Interpretations of Probability, Nonstandard Analysis and Confirmation Theory

The rst chapter presents Bayesian con rmation theory. We then construct in nitesimal 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 con rmation are criticised. It is shown that Popper conates total con rmation 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 modi ed propensity interpretation is presented where probabilities are de ned relative to descriptions of generating conditions. The logical interpretation is brie y 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 o er a more elegant representation of the impact of evidence. Both measures are then discussed in relation to various 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 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 rst 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. 4