School of Historical and Philosophical Studies - Theses

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End date: 2006 × Start date: 2006 × Type: Masters Research thesis × Subject: Bayesian statistical decision theory ×

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    Using a non-universal logic as a foundation for statistical inference and induction
    Pescod, Neil Richard ( 2006)
    The aim of this thesis is to develop a probabilistic form of inference that can treat the problem of induction in a statistically rigorous manner. It will be shown that this requirement can be achieved through the use of a non-universal algebra, in which conditional statements are considered to be meaningful only for those conditions in which the antecedent is true. This means that conditional statements within the logic are treated as being undefined in those situations in which the antecedent is false. With such a non-universal algebra, it will be shown how a non-trivial set of conditional probabilities can be generated by assuming that they are equal to the probabilities of non universal conditionals (known as Adams' hypothesis). That is, using the binary corrector "->" to denote the conditional, Adams' hypothesis can be written explicitly as R(B/A) = R(A-> B). There is, however, a problem associated with this type of interpretation in the case that the conditional is defined universally. For if P(B/A) is given by the classical Kolmogorov expression for conditional probabilities, then Lewis has shown that R(A-> B) can take on only a trivial number of values. But I will show that, for the finite case, that Adams' hypothesis can be retained for a non-universal conditional when A->B is defined only for situations when A is true. This necessitates developing a three-state logic containing non universal propositions that can take on the three logical values: true, false or undefined. There are two ways of representing the probability within such a framework. One method of doing this is to make use a conditional of the type originally proposed by Reichenbach as part of his "quantum logic". But the recommended option allows a symbolic representation of the type of "if - then - else" structures that are a feature of most high-level programming languages., Having shown that, by using such a system of non-universal logic, one can avoid the undesirable consequences of Lewis' triviality results, its implications for objective Bayesianism will be examined. It is concluded that the proposed system provides an improved theoretical account of the procedures used for finite implementations of objective Bayesianism. However a complete formalisation that also covers the continuous case would require much further theoretical work, and is beyond the scope of this thesis. But it is shown that the formulation presented for the finite case can provide a more adequate framework for resolving Hempel's paradox of confirmation than the subjective approach given by Howson and Urbach.