We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts (...) are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts (...) are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

An algebra is effective if its operations are computable under some numbering. When are two numberings of an effective partial algebra equivalent? For example, the computable real numbers form an effective field and two effective numberings of the field of computable reals are equivalent if the limit operator is assumed to be computable in the numberings . To answer the question for effective algebras in general, we give a general method based on an algebraic analysis of approximations by elements of (...) a finitely generated subalgebra. Commonly, the computable elements of a topological partial algebra are derived from such a finitely generated algebra and form a countable effective partial algebra. We apply the general results about partial algebras to the recursive reals, ultrametric algebras constructed by inverse limits, and to metric algebras in general. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

forthcoming in S. Lindström, E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen (eds) Logicism, Intuitionism, and Formalism — What has Become of Them?, Synthese Library, Springer. Pdf file.

In the past few years there has been a turn towards evaluating the empirical foundation of epistemic contextualism using formal (rather than armchair) experimental methods. By-and-large, the results of these experiments have not supported the original motivation for epistemic contextualism. That is partly because experiments have only uncovered effects of changing context on knowledge ascriptions in limited experimental circumstances (when contrast is present, for example), and partly because existing experiments have not been designed to distinguish between contextualism and one of (...) its main competing theories, subject-sensitive invariantism. In this paper, we discuss how a particular, "third-person", experimental design is needed to provide evidence that would support contextualism over subject-sensitive invariantism. In spite of the theoretical significance of third-person knowledge ascriptions for debates surrounding contextualism, no formal experiments evaluating such ascriptions that assess contextualist claims have previously been conducted. In this paper, we conduct an experiment specifically designed to examine that central gap in contextualism’s empirical foundation. The results of our experiment provide crucial support for epistemic contextualism over subject-sensitive invariantism. (shrink)

Richard Whately’s views of arguments involving authority are very different in his Elements of Rhetoric and his Elements of Logic. This essay begins by documenting these differences and wondering why they are. It then proceeds to take a broader and more historical view of Whately’s discussions of authority and finds him occupying an important developmental ground between his predecessor Locke and contemporary views of the argument from authority. In fact, some of the things we now think are important in a (...) good argument from authority are anticipated by Whately. (shrink)

In this paper we define a family of many-valued semantics for hybrid logic, where each semantics is based on a finite Heyting algebra of truth-values. We provide sound and complete tableau systems for these semantics. Moreover, we show how the tableau systems can be made terminating and thereby give rise to decision procedures for the logics in question. Our many-valued hybrid logics turn out to be "intermediate" logics between intuitionistic hybrid logic and classical hybrid logic in a specific sense explained (...) in the paper. Our results show that many-valued hybrid logic is indeed a natural enterprise. (shrink)

In this paper we define a family of many-valued semantics for hybrid logic, where each semantics is based on a finite Heyting algebra of truth-values. We provide sound and complete tableau systems for these semantics. Moreover, we show how the tableau systems can be made terminating and thereby give rise to decision procedures for the logics in question. Our many-valued hybrid logics turn out to be "intermediate" logics between intuitionistic hybrid logic and classical hybrid logic in a specific sense explained (...) in the paper. Our results show that many-valued hybrid logic is indeed a natural enterprise. (shrink)

This book is a collection of papers on modals in the languages of Europe, written by experts in the area of modality. It provides readers with a wealth of data and addresses the issues of under which circumstances modals are borrowed, from which linguistic materials they typically arise in these languages, and whether and how modals form a system which is separate from other word classes.

Zusammenfassung In diesem Beitrag werden Ergebnisse einer empirischen Studie zum Lernen in Sportvereinen vorgestellt. Es kann gezeigt werden, dass Mitglieder durch ihre Vereinsaktivitäten eine Reihe von Kompetenzen erwerben können, die ihnen nicht nur das Handeln innerhalb des Vereins ermöglichen, sondern auch in anderen Kontexten, z.B. der Berufswelt, anwendbar sind. Die Ergebnisse, die auf der Selbsteinschätzung von freiwillig Engagierten beruhen, zeigen weiterhin, dass der Verein als ein Lernort angesehen werden kann, an dem vor allem durch konkrete Erfahrungen, durch den Austausch mit (...) anderen Menschen bzw. durch das Ausprobieren der eigenen Fähigkeiten gelernt wird. Bewusste und zielgerichtete Lernaktivitäten scheinen dagegen eher eine untergeordnete Stellung einzunehmen. Sportvereine unterscheiden sich von anderen Vereinen in Bezug auf die erwerbbaren Inhalte sowie die Teilnahme der Mitglieder an formellen Lernveranstaltungen. (shrink)

In his Convention David Lewis defined conventions as behavioural regularities instantiating proper coordination equilibria made salient by precedent and operational by this being common knowledge. While later proponents of game theoretical approaches in the study of convention have agreed on dropping Lewis’ eccentric ‘coordination’ requirement as well as that of common knowledge, they are confused as to whether conventions should be regarded as proper thereby precluding mixed equilibria. In this paper I argue that mixed equilibria may not be conventions, but (...) also suggest that the reason for this reveals that though common knowledge is not necessary for a convention to operate, it may be utilized to identify the conventional aspect of a given practice. (shrink)