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Foundations Of Nonstandard Analysis  A Gentle Introduction to Nonstandard Extemsions
 In Nonstandard analysis (Edinburgh
"... this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field ..."
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this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field
Interpretations of Probability, Nonstandard Analysis and Confirmation Theory
, 1991
"... The first chapter presents Bayesian confirmation theory. We then construct infinitesimal numbers and use them to represent the probability of unrefuted hypotheses of standard probability zero. Popper's views on the nature of hypotheses, of probability and confirmation are criticised. It is shown tha ..."
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The first chapter presents Bayesian confirmation theory. We then construct infinitesimal numbers and use them to represent the probability of unrefuted hypotheses of standard probability zero. Popper's views on the nature of hypotheses, of probability and confirmation are criticised. It is shown that Popper conflates total confirmation with weight of evidence. It is argued that Popper's corroboration can be represented in a Bayesian formalism. Popper's propensity theory is discussed. A modified propensity interpretation is presented where probabilities are defined relative to descriptions of generating conditions. The logical interpretation is briefly discussed and rejected. A Bayesian account of estimating the values of objective probabilities is given, and some of its properties are proved. Belief functions are then compared with probabilities. It is concluded that belief functions offer a more elegant representation of the impact of evidence. Both measures are then discussed in re...