First published in 1818, The World as Will and Representation contains Schopenhauer's entire philosophy, ranging through epistemology, metaphysics, philosophy of mind and action, aesthetics and philosophy of art, to ethics, the meaning of life and the philosophy of religion, in an attempt to account for the world in all its significant aspects. It gives a unique and influential account of what is and is not of value in existence, the striving and pain of the human condition and the possibility of (...) deliverance from it. This new translation of the first volume of what later became a two-volume work reflects the eloquence and power of Schopenhauer's prose and renders philosophical terms accurately and consistently. It offers an introduction, glossary of names and bibliography, and succinct editorial notes, including notes on the revisions of the text which Schopenhauer made in 1844 and 1859. (shrink)

Leibniz was a philosopher of principles: the principles of Contradiction, of Sufficient Reason, of Identity of Indiscernibles, of Plenitude, of the Best, and of Continuity are among the most famous Leibnizian principles. In this article I shall focus on the first three principles; I shall discuss various formulations of the principles (sect. 1), what it means for these theses to have the status of principles or axioms in Leibniz’s philosophy (sect. 2), the fundamental character of the Principles of Contradiction and (...) Sufficient Reason (sect. 3), some attempts to demonstrate the Principles of Contradiction and Sufficient Reason (sect. 4), and one attempt to demonstrate the Principle of Identity of Indiscernibles (sect. 5). The main results of the chapter are summarized in a short conclusion (sect. 6). (shrink)

. In logic, diagrams have been used for a very long time. Nevertheless philosophers and logicians are not quite clear about the logical status of diagrammatical representations. Fact is that there is a close relationship between particular visual (resp. graphical) properties of diagrams and logical properties. This is why the representation of the four categorical propositions by different diagram systems allows a deeper insight into the relations of the logical square. In this paper I want to give some examples.

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams (...) are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the inference steps of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive -rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here. (shrink)

Ein Beispiel für die Rezeption und Fortführung der schopenhauerschen Logik findet man in den Mathematiklehrbüchern Friedrich Adolph Wilhelm Diesterwegs (1790–1866), In diesem Aufsatz werden die historische und systematische Dimension dieser Anwendung von Logikdiagramme auf die Mathematik skizziert. In Kapitel 2 wird zunächst die frühe Rezeption der schopenhauerschen Logik und Philosophie der Mathematik vorgestellt. Dabei werden einige oftmals tradierte Vorurteile, die das Werk Schopenhauers betreffen, in Frage gestellt oder sogar ausgeräumt. In Kapitel 3 wird dann die Philosophie der Mathematik und der (...) Logik Schopenhauers vorgestellt. Es wird gezeigt, dass Schopenhauers Logikdiagramme auf Ideen der Mathematiker Leonhard Euler (1707–1783), Bernhard Friedrich Thibaut (1775–1832) und Franz Ferdinand Schweins (1780–1856) zurückgehen und dass Schopenhauer diese Logikdiagramme auch bereits zur Analyse mathematischer Begriffe verwendet. Kapitel 4 beschäftigt sich mit der Schopenhauer-Rezeption Diesterwegs. Es wird die Frage behandelt, inwieweit Diesterweg in der Mathematik von Schopenhauer beeinflusst war und welche Rolle dieser Einfluss auf sein Verständnis von Logik und Mathematik hatte. In Kapitel 5 wird schließlich gezeigt, dass Diesterweg die schopenhauersche Begriffsanalyse der Mathematik mit Hilfe von Logikdiagrammen aufgreift und weiterführt. Da Diesterweg einer der einflussreichsten Pädagogen und Mathematikdidaktiker des 20. Jahrhunderts war, ist anzunehmen, dass sich Spuren dieser Methode noch heute finden lassen. Inwiefern dies wahrscheinlich ist, wird in der Zusammenfassung (Kapitel 6) thematisiert. (shrink)

In this paper we argue that there were several currents, ideas and problems in 19th-century logic that motivated John Venn to develop his famous logic diagrams. To this end, we first examine the problem of uncertainty or over-specification in syllogistic that became obvious in Euler diagrams. In the 19th century, numerous logicians tried to solve this problem. The most famous was the attempt to introduce dashed circles into Euler diagrams. The solution that John Venn developed for this problem, however, came (...) from a completely different area of logic: instead of orienting to syllogistic like Euler diagrams, Venn applied Boolean algebra to improve visual reasoning. Venn’s contribution to solving the problem of elimination also played an important role. The result of this development is still known today as the ‘Venn Diagram’. (shrink)

In this paper, we analyze and discuss Schopenhauer’s n-term diagrams for eristic dialectics from a graph-theoretical perspective. Unlike logic, eristic dialectics does not examine the validity of an isolated argument, but the progression and persuasiveness of an argument in the context of a dialogue or even controversy. To represent these dialogue situations, Schopenhauer created large maps with concepts and Euler-type diagrams, which from today’s perspective are a specific form of graphs. We first present the original method with Euler-type diagrams, then (...) give the most important graph-theoretical definitions, then discuss Schopenhauer’s diagrams graph-theoretically and finally give an example of how the graphs or diagrams can be used to analyze dialogues. (shrink)

In Baghdad in the mid twelfth century Abū al-Barakāt proposes a radical new procedure for finding the conclusions of premise-pairs in syllogistic logic, and for identifying those premise-pairs that have no conclusions. The procedure makes no use of features of the standard Aristotelian apparatus, such as conversions or syllogistic figures. In place of these al-Barakāt writes out pages of diagrams consisting of labelled horizontal lines. He gives no instructions and no proof that the procedure will yield correct results. So the (...) reader has to work out what his procedure is and whether it is correct. The procedure turns out to be insightful and entirely correct, but this paper may be the first study to give a full description of the procedure and a rigorous proof of its correctness. (shrink)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

We define a "representation" as an external object with which we denote another object and present information about it to ourselves or others. All the following objects count as representations in our sense: a set of Japanese declarative sentences describing Mount Fuji, a time table of the Boston subway system, a geometry diagram used to demonstrate the Pythagorean theorem, a state map of the United States, a relief map of a Rocky terrain, a ball-and-stick model of a molecular, and a (...) scale model of the Rockefeller center. ;The dissertation studies how different modes of presenting information exhibit different degrees and kinds of efficacy as an aid for human reasoning. The main hypothesis, which we dub the "Constraint Hypothesis," is that the variance in inferential potentials of different modes of representation is largely attributable to particular ways in which structural constraints on representations match and mismatch with constraints on their targets. The hypothesis is intended to provide a definite perspective for the general study of information representation, which is gradually taking the shape of an independent branch of science. ;As the basis for our analysis, we adopt the mathematical framework of the "qualitative information theory" being developed by Dretske, Barwise, Perry, and Seligman. We analyze several key properties that account for the efficacy of particular modes of representation, including the capacity of providing "free rides" in inference, the property of being over-specific in representing information, and the property of being self-consistent in information content. We also propose an analysis of the conceptual boundary between the "linguistic" modes and the "graphical" modes of representation, and explain some of the important differences in their cognitive potentials. All of these analyses crucially refer to structural constraints on representations and their match and mismatch with constraints on the targets, and hence lend piecemeal yet sturdy supports to our main hypothesis. (shrink)

Bei Kant tauchen viele antike Autoren und Theorien auf. In dieser ersten Gesamtbearbeitung zum Thema wird gegen eine langjährige Forschungsmeinung gezeigt, dass nicht so sehr Platon und Aristoteles als vielmehr der hellenistischen Philosophie die entscheidende Rolle für sein Denken zukommt. Anhand der drei Kritiken werden Konstanzen und Umbrüche seines Antikeverständnisses herausgearbeitet und in ihren Konsequenzen für die Kantdeutung beleuchtet.

Many systems of logic diagrams have been offered both historically and more recently. Each of them has clear limitations. An original alternative system is offered here. It is simpler, more natural, and more expressively and inferentially powerful. It can be used to analyze not only syllogisms but arguments involving relational terms and unanalyzed statement terms.

The 18th century was a wealth of knowledge, exploration and rapidly growing technology and expanding record-keeping made possible by advances in the printing press. In its determination to preserve the century of revolution, Gale initiated a revolution of its own: digitization of epic proportions to preserve these invaluable works in the largest archive of its kind. Now for the first time these high-quality digital copies of original 18th century manuscripts are available in print, making them highly accessible to libraries, undergraduate (...) students, and independent scholars. The Age of Enlightenment profoundly enriched religious and philosophical understanding and continues to influence present-day thinking. Works collected here include masterpieces by David Hume, Immanuel Kant, and Jean-Jacques Rousseau, as well as religious sermons and moral debates on the issues of the day, such as the slave trade. The Age of Reason saw conflict between Protestantism and Catholicism transformed into one between faith and logic -- a debate that continues in the twenty-first century. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to insure edition identification: ++++ British Library T112862 Pp.233/234 misnumbered 133/134. London: printed, and sold by J. Downes, 1796. [4],134[i.e.234]p.; 8°. (shrink)