Fuzzy Natural Logic in IFSA-EUSFLAT 2021

The present book contains five papers accepted and published in the Special Issue, “Fuzzy Natural Logic in IFSA-EUSFLAT 2021”, of the journal Mathematics (MDPI). These papers are extended versions of the contributions presented in the conference “The 19th World Congress of the International Fuzzy Systems Association and the 12th Conference of the European Society for Fuzzy Logic and Technology jointly with the AGOP, IJCRS, and FQAS conferences”, which took place in Bratislava (Slovakia) from September 19 to September 24, 2021. Fuzzy Natural Logic (FNL) is a system of mathematical fuzzy logic theories that enables us to model natural language terms and rules while accounting for their inherent vagueness and allows us to reason and argue using the tools developed in them. FNL includes, among others, the theory of evaluative linguistic expressions (e.g., small, very large, etc.), the theory of fuzzy and intermediate quantifiers (e.g., most, few, many, etc.), and the theory of fuzzy/linguistic IF–THEN rules and logical inference. The papers in this Special Issue use the various aspects and concepts of FNL mentioned above and apply them to a wide range of problems both theoretically and practically oriented. This book will be of interest for researchers working in the areas of fuzzy logic, applied linguistics, generalized quantifiers, and their applications

Approximate reasoning with fuzzy-syllogistic systems

The well known Aristotelian syllogistic system consists of 256 moods. We have found earlier that 136 moods are distinct in terms of equal truth ratios that range in τ=[0,1]. The truth ratio of a particular mood is calculated by relating the number of true and false syllogistic cases the mood matches. A mood with truth ratio is a fuzzy-syllogistic mood. The introduction of (n-1) fuzzy existential quantifiers extends the system to fuzzy-syllogistic systems nS, 1<n, of which every fuzzy-syllogistic mood can be interpreted as a vague inference with a generic truth ratio that is determined by its syllogistic structure. We experimentally introduce the logic of a fuzzy-syllogistic ontology reasoner that is based on the fuzzy-syllogistic systems nS. We further introduce a new concept, the relative truth ratio rτ=[0,1] that is calculated based on the cardinalities of the syllogistic cases

Approximate syllogistic reasoning: a contribution to inference patterns and use cases

In this thesis two models of syllogistic reasoning for dealing with arguments that involve fuzzy quantified statements and approximate chaining are proposed. The modeling of quantified statements is based on the Theory of Generalized Quantifiers, which allows us to manage different kind of quantifiers simultaneously, and the inference process is interpreted in terms of a mathematical optimization problem, which allows us to deal with more arguments that standard deductive ones. For the case of approximate chaining, we propose to use synonymy, as used in a thesaurus, for calculating the degree of confidence of the argument according to the degree of similarity between chaining terms. As use cases, different types of Bayesian reasoning (Generalized Bayes' Theorem, Bayesian networks and probabilistic reasoning in legal argumentation) are analysed for being expressed through syllogisms

Mathematical Logic in High School: Hints and Proposals

In school, students learn how to reason and argue, and logic is the art of reasoning. Aristotle, who first developed it, held it to be so, i.e., the foundation of all science. But one certainly cannot impose on girls and boys an institutional course in logic as a prerequisite to all other knowledge. Not even in high school, when the matu- ration process of the students allows the teacher some more abstraction. In truth, in the Archive of Public Education - Cultural, Educational and Professional Pro- file of High Schools [129], it is stated that the logical-argumentative area assumes a central role, because it contributes to the formation of a citizen who “supports her/his own convictions, bringing adequate examples and counterexamples and us- ing concatenations of statements; accepts to change her/his opinion by recognising the logical consequences of a correct argumentation”. But education in logic must be done prudently, in the right formative ways. Some would argue that the practice of mathematics, often based on reasoning, in particular the model of Euclidean geometry, is in itself a cue to progressively in- sinuate logical mechanisms. Unfortunately, in recent times, however, Euclidean geometry seems to be a subject in disgrace, often neglected or forgotten. Instead, there are those who emphasise, in mathematics, the importance of intuition, dis- covery, experience and error, contrasting it with the excessive rigour of too many proofs. The purpose of this thesis is to propose various ideas that, within the fun- damental programmes of high school, specifically of Italian Liceo Classico and Liceo Scientifico, attempt to insinuate logic and accustom the students to logic in a way that we hope is light, clear and pleasant. We therefore do not propose a systematic treatment. We prefer to recall basic logic and then to give scattered ideas rather than a structured and definitive theory. But, as mentioned, we are confident that these hints can best prepare students of high school for logic. Indeed we address ourselves primarily to teachers and we believe that their knowl-edge of logic is useful and indeed necessary. But through them we also wish to address students. The thesis is organised as follows. The first chapter introduces and discusses the whole topic and explains why in our opinion logic is important in high schools. We also discuss how and when to propose it to students. The next two chapters introduce basic logic to teachers and students. The second illustrates the simplest logic, the Boolean one, recapitulating its essential points and emphasising in particular the use of connectives. The third deals with first-order logic, which we may consider the most classical of logics. Here we highlight in par- ticular the function of quantifiers. The following chapters propose several topics, belonging to logic or related to logic, that seem very intriguing and could be considered in high school. First, in chapter four, we treat Aristotelian syllogistics, which, even in recent times, frequently appears in various access tests. We will present some amusing introduc- tions to it, such as those of Lewis Carroll [25] or Pagnan-Rosolini [86]. Chapter five is dedicated to proofs without words. Relying on various examples from geometry, number theory and combinatorial calculus, it illustrates how reason- ing can sometimes be successfully expressed and developed through the images and intuitions they suggest. In this chapter, we also discuss the logic of images proposed by Leonardo in the Codex Atlanticus [73] to address and solve geometric questions often linked to the Pythagorean theorem. The sixth chapter is dedicated to what Henri Poincaré called the “mathematical reasoning” par excellence, namely the principle of induction. This law governs nat- ural numbers and is often used as a powerful demonstrative tool in various exercises. However, students seldom learn it and above all understand it properly. Drawing on the history of the principle of induction, from its primitive intuitions to its for- malisation by Peano and Dedekind, we attempt to approach it in what we hope will be a pleasant and appealing way, also o ering a wide range of examples. Logical paradoxes are another logical theme that is impossible to forget: mental games that not only disorientate but also intrigue and amuse, which are the heart of the seventh chapter. To the classical logics considered at the beginning of the thesis, based on two truth values, yes or no, we then contrast multi-valued and fuzzy logics, which are better suited to analysing situations of uncertainty. We link them, in chapter eight, to the Rényi-Ulam game, which searches for truth in contexts in which the information received may be lying and deceptive. The final chapter takes up a basic topic of high school mathematics: equations. Diophantine games show us how they can be an opportunity for challenge and fun, as well as suggesting intriguing insights into fundamental themes of modern math- ematics: not only number theory and algebra, but also game theory and the theory of computability and computational complexity. For a general and in-depth overview of mathematical logic, we refer to [10] and [114]. For the theory of computability and computational complexity we refer the reader to [37] and [83], for number theory to [60]

Argumentative Skills: A Systematic Framework for Teaching and Learning

In this paper, we propose a framework for fostering argumentative skills in a systematic way in Philosophy and Ethics classes. We start with a review of curricula and teaching materials from the German-speaking world to show that there is an urgent need for standards for the teaching and learning of argumentation. Against this backdrop, we present a framework for such standards that is intended to tackle these difficulties. The spiral-curricular model of argumentative competences we sketch helps teachers introduce the relevant concepts and skills to students early on in their school career. The focus is on secondary schools, but the proposal can also be of use for learning and teaching in universities, especially in introductory classes

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Possible models diagrams: a new approach to teaching propositional logic.

Masters Degree. University of KwaZulu-Natal, Pietermaritzburg.Abstract available in PDF.Quality of scanned PDF has been compromised owing to poor condition of original document