Gödel’s Natural Deduction

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

General proof theory studies how proofs are structured and how they relate to each other, and not primarily what can be proved in particular formal systems. It has been developed within the framework of Gentzen-style proof theory, as well as in categorial proof theory. As Dag Prawitz's monograph "Natural Deduction" (1965) paved the way for this development (he also proposed the term "General Proof Theory"), it is most appropriate to use this topic to celebrate 50 years of this work. The conference took place 27-29 November, 2015 in Tübingen at the Department of Philosophy. The proceedings collect abstracts, slides and papers of the presentations given, as well as contributions from two speakers who were unable to attend.

ADVERTISEMENT RETURN TO ISSUEPREVReviewNEXTBeyond Photovoltaics: Semiconductor Nanoarchitectures for Liquid-Junction Solar CellsPrashant V. Kamat*Prashant V. KamatRadiation Laboratory and Departments of Chemistry & Biochemistry and Chemical & Biomolecular Engineering, University of Notre Dame, Notre Dame, Indiana 46556, United States* To whom correspondence should be addressed. E-mail: [email protected]More by Prashant V. KamatView Biography, Kevin TvrdyKevin TvrdyRadiation Laboratory and Departments of Chemistry & Biochemistry and Chemical & Biomolecular Engineering, University of Notre Dame, Notre Dame, Indiana 46556, United StatesMore by Kevin Tvrdy, David R. BakerDavid R. BakerRadiation Laboratory and Departments of Chemistry & Biochemistry and Chemical & Biomolecular Engineering, University of Notre Dame, Notre Dame, Indiana 46556, United StatesMore by David R. Baker, and Emmy J. RadichEmmy J. RadichRadiation Laboratory and Departments of Chemistry & Biochemistry and Chemical & Biomolecular Engineering, University of Notre Dame, Notre Dame, Indiana 46556, United StatesMore by Emmy J. RadichCite this: Chem. Rev. 2010, 110, 11, 6664–6688Publication Date (Web):October 25, 2010Publication History Received30 July 2010Published online25 October 2010Published inissue 10 November 2010https://doi.org/10.1021/cr100243pCopyright © 2010 American Chemical SocietyRIGHTS & PERMISSIONSArticle Views18304Altmetric-Citations679LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit Read OnlinePDF (3 MB) Get e-AlertsSupporting Info (1)»Supporting Information Supporting Information SUBJECTS:Charge transfer,Electrodes,Oxides,Quantum dots,Semiconductors,Solar cells,Thin films Get e-Alerts

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three dogmas of proof-theoretic semantics are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.

Reprint of: A more general general proof theory

Solving the Beck and Wieland model with optimal experimentation in DualPC

Journal Article TEM Analysis of CsPbBr3 Nanocrystals: Challenges and Perspectives. Get access M Brennan, M Brennan Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN, USA Search for other works by this author on: Oxford Academic Google Scholar Masaru Kuno, Masaru Kuno Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN, USA Search for other works by this author on: Oxford Academic Google Scholar S Rouvimov S Rouvimov Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA Search for other works by this author on: Oxford Academic Google Scholar Microscopy and Microanalysis, Volume 23, Issue S1, 1 July 2017, Pages 2096–2097, https://doi.org/10.1017/S143192761701114X Published: 04 August 2017

The observation of substantially different retention rates of men versus women from initial enrollment as first-year engineering intents to enrollment as engineering majors at the beginning of the sophomore year at the University of Notre Dame has motivated an examination of demographic data to assess retention patterns. Based on the numbers of students enrolling in a required first-year engineering course, which has an initial population of approximately 360 students per year, women have exhibited a lower retention rate to the sophomore year (~50%) than have men (~65%). Demographic data discussed in the present manuscript include: gender; SAT scores; course grades; and intended major. As evidenced by national studies on the retention of women in engineering programs, and as corroborated at Notre Dame, many demographic factors such as SAT scores and course grades do not accurately predict which students will remain in engineering. This necessitated a more thorough examination of student records to look for potential indicators of retention. Most interestingly, the student’s “intended major,” as indicated on the application for admission to Notre Dame, provides a significant indicator of retention. Students who, in their senior year of high school, had indicated an engineering discipline as their intended major on their application for admission remained in the program at a higher rate (~68%) than students who selected a nonengineering discipline as their intended major (~41%). While male students in each category remained in the program at a higher rate than female students in each category, the retention differential between male and female students who selected engineering as their intended was smaller (~9%) than the retention differential between male and female students who selected something other than engineering as their intended major (~20%). Since students who select a major other than engineering on their application for admission comprise approximately 25 percent of first-year students who initially enroll in engineering at Notre Dame, this information is influencing Notre Dame’s efforts to increase retention, particularly among women. Of particular importance, the College now identifies the nonengineering admits, and particularly the women in this group, as at high risk for leaving engineering. The College is, therefore, designing special activities to increase communication with this group to aid them in making appropriate decisions regarding engineering as a potential major.

Full classical S5 in natural deduction with weak normalization

Several groups from the University of Notre Dame are playing an important role in developing the research program at ATLAS. One of their main interests is the study, in collaboration with ANL staff members, of the behavior of nuclei at high spin in the transitional region near A=180 (i.e. the Hg-Pt-Os nuclei), and A=100 (i.e. the Ru-Tc nuclei) with emphasis on shape coexistence and configuration mixing. This group has also participated in many other experiments performed with the BGO gamma-ray facility, especially in the investigation of superdeformation. The {gamma}-ray groups at ANL and Notre Dame have also had collaborative experiments at Gammasphere. The Notre Dame group has built, tested and extensively used a state-of-the-art plunger device for lifetime measurements in conjunction with the ATLAS {gamma}-ray facility. An adapted version of this device is now under construction at Notre Dame, under contract with ANL, for use at the Gammasphere facility.

The Notre Dame Radiation Laboratory is a facility of the Department of Energy operated for DOE by the University of Notre Dame. The majority of the programs within the Laboratory are supported by the Office of Basic Energy Sciences of DOE and, unless otherwise noted in the following, support can be attributed to the Office of Basic Energy Sciences. Certain additional programs, so indicated, are supported by the Office of Standard Reference Data of the National Bureau of Standards. Research programs are briefly described.

Studium Generale: Studies Offered to Astrik L. Gabriel by his Former Students at the Mediaeval Institute, University of Notre Dame, on the Occasion of his Election as an Honorary Doctor of the Ambrosiana in Milan. Edited by <italic>L. S. Domonkos</italic> and <italic>R. J. Schneider</italic>. [Texts and Studies in the History of Mediaeval Education, Number 11.] (Notre Dame, Ind.: Mediaeval Institute,

TributeJohn Thomas Ford, C.S.C. (1932–2021)* Dr. Kenneth L. Parker The field of newman studies suffered the loss of a great scholar and mentor when Fr. John Ford, CSC passed from this life. He died peacefully on 29 December 2021 at Holy Cross House in Notre Dame, Indiana, after a protracted struggle with cancer, and a short bout with COVID, from which he was thought to have recovered. We do not begrudge him the next stage of his journey toward the Beatific Vision; yet, we mourn the loss of his encouraging presence, even as we celebrate the achievements of his life. Though he is best known among us as a leading North American scholar of the life, work, and influence of John Henry Newman, his life story reflects a diversity of interests and the many ways he influenced those who came into contact with him. While this tribute can only touch the surface of his rich and varied life, it is an effort to share with you some of the reasons to be thankful for Fr.John Ford's well-lived life, and to remember with gratitude his generosity of spirit and Christian example. John Thomas Ford was born on 21 November 1932 in Dallas, Texas, and was the only child of Thomas and Leonara (Senn) Ford. He attended elementary school at St. Vincent's School, led by the Sisters of the Holy Cross in Logansport, Indiana, and graduated from Logansport High School in 1950. While attending the University of Notre Dame, Fr. Ford received his call to serve God as a Holy Cross priest and entered the Holy Cross Novitiate in Jordan, Minnesota on 15 August 1951, professed final vows on 16 August 1955, and was ordained a Holy Cross priest on 10 June 1959. After graduating from the University of Notre Dame in 1955 with a bachelor's degree in philosophy, he went on to Holy Cross College in Washington, DC to earn his master's degree in theology in 1959. In 1962, Fr. Ford received his doctorate in theology from the Gregorian University in Rome. In 1962, Fr. Ford taught theology at the University of Notre Dame and served as Prefect in Cavanaugh Hall before he left that fall to teach at Holy Cross College in Washington, DC until 1967. In 1964, he was appointed Assistant Superior at the Foreign Mission Seminary and later became Superior until 1968. From 1968 to 2018, Fr. Ford served on the faculty at Catholic University of America (CUA), in Washington, DC. [End Page 101] I first came to know Fr. Ford toward the end of his years as Chair of the Department of Theology at CUA (1977–1983). I was writing my PhD thesis under the supervision of Eamon Duffy at Cambridge University and anxious to find academic employment. Having applied unsuccessfully for a position in the history of Christian spirituality in his department, I wrote to Fr. Ford to ask advice on how to improve my application materials. He responded with a generous suggestion that we meet when he was in England later that year. He spent a day with me, offering sage advice and encouragement that saw me through the difficult process of beginning an academic career. My experience is but a single example of many lives he touched in that way. As the director of 63 dissertations, 16 on John Henry Newman, he knew the fears and frustrations of the young and managed their struggles with a pastoral touch. A notable dimension of Fr. Ford's life work was his passion for ecumenical dialogue. In the post-conciliar era, he served as a member of the United Methodist/Roman Catholic Bilateral Dialogue (1971–1977), the Faith and Order Commission of the National Council of Churches (1981–2011), the Roman Catholic observer-consultant for the Consultation on Church Union (1974–1988) and its Theology Commission (1982–1985), and the Reformed/Roman Catholic International Consultation (1987–1988). His work in the field of ecumenism was recognized in 2014 by the Franciscan Friars of the Atonement with their Paul Wattson Christian Unity Award. His graduate students benefited from his ecumenical work through his course on The History...

SSNAP: The Solenoid Spectrometer for Nuclear AstroPhysics

Purpose – One of the biggest challenges facing American librarians at present is a mass retirement of highly trained professionals, which will result in a significant loss of expertise. When libraries look at universities for examples of succession planning, they find, for the most part, that they are behind, not ahead of them. A notable exception is the University of Notre Dame, which, as a Catholic institution led by priests, has a very small pool of candidates for high-level positions, so it is forced to select and prepare its top leaders through succession planning. This article aims to examine the work of one such leader. Design/methodology/approach – This article examines the work of one president, Father Theodore M. Hesburgh, whose succession planning efforts resulted in the preparation not only of his successor but also of the members of the administrative team. Findings – This case offers three important lessons for librarians. First, the leadership must be on board with the training program. Second, the training program must go beyond individuals and involve entire cohorts. Finally, the goal of training must be the development of as much talent as possible. Originality/value – The Notre Dame case demonstrates that when an organization looks for talent it finds it, indicating that much talent is lost because it is never recognized, let alone cultivated, and that is the most important lesson of all.