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Extension of relatively σ-additive probabilities on Boolean algebras of logic1

Published online by Cambridge University Press:  12 March 2014

Mohamed A. Amer
Mohamed A. Amer*
Affiliation:
Department of Mathematics, Cairo University, Giza, Egypt
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Abstract

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability ℬ (where ℬ and C are Boolean algebras, and ℬ ⊆ C) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) can be extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 3 , September 1985 , pp. 589 - 596
Copyright
Copyright © Association for Symbolic Logic 1985

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Footnotes

1

An earlier version of this paper was presented at Logic Colloquium '77, and [1] is an abstract of that version. A part of the work on this paper was done while the author was a visiting scholar at the University of California, Berkeley, on leave from Cairo University, sponsored by Professor L. Henkin, and indirectly supported by the International Development Research Center of Canada. Thanks to all of them.

References

REFERENCES

[1]Amer, M., On probability logic, this Journal, vol. 44 (1979), p. 441 (abstract).Google Scholar
[2]Amer, M., Classification of Boolean algebras of logic and probabilities defined on them by classical models, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986) (to appear).Google Scholar
[3]Dunford, N. and Schwartz, J. T., Linear operators. I, Pure and Applied Mathematics, vol. 7, Interscience, New York, 1958.Google Scholar
[4]Halmos, P., Lectures on Boolean algebras, Van Nostrand Mathematical Studies, no. 1, Van Nostrand, Princeton, New Jersey, 1963.Google Scholar
[5]Hanf, W., The Boolean algebra of logic, Bulletin of the American Mathematical Society, vol. 81 (1975), pp. 587–589.CrossRefGoogle Scholar
[6]Hanf, W., Private communications, 1975.Google Scholar
[7]Keisler, H. J., Model theory for infinitary logic, Studies in Logic and the Foundations of Mathematics, vol. 62, North-Holland, Amsterdam, 1971.Google Scholar
[8]Scott, D. and Krauss, P., Assigning probabilities to logical formulas, Aspects of inductive logic (Hintikka, J. and Suppes, P., editors), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1966, pp. 219–264.CrossRefGoogle Scholar
[9]Sikorski, R., Boolean algebras, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 25, Springer-Verlag, Berlin, 1964.Google Scholar