In dieser Marburger Vorlesung aus dem Wintersemester 1924/25 stellt sich Heidegger die Aufgabe, Platons Spatdialog "Sophistes" im Ausgang von Aristoteles verstandlich zu machen. Zentrum des einleitenden Aristoteles-Teils ist die Folge der dianoethischen Tugenden im VI. Buch der "Nikomachischen Ethik", in der Heidegger die sich aufsteigernde Stufenfolge eines Entbergens erkennt und demgemass den Primat der "Physis" aus der Uberlegenheit ihres Entbergens begrundet. Damit legt Heidegger die Zusammengehorigkeit von Sein und Wahrheit als Horizont des aristotelisch-griechischen Philosophierens frei und gewinnt so den "Boden", (...) aus dem die platonische Seinsforschung, wie sie im "Sophistes" vorliegt, erwachsen ist. Dementsprechend stellt sich Heidegger im Hauptteil die Aufgabe, in fortlaufender Interpretation des "Sophistes" konkret zu zeigen, dass und wie Platons Ontologie aus dem Entbergen erwachsen ist. Die Vorlesung bezeugt, dass Heidegger die in seinem fruhen Hauptwerk "Sein und Zeit" gestellte Frage nach dem "Sinn von Sein", d.h. nach der Un-verborgenheit des Seins in der Auseinandersetzung mit der philosophischen Uberlieferung gewonnen hat. (shrink)

In this article, I present and analyze Dietrich von Hildebrand’s explanation of how acting against one’s better knowledge is possible. I do so by comparing it to Plato’s analysis of the same problem. By this comparison, I seek to show the specificity of von Hildebrand’s approach to the phenomenon which, since Aristotle’s time, has been known as “akrasia.”.

English summary: Friedrich Schleiermacher's work as a theologian and Plato scholar marked the start of a new epoch: as a system-forming philosopher, he claimed independent validity, and as church politician, educational policy-maker, and academic theorist, he was one of the most important figures during the Prussian reforms, whose contributions to pedagogy remain influential to this day. This volume provides a clear-sighted overview of the various stages in Schleiermacher's life (1768-1834), with each contribution portraying his fields of work and their (...) contexts, placing his literary work and conceptions in the framework of the respective contemporary debates. Important stages in Schleiermacher reception and impact history are likewise considered. The authors' diverse approaches to interpreting and understanding his output mean that the Guide to Schleiermacher offers concentrated insight into the consensus on and controversies in current Schleiermacher research. German description: Friedrich Schleiermacher (1768-1834) hat als Theologe, aber auch als Plato-Forscher Epoche gemacht; als systembildender Philosoph erhob er selbstandigen Geltungsanspruch, als Kirchen- und Bildungspolitiker sowie als Theoretiker der Universitat gehorte er zu den wichtigsten Akteuren der preussischen Reformen, und seine Beitrage zur Padagogik wirken bis heute nach. Die Gliederung des Handbuchs folgt grob Schleiermachers Lebensstationen. Die einzelnen Beitrage schildern seine Arbeitsfelder und -kontexte und stellen seine literarischen Arbeiten und Entwurfe in den jeweiligen zeitgeschichtlichen Debattenlagen dar. Auch wichtige Stationen seiner Wirkungs- und Rezeptionsgeschichte werden thematisiert.Die Autorinnen und Autoren der Beitrage verfolgen ganz unterschiedliche Interpretations- und Deutungsansatze. Deshalb bietet das Handbuch auch einen konzentrierten Einblick in den gegenwartigen Stand der Schleiermacher-Forschung mit ihren Konsensen und ihren Kontroversen. Mit Beitragen von: Christian Albrecht, Andreas Arndt, Bernd Auerochs, Ulrich Barth, Albrecht Beutel, Georg Eckert, Hermann Fischer+, Ursula Frost, Albrecht Geck, Simon Gerber, Wilhelm Grab, Peter Grove, Matthias Heesch, Eilert Herms, Kirsten Huxel, Walter Jaeschke, Lutz Kappel, Hans-Martin Kirn, Dietz Lange, Inken Madler, Helmut Merkel, Martin Ohst, Claus-Dieter Osthovener, Hermann Patsch, Reiner Preul, Andreas Reich, Joachim Ringleben, Jan Rohls, Martin Rossler, Dirk Schmid, Friedemann Voigt, Folkart Wittekind, Matthias Wolfes. (shrink)

"Freiburger Vorlesungen Sommersemester 1933 und Wintersemester 1933/34 / herausgegeben von Hartmut Tietjen"--T.p. verso.

Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions to (...) geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.

The standard rule of necessitation in systems of natural deduction for the modal logic S4 concludes □A from A whenever all assumptions A depends on are modal formulas. This condition prevents the composability and normalization of derivations, and therefore modifications of the rule have been suggested. It is shown that both properties hold if, instead of changing the rule of necessitation, all elimination rules are formulated in the manner of disjunction elimination, i.e. with an arbitrary consequence.

Gentzen's original proof of the Hauptsatz used a rule of multicut in the case that the right premiss of cut was derived by contraction. Cut elimination is here proved without multicut, by transforming suitably the derivation of the premiss of the contraction.

This paper discusses different interpretations of probability in relation to determinism. It is argued that both objective and subjective views on probability can be compatible with deterministic as well as indeterministic situations. The possibility of a conceptual independence between probability and determinism is argued to hold on a general level. The subsequent philosophical analysis of recent advances in classical statistical mechanics (ergodic theory) is of independent interest, but also adds weight to the claim that it is possible to justify an (...) objective interpretation of probabilities in a theory having as a basis the paradigmatically deterministic theory of classical mechanics. (shrink)

Gentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have (...) at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933. (shrink)

Friedrich Schiller’s notion of moral virtue includes self-determination through practical rationality as well as sensual self-determination through the pursuit of aesthetic value, i.e., through beauty. This paper surveys conceptual assumptions behind Schiller’s notions of moral and aesthetic perfections that allow him to ground both, moral virtue and beauty on conceptions of freedom. While Schiller’s notions of grace and dignity describe relations between the aesthetic and the moral aspects of certain determining actions, the ‘aesthetic condition’ conceptualises human beings from the perspective (...) of aesthetic self-determinability. Schiller thereby provides a normative aesthetic standard that not only affects the moral nature of our motives and actions, but also of what we, as human beings, want to and should be conceived of in the first place. As I argue in this paper, considering this aesthetic self-determinability from a moral perspective results in an aesthetic constitution of moral virtue, which in turn justifies aesthetic obligations. Schiller thereby merges the perfections of the two normative domains for an extended anthropological conception of aesthetically infused notion of moral virtue while assuring the conceptual autonomy of each normative domain. Giving aesthetic demands a practical normative role by partly constituting moral virtue and thereby still maintain their aesthetic normative source is a move that opens up many resources for current research on the interactions between various normative demands, such as aesthetic reasons for moral or legal judgements and action, aesthetic obligations or weighing varying sources and elements of normative authority and hegemony against each other. (shrink)

Gentzens natural deduction. Thereby the appearance of the cuts in translation is explained.

Die 'Beiträge zur Philosophie. Vom Ereignis' dürfen mit Fug und Recht nach 'Sein und Zeit' als das zweite Hauptwerk Heideggers bezeichnet werden. Denn sie enthalten nach der fundamentalontologischen Ausarbeitung der Seinsfrage in 'Sein und Zeit' die erste Durchgestaltung des seinsgeschichtlichen Weges derselben Frage. Das seinsgeschichtliche Denken entfaltet das Wesen des Seins nicht mehr im Überstieg (Transzendenz) über das Seiende auf dessen Wahrheit hin (Horizont), sondern im Vollzug des ereigneten Entwurfs aus dem ereignenden Zuwurf der Wahrheit des Seins. 'Ereignis' ist der (...) Name für die Zusammengehörigkeit der ereignend sich zuwerfenden Wahrheit und des ereigneten Seinsentwurfes das Da-seins. Die 'Beiträge zur Philosophie' sind das Werk, in dem sich das Denken der im 'Brief über den Humanismus' genannten 'Kehre' vollzieht. (shrink)

In the summer of 1928, Kurt Gödel (1906–1978) embarked on his logical journey that would bring him world fame in a mere three years. By early 1929, he had solved an outstanding problem in logic, namely the question of the completeness of the axioms and rules of quantificational logic. He then went on to extend the result to the axiom system of arithmetic but found, instead of completeness, his famous incompleteness theorem that got published in 1931. It belongs to the (...) most iconic achievements of 20th century science and has been instrumental in the development of theories of formal languages and algorithmic computability – two essential components in the birth of the information society. This book explores Gödel’s way from an exceptional high-school student to a firmly established young logician. Essays in Gödel’s hand from the high school show that his central philosophical and scientific convictions were formed early on, before his university studies. Particular emphasis is laid on the course that made Gödel one of the foremost logicians of all times. The scientific biography of young Gödel is followed by English translations from Gödel’s German Gabelsberger shorthand of all his early preserved notebooks on logic and related topics. (shrink)

Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.

The way from linearly written derivations in natural deduction, introduced by Jaskowski and often used in textbooks, is a straightforward root-first translation. The other direction, instead, is tricky, because of the partially ordered assumption formulas in a tree that can get closed by the end of a derivation. An algorithm is defined that operates alternatively from the leaves and root of a derivation and solves the problem.

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of (...) these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934—35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus. (shrink)

The structure of derivations in natural deduction is analyzed through isomorphism with a suitable sequent calculus, with twelve hidden convertibilities revealed in usual natural deduction. A general formulation of conjunction and implication elimination rules is given, analogous to disjunction elimination. Normalization through permutative conversions now applies in all cases. Derivations in normal form have all major premisses of elimination rules as assumptions. Conversion in any order terminates.Through the condition that in a cut-free derivation of the sequent Γ⇒C, no inactive weakening (...) or contraction formulas remain in Γ, a correspondence with the formal derivability relation of natural deduction is obtained: All formulas of Γ become open assumptions in natural deduction, through an inductively defined translation. Weakenings are interpreted as vacuous discharges, and contractions as multiple discharges. In the other direction, non-normal derivations translate into derivations with cuts having the cut formula principal either in both premisses or in the right premiss only. (shrink)

Bruno de Finetti's earliest works on the foundations of probability are reviewed. These include the notion of exchangeability and the theory of random processes with independent increments. The latter theory relates to de Finetti's ideas for a probabilistic science more generally. Different aspects of his work are united by his foundational programme for a theory of subjective probabilities.

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

Gentzen's “Untersuchungen” [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut-free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut-free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts in (...) Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination. (shrink)

History and Philosophy of Logic, Volume 32, Issue 4, Page 402-404, November 2011.

Recently Ekman gave a derivation in natural deduction such that it either contains a substantial redundant part or else is not normal. It is shown that this problem is caused by a non-normality inherent in the usual modus ponens rule.

The paper focuses on the inferential role of quantifiers in Frege, Peano and Russell. Two aspects of the early years of mathematical logic are discussed: the gradual perfection of the principles of reasoning with quantifiers, and the presumed conceptual impossibility of posing metatheoretical questions, as embodied in Jean van Heijenoort’s well-known dictum about “logic as calculus and logic as language.”.

"Hannah Arendt war der schwierigste Film, den ich je gemacht habe", sagt Margarethe von Trotta. Die grosse Denkerin Hannah Arendt fühlte sich.

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

Die im Wintersemester 1929/30 an der Freiburger Universität vierstündig gehaltene Vorlesung ist in zweierlei Hinsicht erregend. Zum einen enthält sie eine breit durchgeführte Analyse der in der Freiburger Antrittsvorlesung 'Was ist Metaphysik?' nur gestreiften Langeweile, zum anderen bietet sie eine ebenso ausführliche Wesensbestimmung des Organismus und des Lebens - eine Thematik, die Heidegger in 'Sein und Zeit', § 12, nur benennt. 'Diese Vorlesung ist wohl die bedeutendste, die Heidegger gehalten hat, und schon fast ein zweites Hauptwerk.' Rüdiger Safranski.

Fabio Acerbi, Il silenzio delle sirene: La matematica greca antica. Rome: Carocci editore, 2010. 445 pp. € 44. ISBN 978-88-430-5579-1.Fabio Acerbi's recent book Il silenzio delle sirene: La matemat...

A formulation of probabilistic causality is given in terms of the theory of abstract dynamical systems. Causal factors are identified as invariants of motion of a system. Repetition of an experiment leads to the notion of stationarity, and causal factors yield a decomposition of the stationary probability law of the experiment into ergodic components. In these, statistical behaviour is uniform. Control of identified causal factors leads to a corresponding statistical law for the events, which is offered as a notion of (...) probabilistic causality. After a suggestion by Feller, randomization is identified as mixing, formulated in above terms. (shrink)

This chapter focuses on the development of Gerhard Gentzen's structural proof theory and its connections with intuitionism. The latter is important in proof theory for several reasons. First, the methods of Hilbert's old proof theory were limited to the “finitistic” ones. These methods proved to be insufficient, and they were extended by infinitistic principles that were still intuitionistically meaningful. It is a general tendency in proof theory to try to use weak principles. A second reason for the importance of intuitionism (...) for proof theory is that the proof-theoretical study of intuitionistic logic has become a prominent part of logic, with applications in computer science. (shrink)

What seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher-order logic, with formal (...) derivations closer to one hundred steps in length and up to four nested temporary assumptions with their scope indicated by vertical intermittent lines. (shrink)

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates . This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those (...) known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry. (shrink)

The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation is (...) given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved. The forgotten parts of Skolem's first paper on the Löwenheim-Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results. (shrink)