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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
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[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