We show that arbitrary tautologies of Johansson’s minimal propositional logic are provable by “small” polynomial-size dag-like natural deductions in Prawitz’s system for minimal propositional logic. These “small” deductions arise from standard “large” tree-like inputs by horizontal dag-like compression that is obtained by merging distinct nodes labeled with identical formulas occurring in horizontal sections of deductions involved. The underlying geometric idea: if the height, h(∂), and the total number of distinct formulas, ϕ(∂), of a given tree-like deduction ∂ of a minimal (...) tautology ρ are both polynomial in the length of ρ, |ρ|, then the size of the horizontal dag-like compression ∂ᶜ is at most h(∂)×ϕ(∂), and hence polynomial in |ρ|. That minimal tautologies ρ are provable by tree-like natural deductions ∂ with |ρ|-polynomial h(∂) and ϕ(∂) follows via embedding from Hudelmaier’s result that there are analogous sequent calculus deductions of sequent ⇒ρ. The notion of dag-like provability involved is more sophisticated than Prawitz’s tree-like one and its complexity is not clear yet. Our approach nevertheless provides a convergent sequence of NP lower approximations of PSPACE-complete validity of minimal logic. (shrink)

The introduction and elimination rules for material implication in natural deduction are not complete with respect to the implicational fragment of classical logic. A natural way to complete the system is through the addition of a new natural deduction rule corresponding to Peirce's formula → A) → A). E. Zimmermann [6] has shown how to extend Prawitz' normalization strategy to Peirce's rule: applications of Peirce's rule can be restricted to atomic conclusions. The aim of the present paper is to extend (...) Seldin's normalization strategy to Peirce's rule by showing that every derivation Π in the implicational fragment can be transformed into a derivation Π' such that no application of Peirce's rule in Π' occurs above applications of →-introduction and →-elimination. As a corollary of Seldin's normalization strategy we obtain a form of Glivenko's theorem for the classical {→}-fragment. (shrink)

A lógica não contém teoremas puramente existenciais: as únicas sentenças existenciaisválidas são aquelas com análogas universais válidas. Aqui, mostramos que istorealmente é assim quando corretamente interpretado: toda validade ex- istencial possuiuma análoga universal simples, que também é válida. Também caracterizamos validadesuniversais e existenciais em termos de tautologias.

To the historian of art, it is evident that the two authors’ notion of ‘art’ is of very recent date, and not shared by everybody. They claim: ‘The purpose of art, surely, is not merely to depict or represent reality -- for that can be accomplished very easily with a camera -- but to enhance, transcend, or even to distort reality’ . They do not explain how one could photograph Paradise or Hell, the Creation of the World, the Passion of (...) Christ, or the escapades of the ancient gods -- all subjects that can be found represented in our museums. Nor is it more legitimate to generalize from certain Indian conventions of representing the female nude than it is for the academic tradition to take the Venus de Medici for the same purpose. Even a fleeting visit to one of the great museums might serve to convince the authors that few of the exhibits conform to the laws of art they postulate. [followed by response from V.S. Ramachandran]. (shrink)

This article proposes a geostatistical solution for area-to-point spatial prediction (downscaling) taking into account boundary effects. Such effects are often poorly considered in downscaling, even though they often have significant impact on the results. The geostatistical approach proposed in this article considers two types of boundary conditions (BC), that is, a Dirichlet-type condition and a Neumann-type condition, while satisfying several critical issues in downscaling: the coherence of predictions, the explicit consideration of support differences, and the assessment of uncertainty regarding the (...) point predictions. An updating algorithm is used to reduce the computational cost of area-to-point prediction under a given BC. In a case study, area-to-point prediction under a Dirichlet-type BC and a Neumann-type BC is illustrated using simulated data, and the resulting predictions and error variances are compared with those obtained without considering such conditions. (shrink)

In the fourth section of Goethe’s Zahme Xenien we find the quatrain from which I have taken the theme of such an old and new controversy, which, as I hope, concerns both Germanic studies and the other humanities: “What was it that kept you from us so apart?” I always read Plutarch again and again. “And what was the lesson he did impart?” “They were all human beings—so much is plain.”1 In the very years when Goethe wrote these lines, that (...) is in the 1820s, Hegel repeatedly gave his lectures on the philosophy of history. Right at the beginning he formulated the opposite view which I should like briefly to characterize as “cultural relativism.”Every age has such peculiar circumstances, such individual conditions that it must be interpreted, and can only be interpreted, by reference to itself…. Nothing is shallower in this respect than the frequent appeal to Greek and Roman example which so often occurred among the French at the time of their Revolution. Nothing could be more different than the nature of these peoples and the nature of our own times.2 What is at issue here is not, of course, Hegel’s assertion that ages and peoples differ from each other. We all know that, and Goethe, the attentive reader and traveler, also knew, for instance, that the Roman carnival differed in its character from the celebrations of the Feast of Saint Rochus at Bingen, both of which he had described so lovingly. What makes the cultural historian into a cultural relativist is only the conclusion which we saw Hegel draw, that cultures and styles of life are not only different but wholly incommensurable, in other words that it is absurd to compare the peoples of a region or an age with human beings of other zones because there is no common denominator that would justify us in doing so. 1. ‘Was hat dich nun von uns entfernt?’ Hab immer den Plutarch gelesen. ‘Was has du den dabei gelernt?’ Sind eben alles Menschen gewesen.’ Johann Wolfgang von Goethe, Sämtlich Werke. Jubiläums-ausgabe in 40 Bänden 4:73; with commentary.2. See Georg Wilhelm Friedrich Hegel, Vorselungen über die Philosophie der Geschichte, Werke, 20 vols. , 12:17. E. H. Gombrich was director of the Warburg Institute and Professor of the History of the Classical Tradition at the University of London from 1959 to 1976. His many influential works include The Story of Art, Art and Illusion, Meditations on a Hobby Horse, The Sense of Order, Ideals and Idols, The Image and the Eye, Tributes, Aby Warburg, and New Light on Old Masters. His previous contributions to Critical Inquiry include “The Museum: Past, Present and Future” , “Standards of Truth: The Arrested Image and the Moving Eye” , and “Representation and Misrepresentation”. (shrink)

Professor Gombrich's last book and first narrative work in over 20 years.

Neste livro Gombrich discute as idéias e as posturas específicas que tiveram influência decisiva na prática da arte renascentista. Todos os estudos aqui reunidos tratam de algo a que se pode chamar clima renascentista de opiniões sobre a arte, além da influência desse clima sobre a prática e a crítica da arte.

It is a thankless task to have to reply to Professor Murray Krieger’s “Retrospective.” Qui s’excuse, s’accuse, and since I cannot ask my readers to embark on their own retrospective of my writings and test them for consistency, I have little chance of restoring my reputation in their eyes. Hence I would have been happier to leave Professor Krieger to his agonizing, if he did not present himself the “spokesman” for a significant body of theorists who appear to have acclaimed (...) my book on Art and Illusion without ever having read it. The followers of this school of criticism—of which Professor Krieger is a prominent member—had apparently convinced themselves that the book lent support to an aesthetics in which the notions of reality and of nature had no place. They thought that I had subverted the old idea of mimesis and that all that remained were different systems of conventional signs which were made to stand for an unknowable reality. True, professor Krieger admits that I never endorsed such an interpretation of my views, and he even concedes that there are passages in Art and Illusion which contradict such an out-and-out relativism, but he wants to convince his readers that these contradictions lead precisely to the ambiguities he now proposes to analyse.If he were right that the book encourages such a misreading, all I could do would be to express my regrets for having failed to make myself sufficiently clear. Luckily I can draw comfort from the fact that unlike these literary critics, the leading archaeologist of this country, Professor Stuart Piggott, had no difficulty at all in discerning my meaning and profiting from my arguments. In his Walter Neurath Memorial Lecture of 1978, entitled Antiquity Depicted: Aspects of Archaeological Illustration, the author did me the honour of taking a statement from my book as his starting point. It is the passage at the end of part I in which I recapitulate the content of the first two chapters:What matters to us is that the correct portrait, like the useful map, is an end product on a long road through schema and correction. It is not a faithful record of a visual experience but the faithful construction of a relational mode.Neither the subjectivity of vision nor the sway of conventions need lead us to deny that such a model can be constructed to any required degree of accuracy. What is decisive here is clearly the word “required.” The form of a representation cannot be divorced from its purpose and the requirements of the society in which the given visual language gains currency.1 1. E. H. Gombrich, Art and Illusion: A Study in the Psychology of Pictorial Representation, A. W. Mellon Lectures in the Fine Arts, 1956 , p. 90. E. H. Gombrich was director of the Warburg Institute and Professor of the History of the Classical Tradition at the University of London from 1959 to 1976. His many influential works include The Story of Art, Art and Illusion, Meditations on a Hobby Horse, The Sense of Order, Ideals and Idols, The Image and the Eye, and, most recently, Tributes. His previous contributions to Critical Inquiry include “The Museum: Past, Present and Future and “Standards of Truth: The Arrested Image and the Moving Eye”. (shrink)

I have stressed here and elsewhere that perspective cannot and need not claim to represent the world "as we see it." The perceptual constancies which make us underrate the degree of objective diminutions with distance, it turns out, constitute only one of the factors refuting this claim. The selectivity of vision can now be seen to be another. There are many ways of "seeing the world," but obviously the claim would have to relate to the "snapshot vision" of the stationary (...) single eye. To ask, as it has so often been asked, whether this eye sees the world in the form of a hollow sphere or of a projection plane makes little sense, for it sees neither. The one point in focus can hardly be said to be either curved or flat, and the remainder of the field of vision is too indistinct to permit a decision. True, we can shift the point of focus at will, but in doing so we lose the previous perception, and all that remains is its memory. Can we, and do we, compare the exact extension of these changing percepts in scanning a row of columns extended at right angles from the central line of vision—to mention the most recalcitrant of the posers of perspectival theory?1 I very much doubt it. The question refers to the convenient choice of projection planes, not to the experience of vision.· 1. I now prefer this formulation to my somewhat laboured discussion in Art and Illusion, chap. 8, sec. 4.E.H. Gombrich was director of the Warburg Institute and Professor of the History of the Classical Tradition at the University of London from 1959 to 1976. His many influential works include The Story of Art, Art and Illusion, Meditations on a Hobby Horse, The Sense of Order, and Ideals and Idols. An early version of "Standards of Truth" was presented at Swarthmore College in October 1976 at a symposium to mark the retirement of Professor Hans Wallach. His contributions to Critical Inquiry include "The Museum: Past, Present, and Future" , "Notes and Exchanges" , and, with Quentin Bell, "Canons and Values in the Visual Arts: A Correspondence". (shrink)