Areas of Mathematics

Mục tiêu của học phần: Sau khi nghiên cứu môn học, sinh viên vừa được trang bị các kiến thức cơ bản về bản chất và nguyên lý tính toán trong các nghiệp vụ tài chính, vừa biết vận dụng các kiến thức đó để xây dựng các bài toán tài chính trong những hoàn cảnh riêng với môi trường và các điều kiện khác nhau.

This book provides an overview of the various biometric technologies, decision-making algorithms and the subsequent market expansion opportunity. Further, this book proposes a Throughput Model, which draws from computer science, economic and psychology literatures to model perceptual, informational sources, judgmental processes and decision choice algorithms. This approach describes how biometrics might be implemented to reduce risks to individuals and organizations, especially when dealing with digital based mediums.

In diesem Buch werden Aspekte der Aussagenlogik und der Prädikaten­ logik der ersten Stufe behandelt. Eine Mathematisierung und Kalkü­ lisierung der Logik kann natürlich ganz verschieden ausfallen, je nach dem, von welchen Motiven man sich primär leiten läßt. Wir stellen drei Gesichtspunkte, die uns auch als die wesentlichsten erscheinen, in den Vordergrund: Die Formalisierung des Wahrheits­ begriffes, die Formalisierung des Beweisbarkeitsbegriffes und das Problem des Suchens nach Beweisen. Diese drei Aspekte führen zu drei verschiedenen Arten von Kalkülen. Die Betonung des (...) Wahrheitsbegriffes führte auf die untersuchung der Hilberttypkalküle von einem Standpunkt, wie er etwa auch im Buch von Rasiowa-Sikorski [Ra-Si] eingenommen wird. Hierbei wurde besonderen Wert auf die algebraischen Techniken gelegt, denn die Natur der Vollständigkeitsbeweise in diesen Kalkülen läßt sich u. E. eigentlich nur algebraisch verstehen. Etwas überspitzt könnte man formulieren, daß die Vollständigkeitsbeweise in Hilberttypkal­ külen Korollare zu Betrachtungen über Kongruenzrelationen in ge­ wissen Boole'schen Algebren sind. Bei den modelltheoretischen Be­ trachtungen haben wir uns kurz ge faßt und nur einige grundlegende Begriffe vorgestellt. (shrink)

Wenn es eine Aufgabe der Philosophie ist, die Herrschaft des Wortes über den menschlichen Geist zu brechen (Frege), dann ist zu fragen, ob die Prädikatenlogik die Wörter «für alle» und «es gibt» angemessen formalisiert. Übersetzt man «alle Menschen sind sterblich» mit «für alle x : wenn x ein Mensch ist, dann ist x sterblich», so hat man zwar die Syllogistik des Aristoteles überwunden, dafür tritt jetzt aber eine Variable x auf, die in der Aussage vorher nicht vorkam.

Ce livre est un recueil des articles que l'auteur a publiés dans la rubrique Logique et Calcul du magazine Pour la Science. La sélection a pour thème la logique dans toute sa diversité. Par exemple, elle peut concerner les mathématiques pures : L'infini est-il paradoxal en mathématiques, Libre arbitre et mécanique quantique ou Les limites logiques des mathématiques. Mais on la trouve aussi au coeur d'applications concrètes. Ainsi, L'étonnante loi de Benford, selon laquelle un nombre pris au hasard, par exemple (...) dans le journal, commence plus souvent par un " 1 " plutôt que par tout autre chiffre, permet de dépister les tricheurs. Ou encore, la logique qui aide à répondre à cette question d'actualité : La répartition idéale des biens existe-t-elle? Jean-Paul Delahaye, avec son style et ses talents reconnus de vulgarisateur, nous fait même réfléchir aux liens qui unissent La belle au bois dormant, les extraterrestres et la fin du monde! Idéal en cette année 2012. (shrink)

Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems culminating in the well-known and powerful Calculus of Constructions. The book (...) also covers the essence of proof checking and proof development, and the use of dependent type theory to formalize mathematics. The only prerequisites are a good knowledge of undergraduate algebra and analysis. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarize themselves with the material. (shrink)

L'ouvrage invite à penser l'épistémologie non classique des mathématiques comme une construction épistémique des mathématiques quasi empirique qu'Imré Lakatos a réalisé au moyen des présupposés ontologiques (comme la redécouverte du temps, le passage du déterminisme à l'indéterminisme, etc.) et qu'il réussit grâce à l'implémentation du principe de complexité dialectique de l'esprit mathématique. L'auteur étudie la thèse selon laquelle, depuis le début du XXe siècle, la science est aussi bien objective que subjective. Et c'est à partir de celle-ci que le principe (...) de "la complexité dialectique de l'esprit mathématique" est mis au jour pour dévoiler le constructivisme de Lakatos. (shrink)

Originally published in 1964. This book is concerned with general arguments, by which is meant broadly arguments that rely for their force on the ideas expressed by all, every, any, some, none and other kindred words or phrases. A main object of quantificational logic is to provide methods for evaluating general arguments. To evaluate a general argument by these methods we must first express it in a standard form. Quantificational form is dealt with in chapter one and in part of (...) chapter three; in the remainder of the book an account is given of methods by which arguments when formulated quantificationally may be tested for validity or invalidity. Some attention is also paid to the logic of identity and of definite descriptions. Throughout the book an attempt has been made to give a clear explanation of the concepts involved and the symbols used; in particular a step-by-step and partly mechanical method is developed for translating complicated statements of ordinary discourse into the appropriate quantificational formulae. Some elementary knowledge of truth-functional logic is presupposed. (shrink)

Originally published in 1967. An introduction to the literature of nonstandard logic, in particular to those nonstandard logics known as many-valued logics. Part I expounds and discusses implicational calculi, modal logics and many-valued logics and their associated calculi. Part II considers the detailed development of various many-valued calculi, and some of the important metathereoms which have been proved for them. Applications of the calculi to problems in the philosophy are also surveyed. This work combines criticism with exposition to form a (...) comprehensive but concise survey of the field. (shrink)

The term "fuzzy logic," as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic---the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediate truth values between true and false are allowed, which are interpreted as degrees of truth. This opens a (...) new way of thinking---thinking in terms of degrees rather than absolutes. For example, it leads to the definition of a new kind of sets, referred to as fuzzy sets, in which membership is a matter of degree. The book examines the genesis and development of fuzzy logic. It surveys the prehistory of fuzzy logic and inspects circumstances that eventually lead to the emergence of fuzzy logic. The book explores in detail the development of propositional, predicate, and other calculi that admit degrees of truth, which are known as fuzzy logic in the narrow sense. Fuzzy logic in the broad sense, whose primary aim is to utilize degrees of truth for emulating common-sense human reasoning in natural language, is scrutinized as well. The book also examines principles for developing mathematics based on fuzzy logic and provides overviews of areas in which this has been done most effectively. It also presents a detailed survey of established and prospective applications of fuzzy logic in various areas of human affairs, and provides an assessment of the significance of fuzzy logic as a new paradigm. (shrink)

The well-known Ratio Studiorum of 1599 states that logical instruction should follow F. Toletus or P. Fonseca. The latter authored the famous Institutionum Dialecticarum Libri Octo, the former a similar manual, Introductio in Dialecticam Aristotelis. As is often observed, the contrast between the Aristotelian and present symbolic logics is perhaps most striking in their analysis of relational statements. Both authors recognize the relational logical form as independent from the traditional subject-predicate form and see the need to recognize relational inferential rules. (...) They differ in their specific rules, however, so neither of the authors has captured the system of relational syllogism in its entirety. (shrink)

This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on (...) the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike. (shrink)

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

James Walmsley in “Categoricity and Indefinite Extensibility” argues that a realist about some branch of mathematics X (e.g. arithmetic) apparently cannot use the categoricity of an axiomatisation of X to justify her belief that every sentence of the language of X has a truth-value. My discussion note first corrects Walmsley’s formulation of his claim. It then shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping that (...) there is a unique such model. (shrink)

We often, in effect, take it for granted that some word or phrase is what is called ‘the name of a class, be that class empty or non-empty’. We do so whenever in effect we either wonder about, or mean to be taking a view on, the number of members a certain suppositious class has, on the ultrasimple number scale: ‘None, more than none’.

This paper is an introduction to the new field of biorhetorics. Biorhetorics is an applied form of rhetoric that evolved from the study of classical rhetoric, particularly Aristotelian. The author illustrates the stages of development necessary for the creation of a species-specific rhetoric: by (1) formalising rhetoric so as to create a functional rhetoric, (2) then reducing this to a symbolic rhetoric that can be used in conjunction with the collected data of an organism’s Umwelt (including its genome) to form (...) (3) a species-specific rhetoric. The paper draws upon the latest research on bacterial and viral communication to show the possibilities of biorhetorics. In the course of discussing the nature of biorhetorics the author distinguishes it from argumentation theory and rhetoric/s of biology, and positions alongside other fields used in the life sciences such as biosemiotics, information theory, game theory, etc. (shrink)

Logical theory – and philosophical theory generally – is just that, theory. Generations of logic students felt a sort of unease about it without knowing what to do about it. Nowadays, students of mathematical logic feel a similar unease when faced with the fact that in standard predicate calculus, “All unicorns are sneaky” is true precisely because there are no unicorns. Blanché’s analysis reminds us that such feelings of unease may indicate a shortcoming in the theory rather than in the (...) student’s understanding. (shrink)

The law of excluded middle is usually considered as intrinsically connected with the realistic standpoint and incompatible with the idealistic position. This is just what Ajdukiewicz claims in his critique of transcendental idealism. The analysis of Ajdukiewicz's argumentation raises the problem of validity of the law of excluded middle for vague (or incomplete) languages. The problem is being solved by differentiating between the logical (or ontological) and the metalogical (or semantical) law of excluded middle: in contrast to the former, the (...) latter is claimed to be invalid for the languages in question, without thereby embracing the idealist position. (shrink)

The paper presents and applies Hilbert's Epsilon Calculus, first describing its standard proof theory, and giving it an intensional semantics. These are contrasted with the proof theory of Fregean Predicate Logic, and the traditional (extensional) choice function semantics for the calculus. The semantics provided show that epsilon terms are referring terms in Donnellan's sense, enabling the symbolisation and validation of argument forms involving E-type pronouns, both in extensional and intensional contexts. By providing for transparency in intensional constructions they support a (...) Model Conceptualism to contrast with traditional intensional logic's Modal Realism. But epsilon-terms also symbolise fictions, and through their difference from iota terms enable the solution of a number of outstanding puzzles about Direct Reference and de re beliefs. (shrink)

In "Truth" Michael Dummett presents a case for intuitionist logic as the logic of ordinary discourse. The case depends on a supposed need to make two intuitions mesh: first, that it is senseless to suppose, of any statement, that it is neither true nor false; second, that there is no guarantee, for every statement, that either there is something in the world to make it true, or there is something to make it false. This paper argues, developing a notion of (...) natural isostheneia, that Dummett's first intuition is wrong as he reads it, and that, consequently, his case for intuitionist logic collapses. (shrink)

Jaakko Hintikka's Kantianism in philosophy of logic and mathematics is known to go further than Kant's own, for he argues that mathematical reasoning involves the "language-games" of seeking and finding. Therefore, logic mirrors the structure of this activity. But Hintikka also pushes the Copemican Revolution further to epistemology and philosophy of science. He agrees that "reason has insight only into what which it produces after a plan of ist own", but gives the idea a new logical turn. Kant thought that (...) reason imposes certain architectonic constraints on the possible outcome of inquiry, but Hintikka's interrogative model of inquiry also emphasizes the activity of and therefore the strategy in, putting questions to Nature. (shrink)

Keith Lehrer's notion of acceptance and its relation to the notion of belief is analyzed in a way that a person only accepts some proposition p if she decides to believe it in order to reach the epistemic aim. This view of acceptance turns out to be untenable: Under the empirical claim that we don't have the power to decide what to beheve it follows that we cannot accept anything. If reaching the truth is the epistemic aim acceptance proves ill-formed, (...) it is impossible to pursue the aim of truth by believing or accepting something because belief itself is a truth-directed attitude. If the epistemic aim is formulated in a weaker sense, combined with other aims, the danger lurks that accepting a proposition p is in the end loosing any connection with the truth of p. (shrink)

The first part of the papaer contains desiderata for a realistic epistemic system as opposed to idealistic ones. One of the main characteristics of idealistic epistemic systems is their deductive infallibility or deductive omniscience. The system presented avoids deductive infallibility though having a strong concept of knowledge. The second part contains the theorems of the system. The system is detailed in so far as it distinguishes between two concepts of belief and one of assumption and interrelates them to the concept (...) of knowledge. Though all concepts satisfy certain consistency criteria the strongest ones hold for the concept of knowledge; whereas a belief in or a assumption (assertion) of a proposition which has inconsistent consequences (not known or believed or assumed by the believer or assumer) does not entail the commitment of believing in (or assuming of) an explicit contradiction. Moreover the system contains a lot of distinctions and details concerning propositions with a second person involved like "a knows that b knows whether p is the case" etc. The third part of the paper contains the semantics of the system which consists of many-valued truth-tables. Since the matrices are finite the system is consistent and decidable. (shrink)