicago Press, Chicago, 1950.Google Scholar
[2]Harper, W. L., Rational belief change, Popper functions and counterfactuals, Foundations of probability theory, statistical inference, and statistical theories of science, vol. I, Reidel, Dordrecht, pp. 73–115, 1976.CrossRefGoogle Scholar
[3]Leblanc, H., On requirements for conditional probability functions, this Journal, vol. 25 (1960), pp. 238–242.Google Scholar
[4]Popper, K. R., The logic of scientific discovery, Basic Books, Inc., New York, 1959.Google Scholar
[5]Stalnaker, R. C., Probability and conditionals, Philosophy of Science, vol. 37 (1970), pp. 64–80.CrossRefGoogle Scholar
[6]van Fraassen, B. C., Representation of conditional probabilities, Journal of Philosophical Logic, vol. 5 (1976), pp. 417–430 (corrigendum 6: p. 365).CrossRefGoogle Scholar