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)
Is the term ‘religion’ otiose or misleading? Is what is commonly understood to be religion the enemy of faith? Can the study of religion be upheld as a discipline of academic respectability? The interrelation of these questions is close, and not without complexity.
"The art object has its own immanent validity," says Hester. By this rather obscure phrase he seeks to dramatize his claim that, modern linguistics and logical positivism notwithstanding, "the poet... succeeds in making the relation between his physical language and its meaning nonconventional." Ostensibly, Hester's book is a discussion and refutation of the claim that meaning is a matter of conventional usage. Poetic metaphor, unlike the literal or technical language he claims Wittgenstein is thinking of, is a "fusion of sense, (...) sensa, and imagery," with sense equivalent to literal meaning through conventional use, sensa equivalent to sound and its onomatopoetic similitudes, and imagery being the "associations" that the writer or reader experiences during the metaphorical "experience-act." Hester holds that poetic metaphors, and their "cousins," scientific models, are "image laden, ambiguous, implicatively full..." but that, unlike the free association inkblots used by the psychologist, they clearly delineate the realm of relevant, meaningful image-association. He believes that Wittgenstein would have accepted metaphor as one of his "language games," but still insists that this language game is useful for communicating meanings, despite the fact that one cannot ostensively identify the conventions of usage for the bounded associations. The book is thus a veritable Pandora's Box of classic problems in the philosophies of mind, language, and aesthetics, and in fact it is the overambitiousness of the work which causes it again and again to leave important problems unresolved.—E. H. W. (shrink)
The title of this work is a somewhat saucy overstatement of its thesis—that perceivers seek in works of art experiences of "discontinuity" and "disorientation," as a kind of "rehearsal" for "real life" situations in which they must negotiate intellectual tensions, resulting from a disparity between what they expect and what actually happens. Art-perceiving, the author asserts, is a "biological, adaptive" mechanism characteristic of the human organism. Peckham, like most thoughtful readers of art history, is irritated by the preposterous assertions that (...) man's perceptions are a mad, disorderly blizzard of phenomena, and the artist alone can bring "order" to the mess. Of course, it is obvious that neither of these notions is very sensible, but the unfortunate truth about the lay psychology of most criticism is that Dr. Peckham's assertions in this connection will probably be regarded as controversial in many departments of literature and fine arts. The author is at his best when barbedly [[sic]] criticizing his colleagues; he is at less than his best, however, when he assumes the mantle of philosophical psychology in order to bring authority to his arguments. Intent upon finding confirmation in both the fashionable and passe schools of behavioral science and philosophy, he masses gluts of aphorisms from Gestalt psychology, Husserl, Heidegger, Susanne Langer, and Paul Ziff (the last pair being very indiscreetly aligned to form notions which are no less intuitive than those of the various art-historians he is admonishing. In the area of psychology, Peckham ignores all of the current approaches, and in the area of philosophy he refers to linguistic analysis or philosophy of science as though each were substantively and methodologically unified, and possessed clear-cut views about the universe. Peckham's central thesis, moreover, leaves one unable to distinguish a work of creative physics from a novel.—E. H. W. (shrink)
People have always been xenophobic, but an explicit philosophical and scientific view of human racial difference only began to emerge during the modern period. Why and how did this happen? Surveying a range of philosophical and natural-scientific texts, dating from the Spanish Renaissance to the German Enlightenment, Nature, Human Nature, and Human Difference charts the evolution of the modern concept of race and shows that natural philosophy, particularly efforts to taxonomize and to order nature, played a crucial role. Smith demonstrates (...) how the denial of moral equality between Europeans and non-Europeans resulted from converging philosophical and scientific developments, including a declining belief in human nature's universality and the rise of biological classification. The racial typing of human beings grew from the need to understand humanity within an all-encompassing system of nature, alongside plants, minerals, primates, and other animals. While racial difference as seen through science did not arise in order to justify the enslavement of people, it became a rationalization and buttress for the practices of trans-Atlantic slavery. From the work of François Bernier to G. W. Leibniz, Immanuel Kant, and others, Smith delves into philosophy's part in the legacy and damages of modern racism. With a broad narrative stretching over two centuries, Nature, Human Nature, and Human Difference takes a critical historical look at how the racial categories that we divide ourselves into came into being. (shrink)
Though it did not yet exist as a discrete field of scientific inquiry, biology was at the heart of many of the most important debates in seventeenth-century philosophy. Nowhere is this more apparent than in the work of G. W. Leibniz. In Divine Machines, Justin Smith offers the first in-depth examination of Leibniz's deep and complex engagement with the empirical life sciences of his day, in areas as diverse as medicine, physiology, taxonomy, generation theory, and paleontology. He shows how these (...) wide-ranging pursuits were not only central to Leibniz's philosophical interests, but often provided the insights that led to some of his best-known philosophical doctrines.Presenting the clearest picture yet of the scope of Leibniz's theoretical interest in the life sciences, Divine Machines takes seriously the philosopher's own repeated claims that the world must be understood in fundamentally biological terms. Here Smith reveals a thinker who was immersed in the sciences of life, and looked to the living world for answers to vexing metaphysical problems. He casts Leibniz's philosophy in an entirely new light, demonstrating how it radically departed from the prevailing models of mechanical philosophy and had an enduring influence on the history and development of the life sciences. Along the way, Smith provides a fascinating glimpse into early modern debates about the nature and origins of organic life, and into how philosophers such as Leibniz engaged with the scientific dilemmas of their era. (shrink)