Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G*0 and G0, (...) in which cuts are restricted to quantifier-free formulas, and prove that the Σqj-witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G*0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb0-CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G*0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID. (shrink)

The papers in this volume, first published in 1982, deal with a number of different aspects of Marx's ideas and the varying constructions put on them by later Marxists. Based on a lecture series, they examine Marxist views of the nature of philosophy, of history and historical explanation, the role and importance of politics, and of literature and the place of ethics. Among the Marxists considered are Lukacs, Sartre, Habermas, Althusser and Macherey. A continuous concern through the volume is the (...) relation within Marxist thought of fact and value, with the implications that will have for our actions. (shrink)

A model for the ideal state includes discussions of the nature and application of justice, the role of the philosopher in society, the goals of education, and the effects of art upon character.

In "Realism and Reason" Hilary Putnam has offered an apparently strong argument that the position of metaphysical realism provides an incoherent model of the relation of a correct scientific theory to the world. However, although Putnam's attack upon the notion of the "intended" interpretation of a scientific theory is sound, it is shown here that realism may be formulated in such a way that the realist need make no appeal to any "intended" interpretation of such a theory. Consequently, it can (...) be shown that realism is immune to Putnam's criticism and that attempts at reformulating this criticism are not likely to meet with success. (shrink)

The influence of Nazism on German culture was a key concern for many Anglo-American writers, who struggled to reconcile the many contributions of Germany to European civilization, with the barbarity of the new regime. In _German Literature Through Nazi Eyes_, H.G. Atkins gives an account of how the Nazis undertook a re-evaluation of German literature, making it sub-ordinate to their own interests. All reference to Jewish writers and influence was virtually eliminated, and key writers such as Goethe and Lessing were (...) re-interpreted. What was left was a military history that was avowedly militant and propagandist. (shrink)

1. "The map of logic." Comparatively recently, Kant's words to the effect that in the two thousand years since Aristotle logic had made "not a single step forward and, all things considered, it seems to be a fully finished and completed discipline" used to be quoted widely and not unsympathetically. Today, however, there are works about logic in which the listing of logical disciplines runs into the dozens. In this regard the attempt by the American logician N. Rescher to compile (...) a diagram or "map of logic," in which the list of the branches of logic takes up several pages, is noteworthy. Also characteristic is the following statement by one of the major modern representatives of mathematical logic, S. Kleene: "Since the discovery of non-Euclidean geometries by Lobachevsky and Bolyai , it has become clear that different systems of geometries are conceptually equally possible…. Identically, there are different logics.". (shrink)