Interpretations of Probability, Nonstandard Analysis and Confirmation Theory

The first chapter presents Bayesian confirmation theory. We then construct em 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 em total confirmation with em weight of evidence. It is argued that Popper's em 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 em 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. em 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 relation to various em betting procedures designed to elicit their values from an individual's belief state. De Finetti's arguments concerning `coherence' are discussed. It is then shown that it is not possible to use bets to derive belief function values unless the better is allowed to vary the amount of the stake. Hume's thinking on induction is discussed. It is argued that some of the problems of Popper's philosophy derive from Hume's. The em Popper-Miller argument is presented and criticised. It is concluded that the core of the argument is valid, but of limited applicability. The correspondence between probabilistic support and deductive relations is discussed. There are two appendices. The first criticises Popper's view on the connection between the content and testability of a hypothesis. The second concerns a nonstandard probability measure proposed in 1967.