INTRODUCTION In ita original form the present paper was presented to the American Mathematical Society, April 2k,, as a companion piece to the writer's ...

We investigate the computational complexity of deciding whether a given inference rule is admissible for some modal and superintuitionistic logics. We state a broad condition under which the admissibility problem is coNEXP-hard. We also show that admissibility in several well-known systems (including GL, S4, and IPC) is in coNE, thus obtaining a sharp complexity estimate for admissibility in these systems.

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV₁ + dWPHP(PV).

We develop an arithmetical theory and its variant , corresponding to “slightly nonuniform” . Our theories sit between and , and allow evaluation of log-depth bounded fan-in circuits under limited conditions. Propositional translations of -formulas provable in admit L-uniform polynomial-size Frege proofs.

We prove that all axiomatic extensions of the full Lambek calculus with exchange can be axiomatized by formulas on the level of the substructural hierarchy.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Benedict XVI, A Life: Volume 1, Youth in Nazi Germany to the Second Vatican Council 1927–1965 by Peter SeewaldEmil AntonBenedict XVI, A Life: Volume 1, Youth in Nazi Germany to the Second Vatican Council 1927–1965 by Peter Seewald, translated by Dinah Livingstone (London: Bloomsbury Continuum, 2020), xi + 500 pp.What better way to spend Pope Benedict XVI's ninety-fourth birthday than by reviewing a Ratzinger biography while having Apfelstrudel (...) and some traditional Bavarian beer?German journalist and author Peter Seewald, who in his many interviews has asked Joseph Ratzinger / Pope Benedict XVI a total of two thousand or more questions (ix), has written a splendid biography that surprises even someone who has already read dozens of books by and about Benedict XVI. Written in an extremely enjoyable style and divided into chapters of just the perfect length, this page-turner deserves the highest praise. It will likely take its place alongside George Weigel's famous (and equally massive) biography of Pope John Paul II, Witness to Hope.The German biography has been divided into two volumes in English (vol. 2 was just published in November of 2021). Volume 1 deals with the years 1927–1965, dominated by two historic "Twos": World War II and Vatican II. Theologically, this is the period of the progressive Ratzinger (regardless of whether he "converted" to a more conservative stance later on), or as Seewald puts it, a man formed by "liberal Bavarian Catholicism" (124).Seewald provides lots of new and interesting information about Joseph Ratzinger, both bitter and sweet. One example from the early chapters concerns the origins of Ratzinger's mother. It turns out that she was an illegitimate child and not actually from South Tyrol, contrary to what her children had thought. A local historian has discovered that Ratzinger's mother Maria was actually born in Mühlbach "in the district of Rosenheim,... in a house that specialized in helping unmarried pregnant women to give birth" (15).For the romantic, the sweetest story of the book comes when Seewald [End Page 963] pushes the reserved Pope Emeritus to reveal more about the "painful decisions" that the young Munich seminarian had to make in relation to his vocation as a celibate priest. "'Were you in love with a girl?' 'Perhaps.' 'So, yes?' 'It could be interpreted like that.' 'How long did this passionate time last? A few weeks? A few months?' 'Longer'" (219). Ratzinger was in love! Or at least it could be interpreted like that! Other heart-warming details include the revelation that Ratzinger's childhood teddy from Marktl followed him all the way to the Apostolic Palace, as well as the fact that, when his blind brother Georg (now deceased) needed a housekeeper, Benedict XVI personally organized the job interview in the Vatican gardens.A darker episode, minimized in Ratzinger's autobiographical memoirs, concerns his role in the radar crew of the German anti-aircraft forces in World War II. The future Pope was entrusted with the task of targeting British bombers. Seewald fills in the narrative: "Ratzinger delivered readings for three guns,.... The more precisely Joseph measured, the higher the hit rate. As the war demanded" (101). Years later, Ratzinger confessed that these experiences still occasionally gave him sweaty nightmares.Seewald goes on to describe the exceptional postwar context in which Ratzinger prepared for the priesthood. The future priests knew that they would soon have to hear confessions of ex-Nazis who had shot people in the war (153). Indeed, after the war, many old Nazis turned to the Church and attempted to get de-Nazification certificates. Ratzinger's parish priest even joked that one day "they will say the only Nazis were the priests" (131).As did everyone, Ratzinger also felt the effects of poverty in postwar Germany. Whenever a festive dinner had to be given for a visitor at the seminary, "we went hungry on the most meagre rations for four long weeks in order to gather the ingredients" (189). After a short period of pastoral ministry subsequent to his ordination, the stellar theologian went back to the seminary as a teacher. Barely twenty-five-years-old, he was younger... (shrink)

Vorwort.--Schultz, F. Die philosophisch-weltanschauliche entwicklung der literarhistorischen methode.--Gumbel, H. Dichtung und volkstum.--Cysarz, H. Das periodenprinzip in der literaturwissenschaft.--Petersen, J. Die literarischen generationen.--Medicus, F. Das problem einer vergleichenden geschichte der künste.--Petsch, R. Die analyse des dichtwerkes.--Muschg, W. Das dichterporträt in der literaturgeschichte.--Jung, C. G. Psychologie und dichtung.--Ermatinger, E. Das gesetz in der literaturwissenschaft.--Nadler, J. Das problem der stilgeschichte.--Wundt, M. Literaturwissenschaft und weltanschauungslehre.--Strich, F. Weltliteratur und vergleichende literaturgeschichte.--Sarnetzki, D. H. Literaturwissenschaft und die dichtung und kritik des tages.

Excerpt from Grundlegung der Allgemeinen Kunstwissenschaft Tiefsinn ist ein Anzeichen des Chaos, das echte Wissenschaft in einen Kosmos verwandeln will, in eine einfache, vollig klare, aufgeloste Ordnung. Echte Wissenschaft kennt, soweit ihre wirkliche Lehre reicht, keinen Tiefsinn. Jedes Stuck fertiger Wissenschaft ist ein Ganzes von den Denkschritten, deren jeder unmittelbar einsichtig, also gar nicht tiefsinnig ist. Tiefsinn ist Sache der Weisheit, begriffliche Deutlichkeit und Klarheit Sache der strengen Theorie. Die Ahnungen des Tiefsinns in eindeutige rationale Gestaltungen umzupragen, das ist der (...) wesentliche Prozel 3 der Neukonstitution strenger Wissenschaften Diese Worte Edmund Husserls stelle ich an den Anfang meines Buches; nicht als Versuch einer Rechtfertigung oder gar Entschuldigung, sondern als Programm. Sie sollen zugleich die Schar derer vor Enttauschungen Warnen, die geistschillernde Betrachtungen uber Kunst und Kunstler erwarten; gluhende Hymnen uber das Mysterium der Kunst, uber das unbegreifliche Wunder schopferi scher Kraft. Nicht um Mysterien und Wunder handelt es sich mir, sondern um ernste und exakte wissenschaftliche Erkenntnis, um die Gewinnung von Grundlagen auf einem Gebiet, das - zwischen der Asthetik und den historischen Kunstdisziplinen gelegen bisher fast ganzlich die angemessene Forschungseinstellung entbehrt hat. So wendet sich dieses Werk einerseits an Philosophen und anderseits an die Vertreter der Kunstgeschichte, Literaturgeschichte usw. Beide Parteien mussen verzeihen, wenn manche Ausfuhrungen fur ihre Bedurfnisse zu breit angelegt sind. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.". (shrink)