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We show that given a simple Polygon P it is NP-hard to determine the smallest α ∈ [0, π] such that P can be illuminated by α-vertexlights, if we place exactly one α-vertexlight in each vertex of P.

D. T. Lee and A. K. Lin [2] proved that VERTEX-GUARDING and POINT-GUARDING are NP-hard for simple polygons. We prove that those problems are NP-hard for ortho-polygons, too.

First we give an overview about variants of visibility and related problems. Then we prove that some guarding and covering problems are NP-hard for ortho-polygons with holes, using a so-called vertex cover technique.

For each vertex of a simple polygon P an integer valued weight is given. We consider the path p1, p2, ..., pk in P which is created according to the following strategy: p1 is a designated start vertex s and pi+1 is obtained by choosing the vertex with smallest weight among all vertices visible from pi and different from p1, p2, ..., pi. If there is no such vertex the path is finished. This path is called geometric lexicographic dead end (...) path. We shall prove the problem of determining whether a distinguished vertex t of P is on the geometric lexicographic dead end path or not to be P-complete. (shrink)

We consider the problem of quadrilaterizing an orthogonal polygon P, that is to decompose P into nonoverlapping convex quadrangles without adding new vertices. In this paper we present a CREW-algorithm for this problem which runs in O time using Θ processors if the rectangle decomposition of P is given, and Θ processors if not. Furthermore we will show that the latter result is optimal if the polygon is allowed to contain holes.

Ludwig Wittgenstein is one of the most important, influential, and often-cited philosophers of the twentieth century, yet he remains one of its most elusive and least accessible. The essays in this volume address central themes in Wittgenstein's writings on the philosophy of mind, language, logic, and mathematics. They chart the development of his work and clarify the connections between its different stages. The contributors illuminate the character of the whole body of work by keeping a tight focus on some key (...) topics: the style of the philosophy, the conception of grammar contained in it, rule-following, convention, logical necessity, the self, and what Wittgenstein called, in a famous phrase, 'forms of life'. (shrink)

Undersøgelser af sammenhængen mellem tysk filosofi og nazismens teorier med særlig vægt på Martin Heidegger (1889-1976).

Wittgenstein's remarks about family resemblance in the Philosophical Investigations should not be construed as implying a comprehensive theory of universals. They possess, rather, a defensive function in his exposition. The remarks allow one, nevertheless, to draw certain general conclusions about how Wittgenstein thought about concepts. Reflection on the notion of family resemblance reveals that kinship and similarity considerations intersect in it in a problematic fashion.

Heidegger's Crisis shows not only how the Nazis exploited philosophical ideas and used philosophers to gain public acceptance, but also how German philosophers played into the hands of the Nazis. Hans Sluga describes the growth, from World War I onward, of a powerful right-wing movement in German philosophy, in which nationalistic, antisemitic, and antidemocratic ideas flourished.

Rethinking politics in a new vocabulary, Hans Sluga challenges the firmly held assumption that there exists a single common good which politics is meant to realize. He argues that politics is not a natural but a historical phenomenon, and not a single thing but a multiplicity of political forms and values only loosely related. He contrasts two traditions in political philosophy: a 'normative theorizing' that extends from Plato to John Rawls and a newer 'diagnostic practice' that emerged with (...) Marx and Nietzsche and has found its three most prominent twentieth-century practitioners in Carl Schmitt, Hannah Arendt, and Michel Foucault. He then examines the sources of diagnostic political thinking, analyzes its achievements, and offers a critical assessment of its limitations. His important book will be of interest to a wide range of upper-level students and scholars in political philosophy, political theory, and the history of ideas. (shrink)

_Wittgenstein_ presents a concise, comprehensive, and systematic treatment of Ludwig Wittgenstein's thought from his early work, _Tractatus Logico-Philosophicus,_ to the posthumous publication of _On Certainty_, notes written just prior to his death. A substantial scholarly addition to our understanding of one of the most original and influential thinkers of the twentieth century, by renowned Wittgenstein scholar, Hans Sluga Proposes an original new interpretation of Wittgenstein's work Written to also be accessible to readers unfamiliar with Wittgenstein's thought Includes discussion (...) of the social and political background and contemporary relevance of Wittgenstein's thoughts. (shrink)

Michael Dummett, following an established line of reasoning, has interpreted Frege as a realist. But his claim that Frege was arguing against a dominant idealism is untenable. While there are passages in Frege's writings that seem to support a realistic interpretation, others are irreconcilable with it. The issue can be resolved only by examining the historical context. Frege's thought is, in fact, related to the philosophy of Hermann Lotze. Frege is best regarded as a transcendental idealist in the Lotze-Kant tradition. (...) His contextual principle is a linguistic version of Kant's principle of the transcendental unity of judgment. By ignoring the historical context Dummett has been led to misinterpret the precise role of the contextual principle in Frege's thought. (shrink)

This essay traces the roots of Wittgenstein’s Pyrrhonism to Mauthner, and argues that Wittgenstein’s later views moved even closer to those of Mauthner, although Wittgenstein never became as thoroughgoing a Pyrrhonian as Mauthner had been. It is argued that Mauthner’s neo-Pyrrhonian view of language was “responsible for the linguistic turn in Wittgenstein’s thinking and thereby indirectly also for the whole linguistic turn in 20th-century analytic philosophy”.

The paper assesses Martin’s recent logico-phenomenological account of judgment that is cast in the form of an eclectic history of judging, from Hume and Kant through the 19th century to Frege and Heidegger as well as current neuroscience. After a preliminary discussion of the complex unity and temporal modalities of judgment that draws on a reading of Titian’s “Allegory of Prudence” (National Gallery, London), the remainder of the paper focuses on Martin’s views on Kant’s logic in general and his theory (...) of singular existential judgment in particular. The paper argues against Martin’s key claims of the primacy of formal logic over transcendental logic and of the synthetic nature of judgment in Kant. It also takes issue with each of the four interpretations of singular existential judgment in Kant offered by Martin: existence as logical predicate, as copula, as thesis and as logical subject. (shrink)