Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T23:36:29.044Z Has data issue: false hasContentIssue false
  • English
  • Français

A Characterization of ImplicativeBoolean Rings

Published online by Cambridge University Press:  20 November 2018

A. H. Copeland Sr.  and
Frank Harary
A. H. Copeland Sr.
Affiliation:
University of Michigan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the theory of probability, the conditional can be treated by an operation analogous to division. Many properties of the conditional can best be studied by means of the corresponding multiplication (called the cross-product). An implicative Boolean ring is defined [2] in terms of a cross-product and the usual Boolean operations. The cross-product is the only device yet known in which the events corresponding to conditional probabilities are themselves elements of the Boolean ring. The fact that such advice was not introduced by Boole is probably the reason why Boolean algebra has been very little used in the theory of probability, although probability was one of the principal applications which Boole had in mind.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 465 - 469
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Boole, G., The laws of thought (Cambridge, 1854), 253–398.Google Scholar
2. Copeland, A. H. Sr., Implicative Boolean algebra, Math. Z., 58 (1950), 285–290.Google Scholar
3. Copeland, A. H. Sr., and Harary, F., The extension of an arbitary Boolean algebra to an implicative Boolean algebra, Abstract 128, Bull. Amer. Math. Soc, 58 (1952), 167 (to appear in Proc. Amer. Math. Soc, 1953).Google Scholar
4. Koopman, B. O., The axioms and algebra of intuitive probability, Ann. Math., 41 (1940), 269–292.Google Scholar