scholarly journals Growing into deduction

2020 ◽  
Vol 63 (1) ◽  
pp. 87-106
Author(s):  
Jovana Kostic ◽  
Katarina Maksimovic
Keyword(s):  
Mathematical Logic ◽  
20Th Century ◽  
Natural Deduction ◽  
Deductive Reasoning ◽  
Formal System ◽  
Formal Systems ◽  
Adults And Children ◽  
Logical Connectives

Psychologists have experimentally studied deductive reasoning since the beginning of the 20th century. However, as we will argue, there has not been much improvement in the field until relatively recently, due to how the experiments were designed. We deem the design of the majority of conducted experiments inadequate for two reasons. The first one is that psychologists have, for the most part, ignored the development of mathematical logic and based their research on syllogistic inferences. The second reason is the influence of the view, which is dogmatically still prevalent in semantics and logic in general, that the categorical notions, such as the notion of truth, are more important than the hypothetical notions, such as the notion of deduction. The influence of this dogma has been twofold. In studies concerning logical connectives in adults and children, much more emphasis has been put on the semantical aspects of the connectives - the truth functions, than on the deductive inferences. And secondly, even in the studies that investigated deductive inferences by using formal systems, the dogma still influenced the choice of the formal system. Researchers, in general, preferred the axiomatic formal systems over the systems of natural deduction, even though the systems of the second kind are much more suitable for studying deduction.

Modality and folk psychology

2019 ◽  
Vol 28 (1) ◽  
pp. 19-27
Author(s):  
Ja. O. Petik
Keyword(s):  
Modal Logic ◽  
Brain Activity ◽  
Formal System ◽  
Philosophy Of Logic ◽  
Psychological States ◽  
Formal Systems ◽  
Modern Psychology ◽  
Philosophical Interpretation ◽  
Important Direction

The connection of the modern psychology and formal systems remains an important direction of research. This paper is centered on philosophical problems surrounding relations between mental and logic. Main attention is given to philosophy of logic but certain ideas are introduced that can be incorporated into the practical philosophical logic. The definition and properties of basic modal logic and descending ones which are used in study of mental activity are in view. The defining role of philosophical interpretation of modality for the particular formal system used for research in the field of psychological states of agents is postulated. Different semantics of modal logic are studied. The hypothesis about the connection of research in cognitive psychology (semantics of brain activity) and formal systems connected to research of psychological states is stated.

AN INTENSIONAL LEIBNIZ SEMANTICS FOR ARISTOTELIAN LOGIC

2010 ◽  
Vol 3 (2) ◽  
pp. 262-272 ◽  
Author(s):  
KLAUS GLASHOFF
Keyword(s):  
Natural Deduction ◽  
Predicate Logic ◽  
Formal System ◽  
Aristotelian Logic ◽  
Categorical Logic ◽  
Characteristic Numbers ◽  
Formal Syntax ◽  
Philosophical Standpoint ◽  
Classical Philosophy

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” concepts—corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle’s theory via predicate logic.

scholarly journals Preface

2011 ◽  
Vol 21 (4) ◽  
pp. 671-677 ◽  
Author(s):  
GÉRARD HUET
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
20Th Century ◽  
Theorem Proving ◽  
Propositional Calculus ◽  
Interactive Theorem Proving ◽  
Predicate Calculus ◽  
Special Issue ◽  
Mathematical Structures ◽  
Infinite Domains

This special issue of Mathematical Structures in Computer Science is devoted to the theme of ‘Interactive theorem proving and the formalisation of mathematics’.The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements.

Mathesis

2020 ◽  
Author(s):  
Baylee Brits
Keyword(s):  
Martin Heidegger ◽  
20Th Century ◽  
Edmund Husserl ◽  
Cultural Theory ◽  
Formal System ◽  
Bertrand Russell ◽  
Symbolic Logic ◽  
Universal Method ◽  
Ernst Cassirer ◽  
Mathesis Universalis

Mathesis universalis is perhaps the ultimate formal system. The fact that the concept ties together truth, possibility, and formalism marks it as one of the most important concepts in Western modernity. “Mathesis” is Greek (μάθησις) for “learning” or “science.” The term is sometimes used to simply mean “mathematics”; the planet Mathesis, for instance, is named after the discipline of mathematics. It is philosophically significant when rendered as “mathesis universalis,” combining a Latinized version of the Greek μάθησις (learning) with the Latin universalis (universal). The most significant modern philosophers to develop the term were René Descartes (1596–1650) and Gottfried Leibniz (1646–1716), who used it to name a formal system that could support a project of scientia generalis (Descartes) or the ars combinatoria (Leibniz). In each case, mathesis universalis is a universal method. In this sense it does not constitute the content of the sciences but provides the formal system that undergirds no less than the acquisition and veracity of knowledge itself. Although mathesis universalis is only rarely mentioned in the literature of Descartes and Leibniz, philosophers including Edmund Husserl, Ernst Cassirer, and Martin Heidegger considered it one of the key traits of modernity, breaking with the era of substance (Rabouin) or resemblance (Foucault) to signal a new period defined by formalism and quantification. Thus, in the 20th century, the scant and often contradictory literature on mathesis actually produced by the great philosophers of the Enlightenment comes to take on an importance that far exceeds the term’s original level of systematic elaboration. The term mathesis universalis was rarely used by either Descartes or Leibniz, and the latter used many different terms to refer to the same concept. The complexity and subtlety of the term, combined with difficulties in establishing a rigorous systematic interpretation, has meant that mathesis universalis is often used vaguely or to encompass all scientific method. It is a difficult concept to account for, because although many philosophers and literary theorists will casually refer to it, often in its abbreviated form (Lacan references mathesis in opposition to poesis to contrast the procedures of the sciences and the arts, for instance), there is not a great deal of consistent theoretical elaboration of the term in literary and cultural theory. Although mathesis universalis is not simply an avatar of mathematics, it is difficult to establish exactly where maths ends and mathesis begins, so to speak. The distinction is murky in both Descartes’s and Leibniz’s work, and this ambiguity would become a key controversy surrounding the term in the 20th century, with Bertrand Russell arguing that the significance of symbolic logic to mathesis universalis prevented it from being a “premier” science. Along with Russell, Ernst Cassirer and Louis Couturat would contest the relation between symbolic logic and the symbolic algebra of mathesis universalis, providing the terms of the debate for 20th-century philosophical work on ontology. Mathesis universalis was also a source of debate and controversy in the 20th century because it provided a node from which to examine the status of scientific truth. It is the work of 20th-century philosophers that expanded the significance of the term, using it to exemplify aspects of Enlightenment thought that many philosophers wished to react against, namely the aspiration to a universal science and the privileging of formal systems as avenues to truth. In this respect, the term is associated with Edmund Husserl, Martin Heidegger, and especially Michel Foucault, whose extensive work on the “classical episteme” provided a popular method of characterizing the development and enduring features of Enlightenment science. Although Foucault’s rendering of mathesis universalis as a “science of calculation” in The Order of Things (1970) is the most commonly used definition in literary and cultural studies, debates centering on Leibniz’s work in the early 20th century suggest that critics still took divergent approaches to the definition and significance of the term. It is Foucault who has popularized the contraction of the term to “mathesis.”

Discrete mathematics

2019 ◽  
Author(s):  
Виктор Ходаков ◽  
Viktor Hodakov ◽  
Надежда Соколова ◽  
Nadezhda Sokolova
Keyword(s):  
Mathematical Logic ◽  
Discrete Mathematics ◽  
Historical Information ◽  
Fractal Sets ◽  
Formal Systems ◽  
Teachers And Students ◽  
Basic Concepts ◽  
Theoretical Results

The training guide is devoted to the presentation of the foundations of discrete mathematics. The main sections are presented: theories of sets, mathematical logic, relations, formal systems, algorithms, algebras, combinatorics, graphs, fractal sets. The theoretical material is illustrated by a large number of examples. The guide includes not only basic concepts and theoretical results, but also methods and algorithms for solving applied tasks. Addressed primarily to teachers and students of higher technical universities, but it can be useful to those who are wish to study it independently. With this goal, a large number of tasks are included, for the knowledge control and the system of assessing them. In each section there are two-level test tasks. The first level – tests to check the compulsory minimum of knowledge, the second – tasks of full complexity. The historical information about scientists who contributed to the development of discrete mathematics is given.

Degrees of formal systems

1958 ◽  
Vol 23 (4) ◽  
pp. 389-392 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Sufficient Conditions ◽  
Formal System ◽  
Independent Interest ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Axiomatizable Theory

In this paper we answer some of the questions left open in [2]. We use the terminology of [2]. In particular, a theory will be a formal system formulated within the first-order calculus with identity. A theory is identified with the set of Gödel numbers of the theorems of the theory. Thus Craig's theorem [1] asserts that a theory is axiomatizable if and only if it is recursively enumerable.In [2], Feferman showed that if A is any recursively enumerable set, then there is an axiomatizable theory T having the same degree of unsolvability as A. (This result was proved independently by D. B. Mumford.) We show in Theorem 2 that if A is not recursive, then T may be chosen essentially undecidable. This depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest.Our second result, given in Theorem 3, gives sufficient conditions for a theory to be creative. These conditions are more general than those given by Feferman. In particular, they show that the system of Kreisel described in [2] is creative.

G. Kreisel. On the concepts of completeness and interpretation of formal systems. Fundamenta mathematicae, vol. 39 (for 1952, pub. 1953), pp. 103–127. - G. Kreisel. Applications of mathematical logic to various branches of mathematics. Applications scientifiques de la logique mathématique, Actes du 2e Colloque International de Logique Mathématique, Paris — 25–30 août 1952, Institut Henri Poincaré, Collection de logique mathématique, ser. A no. 5, lithographed (“dactyl-spécial”), Gauthier-Villars, Paris 1954, and E. Nauwelaerts, Louvain 1954, pp. 37–49. - A. Robinson and G. Kreisel. Discussion. Applications scientifiques de la logique mathématique, Actes du 2e Colloque International de Logique Mathématique, Paris — 25–30 août 1952, Institut Henri Poincaré, Collection de logique mathématique, ser. A no. 5, lithographed (“dactyl-spécial”), Gauthier-Villars, Paris 1954, and E. Nauwelaerts, Louvain 1954, p. 50. - G. Kreisel. Models, translations, and interpretations. Mathematical interpretation of formal systems, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1955, pp. 26–50.

1959 ◽  
Vol 24 (3) ◽  
pp. 236-238
Author(s):  
William Craig
Keyword(s):  
Mathematical Logic ◽  
Publishing Company ◽  
Foundations Of Mathematics ◽  
Henri Poincaré ◽  
North Holland Publishing Company ◽  
Formal Systems ◽  
North Holland Publishing ◽  
Systems Studies ◽  
Mathematical Interpretation

scholarly journals Image of Soviet and Russian logic in the West. Latter Half of the XXth Century

2021 ◽  
Vol 27 (2) ◽  
pp. 133-152
Author(s):  
Valentin A. Bazhanov ◽  
Irving H. Anellis
Keyword(s):  
Mathematical Logic ◽  
20Th Century ◽  
Scientific Knowledge ◽  
Russian Language ◽  
History Of Logic ◽  
The West ◽  
Xxth Century ◽  
History Of ◽  
Publishing Houses ◽  
The Russian Language

The article attempts to overview Western scientific knowledge of research in mathematical logic and its history in the USSR and Russia in the first half of the 20th century. We claim that Western scholars followed and were generally aware of the main works of their Soviet and Russian colleagues on mathematical logic and its history. It was possible, firstly, due to the fact that a number of Western scientists knew the Russian language, and, secondly, because Soviet and Russian logicians published their works in English (sometimes in German) in the original journals of mathematical logic or Soviet publishing houses (mainly Mir Publishers) translated Soviet authors into English. Thus, the names of A.G. Dragalin, Yu.L. Ershov, A.S. Karpenko, A.N. Kolmogorov, Z.A. Kuzicheva, Yu.I. Manin, S.Yu. Maslov, F.A. Medvedev, G.E. Mints, V.N. Salii, V.A. Smirnov, A.A. Stolyar, N.I. Styazhkin, V.A. Uspensky, I.M. Yaglom, S.A. Yanovskaya, A.P. Yushkevich, A.A. Zinov’ev were quite known to their Western counterparts. With the dawn of perestroika, contacts of Soviet / Russian logicians expanded significantly. Nevertheless, the analysis of Western works on mathematical logic and the history of logic suggests that by the end of the 20th century the interest of Western scientists in the works of their Russian colleagues had noticeably waned.

EL RAZONAMIENTO LÓGICO MATEMÁTICO EN LOS ESTUDIANTES DEL SÉPTIMO GRADO DE LA ESCUELA “MANUELA CAÑIZARES” CANTÓN SALINAS, 2014 - 2015.

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Héctor Cárdenas Vallejo ◽  
Rosario Cárdenas Vallejo
Keyword(s):  
Mathematical Logic ◽  
Cognitive Skills ◽  
Basic Education ◽  
Deductive Reasoning ◽  
Mathematical Reasoning ◽  
Logical Reasoning ◽  
Jean Piaget ◽  
Seventh Grade ◽  
Ex Post ◽  
Santa Elena

El presente trabajo se orienta a investigar el nivel de conocimientos en el aula y aplicación de las destrezas cognitivas en el contexto social acerca de la comprensión de las matemáticas, en los estudiantes del séptimo grado de la Escuela de Educación Básica Mixta Fiscal “Manuela Cañizares” de la parroquia Santa Rosa, cantón Salinas, provincia de Santa Elena; los resultados encontrados, para el desarrollo del pensamiento lógico matemático en el proceso de enseñanza-aprendizaje, están fundamentados en la aplicación de una evaluación y encuesta, las cuales permitieron valorar el nivel de razonamiento lógico matemático y las dificultades en los estudiantes, tomando como base la teoría planteada por Jean Piaget y los planteamientos expuestos por el Ministerio de Educación (MinEduc), sobre el desarrollo de las destrezas cognitivas. Los aspectos más sustanciales que comprenden este trabajo de investigación son: el pensamiento numérico, espacial, métrico, aleatorio y manejo de dinero, ya que estas tienen relación directa con la psicología, gramática, matemáticas, teoría del conocimiento y epistemología, por su rigor, exactitud, solidez, universalización y sistematización. El estudio es de carácter descriptivo de orden cualitativo y cuantitativo, desarrollado con un diseño expost facto, y la observación directa de las actividades escolares del estudiantado y los maestros. Como resultados de la investigación se puede afirmar que la escasa preparación de los maestros en la aplicación de las destrezas cognitivas en el proceso de enseñanza de las matemáticas, ha provocado que los estudiantes tengan un bajo nivel de razonamiento lógico matemático.   Palabras clave: razonamiento lógico; razonamiento matemático; aprendizaje matemático; pensamiento lógico; razonamiento deductivo.   MATHEMATICAL LOGICAL REASONING IN SEVENTH GRADE STUDENTS OF SCHOOL “MANUELA CAÑIZARES” CANTÓN SALINAS 2014-2015.   ABSTRACT   This paper aims to investigate the level of knowledge in the classroom and application of cognitive skills in the social context on the understanding of mathematics, students (as) the seventh grade Basic Education School Fiscal Mixta “Manuela Canizares “Santa Rosa parish, canton Salinas, Santa Elena; the results found for the development of mathematical logic thinking in the process of teaching and learning is based on the application of an evaluation and survey, which allowed assessing the level of logical mathematical reasoning and difficulties in students, based theory proposed by Jean Piaget and Disclaimers approach presented by the development of cognitive skills. The most significant aspects that comprise this research are: numeric, spatial, metric, random thoughts and money management, as these are directly related to psychology, grammar, mathematics, theory of knowledge and epistemology, for its rigor, accuracy, robustness, and universal systematization. The study is descriptive qualitative and quantitative, developed with an ex post facto design, and direct observation of classroom activities for students and teachers. As research results it can be stated that the data were limited teacher preparation in the application of cognitive skills in the teaching of mathematics, has caused students to have a low level of mathematical logic reasoning.   Keywords: logical reasoning; mathematical reasoning; mathematical learning; logical thinking; deductive reasoning.   Recibido: septiembre de 2014Aprobado: enero de 2015

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