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Landes, Jürgen (2009): The Principle of Spectrum Exchangeability within Inductive Logic. Dissertation, University of Manchester

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We investigate the consequences of the principle of Spectrum Exchangeability in inductive logic over the polyadic fragment of first order logic. This principle roughly states that the probability of a possible world should only depend on how the inhabitants of this world behave with respect toindistinguishability. This principle is a natural generalization of exchangeability principles that have long been investigated over the monadic predicate fragment of first order logic. It is grounded in our deep conviction that in the state of total ignorance all possible worlds that can be obtained from each other by basic symmetric transformations should have the same a priori probability. After first fixing our framework and showing some basic lemmata we prove that the principle of spectrum exchangeability implies several simple principles of exchangeability that are all based on our conviction that the probability functions should be invariant under basic renaming procedures. We then go on and show several representation theorems for the probability functions satisfying spectrum exchangeability. One of these representation results shows that we can represent the probability of sentences of a polyadic language in terms of the probability of sentences of a unary language. The other main representation result is a de Finetti-style result that shows that we can write every probability function satisfying spectrum exchangeability as an integral over some basic probability function weighted by a de Finetti prior µ. After that we use the de Finetti representation results to show a representation result for probability functions satisfying language invariance and spectrum exchangeability. Rather surprisingly it turns out that the notion of language invariance allows us to seamlessly extend our framework to the fragment of first order logic containing the equality symbol and the predicate fragment. Thereafter we study principles that make inductive assertions. We investigate the Paris Conjecture and we prove that in some instances the principle of instantial relevance holds for t−heterogeneous probability functions. However this principle fails for the completely independent function. Furthermore we show that the assumption of the principle of constant exchangeability and Johnsons’ sufficientness principle leads to only two trivial probability functions satisfying both these two principles.

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