Introduction to Mathematical 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.

The purpose of this work is to show how logic has developed in Mexican society from the middle of the last century until the present day. This process presents two characteristics: (1) works of logic were generally written by authors who, in spite of their national and international prestige, never succeeded in forming a school, but limited themselves to follow the trends outlined by others; and (2) the greater part of the work about logic was written to cover educational needs and, therefore, has been subject to national education policy changes. On the other hand, it should be taken into consideration that the authors we are going to review may be divided into two main groups: those who have used logic from the perspective of some philosophy (i.e. positivism, neo-Kantianism and Marxism) and those who have spread it within the reference framework of formal contemporary logic, whether ‘mathematical logic’ or ‘symbolic logic’. We are going to find that those writers forming the first group have been, for several reasons, opposed to the introduction of mathematical or symbolic logic.

Alonzo Church. Mathematical logic. Lectures delivered at Princeton University, October 1935–January 1936. Notes by F. A. Ficken, H. G. Landau, H. Ruja, R. R. Singleton, N. E. Steenrod, J. H. Sweer, F. J. Weyl. Mimeographed. Princeton University Mathematics Department, Princeton, N. J., 1936, iii + 113 pp. - Volume 2 Issue 1

A Petri net can provide a graphical model of a concurrent system. They have been used to model such varied systems as resource control, communication networks and manufacturing processes. Computer Science students will, hopefully, learn the theory of Petri nets in a discrete mathematics course or a course on theoretical foundations of Computer Science. They may even see a graph of a sample net in a textbook, but it is unlikely they will have a chance to experiment with a live net. The Interactive Petri Net Simulator (IPNetS) described in this paper provides an easy point-and-click environment for students to design, modify, examine, and execute a Petri net. In seconds any simple Petri net can be created and animated. IPNetS provides display, file handling, and user control functions while a separate execution unit does the actual execution of the net. This flexible design allows for easy experimentation with different execution units and different types of nets. An Ada and a C++ execution unit are contrasted.

The book examines the current state and problems of bioheuristics development, biographical and memetic algorithms, the issues of finding optimal solutions by trajectory algorithms and bioheuristics of multicriteria optimization. Ant and swarm bioheuristics, their features and modifications are presented. Hybrid formic and hypeuruistic swarm segmentation algorithms of complexly structured images are considered as applications. The monograph is interdisciplinary, it is addressed to masters and graduate students studying the theory and practice of creating intelligent information systems and technologies, as well as specialists in the theoretical foundations of computer science, software engineering, system analysis, information management and processing, information systems and processes, bioinformatics.

The journal of symbolic logic began publishing in 1936. From the outset, the Journal included a Reviews Section, edited by Alonzo Church. The very first issue carried three pages of reviews, written by Bernays, Church, Rosser, and Quine.As the first issue stated, “It is intended that this section of the Journal shall serve as a complete bibliography of current literature in the field of symbolic logic, from January 1, 1936. To this end an effort will be made to include in it, at least by title, all publications in this field, both books and articles in journals, and as far as possible these will be accompanied by signed reviews.” Thus the purpose of the Reviews Section was two-fold. For one, there was its bibliographical function: to provide a complete, suitably indexed, listing of all the publications—both books and journal articles—in symbolic logic, wherever and in whatever language published. And secondly, the reviews were to provide critical, analytical commentary on these publications, whenever possible.The Reviews Section should be viewed as a continuation and expansion of Church's A bibliography of symbolic logic. The Bibliography was a listing, with some annotations, of the publications in symbolic logic prior to 1936. It was published as volume 1, number 4 of the Journal. Church made the listing as complete as he could, of course.

Between the World Wars, a robust research community emerged in the nascent discipline of mathematical logic in Warsaw. Logic in Warsaw grew out of overlapping imperial legacies, launched mainly by Polish-speaking scholars who had trained in Habsburg universities and had come during the First World War to the University of Warsaw, an institution controlled until recently by Russia and reconstructed as Polish under the auspices of German occupation. The intellectuals who formed the Warsaw School of Logic embraced a patriotic Polish identity. Competitive nationalist attitudes were common among interwar scientists – a stance historians have called “Olympic internationalism,” in which nationalism and internationalism interacted as complementary rather than conflicting impulses.One of the School’s leaders, Jan Łukasiewicz, developed a system of notation that he promoted as a universal tool for logical research and communication. A number of his compatriots embraced it, but few logicians outside Poland did; Łukasiewicz’s notation thus inadvertently served as a distinctively national vehicle for his and his colleagues’ output. What he had intended as his most universally applicable invention became instead a respected but provincialized way of writing. Łukasiewicz’s system later spread in an unanticipated form, when postwar computer scientists found aspects of its design practical for working under the specific constraints of machinery; they developed a modified version for programming called “Reverse Polish Notation” (RPN). RPN attained a measure of international currency that Polish notation in logic never had, enjoying a global career in a different discipline outside its namesake country. The ways in which versions of the notation spread, and remained or did not remain “Polish” as they traveled, depended on how readers (whether in mathematical logic or computer science) chose to read it; the production of a nationalized science was inseparable from its international reception.

On Expressive Completeness of Duration and Mean Value Calculi (Extended Abstract)

Traditional logic as a part of philosophy is one of the oldest scientific disciplines and can be traced back to the Stoics and to Aristotle. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, and others to create a logistic foundation for mathematics. It steadily developed during the twentieth century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. This book treats the most important material in a concise and streamlined fashion. The third edition is a thorough and expanded revision of the former. Although the book is intended for use as a graduate text, the first three chapters can easily be read by undergraduates interested in mathematical logic. These initial chapters cover the material for an introductory course on mathematical logic, combined with applications of formalization techniques to set theory. Chapter 3 is partly of descriptive nature, providing a view towards algorithmic decision problems, automated theorem proving, non-standard models including non-standard analysis, and related topics. The remaining chapters contain basic material on logic programming for logicians and computer scientists, model theory, recursion theory, Gdels Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. Each section of the seven chapters ends with exercises some of which of importance for the text itself. There are hints to most of the exercises in a separate file Solution Hints to the Exercises which is not part of the book but is available from the authors website.

INTRODUCTION In the most narrow sense, the Polish school of logic may be understood, as the Warsaw school of mathematical logic with Jan Łukasiewicz, Stanislaw Leśniewski, and Alfred Tarski as the leading figures. However, valuable contributions to mathematical logic were also made outside Warsaw, in particular by Leon Chwistek. Thus, the Polish school of logic sensu largo also comprises logicians not belonging to the Warsaw school of logic. The third interpretation is still broader. If logic is not restricted only to mathematical logic, several Polish philosophers who were strongly influenced by formal logical results, for example Kazimierz Ajdukiewicz and Tadeusz Kotarbinski, can be included in the Polish school of logic sensu largissimo . Polish work on logic can therefore encompass a variety of topics, from the ‘hard’ foundations of mathematics (e.g. inaccessible cardinals, the structure of the real line, or equivalents of the axiom of choice) through formal logic, semantics, and philosophy of science to ideas in ontology and epistemology motivated by logic or analysed by its tools. Since the development of logic in Poland is a remarkable historical phenomenon, I shall first discuss its social history, especially the rise of the Warsaw school. Then I shall describe the philosophical views in question, the most important and characteristic formal results of Polish logicians, their research in the history of logic, and applications of logic to philosophy. My discussion will be selective: in particular I will omit most results in the ‘hard’ foundations of mathematics.

In this study, we address the problem of low retention and high dropout rates of computer science university students in early semesters of the studies. Complex and high abstract mathematical learning materials have been identified as one reason for the dropout rate. In order to support the understanding and practicing of core mathematical concepts, we developed a game-based multitouch learning environment in which the need for a suitable learning environment for mathematical logic was combined with the ability to train cooperation and collaboration in a learning scenario. As application domain, the field of mathematical logic had been chosen. The development process was accomplished along three steps: First, ethnographic interviews were run with 12 students of computer science revealing typical problems with mathematical logic. Second, a multitouch learning environment was developed. The game consists of multiple learning and playing modes in which teams of students can collaborate or compete against each other. Finally, a twofold evaluation of the environment was carried out (user study and cognitive walk-through). Overall, the evaluation showed that the game environment was easy to use and rated as helpful: The chosen approach of a multiplayer game supporting competition, collaboration, and cooperation is perceived as motivating and “fun.”

The article conducts research on the fundamental importance and methodological implementation of interdisciplinary connections between mathematics and computer science in the educational process. Attention is focused on the close interweaving of these two disciplines, creating a powerful synergistic effect where mathematics acts as the theoretical foundation, and computer science provides tools for solving complex problems and visualizing abstract concepts. It is emphasized that such an integrated approach is key to forming a holistic scientific worldview in students, as well as developing critical, algorithmic, and computational thinking. The paper provides a detailed analysis of how exactly key mathematical sections provide the theoretical basis for computer science. Specifically, discrete mathematics is the foundation for algorithms and data structures (arrays, trees), mathematical logic is critically important for programming languages and artificial intelligence, graph theory is applied to network modeling and pathfinding algorithms, and probability theory and statistics lie at the core of machine learning, data analysis, and cryptography. This review confirms the thesis that mathematics is essentially "the language computer science speaks". Particular attention is paid to practical methodological approaches that help the teacher effectively use this connection in lessons. Specific educational examples are provided. In particular, the study of arithmetic progression in a mathematics lesson is proposed to be combined with writing a simple code (a Python example is given) that calculates the $n$-th term or the sum of the first $n$ terms of the sequence. Furthermore, the role of computer science in visualizing complex mathematical concepts, such as function graphs using Matplotlib or GeoGebra, as well as the creation of fractals and 3D modeling of geometric bodies using SketchUp or Tinkercad, is highlighted.

Donald Davidson (1984g) famously proposed that what we want from a theory of meaning we may get from a theory of truth. What we want from a theory of meaning for a natural language, that being the case at issue, includes at least an account of the phenomena that give rise to the principle of compositionality: We want an explanation of how the meaning of a complex expression, such as a sentence, depends upon and is determined by the meanings of its syntactic parts. One central advantage of Davidson’s proposal, as he saw it, is that we know from the work of Tarski (1958) how to construct theories of truth for languages with certain sorts of structures, and such a theory does help us to understand how at least some of the semantic properties of complex expressions are determined by those of their parts. Davidson’s practice suggests he may have hoped that all sentences of natural language could be assigned logical forms of the variety familiar from mathematical logic. If so, then semantics would all but reduce to syntax: The only outstanding problem would be to uncover the ‘logical forms’ of the sentences of, say, English. This hope does not now seem reasonable. But the methods available for investigating logical form—that is, for investigating syntax—have become so powerful that it is not at all unreasonable to suppose that the syntactic part of the problem is solvable, at least in principle, and Tarski’s work can be adapted to many of the resulting structures. The work that is left for the semanticist proper then becomes to solve the problems syntax and Tarski do not solve: These include, for example, discovering what sorts of semantic values need to be assigned to novel semantic primitives, such as mass terms, and explaining the semantic significance of novel modes of composition, such as [V PV [CPS]]. It is unfortunate that a proposal with such great advantages faces such grave difficulties. Indeed, it is probably fair to say that most philosophers of language, and perhaps even most working semanticists, regard truththeoretic semantics as a non-starter, for the simple reason that it seems to have the wrong subject-matter. We will not get what we want from a theory of meaning from any theory that does not specify meanings: No theory that does not specify what the sentence ‘snow is white’ means, for example, can possibly explain how this sentence’s meaning what it does is determined by its parts’ meaning what they do. But a theory of truth for English does not specify what even one sentence of English means, since any such theory is consistent with infinitely many different suppositions concerning what any sentence of English means. For example, a theory of truth that proves this T-sentence2 for ‘snow is white’:

After years of discussion, the consensus among academic computer scientists seems to be that undergraduate computer science programs need more mathematical content. This mathematical content is not to be found in areas such as calculus and differential equation but in areas such as discrete mathematics and mathematical logic. The objective of this article is to re‐examine the central role of mathematics in computer science by reviewing the undergraduate computer science curriculum from Curricula ‘68 to the current Computing Curricula 1991. The trend toward a greater concentration in such courses as discrete mathematics will be explored. Furthermore, we collected and put together comments and concerns of three prominent computer scientists about the importance of mathematics in computing science and lack of mathematical content in computer science programs. We then look at one area of computer science amenable to mathematical treatment. This area, software specification, is one of the important phases of the software development. The use of formal methods based on discrete mathematics has been increasingly emphasized recently for software specifications in an effort to produce precise and concise descriptions. We will provide an example to demonstrate the importance of a mathematical approach to arrive at precise specifications and argue against claims that mathematical notations introduce unnecessary complication by pointing out the advantages of and misconceptions about the use of formal notations. The article is concluded by once again emphasizing the integration and applications of mathematics in all areas of computing discipline, especially for software development.

Willard Van Orman Quine (b. 1908–d. 2000) was the main agenda setter in post–World War II philosophy of language and related fields. He was also a highly active logician; if his views on formal and mathematical logic did not shape the field in quite the same way as his work on language, epistemology, and metaphysics, the latter largely sprang from his work on logic and the related domain of the foundations of mathematics. His doctoral thesis (“The Logic of Sequences,” 1933) attempted to clean up and generalize yet boil down the great Principia Mathematica of Bertrand Russell and Alfred North Whitehead (Cambridge, UK: Cambridge University Press, 1910, 1927). Quine’s subsequent works—including A System of Logistic (Cambridge, MA: Harvard University Press, 1934), Mathematical Logic (1940, 1951; Quine 1951, cited under Quine’s Monographs), “New Foundations for Mathematical Logic” (1937, 1953, 1961; Quine 1961, cited under Logic and Set Theory), and Set Theory and Its Logic (1963, 1969; Quine 1969, cited under Quine’s Textbooks)—developed more refined and original ideas on largely the same topics. In philosophy he is known most generally for his commitment to extensionalism, which, for scientific purposes, rules out as inadmissible that two expressions could have the same truth-value or extension or refer to the same object yet not be interchangeable in containing sentences without affecting the truth-values of the containing sentences. Closely related are his celebrated criticisms of the unexplicated reliance on the concepts “meaning,” the “a priori,” and “analyticity” among empiricist philosophers; pressed harder, Quine believed, those notions cannot support the edifices empiricists had envisaged. A scientifically rigorous attitude toward language shows that the notion of the meaning of a sentence, in the intuitive sense that sustains the idea that two correct translations of a given foreign sentence must be in some clear sense be equivalent, must be discarded; meaning in that sense is “indeterminate.” In metaphysics, specifically ontology, he is celebrated for formulating a clear and seemingly rigorous test for ontological commitment in his “On What There Is” (1948; Quine 1961, cited under Ontology): existence is equated with being the value of variable in the first-order predicate calculus. Less well known but equally interesting are his later views concerning the question of what exists; he held that certain considerations undermine the idea that there is a single answer to the question “What is there?” Finally, the notion that epistemology should be “naturalized” and more generally that philosophy itself should be thought of as taking its place within natural science has caused considerable commentary and indeed controversy.