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On Inductive Generalization in Monadic First-Order Logic With Identity

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Publisher Summary The chapter examines the results obtained by means of a system when the relation of identity is used in addition to monadic predicates. The chapter compares the new system of inductive logic sketched by Jaakko Hintikka with Carnap's system. The main advantage of Hintikka's system is that it gives natural degrees of confirmation to inductive generalizations, whereas Carnap's confirmation function c * enables one to deal satisfactorily with singular inductive inference only. According to Carnap's system, general sentences that are not logically true receive nonnegligible degrees of confirmation only if the evidence contains a large part of the individuals in the whole universe. In infinite domains of individuals, a system of inductive logic based on c * gives all general sentences that are not logically true a zero probability independent of the amount of evidence. In Hintikka's system of inductive logic, a priori probabilities are first distributed among constituents. The probability of each constituent is then divided evenly among the state-descriptions that make the constituent in question true.