Logic as the Universal Science : Bertrand Russell's Early Conception of Logic and Its Philosophical Context

Bertrand Russell (1872 1970) introduced the English-speaking philosophical world to modern, mathematical logic and foundational study of mathematics. The present study concerns the conception of logic that underlies his early logicist philosophy of mathematics, formulated in The Principles of Mathematics (1903). In 1967, Jean van Heijenoort published a paper, Logic as Language and Logic as Calculus, in which he argued that the early development of modern logic (roughly the period 1879 1930) can be understood, when considered in the light of a distinction between two essentially different perspectives on logic. According to the view of logic as language, logic constitutes the general framework for all rational discourse, or meaningful use of language, whereas the conception of logic as calculus regards logic more as a symbolism which is subject to reinterpretation. The calculus-view paves the way for systematic metatheory, where logic itself becomes a subject of mathematical study (model-theory). Several scholars have interpreted Russell s views on logic with the help of the interpretative tool introduced by van Heijenoort,. They have commonly argued that Russell s is a clear-cut case of the view of logic as language. In the present study a detailed reconstruction of the view and its implications is provided, and it is argued that the interpretation is seriously misleading as to what he really thought about logic. I argue that Russell s conception is best understood by setting it in its proper philosophical context. This is constituted by Immanuel Kant s theory of mathematics. Kant had argued that purely conceptual thought basically, the logical forms recognised in Aristotelian logic cannot capture the content of mathematical judgments and reasonings. Mathematical cognition is not grounded in logic but in space and time as the pure forms of intuition. As against this view, Russell argued that once logic is developed into a proper tool which can be applied to mathematical theories, Kant s views turn out to be completely wrong. In the present work the view is defended that Russell s logicist philosophy of mathematics, or the view that mathematics is really only logic, is based on what I term the Bolzanian account of logic . According to this conception, (i) the distinction between form and content is not explanatory in logic; (ii) the propositions of logic have genuine content; (iii) this content is conferred upon them by special entities, logical constants . The Bolzanian account, it is argued, is both historically important and throws genuine light on Russell s conception of logic.

Beatific Enjoyment in Scholastic Philosophy and Theology: 1240-1335

This dissertation examines the concept of beatific enjoyment (fruitio beatifica) in scholastic theology and philosophy in the thirteenth and early fourteenth century. The aim of the study is to explain what is enjoyment and to show why scholastic thinkers were interested in discussing it. The dissertation consists of five chapters. The first chapter deals with Aurelius Augustine's distinction between enjoyment and use and the place of enjoyment in the framework of Augustine's view of the passions and the human will. The first chapter also focuses upon the importance of Peter Lombard's Sentences for the transmission of Augustine's treatment of enjoyment in scholastic thought as well as upon Lombard's understanding of enjoyment. The second chapter treats thirteenth-century conceptions of the object and psychology of enjoyment. Material for this chapter is provided by the writings - mostly Sentences commentaries - of Alexander of Hales, Albert the Great, Bonaventure, Thomas Aquinas, Peter of Tarentaise, Robert Kilwardby, William de la Mare, Giles of Rome, and Richard of Middleton. The third chapter inspects early fourteenth-century views of the object and psychology of enjoyment. The fourth chapter focuses upon discussions of the enjoyment of the Holy Trinity. The fifth chapter discusses the contingency of beatific enjoyment. The main writers studied in the third, fourth and fifth chapters are John Duns Scotus, Peter Aureoli, Durandus of Saint Pourçain, William of Ockham, Walter Chatton, Robert Holcot, and Adam Wodeham. Historians of medieval intellectual history have emphasized the significance of the concept of beatific enjoyment for understanding the character and aims of scholastic theology and philosophy. The concept of beatific enjoyment was developed by Augustine on the basis of the insight that only God can satisfy our heart's desire. The possibility of satisfying this desire requires a right ordering of the human mind and a detachment of the will from the relative goals of earthly existence. Augustine placed this insight at the very foundation of the notion of Christian learning and education in his treatise On Christian Doctrine. Following Augustine, the twelfth-century scholastic theologian Peter Lombard made the concept of enjoyment the first topic in his plan of systematic theology. The official inclusion of Lombard's Sentences in the curriculum of theological studies in the early universities stimulated vigorous discussions of enjoyment. Enjoyment was understood as a volition and was analyzed in relation to cognition and other psychic features such as rest and pleasure. This study shows that early fourteenth-century authors deepened the analysis of enjoyment by concentrating upon the relationship between enjoyment and mental pleasure, the relationship between cognition and volition, and the relationship between the will and the beatific object (i.e., the Holy Trinity). The study also demonstrates the way in which the idea of enjoyment was affected by changes in the method of theological analysis - the application of Aristotelian logic in a Trinitarian context and the shift from virtue ethics to normative ethics.

Um estudo sobre as origens da Lógica Matemáitca

The present study has as objective to explaining about the origins of the mathematical logic. This has its beginning attributed to the autodidactic English mathematician George Boole (1815-1864), especially because his books The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854) are recognized as the inaugural works of the referred branch. However, surprisingly, in the same time another mathematician called Augutus of Morgan (1806-1871) it also published a book, entitled Formal Logic (1847), in defense of the mathematic logic. Even so, times later on this same century, another work named Elements of Logic (1875) it appeared evidencing the Aristotelian logic with Richard Whately (1787-1863), considered the better Aristotelian logical of that time. This way, our research, permeated by the history of the mathematics, it intends to study the logic produced by these submerged personages in the golden age of the mathematics (19th century) to we compare the valid systems in referred period and we clarify the origins of the mathematical logic. For that we looked for to delineate the panorama historical wrapper of this study. We described, shortly, biographical considerations about these three representatives of the logic of the 19th century formed an alliance with the exhibition of their point of view as for the logic to the light of the works mentioned above. In this sense, we aspirated to present considerations about what effective Aristotelian´s logic existed in the period of Boole and De Morgan comparing it with the new emerging logic (the mathematical logic). Besides of this, before the textual analysis of the works mentioned above, we still looked for to confront the systems of Boole and De Morgan for we arrive to the reason because the Boole´s system was considered better and more efficient. Separate of this preponderance we longed to study the flaws verified in the logical system of Boole front to their contemporaries' production, verifying, for example, if they repeated or not. We concluded that the origins of the mathematical logic is in the works of logic of George Boole, because, in them, has the presentation of a new logic, matematizada for the laws of the thought similar to the one of the arithmetic, while De Morgan, in your work, expand the Aristotelian logic, but it was still arrested to her

Uma reavaliação do pensamento lógico de George Boole à luz da história da matemática

The aim of the present study is to reevaluate the logical thought of the English mathematician George Boole (1815 - 1864). Thus, our research centers on the mathematical analysis of logic in the context of the history of mathematics. In order to do so, we present various biographical considerations about Boole in the light of events that happened in the 19th century and their consequences for mathematical production. We briefly describe Boole's innovations in the areas of differential equations and invariant theory and undertake an analysis of Boole's logic, especially as formulated in the book The Mathematical Analysis of Logic, comparing it not only with the traditional Aristotelian logic, but also with modern symbolic logic. We conclude that Boole, as he intended, expanded logic both in terms of its content and also in terms of its methods and formal elaboration. We further conclude that his purpose was the mathematical modeling of deductive reasoning, which led him to present an innovative formalism for logic and, because the different ways it can be interpreted, a new conception of mathematics

El uso aristotélico de variables en lógica y sus supuestos ontológicos

A logical reading on Aristotle’s Organon discovers some inconsistencies in the text which have to be solved by reducing them to metaphysical decisions of the author, if they are not just identified as deficiencies in the exposition that should be corrected. The present article tries to display a line of reading paying attention to those so-called inconsistencies, in an attempt to understand them as specific steps in Aristotle’s research. In order to this goal it focuses on the exposition procedure of the Aristotelian figures: the use of variables, whose introduction by Aristotle has been celebrated all over logical tradition. An analysis of the distinctive and internal features in this procedure will allow us to link Aristotle’s logos research and the “being qua being” investigation, and to determine also –though in a negative way– the connection between this reading and the logical-traditional one on Aristotle’s Organon.

El daño y el nexo causal en la perdida de oportunidad

El presente escrito se ocupa de estudiar el nexo de causalidad como elemento estructural de la responsabilidad cuando éste es difuso. Para ello, se pretende abordar la pérdida de la oportunidad como una teoría especial de causalidad que tiene lugar cuando el nexo causal no resulta claro, lo cual contradice la tesis preponderante de la doctrina y la jurisprudencia tradicional según la cual, la pérdida de la oportunidad es un criterio autónomo del daño. En su contenido se realiza una explicación del por qué se entiende la pérdida de la oportunidad como una teoría especial de causalidad y no como un criterio autónomo de daño, haciendo énfasis en el elemento de certeza que caracteriza al daño. Posteriormente, se advierte del tratamiento que la jurisprudencia le ha dado a la pérdida de la oportunidad. A su turno, el presente documento, indica la naturaleza jurídica de la pérdida de la oportunidad, afirmando que es una inferencia lógica que realiza el juez y no un hecho que altere el estado de las cosas como si sucede con el daño. Finalmente, se aborda la prueba de la teoría de la pérdida de la oportunidad mediante un cálculo de probabilidades y se identifican los pasos para realizar una adecuada reparación integral.

Aristotelian predicables, universality and realism: the logic of comparison in topics as denying the view that Aristotle was a realist

Aristotle is reportedly held to have been a Moderate Realist in that he would maintain that a concept derives from an act of grasping a mind-independent universal object that exists somehow inside of the many different things which the concept is predicated of. As far as a universal is independent of mind, it would stand for the proper object of a concept that subsumes a given number of things as its own instantiations. But we claim that Aristotle rejected such a view and instead did perceive and comprehend universality as a feature of thought rather than as a feature of reality in its own right. As showed in the chapters of Topics regarding the so-called logic of comparison (with the support of Albert the Great’s commentary), each predicate can be more or less consistent with the attribute of the subject of which it may be predicated. Both essential and accidental attributes assume a definite degree of being related to the degree of belonging to substance. Unlike particular things, the universality of a concept is to be understood always in comparison with another concept according to a hierarchy of predicates in terms of universality degree arranged by comparative terms such as ‘more’, ‘less’, and ‘likewise’. What is really mind-independent are the truth conditions which make a universal true when exclusively referring to a set of things identically meant by the same predicate whose universality is given by the place occupied in the hierarchy of predicates.

Medical education as moral formation: an Aristotelian account of medical professionalsim.

The medical professionalism movement, bolstered by many influential medical organizations and institutions, has in the last decade produced a number of conceptual definitions of professionalism and a number of concrete proposals for its measurement and teaching. These projects, however laudable, are misguided when they treat professionalism as a unitary descriptive concept rather than as a contested and therefore primarily evaluative one; when they conceive professionalism as a domain of medical practice separable in principle from other domains; and when they treat professionalism as, in principle, a specifiable goal or product of sufficiently well designed educational curricula. The logic of professionalism-as-product corresponds to the logic of techne (art or practical skill) in Aristotle's Nicomachean Ethics. Aristotle provides a cogent argument, however, that the moral excellences denoted by "professionalism" cannot be "produced" or even prespecified in the concrete; rather, they must be acquired through long practice under the careful concrete guidance of teachers who themselves embody these moral excellences. Phronesis (practical wisdom) rather than techne must therefore be the guiding logic of educational initiatives in medical professional formation, with particular emphasis on close mentorship and on the moral character both of students and of those who teach them.

The logic of ADHD : a brief review of fallacious reasoning

This paper has two central purposes: the first is to survey some of the more important examples of fallacious argument, and the second is to examine the frequent use of these fallacies in support of the psychological construct: Attention Deficit Hyperactivity Disorder (ADHD). The paper divides 12 familiar fallacies into three different categories—material, psychological and logical—and contends that advocates of ADHD often seem to employ these fallacies to support their position. It is suggested that all researchers, whether into ADHD or otherwise, need to pay much closer attention to the construction of their arguments if they are not to make truth claims unsupported by satisfactory evidence, form or logic.