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Colin Howson - Should Probabilities by Countably Additive

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Published on Mar 7, 2011

Though it may not sound like a very exciting question, a good deal of both the mathematics and the interpretation of probability depend on the answer to it. Many of the striking theorems of mathematical probability, like the celebrated 'with probability 1' convergence theorems, depend on the axiom of countable additivity, as does the use of the techniques of modern measure theory generally. The axiom is also widely supposed to be demanded by de Finetti's well-known theory of coherence, despite the fact that de Finetti repudiated it. I argue that nearly all the standard arguments for countable additivity are invalid or question-begging, that it is not presupposed by de Finetti's theory of coherence, and that his repudiation of it arose from a very logical view of subjective probability. I endorse that view, and show that de Finetti's theory of coherence/consistency for assignments of probability-values differs formally very little from the definition of consistency for truth-value assignments implicit in classical propositional logic.

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