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)

We prove that the standard cut is definable in each existentially closed model ofIΔ0+ exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.

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)

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)

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.

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 $${\Delta_1}$$ 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 $${I\Delta_0 + \neg \mathit{exp}}$$ implies $${B\Sigma_1}$$ to a purely recursion-theoretic question.

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.

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)

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)

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)

Disjunctivism about perceptual appearances, as I conceive of it, is a theory which seeks to preserve a na.

The problems of perception feature centrally in work within what we now think of as different traditions of philosophy in the early part of the twentieth century, most notably in the sense-datum theories of early analytic philosophy together with the vigorous responses to them over the next forty years, but equally in the discussions of pre-reflective consciousness of the world characteristic of German and French phenomenologists. In the English-speaking world one might mark the beginning of the period with Russell’s The (...) Problems of Philosophy (Russell 1912) and its nemesis in Austin’s Sense and Sensibilia (Austin 1962 – published posthumously, but given originally as lectures at the end of our period in 1947). On the continent, a corresponding route takes us from Husserl’s Logical Investigations (Husserl 1900/1) to Merleau-Ponty’s Phenomenology of Perception.. (shrink)

In this paper I discuss Hume’s theory of pride and the ‘remarkable mechanism’ of sympathy. In the first part of the paper I outline the ways in which Hume’s theory can accommodate the sense in which the passions are directed on things or possess intentionality while still holding to his view that passions are simple feelings. In the second part of the paper I consider a problem internal to Hume’s account of pride which arises in his discussion of the love (...) of fame and the functioning of sympathy; I explain how the tensions can be reconciled by recognising that Hume’s theory of sympathy is more nuanced than has commonly been recognized. In the third part I turn back to the evaluation of Hume’s theory of pride and argue that while it is unfair to complain that Hume does not make self-evaluation a central component of pride, Hume’s treatment of the idea of self in his theory of the passions is inadequate because he can make no proper room for the phenomenon of vicarious pride. (shrink)

Listening to someone from some distance in a crowded room you may experience the following phenomenon: when looking at them speak, you may both hear and see where the source of the sounds is; but when your eyes are turned elsewhere, you may no longer be able to detect exactly where the voice must be coming from. With your eyes again fixed on the speaker, and the movement of her lips a clear sense of the source of the sound will (...) return. This ‘ventriloquist’ effect reflects the ways in which visual cognition can dominate auditory perception. And this phenomenological observation is one what you can verify or disconfirm in your own case just by the slightest reflection on what it is like for you to listen to someone with or without visual contact with them. (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)

Agroecology has been criticized for being more labor-intensive than other more industrialized forms of agriculture. We challenge the assertion that labor input in agriculture has to be generally minimized and argue that besides quantity of work one should also consider the quality of work involved in farming. Early assessments on work quality condemned the deskilling of the rural workforce, whereas later criticisms have concentrated around issues related to fair trade and food sovereignty. We bring into the discussion the concept of (...) contributive justice to welcome the added labor-intensity of agroecological farming. Contributive justice demands a work environment where people are stimulated to develop skills and learn to be productive. It also suggests a fairer distribution of meaningful work and tedious tasks. Building on the notion of contributive justice we explore which capabilities and types of social relationships are sustainably promoted and reinforced by agroecological farming practices. We argue that agroecological principles encourage a reconceptualization of farm work. Farmers are continuously stimulated to develop skills and acquire valuable experiential knowledge on local ecosystems and agricultural techniques. Further, generalized ecological studies recognize the significance of the farmer’s observations on natural resources management. This contributes to the development of a number of capabilities and leads to more bargaining power, facilitating self-determination. Hereby farm work is made more attractive to a younger generation, which is an essential factor for safeguarding the continuity of family farms. (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)