It is well known that the hypothesis that all real numbers are constructible in the sense of Gödel [1] implies the existence of a Σ21well-ordering of the Baire space [1, p. 67]. We are concerned with the converse to this theorem. From the assumption of the existence of a Σ21well-ordering with total domain, we derive various consequences which in the presence of a nonconstructible real seem highly pathological. However, while several of these consequences are obviously absurd, none have as yet (...) been disproven. Indeed some of the stranger consistency proofs of Jensen seem to indicate that there is a possibility that they may be consistent. I am referring to such results as existence of a model for ZF set theory in which the degrees of constructibility form a sequence of typeω+ 1 with the st degree being the only one which contains a nonconstructible real, and that degree being the degree of a Δ31nonconstructible real.In what follows we shall use the small Greek lettersα, β, γto range over the Baire spaceNNwhereNis the set of natural numbers. We assume that the reader is familiar with the use of trees to encode closed subsets of this space. The class Σ21is the collection of all those subsets of the Baire space which can be defined by a formula which is Σ21in a constructible parameter. Correspondingly Π21will be taken to mean Π11in a constructible parameter. The proof of the first lemma is left as an exercise. It involves nothing more than coding perfect sets by trees and then counting quantifiers. (shrink)

This book is a collection of secondary essays on America's most important philosophic thinkers—statesmen, judges, writers, educators, and activists—from the colonial period to the present. Each essay is a comprehensive introduction to the thought of a noted American on the fundamental meaning of the American regime.

Uniting thirty years of authoritative scholarship by a master of textual detail, _Machiavelli's Virtue_ is a comprehensive statement on the founder of modern politics. Harvey Mansfield reveals the role of sects in Machiavelli's politics, his advice on how to rule indirectly, and the ultimately partisan character of his project, and shows him to be the founder of such modern and diverse institutions as the impersonal state and the energetic executive. Accessible and elegant, this groundbreaking interpretation explains the puzzles and (...) reveals the ambition of Machiavelli's thought. "The book brings together essays that have mapped [Mansfield's] paths of reflection over the past thirty years.... The ground, one would think, is ancient and familiar, but Mansfield manages to draw out some understandings, or recognitions, jarringly new."—Hadley Arkes, _New Criterion_ "Mansfield's book more than rewards the close reading it demands."—Colin Walters, _Washington Times_ "[A] masterly new book on the Renaissance courtier, statesman and political philosopher.... Mansfield seeks to rescue Machiavelli from liberalism's anodyne rehabilitation."—Roger Kimball, _The Wall Street Journal_. (shrink)

This study reconstructs the relationship between philosophy and politics in the way in which Heidegger's failure as a politician influenced the redevelopment of philosophy in the 1930s. The author also explains how Heidegger's failure influenced the content and direction of his later work.

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...) expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

Studies the politics of Heidegger in terms of "thrownness" or "existential contingency". Attempts to think through Heidegger's philosophy in a manner that parallels his own dialogue with other key western thinkers.

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order (...) logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

The global pesticide complex has transformed over the past two decades, but social science research has not kept pace. The rise of an enormous generics sector, shifts in geographies of pesticide production, and dynamics of agrarian change have led to more pesticide use, expanding to farm systems that hitherto used few such inputs. Declining effectiveness due to pesticide resistance and anemic institutional support for non-chemical alternatives also have driven intensification in conventional systems. As an inter-disciplinary network of pesticide scholars, we (...) seek to renew the social science research agenda on pesticides to better understand this suite of contemporary changes. To identify research priorities, challenges, and opportunities, we develop the pesticide complex as a heuristic device to highlight the reciprocal and iterative interactions among agricultural practice, the agrochemical industry, civil society-shaped regulatory actions, and contested knowledge of toxicity. Ultimately, collaborations among social scientists and across the social and biophysical sciences can illuminate recent transformations and their uneven socioecological effects. A reinvigorated critical scholarship that embraces the multifaceted nature of pesticides can identify the social and ecological constraints that drive pesticide use and support alternatives to chemically driven industrial agriculture. (shrink)

Subjectivity and identity are newly configured within cyberspace and technologically mediated environments. The global musical subject is thus defined and framed within global empires and techno‐culture in ways not unrelated to political interests. ‘Being musical’ becomes a critical issue. The New Zealand music curriculum resonates with reflections of global ‘progress’, and music educators, as cultural workers, therefore require an awareness of political and strategic conceptions of musical knowledge as well as a familiarity with the discourses through which the work of (...) music education is enframed. Where technology enters hegemonic discourses of musical knowledge, the relationship between global communication and technology requires interrogation, and unquestioned past and present curriculum ‘certainties’ come under scrutiny. (shrink)

Subjectivity and identity are newly configured within cyberspace and technologically mediated environments. The global musical subject is thus defined and framed within global empires and techno‐culture in ways not unrelated to political interests. ‘Being musical’ becomes a critical issue. The New Zealand music curriculum resonates with reflections of global ‘progress’, and music educators, as cultural workers, therefore require an awareness of political and strategic conceptions of musical knowledge as well as a familiarity with the discourses through which the work of (...) music education is enframed. Where technology enters hegemonic discourses of musical knowledge, the relationship between global communication and technology requires interrogation, and unquestioned past and present curriculum ‘certainties’ come under scrutiny. (shrink)

Despite their differences in origin, the three influential schools of twentieth-century continental cultural criticism--the Frankfurt School, existentialism, and poststructuralism--have long been treated as an ensemble and with critical hesitancy. Examining these schools as responses to the apparent collapse of Western civilization in the twentieth-century and as formidable intellectual challenges to the cultural legacies of the Enlightenment, this book provides a productive base for criticism and broadens our understanding of their histories and reception.

The Curry-Howard isomorphism is a proof-theoretic result that establishes a connection between derivations in natural deduction and terms in typed lambda calculus. It is an important proof-theoretic result, but also underlies the development of type systems for programming languages. This fact suggests a potential importance of the result for a philosophy of code.

This classic, provocative introduction to classical metaphysical questions focuses on appreciating the problems, rather than attempting to proffer answers.

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

Having visited Ireland regularly during the 1930s, Ludwig Wittgenstein resigned his Cambridge philosophy professorship in 1947 and moved there, living in a fishing village on the Atlantic coast and hotels in Dublin and the Wicklow Mountains. Although Wittgenstein spent some time out of the country, Ireland was effectively his base for three very productive years during which he worked on what would become one of his key books, the posthumously published Philosophical Investigations. Wittgenstein in Ireland represents the first sustained account (...) of Wittgenstein's time in Ireland, placing it in its historical context. Wall pays a good deal of attention to the representation of the Irish landscape in which the Austrian philosopher found himself able to work; a large part of his writings were produced in the bleak landscapes of Ireland and Norway. (shrink)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of (...) the connectives in question. (shrink)