Familial clustering of a disease is defined as the occurrence of the disease within some families in excess of what would be expected from the occurrence in the population. It has been demonstrated for several cancer types, ranging from rare cancers as the adenomatosis-coli-associated colon cancer or the Li-Fraumeni syndrome to more common cancers as breast cancer and colon cancer. Familial clustering, however, is merely an epidemiological pattern, and it does not tell whether genetic or environmental causes or both in (...) combination are responsible for the familial clustering. Familial clustering may be due to genetic predisposition to the disease, but exposure to environmental factors — shared by members of some families, but not by members of other families — may also cause familial clustering and hence mimic genetic inheritance in the study of nuclear families. Based on assumptions regarding the individual steps in the biological process starting with exposure to carcinogens and ending with death from disseminated cancer we suggest that genetic and environmental factors may both be involved in most of these steps. The present paper focuses on research methodologies necessary to discriminate between the effect of genes and family environment in the development of cancer. (shrink)

SummaryA health facility-based approach to delivering anthelminthic drugs to children aged 12–59 months in Zambia was compared with an approach where community-directed treatment was added to the HF approach. This paper reports on the socio-demographic factors associated with treatment coverage in the HF+ComDT and HF areas after 18 months of implementation. Data were collected by interviewing 288 and 378 caretakers of children aged 12–59 months in the HF+ComDT and HF areas, respectively. Bivariate and multivariate logistic regression analyses were used for (...) data analysis. Statistically significant predictors of a child being treated were: a child coming from the HF+ComDT area, being 12–36 months old, the family having lived in the area for >20 years, coming from a household with only one under-five child and living ≤3 km from the health facility. It is concluded that socio-demographic factors are of public health relevance and affect treatment coverage in both the HF+ComDT and the HF approaches. The implementation and strengthening of interventions like ComDT that bring treatment closer to households will enable more children to have access to treatment. (shrink)

For over a century, the Danish thinker Søren Kierkegaard has been at the center of a number of important discussions, concerning not only philosophy and theology, but also, more recently, fields such as social thought, psychology, and contemporary aesthetics, especially literary theory. Despite his relatively short life, Kierkegaard was an extraordinarily prolific writer, as attested to by the 26-volume Princeton University Press edition of all of his published writings. But Kierkegaard left behind nearly as much unpublished writing, most of which (...) consists of what are called his "journals and notebooks." Kierkegaard has long been recognized as one of history's great journal keepers, but only rather small portions of his journals and notebooks are what we usually understand by the term "diaries." By far the greater part of Kierkegaard's journals and notebooks consists of reflections on a myriad of subjects--philosophical, religious, political, personal. Studying his journals and notebooks takes us into his workshop, where we can see his entire universe of thought. We can witness the genesis of his published works, to be sure--but we can also see whole galaxies of concepts, new insights, and fragments, large and small, of partially completed but unpublished works. Kierkegaard's Journals and Notebooks enables us to see the thinker in dialogue with his times and with himself. Volume 6 of this 11-volume series includes four of Kierkegaard's important "NB" journals, covering the months from early May 1849 to the beginning of 1850. At this time Denmark was coming to terms with the 1848 revolution that had replaced absolutism with popular sovereignty, while the war with the German states continued, and the country pondered exactly what replacing the old State Church with the Danish People's Church would mean. In these journals Kierkegaard reflects at length on political and, especially, on ecclesiastical developments. His brooding over the ongoing effects of his fight with the satirical journal Corsair continues, and he also examines and re-examines the broader personal and religious significance of his broken engagement with Regine Olsen. These journals also contain reflections by Kierkegaard on a number of his most important works, including the two works written under his "new" pseudonym Anti-Climacus and his various attempts at autobiographical explanations of his work. And, all the while, the drumbeat of his radical critique of "Christendom" continues and escalates. Kierkegaard wrote his journals in a two-column format, one for his initial entries and the second for the extensive marginal comments that he added later. This edition of the journals reproduces this format, includes several photographs of original manuscript pages, and contains extensive scholarly commentary on the various entries and on the history of the manuscripts being reproduced. (shrink)

"The day will come when not only my writings, but precisely my life--the intriguing secret of all the machinery--will be studied and studied." Søren Kierkegaard's remarkable combination of genius and peculiarity made this a fair if arrogant prediction. But Kierkegaard's life has been notoriously hard to study, so complex was the web of fact and fiction in his work. Joakim Garff's biography of Kierkegaard is thus a landmark achievement. A seamless blend of history, philosophy, and psychological insight, all conveyed with (...) novelistic verve, this is the most comprehensive and penetrating account yet written of the life and works of the enigmatic Dane who changed the course of intellectual history. Garff portrays Kierkegaard not as the all-controlling impresario behind some of the most important works of modern philosophy and religious thought--books credited with founding existentialism and prefiguring postmodernism--but rather as a man whose writings came to control him. Kierkegaard saw himself as a vessel for his writings, a tool in the hand of God, and eventually as a martyr singled out to call for the end of "Christendom." Garff explores the events and relationships that formed Kierkegaard, including his guilt-ridden relationship with his father, his rivalry with his brother, and his famously tortured relationship with his fiancée Regine Olsen. He recreates the squalor and splendor of Golden Age Copenhagen and the intellectual milieu in which Kierkegaard found himself increasingly embattled and mercilessly caricatured. Acclaimed as a major cultural event on its publication in Denmark in 2000, this book, here presented in an exceptionally crisp and elegant translation, will be the definitive account of Kierkegaard's life for years to come. (shrink)

Søren Kierkegaard published an extraordinary number of works during his lifetime, but he left behind nearly as much unpublished writing, most of which consists of what are called his "journals and notebooks." Volume 3 of this 11-volume edition of Kierkegaard's Journals and Notebooks includes Kierkegaard's extensive notes on lectures by the Danish theologian H. N. Clausen and by the German philosopher Schelling, as well as a great many other entries on philosophical, theological, and literary topics. In addition, the volume includes (...) many personal reflections by Kierkegaard, notably those in which he provides an account of his love affair with Regine Olsen, his onetime fiancée. (shrink)

A work that "not only treats of irony but is irony," wrote a contemporary reviewer of The Concept of Irony, with Continual Reference to Socrates. Presented here with Kierkegaard's notes of the celebrated Berlin lectures on "positive philosophy" by F.W.J. Schelling, the book is a seedbed of Kierkegaard's subsequent work, both stylistically and thematically. Part One concentrates on Socrates, the master ironist, as interpreted by Xenophon, Plato, and Aristophanes, with a word on Hegel and Hegelian categories. Part Two is a (...) more synoptic discussion of the concept of irony in Kierkegaard's categories, with examples from other philosophers and with particular attention given to A. W. Schlegel's novel Lucinde as an epitome of romantic irony.The Concept of Irony and the Notes of Schelling's Berlin Lectures belong to the momentous year 1841, which included not only the completion of Kierkegaard's university work and his sojourn in Berlin, but also the end of his engagement to Regine Olsen and the initial writing of Either/Or. (shrink)

With this first part of the eleventh volume, Bruce Kirmmse et al.’s monumental task of translating Søren Kierkegaard’s journals and notebooks begins to draw to a close. The journals and notebooks t...

In lieu of an abstract, here is a brief excerpt of the content:Alan Collett LITERATURE, CRITICISM, AND FACTUAL REPORTING Novels frequently deal with real events. How is it that some theorists have been able to argue that, regarded as literature, such novels are always fictional? The answer is that it is usually possible to show that a work which we are prepared to call "literary" creates an imaginary world possessing its own properties. Itcan then be maintained that this imaginary world (...) constitutes the aesthetic object of the work and that only those properties internal to it are relevant to aesthetic judgment. It then follows that the meaning ofthe literary work is this imaginary world and that criticism is properly concerned with its elucidation. Or, to say the same thing in a different idiom: we should distinguish in the things said by critics between those sorts of remarks which help to establish a text's meaning solely by reference to internal features of the text and those sorts of remarks which, if they are aimed at establishing meaning, do so by reference to external evidence. In the case of novels dealing with real events, those sentences which describe particular events, situations, characters, and places (reports)1 play a role in the creation of the imaginary world of the work. The real events described will, of course, often be known to the reader, and indeed the effectiveness of a work may depend to a large extent upon the fact that it deals with known historical events. But in order to avoid treating such texts as factual works rather than as works of literature we should, the theory goes, stress plausibility and coherence above truth to the world. Aristode's rule that "a convincing impossibility is preferable to an unconvincing possibility" sums up the way in which critical assessment can guard against confusing literary 282 Alan Collett283 works with informative works. Literary criticism concerns itselfwith the plausibility and coherence of accounts of real events at the expense of their truth. A literary artist will, therefore, be judged to be better the more his descriptions seem true in contrast with the historian whose descriptions must actually be true. The literary artist, on this traditional view, endows the events he describes with meaning. When those events are real events their meaning will be intelligible to the extent to which they reflect a convincingly possible reality. The accurate reporting of real events is perfecdy permissible always provided the events chosen make an artistic point by functioning in the narrative in a way which is probable or convincing. Accuracy, as such, is never a criterion of literary worth. The fictionality of literary works dealing with real events, then, is connected with the skill the author exhibits in usingthe events he chooses to describe in such a way that a meaningful imaginary world is created. This fictional world will mirror the real world via its internal coherence which excludes the contingendy true in order to make a plausible artistic whole, and one function of literary criticism proper is the fixing of the correct interpretation of the meaning of this fictional world. A recent reformulation ofthis traditional view, offered by S. H. Olsen in TL· Structure ofLiterary Understanding? contains some suggestions as to how it is possible to interpret literary practice so that report sentences can be understood to function other than as the bearers of truth. On this view, a literary work, as a peculiar kind of utterance, cannot be interpreted without reference to some context. And although a literary work may be pardy distinguished by its textual or structural features, the mere possession of such features is not in itself sufficient to make a text a literary work. Textual or structural features, Olsen argues, will serve an aesthetic function only when the decision is already taken that the text to which they belong is a literary work. Since the function of sentences cannot be determined simply on the basis of their linguistic features, they will need to be interpreted in order to be correctly understood. Interpretation will involve making assumptions about the purpose of the utterances in question and the assumed purpose will then determine how their linguistic and structural features are to be understood. Thus, in one of... (shrink)

Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component (...) systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category. (shrink)

This collection introduces readers to some of the most respected Pre-Socratic scholarship of the twentieth century. It includes translations of important works from European scholars that were previously unavailable in English and incorporates the major topics and approaches of contemporary scholarship. Here is an essential book for students and scholars alike. "Students of the Pre-Socratics must be grateful to Mourelatos and his publishers for making these essays available to a wider public."--T. H. Irwin, American Journal of Philology "Mourelatos is a (...) superb editor, and teaching Pre-Socratics in the future with this collection on the reading list will not only be easier but also better."--Jorgen Mejer, The Classical World "The editor has done his work judiciously. It would be difficult to devise a better balance between different parts of the subject."--Edward Hussey, Archives internationales d'histoire des sciences "[This book] will undoubtedly become an indispensable aid for beginning and advanced students of the Pre-Socratics."--David E. Hahm, Isis. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:184 BOOK REVIEWS knower, one may avoid undercutting the position that the cognitive powers are passive, without failing to do justice to the fact that aware· ness and discrimination are activities of the knower {pp. 71-72; 148· 49, n. 6). Second, Kai holds that the individual human being cannot really he said to have intuitive mind in himself: "Man has mind; hut only to a certain degree and without (...) mind losing its independence from the human whole.... the divine mind is present in man without los· ing its character of divine mind " (p. 75). If only God has intuitive mind {pp. 155, n. 32; 156-57, n. 13), how can the individual human being really know, even in the sense of having discursive thought? Since knowledge depends on first principles, what does not possess the intuitive mind which knows the first principles would, it seems, not properly have knowledge. The only hope for a definitive interpretation of the disconcertingly brief De Anima, 3. 5, lies in the possibility that one can establish a definite connection between this chapter and other elements in the Aris· totelian corpus, a connection that would elucidate the otherwise in· scrutable teaching on active mind. It is the great merit of Kai to have made an excellent case for a connection between De Anima, 3. 5, and two other elements in the Aristotelian corpus: nous as it functions in the logic and epistemology, and the unmoved mover of Metaphysics, Lambda. Exploring this connection in a penetrating and thorough analysis of a great body of Aristotelian and secondary te~ts, Kai has not merely taken a position on a more than two-thousand-year-old con· troversy, hut also has decisively advanced our understanding of active mind. St. John's University Jamaica, New York JAMES T. H. MARTIN Anselm Studies: An Occasional Journal. Vol. 2. Edited by JosEPH ScHNAUBELT, OSA, et al., White Plains, New York: Kraus Inter· national Publications, 1988. Pp. 634. $55.00. This volume contains the proceedings of the Fifth International Saint Anselm Conference (Villanova University, September 16-21, 1985). Held in conjunction with the Tenth International Patristic, Medieval, and Renaissance Conference, the meeting occasioned papers concen· trating on areas of "influence and juncture" in the life and thought of St. Anselm and St. Augustine. Given the dependence of much of Anselm's work on the thought of St. Augustine, the Conference took as its particular objective the understanding of the relationship between BOOK REVIEWS 185 faith and reason. While the focus of the papers in this volume includes both saints, the principal emphasis falls on Anselm as inheritor and interpretor of the methods and ideas of his intellectual ancestor. A brief introduction provides cameo presentations of Anselm's character as abbot, monk, and archbishop, together with a conspectus on Anselm's position vis-a-vis the canonical and ecclesiological issues of his day. The papers are then divided into five chapters: "Anselm's Monasticism." "The Proslogion Argument," "Truth: Anselm and Augustine," "Theological Method," and "The Anselmian Worldview." While the papers are of use to scholarly readers desiring detailed treatment of central issues in Anselm's logic and epistemology, the un· doubted strength of the collection lies in its well-calculated effort to integrate an understanding of Anselm's intellectual and spiritual out· look with the theological, spiritual, and literacy currents of his time. Especially for the historian of intellect and society, the essays in this volume provide analysis that advances our understanding of the cultural context of the 11th and 12th centuries. The contribution to our understanding of Anselm's culture is par· ticularly clear in the first and fourth divisions of the volume, the sec· tions dealing with monasticism and Anselm's worldview. In the former division, Glenn Olsen's assessment of " Anselm and Homosexuality" neatly evaluates the evidence for Anselm's sexual orientation made controversial by John Boswell's Christianity, Social Tolerance, and Homosexuality. In fact, the present article, together with its notes, provides a useful summary of the critical literature which has grown up of late around the Boswellian thesis as well as the study of literary expressions of affection in the 11th and 12th centuries. Olsen grasps the... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Reading Aristotle with Thomas Aquinas: His Commentaries on Aristotle’s Major Works by Leo J. EldersEfrem Jindráček O.P.Reading Aristotle with Thomas Aquinas: His Commentaries on Aristotle’s Major Works. By Leo J. Elders. Edited by JÖrgen Vijgen. Washington, D.C.: The Catholic University of America Press, 2023. Pp. xi + 560. $75.00 (hardcover). ISBN: 978-0-8132-3579-0.The prolific Thomistic scholar Jörgen Vijgen has edited a new book by the well-known and recently deceased (...) Thomistic Dutch philosopher, university [End Page 718] professor, and religious priest Leo J. Elders, S.V.D. (1926–2019). According to the bookplate, the book is based on the French original Aristote et Thomas d’Aquin (Paris, 2018), but according to the editor’s preface it is the original English version: “while originally written in English, a French translation appeared already in 2018” (vii). In any case, this is a very precise and interesting panoramic presentation of all twelve of Thomas’s commentaries on Aristotle’s books. The presentations are arranged not according to the chronological sequence of Thomas’s writing, but rather according to the order in which we usually arrange the treatises of Aristotle (i.e., from On Interpretation to Politics). The text is based on Elders’s lecturing activities in Houston in the 1980s, which he himself revised and added to towards the end of his life. It is also obviously a thematic follow-up to several of his articles continuously published on the subject (“Saint Thomas d’Aquin et Aristote,” Revue Thomiste 88 [1988]: 357–76; “St. Thomas Aquinas’s Commentary of Aristoteles’s Physics,” Doctor Angelicus 7 [2007]: 17–51; “The Aristotelian Commentaries of St. Thomas Aquinas,” The Review of Metaphysics 63 [2009]: 29–53).The author has, of course, chosen the best existing Latin editions of Thomas’s texts (Leonine and Marietti), but for the sake of easier editing and citation he has kept the numbering of the paragraphs according to the Marietti edition. Each chapter devoted to one of Thomas’s commentaries has almost the same structure (the presentation of Aristotelian text, Thomas’s intention and commentary according to uniform books, concluding remarks). The book concludes with an extremely comprehensive bibliography (511–25), index of names (527–31) and a subject index (533–36).Elders articulates the basic philosophical question that drives this work in the preface: “if they [Thomas’s commentaries] contained Thomas’s own philosophy and if they were a reliable presentation of Aristotle’s thought, or, on the contrary, if they deformed it in some place by introducing some of Thomas’s Christian faith and theology” (ix). After the analysis of Thomas’s twelve commentaries, Elders arrived at the conclusion that “the commentaries are a masterful and faithful presentation of Aristotle’s thought and of that of Thomas himself. Thomas’s Christian faith does not falsify or alternate the text, but give occasionally an outlook at what lies behind philosophical thought” (ibid.).In the Introduction (1–13), Elders discusses the objections of other contemporary scholars (É. Gilson, J. Owens, M. D. Jordan, H. V. Jaffa) who have reached different positions, even with the help of texts that are, on the contrary, close to the author’s view (D.-M. Chenu, C. Kaczor, F. del Punta). And not only that: “In order to trace the relatively few texts where Thomas disagrees with certain doctrine of Aristotle or completes certain statements, my method is to go through each chapter of the different commentaries and to note where Thomas makes observations and advances a different explanation. In order to do so, it seemed best to present a summary of the contents of the different chapters of each treatise, noting those places where Thomas makes [End Page 719] corrections and additions, or brings in sentences expressing truths of the Christian faith” (13).In order to examine Elders’s analytical reading of Thomas’s commentaries, we will consider three representative chapters: namely, his treatment of Thomas’s commentaries on Peri hermeneias (14–58), Meteorologica (205–32), and Metaphysics (311–400). These three commentaries are unequal in several respects: not all were completed by Thomas himself, and not all... (shrink)

For over a century, the Danish thinker Søren Kierkegaard (1813-55) has been at the center of a number of important discussions, concerning not only philosophy and theology, but also, more recently, fields such as social thought, psychology, and contemporary aesthetics, especially literary theory. Despite his relatively short life, Kierkegaard was an extraordinarily prolific writer, as attested to by the 26-volume Princeton University Press edition of all of his published writings. But Kierkegaard left behind nearly as much unpublished writing, most of (...) which consists of what are called his "journals and notebooks." Kierkegaard has long been recognized as one of history's great journal keepers, but only rather small portions of his journals and notebooks are what we usually understand by the term "diaries." By far the greater part of Kierkegaard's journals and notebooks consists of reflections on a myriad of subjects--philosophical, religious, political, personal. Studying his journals and notebooks takes us into his workshop, where we can see his entire universe of thought. We can witness the genesis of his published works, to be sure--but we can also see whole galaxies of concepts, new insights, and fragments, large and small, of partially (or almost entirely) completed but unpublished works. Kierkegaard's Journals and Notebooks enables us to see the thinker in dialogue with his times and with himself. Volume 7 of this 11-volume series includes six of Kierkegaard's important "NB" journals (Journals NB15 through NB20), covering the months from early January 1850 to mid-September of that year. By this time it had become clear that popular sovereignty, ushered in by the revolution of 1848 and ratified by the Danish constitution of 1849, had come to stay, and Kierkegaard now intensified his criticism of the notion that everything, even matters involving the human soul, could be decided by "balloting." He also continued to direct his barbs at the established Danish Church and its clergy (particularly Bishop J. P. Mynster and Professor H. L. Martensen), at the press, and at the attempt by modern philosophy to comprehend the incomprehensibility of faith. Kierkegaard's reading notes include entries on Augustine, the Stoics, German mystics, Luther, pietist authors, and Rousseau, while his autobiographical reflections circle around the question of which, if any, of several essays explaining his life and works he ought to publish. Perhaps unsurprisingly, Kierkegaard's more personal reflections return once again to his public feud with M. A. Goldschmidt and his broken engagement to Regine Olsen. Kierkegaard wrote his journals in a two-column format, one for his initial entries and the second for the extensive marginal comments that he added later. This edition of the journals reproduces this format, includes several photographs of original manuscript pages, and contains extensive scholarly commentary on the various entries and on the history of the manuscripts being reproduced. (shrink)

Originally published in 1936, this book presents the ancient Greek text of the paraphrases and quotations of Zeno's philosophical arguments, together with a facing-page English translation and editorial commentary. Detailed notes are incorporated throughout and a bibliography is also included. This book will be of value to anyone with an interest in Zeno and ancient philosophy.

This article is the text of an interview with D. H. Mellor conducted by Andrew Pyle and first published in the Spring 1993 issue of the philosophical journal Cogito.

By the death, last summer, of Jack Robson, the world of utilitarian studies and a wider world of scholarship on both sides of the Atlantic lost one of their most distinguished figures. It would not be appropriate here, even if it were possible now, to attempt a full and measured assessment of his work. Writing only a few months after the news of his death, while the sense of loss is still so sharp for all his many friends, two things (...) are possible. Something can and should be said to acknowledge and celebrate Robson's achievement as a scholar; and to this can be added some personal recollections of one whose human qualities were as outstanding as his scholarship. (shrink)

Søren Kierkegaard published an extraordinary number of works during his lifetime, but he left behind nearly as much unpublished writing, most of which consists of what are called his "journals and notebooks." Volume 3 of this 11-volume edition of Kierkegaard's Journals and Notebooks includes Kierkegaard's extensive notes on lectures by the Danish theologian H. N. Clausen and by the German philosopher Schelling, as well as a great many other entries on philosophical, theological, and literary topics. In addition, the volume includes (...) many personal reflections by Kierkegaard, notably those in which he provides an account of his love affair with Regine Olsen, his onetime fiancée. (shrink)

This new edition emphasizes Emerson's philosophy and thoughts on such issues as freedom and fate; creativity and established culture; faith, experience, and evidence; the individual, God, and the world; unity and dualism; moral law, grace, and compensation; and wealth and success. Emerson's text has been fully annotated to explain difficult words and to clarify his references. The Introduction, Notes, Bibliography, Index, and Chronology of Emerson's life help the reader understand his distinctive outlook, his contributions to philosophy, and his place in (...) American culture and society. (shrink)

Selected response items and constructed response items are often found in the same test. Conventional psychometric models for these two types of items typically focus on using the scores for correctness of the responses. Recent research suggests, however, that more information may be available from the CR items than just scores for correctness. In this study, we describe an approach in which a statistical topic model along with a diagnostic classification model was applied to a mixed item format formative test (...) of English and Language Arts. The DCM was used to estimate students’ mastery status of reading skills. These mastery statuses were then included in a topic model as covariates to predict students’ use of each of the latent topics in their written answers to a CR item. This approach enabled investigation of the effects of mastery status of reading skills on writing patterns. Results indicated that one of the skills, Integration of Knowledge and Ideas, helped detect and explain students’ writing patterns with respect to students’ use of individual topics. (shrink)

The Peanese convention for the use of dots as brackets has the disadvantage that it gives only an awkward method for representing chains of indefinite length, such as the compound implicationSuch chains occur frequently in logical investigations of a metatheoretic nature, and it is convenient to have a systematic method of abbreviating them. The most obvious method of doing this would be to leave the parentheses out entirely, and to understand that in such cases the implication sign or other operation (...) appearing on the extreme left is the most inclusive; but this method, which has been followed by Quine, is at variance with our ordinary algebraic usage, in which we write a − b + c − d for ((a − b) + c) − d, and consequently leads to confusion. It is desirable to have a modification of the Peanese convention which gives a simple method of representing such chain implications, and at the same time avoids this difficulty.Such a modification is easily obtained by generalizing somewhat a procedure of Church. The latter author uses only a single dot, which he writes on the right of an operator to signify a bracket extending from that point to the end of the formula (or parenthesized expression). If dots are used only on the right of operations this is all that can be desired. But the essential idea of this device can be extended to the case where dots are used also on the left, as follows: let us suppose that a group of dots on the right of an operation or prefix denotes the beginning of a bracket which extends to the right until it meets a group with an equal or larger number of dots on the left of an operation; and that the scope of a group of dots on the left of an operation shall extend to the left until it reaches a larger group of dots on the right of some operation. (shrink)

The Greenberger-Horne-Zeilinger state is the most famous example of a state with multiparticle entanglement. In this article we describe a group theoretic framework we have been developing for understanding the entanglement in general states of two or more quantum particles. As far as entanglement is concerned, two states of n spin-1/2 particles are equivalent if they are on the same orbit of the group of local rotations (U(2)n). We consider both pure and mixed states and calculate the number of independent (...) parameters needed to describe such states up to this equivalence. We describe how the entanglement of states in a given equivalence class may be characterized by the stability group of the action of the group of local rotations on any of the states in the class. We also show how to calculate invariants under the group of local actions for both pure and mixed states. In the case of mixed states we are able to explicitly exhibit sets of invariants which allow one to determine whether two generic mixed states are equivalent up to local unitary transformations. (shrink)