Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from (...) the realm of philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part II supplements (...) the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let [Formula: see text] be the lattice of weak degrees of mass problems associated with nonempty [Formula: see text] subsets of the Cantor (...) space. The lattice [Formula: see text] has been studied in previous publications. The purpose of this paper is to show that [Formula: see text] partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in [Formula: see text] which are indexed by the ordinal numbers less than [Formula: see text] and which correspond to the hyperarithmetical hierarchy. Namely, for each [Formula: see text], let hα be the weak degree of 0, the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p* be the weak degree of the mass problem P* = {Y | ∃X ⊆ BLR )} where BLR is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in [Formula: see text]. We prove that all of the weak degrees [Formula: see text], [Formula: see text], are distinct and belong to [Formula: see text]. In addition, we prove that certain index sets associated with [Formula: see text] are [Formula: see text] complete. (shrink)

Considerable controversy has recently arisen regarding the patenting of medical and surgical processes in the United States. One such patent, viz. for a "chevron" incision used in ophthalmologic surgery, has especially occasioned heated response including a major, condemnatory ethics policy statement from the American Medical Association as well as federal legislation denying patent protection for most uses of a patented medical or surgical procedure. This article identifies and discusses the major legal, ethical and public policy considerations offered by proponents and (...) opponents of such patents. The existing literature divides up into those who favor such patents essentially without qualification, and those who condemn and wish to outlaw them. We advance a compromise position where administrative and legislative action is called for to provide more specific guidelines regarding the patentability of such processes by the Patent and Trademark Office. Our position, in sum, will be that too much is at stake in this complicated area for either the blanket prohibition, or wholesale, uncritical acceptance, of the patenting of medical and surgical processes or techniques. (shrink)

In Making Science, Cole shows how social variables and cognitive variables interact in the evaluation of frontier knowledge.

From roughly 1965 to 1995, Columbia University's Department of Sociology was a leading center for social study of science, both nationally and internationally. It was often referred to as the Merton School or Columbia School, and four scholars paved its way : Robert K. Merton, Harriet Zuckerman, Stephen Cole, and Jonathan Cole. The goal of the Columbia School was to create and legitimate a new sociological specialty focusing on the scientific community and the growth of scientific knowledge (...) and they did this by exploring the norms, values, and structure of science. The core values of the scientific and university system and the norms attached to them consisted of deeply held beliefs as well as principles that determined how individuals would act. They were organizing principles designed to support the institution in meeting its goals and mission. They influenced the types of social structures that developed to carry out the activities of scientists. Edited by Jonathan R. Cole, this collection of individually and co-authored papers and essays by Cole represents some of the work produced by Cole and his close collaborators over the past 50 years. the essays are contextualized with an introduction by Cole. The pieces in this volume represent the major concerns on the Columbia School, whose focus of attention was not on the production of individual pieces of knowledge, but on the social systems that made that production possible. (shrink)

This paper examines ideas which underpin the 'New Literacy Studies', including Street's critique of the 'autonomous model' of literacy. Through a re-examination of the work of Scribner and Cole, the implication that literacy is not related to cognitive development is questioned. Gee's critique of linguistic prescription is discussed. Drawing on action research perspectives, which argue that educational theory concerns the identification of educational problems and values, an approach to literacy for education is proposed.

We show that any coherent complete partial order is obtainable as the fixed-point poset of the strong Kleene jump of a suitably chosen first-order ground model. This is a strengthening of Visser’s result that any finite ccpo is obtainable in this way. The same is true for the van Fraassen supervaluation jump, but not for the weak Kleene jump.

Stephen Pinker sets out over a dozen arguments in The language instinct (Morrow, New York, 1994) for his widely shared view that natural language is inadequate as a medium for thought. Thus he argues we must suppose that the primary medium of thought and inference is an innate propositional representation system, mentalese. I reply to the various arguments and so defend the view that some thought essentially involves natural language. I argue mentalese doesn't solve any of the problems Pinker (...) cites for the view that we think in natural language. So I don't think I think the way he thinks I think. (shrink)

This paper examines ideas which underpin the 'New Literacy Studies', including Street's critique of the 'autonomous model' of literacy. Through a re-examination of the work of Scribner and Cole, the implication that literacy is not related to cognitive development is questioned. Gee's critique of linguistic prescription is discussed. Drawing on action research perspectives, which argue that educational theory concerns the identification of educational problems and values, an approach to literacy for education is proposed.