Visual Theorem Proving with the Incredible Proof Machine

‘Natural deduction’ is natural in virtue of the way that it allows formal proofs to closely mirror the structure of familiar ordinary language arguments.1 For example, the sentences of the ordinary language argument (1) correspond one-to-one with the lines of the proof (2): (1) Whoever committed the murder left by the window. Anyone who left by the window would have had mud on his shoes. Suppose that the butler committed the murder. Then he left by the window. In that case, he has mud on his shoes. So if the butler committed the murder, he has mud on his shoes.

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.

The conception of the very first decision procedures for first-order sentences brought about the need for giving citizenship to function symbols (e.g. Skolem functions). We argue that a closer look at proof procedures for first-order sentences with equality brings about the need for introducing (function) symbols for rewrites. This is appropriately done via the framework of labelled natural deduction which allows to formulate a proof theory for the “logical connective” of propositional equality. The basic idea is that when analysing an equality sentence into (i) proof conditions (introduction) and (ii) immediate consequences (elimination), it becomes clear that we need to bring in identifiers (i.e. function symbols) for sequences of rewrites, and this is what we claim is the missing entity in P. Martin-Löf’s equality types (both intensional and extensional). What we end up with is a formulation of what appears to be a middle ground solution to the ‘intensional’ versus ‘extensional’ dichotomy which permeates most of the work on characterizing propositional equality in natural deduction style. (Part of this material was presented at the Logical Methods in the Humanities Seminar, Stanford University, and the authors would like to thank Solomon Feferman and Grigori Mints for their comments and suggestions.)

This is a companion to a paper by the authors entitled “Godel on deduction”, which examined the links between some philosophical views ascribed to Godel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Godel presented with the help of Gentzen’s sequents, which amounts to Jaśkowski’s natural deduction system of 1934, and which may be found in Godel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Godel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Godel’s Notre Dame course, and is followed by comments concerning Godel’s natural deduction system.

The treatment of propositional logic, and of all following logical systems, is firmly based on a precise and systematic introduction of its language and syntax. Simple as the language may be, the reader will benefit from the systematic use of inductive and recursive procedures employed in this book. It would be hard and inconvenient to establish important properties of the language with a mathematical precision without recourse to the suitable induction principles. As a rule it is a rewarding experience for newcomers to see that what should be the case in formal logic, can indeed be simply proved. The semantics of propositional logic is based on the traditional truth values 0 and 1 (false and true). A rich variety of important and useful principles are established with surprising ease by inductive means. On the whole, the reader will be surprised how much milage one gets out of a well-chosen methodology. Once the notion of truth is made clear, one turns to formal proof methods. The system used here is Gentzen’s Natural Deduction. If ever the adjective “natural” was well chosen, it was here. Natural Deduction allows students to concoct their own derivations without any intricate tricks. Even the completeness theorem—true = derivable—becomes simple in the present setting. Here and in all later chapters, numerous exercises present the reader with occasions to apply the given tools.

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.

In C#, local variables must be given names that allow all variables to be uniquely identified throughout the method.

14 Hybrid logics

Conceptualizations of teacher knowledge have shifted to focusing on the role of experiential rather than theoretical knowledge. There are different approaches to this, but the idea of an image that guides practice is widespread. One approach to images that has not been frequently investigated in studies of second language teachers is through sociocultural theory, specifically in the contribution of activity theory known as orienting activity. In this view, images mediate teaching actions by linking theoretical knowledge and practical experience. This article will use orienting activity to examine the role that images play in an English as a second language teacher’s personal practical knowledge by drawing on classroom observations, informal discussions, simulated recall interviews and documentary evidence collected over a nine month period. Analysis of the data shows that several key images mediated the teacher’s classroom actions. These emerged in the observations and in the teacher’s commentary on planned and spontaneous, routine and novel and successful and unsuccessful activities. The images appeared to have developed through both tacit knowledge associated with practical experience and theoretical knowledge associated with research and theory. This demonstrates the usefulness of orienting activity as a theoretical framework for exploring classroom practices and teacher development through the links it can make between theory and practice.

Currently, there is little evidence about how non-English-speaking students learn computer programming. For example, there are few validated assessment instruments to measure the development of programming skills, especially for the Spanish-speaking population. Having valid assessment instruments is essential to identify the difficulties of the students, make formative decisions, and evaluate the impact of pedagogical interventions. This article presents the outcomes of the early steps toward a validation in Spanish of the SCS1 assessment, derived from the first conceptual inventory to measure programming knowledge decoupled from a particular language. The goal of this initial step is twofold: (1) identifying the questions that are too difficult in this version and might need to be adapted based on empirical evidence of how students answer them and (2) identifying students’ common difficulties with the most challenging questions. This study started with a “think-aloud” protocol to validate whether the Spanish version was similar to the original instrument and if the students understood the translation as intended. Then, the instrument was applied to a sample of 71 university students who had previously taken an introductory programming course using Python. We computed the difficulty index for each question and identified students’ difficulties in answering them. We also propose potential improvements for these items. The main contributions of this study include (1) progress toward the validation of an instrument for evaluating programming skills in Spanish, (2) noting the advantage that experienced students in Python have to complete the instrument, (3) identifying the tendency of students to translate the pseudo-code from Spanish into English, and (4) the identification of common mistakes, like misunderstanding the scope of local variables or the confusion between returning and printing.

Regrettably, it is not the case that anyone with interest in obesity can just set up and start running an NHS-funded obesity service. Services need to be commissioned, and although there is guidance on who should take responsibility for the commissioning of each tier of intervention (Department of Health 2014), together with guidance on what should be included in these services (NHS Commissioning Board 2012), there is a great deal of scope for local variability. However, some of the barriers and challenges to setting up and running an effective weight management service are common to all services however they are commissioned.

Since the nineteenth century, Greece has been involved in efforts to improve the quality of teacher education. However, the salient characteristic of teacher preparation has been a disposition to disengage theory from practice and to disjoin the university from classroom experience. Taking the best of what is scientifically derived and historically understood, reformers of Greek education must develop a new vision for teacher preparation, one that involves isolation neither from the academy nor from everyday pedagogical and practical experience in the classroom. Teachers are simultaneously scholars and practitioners; thus their preparation should develop within a professional community that respects diverse ways of knowing and doing.

Professional knowledge in teaching is broadly based on personal knowledge. Hence, it is important to build teachers’ development out of their initial knowledge. The idea of a sociogenesis of educational knowledge, teacher knowledge and training models as well as teaching models are the basis of this study. It aims to diagnose the knowledge attested by four teachers, how the components of this knowledge relate to the events and experiences of their cultural and musical education, and how they explain their professional knowledge. In-depth interviews with four teachers were carried out. The interview content and some features of the vocabulary were analysed. The results show that two teachers have a wide range of interrelated knowledge, although it is apparently unrelated to their classroom experience. The other two teachers have a ‘closed’ cognitive system, which is centred on their own practical experience.

Reflective practice has potentially positive effects on an organization’s capacity to focus on student learning and teaching practices. In an effort to comply with policy and provide teachers with opportunities to reflect on their practice, districts, schools, and teachers have turned to various models that feature collaborative experiences. One such model is the Critical Friends Group (CFG), focusing on the improvement of practice and learning experiences for children, using protocol-driven structured conversations. This article examines the mechanisms that support and sustain reflective practice within a community of practice and utilizes Wenger’s three dimensions of practice – mutual engagement, joint enterprise, and shared repertoire. The study followed a CFG in an elementary school using ethnographic case study methods to conduct a close examination of CFG meetings, protocol use, and analysis of teacher interviews. The propensity to ‘look out’ at school-wide issues vs. looking in at classroom practices, perceptions about feedback on classroom practice, and using protocols to guide conversation emerged from the data as prominent mechanisms in communities of practice. CFG facilitators and coaches, as well as participants must be explicit about the purposes of their work and protocol use alone does not ensure a focus on student learning and teaching practices.

Logic of agreement: foundations, semantic system and proof theory