A First Course in Logic

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

In this paper, we share our experiences concerning the introduction of the Android-based self-study app AXolotl within the first-semester logic course offered at our university. This course is mandatory for students majoring in Computer Science and Artificial Intelligence. AXolotl was used as part of an optional lab assignment bridging clausal reasoning and SAT solving with classical reasoning, proof construction, and first-order logic. The app provides an intuitive interface for proof construction in various logical calculi and aids the students through rule application. The goal of the lab assignment was to help students make a smoother transition from clausal and decompositional reasoning used earlier in the course to inferential and contextual reasoning required for proof construction and first-order logic. We observed that the lab had a positive influence on students' understanding and end the paper with a discussion of these results.

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.

Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.

We now discuss the model theory of (classical) first-order logic together with partial interpretations. As in the propositional case, the syntax of partial firstorder logic is essentially the same as in the classical case. The main difference from classical first-order logic is the model theory, and the resulting formal axiomatics. The reader is referred e.g. to Smullyan [94] for a comprehensive discussion of classical first-order logic.

This is a textbook based on an interdisciplinary (advanced undergraduate level) course that was originally taught by the three authors at Duke. It is intended for many different disciplines. Even though it covers mathematical and philosophical topics, it occasionally still feels like a textbook whose origins lie more in computer science than in the other disciplines. This isn’t even surprising, for an interdisciplinary focus on logic easily draws one towards computer science and artificial intelligence. In three parts, each written by one of the authors, a three-chapter overview is given of the topics of proof-theory, computability-theory and philosophical logic. Yet, rather than a juxtaposition of logical, mathematical, computational, and philosophical subjects, this textbook isn’t merely intended as an illustration of how logic is relevant for many different disciplines, but forms the basis for a different kind of introduction to logic. This reviewer surely applauds the intent to teach logic to philosophers more like we would teach it to computer scientists—with an emphasis on what it allows us to achieve, but also to teach it to computer scientists more like we would (or ought to) teach it to philosophers—with more attention to the diversity in logic than to the dominant background role of classical logic. While Part 1 is concerned with proof-theory, with chapters on classical propositional logic, first-order logic, and Prolog, it also double-tasks as an introduction to formal logic. Thus, chapter 1 reviews the language and semantics of propositional logic, and even goes on to prove two basic meta-theoretical results by

The role of games in computing science has been given some consideration in the literature. We aim to design games that can be used for teaching logic. Specifically, we build on the Minefield game used in the Stanford Intro to Logic course, and present a version, which we call MineFOL, that incorporates time constraints. We also move beyond the Stanford examples in that we provide a formalisation of the game, with a view to eventually generating game instances automatically as well as from user input, and show how MineFOL can be used as a learning tool. We also developed an online platform supporting learners in at least three ways. First, the learners can practice reasoning in First Order Logic (FOL) and proving strategies such as resolution through various MineFOL games. Second, the learners can practice formalisation in FOL, by allowing learners to build their own games. Third, the learners become aware of the need of interleaving various technologies to solve a logicbased task. In this line, MineFOL is formalised in such way to allow augmenting reasoning in FOL with search strategies.

We asked all of the liberal arts faculty who advise undergraduates on course selection at the 14 colleges in the Associated Colleges of the Midwest a series of questions regarding their perceptions of the personality traits of Computer Science (CS) students, topics they think are covered in CS classes, and their overall impressions of CS. Our goal was to assess empirically the hypotheses that (1) many non-CS faculty do not really know what computer science is, and (2) many non-CS faculty are unaware of the differences between CS and Information Technology (IT). We received over 250 survey responses, which revealed that, among non-CS faculty, only 9% disagree or are neutral that CS should even be part of a liberal arts curriculum, but 32% think CS students learn to administer computers and computer networks, while 34% believe that CS students are taught to use Microsoft products in the classroom. However, over 95% of non-CS faculty also recognize the importance of both programming and algorithms to the study of computer science. The overall data suggests that a majority of non-CS faculty in the Associated Colleges of the Midwest have a a basic understanding of CS, while the remainder have an overly broad definition of CS (i.e. they think that CS includes IT).

This paper outlines the effectiveness of an undergraduate mentoring program offered through the Department of Computer Science at the University of Maryland in retaining historically marginalized undergraduate students in science, technology, engineering, mathematics, and computer science (STEM-CS) majors. Each undergraduate teacher/mentor (called an Ambassador) is responsible for providing support for current university students or for K-12 computing outreach programs. Enrollment and survey data show that the Ambassador program is successful in retaining undergraduate computing majors in STEM-CS. Ambassadors indicated that they appreciated the community aspect of the Ambassador program and they felt empowered to solve diversity problems in STEM-CS.

Many researchers regarded that problem solving skill is one of the most important learning outcomes to help students adapt to the learning and working environment (Xiao et al, 2019). Through reading a large number of articles and experimental research, this study finds that currently the development of and research on problem-solving skill in university students who are major in computer science in digital logic course in Inner Mongolia is limited, especially in the context of STEM education. STEM education is currently widely valued in the K12 stage, but the STEM literacy of college students also needs to be paid attention to. There is an important need for developing university students’ problem-solving skill systematically and effectively, revealing the importance of this research. In the research field, process of engineering design was designed to developing students’ problem-solving skill. In order to have an in-depth understanding of the research questions, mixed methods were adopted. Students were divided into 2 groups: Experimental Group and the Control Group. The participants were 1 teacher and 140 students at university second-year at a university in Inner Mongolia. To collect reliable and valid data, standardized test of problem-solving skill, teacher interview form was designed. Independent T-test was used to analyze the quantitative data.

In the context of higher education strongly transforming itself under the impact of technology, Artificial Intelligence (AI) has become an important support tool in the learning process, especially for Computer Science (CS) students. Integrating a structured literature review and conceptual analysis, this paper evaluates the role of AI as a self-learning assistant, focusing on outstanding benefits such as personalized learning, source code debugging, and explaining complex algorithms, as well as challenges related to academic integrity, ethics, and independent thinking skills of students. Through analysis, the paper proposes a comprehensive recommendation framework, including: (1) an in-depth analysis of the five most effective AI tools; (2) smart strategies for AI-use through Prompt Engineering; (3) new pedagogical approaches for instructors; and (4) support policies from institutions. The results indicate that although AI cannot substitute human instructors completely, it can act as a powerful lever to enhance self-learning capabilities. When applied appropriately, it contributes to elevating the quality of Computer Science education in universities. Keywords: AI-assisted learning, Artificial Intelligence, Computer Science students, higher education, self-learning assistant.

Full classical S5 in natural deduction with weak normalization

Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL. Our formalization is based on work by Stefan Berghofer, which we have since updated to use Isabelle's declarative proof style Isar (Archive of Formal Proofs, Entry FOL-Fitting, August 2007 / July 2018). We represent variables with de Bruijn indices; this makes substitution under quantifiers less intuitive for a human reader. However, the nature of natural numbers yields an elegant solution when compared to implementations of substitution using variables represented by strings. The sequent calculus considered has the special property of an always empty antecedent and a list of formulas in the succedent. We obtain the proofs of soundness and completeness for the sequent calculus as a derived result of the inverse duality of its tableau counterpart. We strive to not only present the results of the proofs of soundness and completeness, but also to provide a deep dive into a programming-like approach to the formalization of first-order logic syntax, semantics and the sequent calculus. We use the formalization in a bachelor course on logic for computer science and discuss our experiences.

Abstract. This article explores the possibilities and advantages of using the Python programming language together with Arduino robotics kits in the process of training future computer science teachers in pedagogical educational institutions. Considerable emphasis is placed on analyzing the principles of operation of the simplest programs and devices that can be used to visualize the possibilities of using the Arduino platform and Python libraries to create robots that can be implemented in the educational process and everyday life. The authors have developed the program code and presented a working model of a robot assembled and programmed on the basis of Arduino components to measure humidity in computer laboratories and create automatic plant irrigation systems to maintain proper conditions in classrooms. The article presents an example of integrating the learning of the Python programming language with the use of Arduino robotics kits. This method is aimed at improving the quality of training of future computer science teachers, expanding their methodological arsenal and ensuring the ability to teach students using innovative means. The results of the study and the developed teaching and learning materials will help to interest students in STEM education and prepare a new generation of computer science teachers for the challenges of the modern technological world, expand their methodological arsenal and develop the ability to use integrated technical, engineering and mathematical solutions to solve theoretical and practical problems. In addition, this study can serve as a guide to popularize the use of Arduino and Python in educational institutions, and encourage future computer science teachers to implement STEM and hands-on teaching methods, which will contribute to better student learning and improve the quality of professional education in the field of information technology in general. In general, the study opens up a wide range of opportunities for using modern digital technologies in professional education and stimulates interest in STEM education among computer science teachers and students, thereby contributing to the development of a new generation of engineers and technology leaders. Further research could be aimed at developing specific training courses and methods for integrating the Python programming language and Arduino robotics kits into STEM education. It is important to create interdisciplinary STEM courses with the participation of teachers of computer science, physics, mathematics, vocational education in the IT industry, etc.