A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in (...) the limit if, after some finite point in the sequence, all the guesses are correct and the same. Functions, sets, and functionals which are decidable by such infinite algorithms will be calledlimiting recursive. These, together with classes of objects which can beidentified in the limit, are the subjects of this report.Without qualification,setwill mean set of numbers;functionwill mean number-theoretic function of 1 variable, possibly partial;functionalswill take numerical values and have any number of numerical and/or function variables, the latter ranging solely over total functions of 1 variable. Thus a function is a special case of a functional,xwill invariably stand for a numerical variable;φfor a function variable;gfor a guess ;nfor the numerical variable which indexes the guesses, referred to as thetime. (shrink)

then essentially characterized the hypotheses that mechanical scientists can successfully decide in the limit in terms of arithmetic complexity. These ideas were developed still further by Peter Kugel [4]. In this paper, I extend this approach to obtain characterizations of identification in the limit, identification with bounded mind-changes, and identification in the short run, both for computers and for ideal agents with unbounded computational abilities. The characterization of identification with n mind-changes entails, as a corollary, an exact arithmetic characterization of (...) Putnam's n-trial predicates, which closes a gap of a factor of two in Putnam's original characterization [12]. (shrink)

Charles Sanders Peirce has been characterized as the greatest American philosophic genius. He is the creator of pragmatism and one of the founders of modern logic. James, Royce, Schroder, and Dewey have acknowledged their great indebtedness to him. A laboratory scientist, he made notable contributions to geodesy, astronomy, psychology, induction, probability, and scientific method. He introduced into modern philosophy the doctrine of scholastic realism, developed the concepts of chance, continuity, and objective law, and showed the philosophical significance of the theory (...) of signs and mathematical logic. The present series is the first published edition of his systematic works. (shrink)

We must restrict to mere probability not only statements of comparatively great uncertainty, like predictions about the weather, where we would cautiously ...

I have applied a fairly general, learning theoretic perspective to some questions raised by Reichenbach's positions on induction and discovery. This is appropriate in an examination of the significance of Reichenbach's work, since the learning-theoretic perspective is to some degree part of Reichenbach's reliabilist legacy. I have argued that Reichenbach's positivism and his infatuation with probabilities are both irrelevant to his views on induction, which are principally grounded in the notion of limiting reliability. I have suggested that limiting reliability is (...) still a formidable basis for the formulation of methodological norms, particularly when reliability cannot possibly be had in the short run, so that refined judgments about evidential support must depend upon measure-theoretic choices having nothing to do in the short run with the truth of the hypothesis under investigation. To illustrate the generality of Reichenbach's program, I showed how it can be applied to methods that aim to solve arbitrary assessment and discovery problems in various senses. In this generalized Reichenbachian setting, we can characterize the intrinsic complexity of reliable inductive inference in terms of topological complexity. Finally, I let Reichenbach's theory of induction have the last say about hypothetico-deductive method. (shrink)

This book is devoted to a different proposal--that the logical structure of the scientist's method should guarantee eventual arrival at the truth given the scientist's background assumptions.

"Judicious in every respect: selection, translation and structuring of the texts, footnotes, bibliography, and index.... The book of choice for undergraduate courses." --Edward M. Galligan, University of North Carolina.

Clark Glymour and Kevin T. Kelly. Thoroughly Modern Meno.