Behavioral ecologists often assume that natural selection will produce organisms that make optimal decisions. In the context of information processing, this means that the behavior of animals will be consistent with models from fields such as signal detection theory and Bayesian decision theory. We discuss work that applies such models to animal behavior and use the case of Bayesian updating to make the distinction between a description of behavior at the level of optimal decisions and a mechanistic account of how (...) decisions are made. The idea of ecological rationality is that natural selection shapes an animal's decision mechanisms to suit its environment. As a result, decision-making mechanisms may not perform well outside the context in which they evolved. Although the assumption of ecological rationality is plausible, we argue that the exact nature of the relationship between ecology and cognitive mechanism may not be obvious. (shrink)

Church history is dead, long live historical theology! This restatement of the monarchical law of le mort saisit le vif is at once a statement of irreparable discontinuity and assumed continuity. The old monarch is no more, yet a new and different monarch ascends to fill the same vacant throne. This is the paradox of church history becoming historical theology. Reviewing the work of W.A Dreyer and J. Pillay on the re-imagining of church history as historical theology, this article explores (...) the tension between the demise of church history as a subject in South Africa and the emerging understanding and application of historical theology, arguing that more can be made of trans-disciplinary dialogues. (shrink)

This book offers both theoretical overviews and practical approaches for educators, academics, education students and parents who are interested in transforming schools. It encourages reinvigorating approaches to learning and teaching that can easily be integrated into both public and private K-12 school classrooms, with many ideas also applicable to higher education. It supports an educational system based on the beliefs that heart and spirit are intertwined with mind and intellect, and that inner peace, wisdom, compassion, and conscience can be developed (...) together with academic content and skills. (shrink)

Over the last three years Hume’s use of the term “a priori” has suddenly become very topical. Three discussions, by Stephen Buckle, myself, and Houston Smit, all focusing on Hume’s argument concerning induction in Section IV of the Enquiry, have independently picked up on this question, which seems previously to have gone almost unnoticed.1 That there is an issue here can be seen by examining what Hume says when considering the foundation of our inferences concerning matter of fact; why, (...) for example, we expect a billiard ball to move in a particular way when struck by another: “I shall venture to affirm, as a general proposition, which admits of no exception, that the knowledge of [cause and effect] is not, in any instance, attained by reasonings a priori; but arises entirely from experience … Let an object be presented to a man of ever so strong natural reason and abilities; if that object be entirely new to him, he will not be able, by the most accurate examination of its sensible qualities, to discover any of its causes or effects. … No object ever discovers, by the qualities which appear to the senses, either the causes, which produced it, or the effects, which will arise from it; nor can our reason, unassisted by experience, ever draw any inference concerning real existence and matter of fact.” (E 27, 4.6) In this passage Hume is clearly allowing an object’s “qualities which appear to the senses” as being available to Reason “unassisted by experience”. So what he counts as “experience” seems essentially to involve some memory of prior experience, rather than merely current experience. (shrink)

ABSTRACT: So far in this book, we have examined algorithmic decision systems from three autonomy-based perspectives: in terms of what we owe autonomous agents (chapters 3 and 4), in terms of the conditions required for people to act autonomously (chapters 5 and 6), and in terms of the responsibilities of agents (chapter 7). -/- In this chapter we turn to the ways in which autonomy underwrites democratic governance. Political authority, which is to say the ability of a government to exercise (...) power, may be justifiable or not. Whether it is justified and how it can come to be justified is a question of political legitimacy. Political legitimacy is another way in which autonomy and responsibility are linked. This relationship is the basis of the current chapter, and it is important in understanding the moral salience of algorithmic systems. We will draw the connection as follows. We begin, in section 8.1, by describing two uses of technology: crime predicting technology used to drive policing practices and social media technology used to influence elections (including by Cambridge Analytica and by the Internet Research Agency). In section 8.2 we consider several views of legitimacy and argue for a hybrid version of normative legitimacy based on one recently offered by Fabienne Peter. In section 8.3 we will explain that the connection between political legitimacy and autonomy is that legitimacy is grounded in legitimating processes, which are in turn based on autonomy. Algorithmic systems—among them PredPol and the Cambridge Analytica-Facebook-Internet Research Agency amalgam—can hinder that legitimation process and conflict with democratic legitimacy, as we argue in section 8.4. We will conclude by returning to several cases that serve as through-lines to the book: Loomis, Wagner, and Houston Schools. -/- The link below is to an open-access copy of the chapter. (shrink)

In this systematic study, Professor Achinstein analyzes such concepts as definitions, theories, and models, and contrasts his view with currently held positions that he finds inadequate.

Al-Kindi was the first philosopher of the Islamic world. He lived in Iraq and studied in Baghdad, where he became attached to the caliphal court. In due course he would become an important figure at court: a tutor to the caliph's son, and a central figure in the translation movement of the ninth century, which rendered much of Greek philosophy, science, and medicine into Arabic. Al-Kindi's wide-ranging intellectual interests included not only philosophy but also music, astronomy, mathematics, and medicine. Through (...) deep engagement with Greek tradition al-Kindi developed original theories on key issues in the philosophy of religion, metaphysics, physical science, and ethics. He is especially known for his arguments against the world's eternity, and his innovative use of Greek ideas to explore the idea of God's unity and transcendence. Despite al-Kindi's historical and philosophical importance no book has presented a complete, in-depth look at his thought until now. In this accessible introduction to al-Kindi's works, Peter Adamson surveys what is known of his life and examines his method and his attitude towards the Greek tradition, as well as his subtle relationship with the Muslim intellectual culture of his day. Above all the book focuses on explaining and evaluating the ideas found in al-Kindi's wide-ranging philosophical corpus, including works devoted to science and mathematics. Throughout, Adamson writes in language that is both serious and engaging, academic and approachable. This book will be of interest to experts in the field, but it requires no knowledge of Greek or Arabic, and is also aimed at non-experts who are simply interested in one of the greatest of Islamic philosophers. (shrink)

In this book, Peter Achinstein proposes and defends several objective concepts of evidence. He then explores the question of whether a scientific method, such as that represented in the four "Rules for the Study of Natural Philosophy" that Isaac Newton invoked in proving his law of gravity, can be employed in demonstrating how the proposed definitions of evidence are to be applied to real scientific cases.

This volume brings together eleven essays by the distinguished philosopher of science, Peter Achinstein. The unifying theme is the nature of the philosophical problems surrounding the postulation of unobservable entities such as light waves, molecules, and electrons. How, if at all, is it possible to confirm scientific hypotheses about "unobservables"? Achinstein examines this question as it arose in actual scientific practice in three nineteenth-century episodes: the debate between particle and wave theorists of light, Maxwell's kinetic theory of gases, and (...) J.J. Thomson's discovery of the electron. The book contains three parts, each devoted to one of these topics, beginning with an essay presenting the historical background of the episode and an introduction to the philosophical issues. There is an illuminating evaluation of various scientific methodologies, including hypothetico-deductivism, inductivism, and the method of independent warrant which combines features of the first two. Achinstein assesses the philosophical validity of both nineteenth-century and modern answers to questions about unobservables, and presents and defends his own solutions. (shrink)

P.F. Strawson's essay traces some formal characteristics of logic and grammar to their roots in general features of thought and experience.

La traducción que al inglés hace el ilustre profesor norteamericano Harold Raley (cuya filosofía sigue a Ortega y a Marías) resulta excelente. El profesor Harold Cecil RALEY nació el año 1934 en el Estado de Alabama (USA), en cuya Universidad se doctoró en Lenguas y Literaturas Románicas, y quiso especializarse en la obra de José Ortega y Gasset. Casado y padre de varios hijos, ha sido catedrático de Lengua y Literatura españolas de la Universidad del Estado de Oklahoma desde 1964; (...) también catedrático y decano de Bellas Artes y Humanidades en la Houston Baptist University (nombrado por ella Distinguished Professor). Entre otros libros, es autor de los siguientes: "Ortega y Gasset, filósofo de la unidad europea" (Prólogo de Julián Marías. Revista de Occidente. Madrid, 1977; edición inglesa: "José Ortega y Gasset, the Philosopher of European Unity". The University of Alabama Press. USA, 1971); "La visión responsable. La filosofía de Julián Marías" (Prólogo de José Luis Pinillos. Espasa-Calpe. Madrid, 1977; edición inglesa: "Responsible Vision: The Philosophy of Julián Marías". The American Hispanist. Clear Creek, Indiana. USA, 1980); "Julián Marías. Una filosofía desde dentro" (Alianza Editorial. Madrid, 1997; edición inglesa: "A Watch over Mortality: the Philosophical Story of Julián Marías". The State University of New York Press. USA, 1997); "El espíritu de España" (Prólogo de Julián Marías. Alianza Editorial. Madrid, 2003; edición inglesa: "The Spirit of Spain". Halcyon Press. Houston. USA, 2002); "The Light of Eden. A Christian Worldview" (John M. Hardy Publishing. Houston. USA, 2008); "Language Oddities. Quirks and Curiosities of English and Other Languages" (Kingsle. Houston. USA, 2011); "Louisiana Rogue. The Life and Times of Pierre Prospère-Tourmoulin translated by Peter Tourmoulin" (Lamar University Press. Beaumont. USA, 2013); "The Unknown God: Mysteries of Deity, Time, Space, and Creation" (Charleston, SC, USA, 2011. También es autor de más de un centenar de estudios y artículos sobre temas filosóficos y literarios. Además ha traducido al inglés algunos de los libros de Julián Marías. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 497 Both theologians and philosophers need to see a completely integrated treatment of both rational and faith aspects of Aquinas's theology of creation. To this end, more work on theology as science also would be helpful. Emery's treatment of the end and subject of a science is not quite neoplatonic enough. His presentation of the subject of theology forces God, its subject in the Summa theologiae, on (...) earlier texts of Albert (49) and Aquinas (302). In I Sent. Aquinas says the subject is ens divinum, "being as related to the divine," which is much wider than God, who is the end but not subject oftheology. His model is Avicenna's ens commune as the subject of a metaphysics with God as its end. More work also must be done on the articles of faith, part of the subject of theology for Albert; but theology's proper principles for Aquinas. How primordial the articles are, and the interplay of reason and faith in knowing them, are still unresolved issues. In sum, one hopes this fine book by Fr. Emery will be the first of many helping give Thomism the new look it deserves. Centerfor Tlwmistic Studies Houston, TX R. E. HOUSER Relation als Vergleich: Die Relationstheorie des Johannes Buridan im Kontext seines Denkens und der Scholastik. By ROLF SCHONBERGER. Studien und Texte zur Geistesgeschichte des Mittelalters, Band 43. Leiden/New York/Koln: E. J. Brill, 1994. Pp. 489. $108.75 (cloth). This book is a revised version of the author's 1990 Habilitationsschrift at Ludwig-Maximilians-Universitat in Munich. It aims to articulate the theory of relations developed by the noted Parisian philosopher, John Buridan (ca. 1295-1358), "in connection with the extensive scholastic discussion of this problem," and, in particular, to show how the same arguments were advanced or refuted by different philosophers from "fundamentally different starting points" (ix, 59). This stems from the author's working hypothesis that the medieval debate on the nature of relations proceeded dialectically, and that Buridan's contribution to it can be better understood by taking its argumentative context into consideration. The scope of the book is wide-ranging, to say the least. Readers expecting a book about Buridan's theory of relations will find the title somewhat misleading, however, since background material and a survey of the "scholastic context" of Buridan's theory fill up the first 372 pages, with the author's presentation and discussion of the theory itself limited to the final 498 BOOK REVIEWS 75. Furthermore, the author seems to regard the relevance of the former discussion to the latter as self-evident. This is fine where the connections are fairly uncontroversial, as in the case of William of Ockham, but where they are less so, as in the cases 0f Giles of Rome, Henry of Ghent, Meister Eckhart, Durand of St. Pour~ain, and Peter John Olivi, one is left in the dark about why separate chapters are devoted to each. Giles and Olivi are not even mentioned in the chapters on Buridan. Henry and Durand make one appearance each, both in the course of the trivial observation that unlike in their theories, "consideration of the doctrine of the Trinity plays no role in Buridan's theory of relations" (399). Eckhart is mentioned a few times, but in terms so speculative as to be useless for the general reader. For example, the author finds Jan Pinborg's characterization of Buridan as a someone who "took the arbitrariness of language seriously" to be "of course... too microscopic " (393). The reason is that Pinborg allegedly failed to see the radicalization of conventionality in the fourteenth century in terms of an "epochal shift" from "the symbolic and quasi-naturalistic conception of language in the early Middle Ages," a conception that, the author parenthetically notes, "is perhaps still to be found (or further strengthened) in Bonaventure or Meister Eckhart" (ibid.). But even bracketing the question of how this description fits the latter authors, the Buridan story is more complex than this. Buridan does emphasize that spoken and written expressions signify conventionally, but he never abandons the Aristotelian/Boethian tripartite conception of language, according to which concepts are said to... (shrink)

Confirmation theorists seek to define a function that will take into account the various factors relevant in determining the degree to which an hypothesis is confirmed by its evidence. Among confirmation theorists, only Rudolf Carnap has constructed a system which purports to consider factors in addition to the number of instances, viz. the variety manifested by the instances and the amount of analogy between the instances. It is the purpose of this paper to examine the problem which these additional factors (...) raise for confirmation theory, and to prove that, despite Carnap's claim, no confirmation function satisfying the requirements he has specified can take account of variety and analogy. This result is first proved for a special case, and then, in a subsequent section, is generalized through the introduction of a theorem (the proof of which is given in Appendix I). In the final section of the paper it is shown that, contrary to a claim which Carnap has made, not even the concept of the "logical width" of a predicate will enable confirmation functions satisfying his requirements to take adequate account of analogies between instances. (shrink)

The latest in the series based on the popular History of Philosophy podcast, this volume presents the first full history of philosophy in the Islamic world for a broad readership. It takes an approach unprecedented among introductions to this subject, by providing full coverage of Jewish and Christian thinkers as well as Muslims, and by taking the story of philosophy from its beginnings in the world of early Islam all the way through to the twentieth century. Major figures like Avicenna, (...) Averroes, and Maimonides are covered in great detail, but the book also looks at less familiar thinkers, including women philosophers. Attention is also given to the philosophical relevance of Islamic theology (kalam) and mysticism--the Sufi tradition within Islam, and Kabbalah among Jews--and to science, with chapters on disciplines like optics and astronomy. The book is divided into three sections, with the first looking at the first blossoming of Islamic theology and responses to the Greek philosophical tradition in the world of Arabic learning. This 'formative period' culminates with the work of Avicenna, the pivotal figure to whom most later thinkers feel they must respond. The second part of the book discusses philosophy in Muslim Spain (Andalusia), where Jewish philosophers come to the fore, though this is also the setting for such thinkers as Averroes and Ibn Arabi. Finally, a third section looks in unusual detail at later developments, touching on philosophy in the Ottoman, Mughal, and Safavid empires and showing how thinkers in the nineteenth to the twentieth century were still concerned to respond to the ideas that had animated philosophy in the Islamic world for centuries, while also responding to political and intellectual challenges from the European colonial powers. (shrink)

SENSATIONALISM 1 1. Introductory 1 2. Mach's Sensationalism 4 3. Developments of Sensationalism 22 II. THE INHERENT WEAKNESS OF SEN- SATIONALISM 25 1. The Point of Sensationalism 25 2. The Ambiguity of 'Sensation' 27 3. The Fundamental Conflict 35 4. Mistakes, Incorrigibility and Simplicity 40 III. DESCRIPTION 51 1. Describing and Descriptions 51 2. Describing in Terms of Sensations 67 IV. THE POSSIBILITY OF 'PURE' DES- CRIPTIONS 79 V. SCIENTIFIC PROBLEMS 99 VI. DESCRIPTIONS AND EXPLANATIONS 111 BIBLIOGRAPHY 142 INDEX 145 (...) -/- -v- . (shrink)

The historian Ibn Khaldūn (d. 1406) is most often treated by historians of philosophy as part of the story of political philosophy in the Islamic world. While this is perfectly legitimate, it may be misleading when it comes to the question of the method he proposes for the historian. This paper argues that that method is in fact based on a different branch of (Aristotelian) science: natural philosophy. After rendering this proposition initially plausible by noting frequent references to "nature" in (...) the _Muqaddima_, the paper discusses a number of parallels between natural philosophy and history as Ibn Khaldūn conceives it. Ultimately it is suggested that this method is legitimated insofar as history is for Ibn Khaldūn a study of human nature. (shrink)

Recently in this journal Sherrilyn Roush (2004) defends positive relevance as a necessary (albeit not a sufficient) condition for evidence by rejecting two of the counterexamples from my earlier (2001) work. In this reply I argue that Roush's critique is not successful.

One can rightly say of Peter Sloterdijk that each of his essays and lectures is also an unwritten book. That is why the texts presented here, which sketch a philosophical physiognomy of Martin Heidegger, should also be characterized as a collected renunciation of exhaustiveness. In order to situate Heidegger's thought in the history of ideas and problems, Peter Sloterdijk approaches Heidegger's work with questions such as: If Western philosophy emerged from the spirit of the polis, what are we (...) to make of the philosophical suitability of a man who never made a secret of his stubborn attachment to rural life? Is there a provincial truth of which the cosmopolitan city knows nothing? Is there a truth in country roads and cabins that would be able to undermine the universities with their standardized languages and globally influential discourses? From where does this odd professor speak, when from his professorial chair in Freiburg he claims to inquire into what lies beyond the history of Western metaphysics? Sloterdijk also considers several other crucial twentieth-century thinkers who provide some needed contrast for the philosophical physiognomy of Martin Heidegger. A consideration of Niklas Luhmann as a kind of contemporary version of the Devil's Advocate, a provocative critical interpretation of Theodor Adorno's philosophy that focuses on its theological underpinnings and which also includes reflections on the philosophical significance of hyperbole, and a short sketch of the pessimistic thought of Emil Cioran all round out and deepen Sloterdijk's attempts to think with, against, and beyond Heidegger. Finally, in essays such as "Domestication of Being" and the "Rules for the Human Park," which incited an international controversy around the time of its publication and has been translated afresh for this volume, Sloterdijk develops some of his most intriguing and important ideas on anthropogenesis, humanism, technology, and genetic engineering. (shrink)

Dieser Band versammelt die wichtigsten Arbeiten, in denen Peter Stemmer die kontraktualistische Moralkonzeption aus Handeln zugunsten anderer weiterentwickelt. Im Zentrum steht die Frage, anhand welcher Kriterien eine Moral, wie sie sein sollte, von falschen oder schlechten Moralen unterschieden werden kann. Besondere Aufmerksamkeit gilt der Rechtfertigung und ihrer Beziehung zum Unterdrückungsverbot. Behandelt wird auch der Zusammenhang von Wollen, praktischen Gründen und Normativität.

In this note a T1 formal space is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a formal space, and prove that the class of points of a weakly set-presentable formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties for constructive topological spaces , strengthening separation properties discussed elsewhere. Finally we relate (...) the properties for ct-spaces with corresponding properties of formal spaces. (shrink)

This article offers an analysis of the argumentative method of two treatises by Alexander of Aphrodisias, On Fate and On Providence, the latter of which is preserved only in Arabic translation. It is argued that both texts use techniques from Aristotelian dialectic, albeit in different ways, with On Fate adhering to methods outlined in Aristotle's Topics whereas On Providence uses the ‘aporetic’ method familiar from texts such as MetaphysicsΒ‎. This represents a revision of a previous study of Alexander's method in (...) On Fate by Jaap Mansfeld, which emphasized parallels between that method and the techniques of ancient scepticism. It is, however, suggested that Alexander does reflect developments in epistemology during the Hellenistic period, especially in so far as he ‘upgrades’ the status of endoxa to play something like the role of common conceptions in the dogmatic Hellenistic schools. (shrink)

At once provocative and inspiring_, Against the Flow_ is a work of polemic from an internationally respected writer and thinker on arts education. Peter Abbs argues that contemporary education ignores the aesthetic and ethical as a result of being in thrall to such forces as the market economy and managerial and functional dictates. He identifies the present education system as being inimical to creativity and authentic learning and instead, narrowly focused on the quantitative measuring of results. This absence of (...) a creative and ethical dimension in education has implications for art making in wider society. Art is shown as emerging from, and appealing to, the ironic postmodernist sensibility and mass media-led culture, while being devoid of philosophical significance. This book opens up a fresh and timely debate about the vital power of creativity in modern education. Drawing on examples from modern poetry, literature and visual art, it is an eloquent and passionate argument for the need to develop ethical and aesthetic energies to confront the growing vacuity of contemporary culture. (shrink)

The first book in the Great Medieval Thinkers series to focus on an Islamic philosopher. It offers a brief, accessible introduction to the thought of the philosopher al -Kindi. His works, though brief, are of great historical importance. Al-Kindi was the first philosopher of the Islamic world. Peter Adamson will survey what is known of al-Kindi's life, examine his thought on a wide range of topics, and consider the relationship of al-Kindi's work to his Greek sources.

In this short book Peter Sloterdijk offers a genealogy of the concept of freedom from Ancient Greece to the present day. This genealogy is part of a broader theory of the large political body, according to which Sloterdijk argues that political communities arise in response to a form of anxiety or stress. Through a highly original reading of Rousseaus late Reveries of a Solitary Walker, Sloterdijk shows that, for Rousseau, the modern subject emerges as a subject free of all (...) stress, unburdened by the cares of the world. Most of modern philosophy, and above all German Idealism, is an attempt to reign back Rousseaus useless and anarchical subject and anchor it in the cares of the world, in the task of having to produce both the world and itself. In the light of this highly original account, Sloterdijk develops his own distinctive account of freedom, where freedom is conceptualized as the availability for the improbable. This important text, in which Sloterdijk develops his account of freedom and the modern subject, will be of great interest to students and scholars in philosophy and the humanities and to anyone interested in contemporary philosophy and critical theory. (shrink)

London-based company ScanLab Projects is a multi-disciplinary commercial collaboration between architect, artist, coders and designers who utilize technologies surrounding 3D laser scanning in their practice. Inherent in the manner their projects are pitched is through reference to the photographic as technological process. Central to their engagement with the light detection and ranging (LiDAR) scanning apparatus is a consideration of the relationality between virtual or digital object and what could be determined as extrinsic or ‘real’ terrain. In Post-lenticular Landscapes, 2017, ScanLab (...) created a series of LiDAR scans of Yosemite National Park. The landscape, presented as a stereoscopic film work where the spectator flies through an ephemeral black and white point cloud, is contextualized relationally to a certain photo-historical context and lineage: As Yosemite is synonymous with the advent of photographic process through the work of Muybridge, Watkins, Woods and Adams, the work revisits an archetypal image of the American sublime. In this text, I unpack ScanLab’s usage and conceptualization of LiDAR scans referentially to an understanding of the photographic. By considering how ScanLab frames our understanding of the project through the technological apparatus, the text attempts to problematize the scans by reading them through a particular art historic heritage. In this context, I posit alternate ways of reading the work, specifically through reference to the image of Yosemite that has proliferated across the desktops of Apple computers since 2014. Furthermore, through a reading of metonymy in the writing of Eelco Runia, the eerie as understood by Mark Fisher and in relation to Lev Manovich’s description of photorealism, I propose that the future of the mediation and understandings of machinic vision are to be thought with a reconsidered notion of the photographic. (shrink)

This paper examines problems of order and periodicity which arise when the attempt is made to define a confirmation function for a language containing elementary number theory as applied to a universe in which the individuals are considered to be arranged in some fixed order. Certain plausible conditions of adequacy are stated for such a confirmation function. By the construction of certain types of predicates, it is proved, however, that these conditions of adequacy are violated by any confirmation function defined (...) for the type of language in question. Various possible solutions to these difficulties are explored and found to be inadequate. In particular, a proposal which stems from the suggestion to restrict a fundamental principle of confirmation to hypotheses containing only non-positional predicates is cited. This proposal, however, is shown to prevent confirmation functions from taking periodicities into account, and so is deemed unsatisfactory. A general theorem is proved to the effect that if non-positional predicates are taken to satisfy the conditions of adequacy which have been formulated, then no periodicity predicates whatsoever (i.e., predicates used in formulating hypotheses which foretell periodicities) can be subject to these conditions, on pain of contradiction. Yet it seems that periodicity predicates must be subject to these conditions of adequacy if a confirmation function is to recognize periodic occurrences. Thus, an impasse seems to be reached. In the final sections we consider the beginnings of one possible solution to these difficulties. Our proposal involves treating sets of individuals, rather than individuals themselves, as instances of an hypothesis which predicts a periodicity. On this basis we formulate new conditions of adequacy which are free from the previous difficulties and which will permit a confirmation function that satisfies them to take periodicities into account. (shrink)