We study the quantifier complexity and the relative strength of some fragments of arithmetic axiomatized by induction and minimization schemes for Δn+1 formulas.

We study the quantifier complexity and the relative strength of some fragments of arithmetic axiomatized by induction and minimization schemes for Δn+1 formulas.

By a result of L.D. Beklemishev, the hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over any $$\Pi _2$$ -axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true $$\Pi _2$$ -sentences, S, we construct a sound $$(\Sigma _2 \! \vee \! \Pi _2)$$ -axiomatized theory T extending S such that the (...) hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over T is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen. (shrink)

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+1-formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first-order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, we obtain (...) results on the quantifier complexity, finite axiomatizability and relative strength of schemes for Δn+1-formulas. (shrink)

A well known theorem proved by J. Paris and H. Friedman states that BΣn +1 is a Πn +2-conservative extension of IΣn . In this paper, as a continuation of our previous work on collection schemes for Δn +1-formulas , we study a general version of this theorem and characterize theories T such that T + BΣn +1 is a Πn +2-conservative extension of T . We prove that this conservativeness property is equivalent to a model-theoretic property relating Πn-envelopes and (...) Πn-indicators for T . The analysis of Σn +1-collection we develop here is also applied to Σn +1-induction using Parsons' conservativeness theorem instead of Friedman-Paris' theorem.As a corollary, our work provides new model-theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos : BΣn +1 and IΣn +1 are Σn +3-conservative extensions of their parameter free versions, BΣ–n +1 and IΣ–n +1. (shrink)

LetTbe a recursive theory in the language of first order Arithmetic. We prove that ifTextends: (a) the scheme of parameter free Δ1-minimization (plusexp). or (b) the scheme of parameter free Π1-induction, then there are no Σ1-maximal models with respect toT. As a consequence, we obtain a new proof of an unpublished theorem of Jeff Paris stating that Σ1-maximal models with respect toIΔ0+expdo not satisfy the scheme of Σ1-collectionBΣ1.

For a theory T, we study relationships among IΔ n +1 (T), LΔ n+1 (T) and B * Δ n+1 (T). These theories are obtained restricting the schemes of induction, minimization and (a version of) collection to Δ n+1 (T) formulas. We obtain conditions on T (T is an extension of B * Δ n+1 (T) or Δ n+1 (T) is closed (in T) under bounded quantification) under which IΔ n+1 (T) and LΔ n+1 (T) are equivalent. These conditions depend (...) on Th Πn +2 (T), the Π n+2 –consequences of T. The first condition is connected with descriptions of Th Πn +2 (T) as IΣ n plus a class of nondecreasing total Π n –functions, and the second one is related with the equivalence between Δ n+1 (T)–formulas and bounded formulas (of a language extending the language of Arithmetic). This last property is closely tied to a general version of a well known theorem of R. Parikh. Using what we call Π n –envelopes we give uniform descriptions of the previous classes of nondecreasing total Π n –functions. Π n –envelopes are a generalization of envelopes (see [10]) and are closely related to indicators (see [12]). Finally, we study the hierarchy of theories IΔ n+1 (IΣ m ), m≥n, and prove a hierarchy theorem. (shrink)

Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.For a set of formulas Γ, define predicative oracle reflection for T over Γ ) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then$$\forall \,\lambda (...) < {\rm{\Lambda }}\,.$$In particular, define predicative oracle consistency ) as Pred–O–RFN{0=1}.Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left.$$We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic.Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles. (shrink)

We characterize the sets of all Π2 and all equation image theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to equation image sentences cannot be extended to Πn + 2 sentences. © 2011 WILEY-VCH Verlag (...) GmbH & Co. KGaA, Weinheim. (shrink)

This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta_1}$$\end{document} provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether IΔ0+¬exp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I\Delta_0 + \neg \mathit{exp}}$$\end{document} implies BΣ1\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B\Sigma_1}$$\end{document} to a purely recursion-theoretic question. (shrink)

In this paper we continue the study of the theories IΔ n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that IΔ n+1 (T) is Π n+2 –axiomatizable. In particular, IΔ n+1 (IΔ n+1 ) gives an axiomatization of Th Π n+2 (IΔ n+1 ) and is not finitely axiomatizable. This fact relates the fragment IΔ n+1 (IΔ n+1 ) to induction (...) rule for Δ n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons’ conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for Π n+2 and Δ n+1 formulas (see [2]). (shrink)

This book applies Augustine's thought to current questions of teaching and learning. The essays are written in an accessible style and is not intended just for experts on Augustine or church history.

The state of computing science and, particularly, software engineering and knowledge engineering is generally considered immature. The best starting point for achieving a mature engineering discipline is a solid scientific theory, and the primary reason behind the immaturity in these fields is precisely that computing science still has no such agreed upon underlying theory. As theories in other fields of science do, this paper formally establishes the fundamental elements and postulates making up a first attempt at a theory in this (...) field, considering the features and peculiarities of computing science. The fundamental elements of this approach are informons and holons, and it is a general and comprehensive theory of software engineering and knowledge engineering that related disciplines (e.g., information systems) can particularise and/or extend to take benefit from it (Lakatos’ concepts of core theory and protective belt theories). (shrink)

Several climate ethicists have recently argued that having children is morally equivalent to over-consumption, and contributes greatly to parents’ personal carbon footprints. We show that these claims are mistaken, for two reasons. First, including procreation in parents’ carbon footprints double-counts children’s consumption emissions, once towards their own, and once towards their parents’ footprints. We show that such double-counting defeats the chief purpose of the concept of carbon footprint, namely to measure the sustainability and equitability of one’s activities and choices. Furthermore, (...) we show that proposals to avoid double-counting have other unacceptable implications. Second, we show that the key arguments for a supposed moral equivalence of procreation and consumption overgenerate and lead to unacceptable consequences in many cases, such as for the work of doctors who save lives or enable procreation. Finally, we propose that rather than counting children’s emissions towards their parents’ carbon footprints, we should consider these emissions as part of the parents’ carbon impact, i.e. the difference that their choices make to the overall global carbon emissions. It is from the perspective of impact that we should think about the ethics of procreation in an age of climate change. (shrink)

Als praktische Philosophie ist der Pragmatismus an Prinzipien und Mechanismen von Gesellschaft interessiert. Es geht im Kern darum, diese Prinzipien und Mechanismen zu verstehen, Handlungsformen und Institutionen mit Blick auf gesellschaftliche Probleme und deren Behebung zu entwickeln und auf diesem Wege Gesellschaft menschlicher, gerechter und demokratischer zu gestalten. Aus diesem Grund scheint der Pragmatismus für die kritische zeitgenössische Sozialwissenschaft von größter Bedeutung. Der vorliegende Band versammelt eine Reihe von Beiträgen, die das Themenfeld pragmatistischer Sozialforschung näher bestimmen. Diese lassen sich in (...) zwei Gruppen einteilen: Beiträge, die pragmatistische Theorien und Methoden mit anderen Denktraditionen oder Forschungsansätzen ins Gespräch bringen, sowie Beiträge, die pragmatistische Perspektiven auf zeitgenössische, gesellschaftliche Problemlagen beziehen. (shrink)

This Festschrift in Professor Kristeller’s honor consists of contributions by scholars who have had some connection with Columbia University, his "intellectual home in the United States for three decades." It also includes a Tabula Gratulatoria listing many other friends from the United States and Europe. The editor’s opening essay provides an interesting and informative account of this scholar’s academic career, and should be read together with the complete annotated bibliography of his publications through 1974. The latter lists 149 "major publications" (...) and 220 "minor publications." Kristeller’s contributions to the history of Renaissance philosophy are well known to historians of philosophy, and deservedly so. Here reference should be made to his groundbreaking studies on Marsilio Ficino and Pomponazzi, and on others such as Pico della Mirandola and Petrarch, as well as on Renaissance Platonism, Aristotelianism in the Renaissance, Thomism in the Renaissance, Paduan Averroism, and Alexandrism. But he has also contributed greatly to the fields of medieval and Renaissance history, and especially to our understanding of Renaissance humanism, Renaissance music, and Renaissance art. He is universally recognized as one of the world’s foremost authorities on manuscript research, as is witnessed, for example, by the cooperative project, Catalogus Translationum et Commentariorum, which he founded, and by his Iter ltalicum. In all of these enterprises he has set an enviable example for other scholars by the exacting standards and the breadth of his expertise. It is only fitting, then, that the many essays in this Festschrift should reflect the breadth and depth of the scholarship so evident in the man to whom they are dedicated. Limitations of space will only permit us to list them here, with a few remarks reserved for those of more special interest to philosophers and historians of philosophy: Eugene F. Rice, Jr., "The De magia naturali of Jacques Lefèvre d’Etaples" ; Donald R. Kelley, "Louis Le Caron Philosophe", on Le Caron’s effort to bring together jurisprudence and classical, especially Platonic, philosophy; Richard H. Popkin, "The Pre-Adamite Theory in the Renaissance", with fascinating material about theories concerning men before Adam in La Peyrère and widely scattered earlier sources; Richard Lemay, "The Fly against the Elephant: Flandinus against Pomponazzi on Fate", on an unedited attack by an Augustinian Bishop against Pomponazzi’s espousal of the Stoic doctrine of fate; Martin Pine, "Pietro Pomponazzi and the Medieval Tradition of God’s Foreknowledge", on Pomponazzi’s solution to the problem of divine foreknowledge and human freedom inlight of his familiarity with earlier discussions by Boethius, Aquinas, Scotus, and Ockham; F. Edward Cranz, "Editions of the Latin Aristotle Accompanied by the Commentaries of Averroes", helpful to all who wish to consult late fifteenth- and sixteenth-century versions of the Latin Aristotle and especially the Latin Averroes; Josef Soudek, "A Fifteenth-Century Humanistic Bestseller: The Manuscript Diffusion of Leonardo Bruni’s Annotated Latin Version of the Aristotelian Economics" ; Edward P. Mahoney, "Nicoletto Vernia on the Soul and Immortality", which details a radical shift on Vernia’s part from an earlier Averroistic reading of Aristotle; Joan Kelly-Gadol, "Tommaso Campanella: The Agony of Political Theory in the Counter-Reformation", which attempts to account for some of the shifts and inconsistencies in Campanella’s political writings by placing them within the troubled personal and political circumstances of his life; Charles Trinkaus, "Protagoras in the Renaissance: An Exploration" ; Maristella de Panizza Lorch, "Voluptas, molle quoddam et non invidiosum nomen: Lorenzo Valla’s Defense of voluptas in the Preface to his De voluptate" ; Neal W. Gilbert, "Richard de Bury and the ‘Quires of Yesterday’s Sophisms"’, with much interesting material on the medieval tradition of sophismata, especially at Oxford; Malcolm Brown, "A Pre-Aristotelian Mathematician on Deductive Order" ; John H. Randall, Jr., "Paduan Aristotelianism Reconsidered", on evidence for influence of the Italian Aristotelian tradition on Galileo; William F. Edwards, "Niccoló Leoniceno and the Origins of Humanist Discussion of Method" ; C. Doris Hellman, "A Poem on the Occasion of the Nova of 1572" ; Edward Rosen, "Kepler’s Mastery of Greek" ; W. T. H. Jackson, "The Politics of a Poet: The Archipoeta as Revealed by his Imagery" ; John Charles Nelson, "Love and Sex in the Decameron" ; George B. Parks, "Pico della Mirandola in Tudor Translation" ; Richard Harrier, "Invention in Tudor Literature: Historical Perspectives" ; Helene Wieruszowski, "Jacob Burckhardt and Vespasiano da Bisticci " ; Morimichi Watanabe, "Gregor Heimburg and Early Humanism in Germany" ; Raymond de Roover, "Cardinal Cajetan on ‘Cambium’ or Exchange Dealings" ; and a series of text editions with introductions including Julius Kirshner, "Conscience and Public Finance: A Questio disputata of John of Legnano on the Public Debt of Genoa" ; John Mundy, "The Origins of the College of Saint-Raymond at the University of Toulouse" ; Charles B. Schmitt, "Girolamo Borro’s Multae sunt nostrarum ignorantionum causae " ; Guido Kisch, "An Unpublished Consiliumof Johannes Sichardus" ; Patricia H. Labalme, "The Last Will of a Venetian Patrician " ; Felix Gilbert, "The Last Will of a Venetian Grand Chancellor" ; Herbert S. Matsen, "Giovanni Garzoni to Alessandro Achillini : An Unpublished Letter and Defense" ; Theodore E. James, "A Fragment of An Exposition of the First Letter of Seneca to Lucilius Attributed to Peter of Mantua". The editor, his collaborators, and the contributors are all to be commended for the high quality of this volume.—J.F.W. (shrink)

The progress of electronic health technologies and biobanks holds enormous promise for efficient research. Evidence shows that studies based on sharing and secondary use of data/samples have the potential to significantly advance medical knowledge. However, sharing of such resources for international collaboration is hampered by the lack of clarity about ethical and legal requirements for transfer of data and samples across international borders. Here, the International Clinical Trial Center Network reports the legal and ethical requirements governing data and sample exchange (...) across four continents. The most recurring requirement is ethical approval, whereas only in specific conditions approval of national health authorities is required. Informed consent is not required in all sharing situations. However, waiver of informed consent is only allowed in certain countries/regions and under certain circumstances. The current legal and ethical landscape appears to be very complex and under constant evolution. Regulations differ between countries/regions and are often incomplete, leading to uncertainty. With this work, ICN illuminates the unmet need for a single international collaborative framework to facilitate DSE. Harmonising requirements for global DSE will reduce inefficiency and waste in research. There are many challenges to realising this ambitious vision, including inconsistent terminology and definitions, and heterogeneous and dynamic legal constraints. Here, we identify areas of agreement and significant difference as a necessary first step towards facilitating international collaboration. We propose the establishment of a working group to continue the comparison across jurisdictions, create a standardised glossary and define a set of basic principles and fundamental requirements for DSE. (shrink)

“Every surgeon carries within himself a small cemetery where from time to time he goes to pray—a place of bitterness and regret, where he must look for an explanation of his failures” René Leriche,...

Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disease, primarily leading to the degeneration of motor neurons. The traditional focus on motor neuron‐centric mechanisms has recently shifted towards understanding the contribution of non‐neuronal cells, such as microglia, in ALS pathophysiology. Advances in induced pluripotent stem cell (iPSC) technology have enabled the generation of iPSC‐derived microglia monocultures and co‐cultures to investigate their role in ALS pathogenesis. Here, we briefly review the insights gained from these studies into the role of microglia in ALS. (...) While iPSC‐derived microglia monocultures have revealed intrinsic cellular dysfunction due to ALS‐associated mutations, microglia‐motor neuron co‐culture studies have demonstrated neurotoxic effects of mutant microglia on motor neurons. Based on these findings, we briefly discuss currently unresolved questions and how they could be addressed in future studies. iPSC models hold promise for uncovering disease‐relevant pathways in ALS and identifying potential therapeutic targets. (shrink)

This work aims to rescue, transcribe and create new artistic and cultural expressions through the use of native peoples’ historical visual recordings, integrating intelligent technologies. For this purpose, a Chilean native peoples’ digital repository is collected, in order to apply a Digital Humanities-based methodology. From the chosen material, portraits are selected, recoloured through a AI-based model; the facial mesh is constructed using a facial landmark detector; the points of the mesh are reconstructed by a Delaunay triangulation; to finally apply an (...) additive manufacturing process. Thus, these physical pieces allow to compare the native people’s own physiognomies, creating new cultural expressions. (shrink)

El trabajo analiza las relaciones comerciales entre España y Latinoamérica y el Caribe utilizando datos del Ministerio de Industria, Comercio y Turismo, la Organización Mundial del Comercio y_ Atlas of Economic Complexity_. Trata de responder a dos cuestiones. De una parte, si las relaciones bilaterales entre Latinoamérica y el Caribe con España se han desarrollado de manera positiva y de otra si el comercio ha evolucionado hacia productos con un índice de complejidad mayor y por tanto hacia actividades de mayor (...) valor añadido. A lo largo del trabajo se comprueba que el porcentaje de importaciones y exportaciones entre España y Latinoamérica y el Caribe se mantiene estable en el rango del 4% y el 6% a lo largo de las últimas décadas y de otra parte se constata que no se ha producido una mayor complejidad de los productos, siendo los sectores extractivos y primarios los que priman en las exportaciones hacia España. (shrink)

In this special issue, we draw on pragmatist political and social theory and philosophy to illustrate the creative potential of this intellectual tradition for thinking about the numerous crises that haunt liberal democratic societies today. The introduction identifies five overlapping problem constellations (demise of public power, lasting consequences of inequality, pluralization of society, return of authoritarian practices and globalization of the world) that have driven the recent rise of undemocratic or authoritarian patterns of social organization and political rule. Against this (...) backdrop, we conclude that the revitalization of certain dimensions of liberal democracy will not suffice to overcome these problems, which means that democratic practices need radical rethinking and reconceptualization. For this intellectual and political endeavour, we argue, pragmatism provides a suitable framework to identify problems that require resolution and define and mobilize collective problem solving capacities from already existing practices. All eight contributions to this special issue draw on pragmatist political and social theory and philosophy to illustrate to what extent, and to what ends, this intellectual tradition can revitalize the political and social discourse on the past, present and future of democracy. The articles are organized in two sections : (1) Pragmatist critique and the critical potential of pragmatism, (2) pragmatist politics and theories of democratic practice. (shrink)

This paper decribes the theoretical and methodological issues involved in the social shaping of technology and work, with particular reference to human centred computer integrated manufacturing (CIM) systems. Conventional approaches to the understanding and shaping of the relationship between technology, work and human development are criticised, and an alternative, human centred approach is outlined. The methods and processes whereby the design of human centred CIM systems may be shaped and evaluated are then described and appraised.

We claim that a notion of rhetic acts can fulfil a useful function in speech act theory. Austin’s examples of rhetic acts are saying that something is so and so, telling someone to do something, and asking whether something is so or so. Though this certainly sounds as if he is talking about the illocutionary acts of asserting, giving directions, and asking questions, we explain why the acts Austin mentions are not illocutionary after all. In short, illocutionary acts are acts (...) that commit the participants in a conversation to various things. The illocutionary force of an utterance is determined by what it commits the interlocutors to. Our suggestion is that what looks like illocutionary force in the cases mentioned by Austin is really the contributions grammatical moods make to the literal meaning of utterances. We argue that this contribution is not part of what characterizes illocutionary acts as such. The semantic contribution of mood already characterizes those speech acts that Austin uses to exemplify rhetic acts. These rhetic acts are not committing in the way that illocutionary acts essentially are. So there must be a sense in which such rhetic acts are not illocutionary ones. In the final section, we suggest that the varieties of rhetic acts indicated by Austin fit neatly into a standard linguistic theoretical framework of conversational scoreboards. (shrink)

From the most prominent thinkers in Latin American philosophy, literature, politics, and social science comes a challenge to conventional theories of globalization. The contributors to this volume imagine a discourse in which revolution requires no temporalized march of progress or takeovers of state power but instead aims at local control and the material conditions for human dignity.