Russell and Frege on the Logic of Functions
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Discrete Mathematics
Discrete Mathematics1 http://lcs.ios.ac.cn/~znj/DM2017 Naijun Zhan March 19, 2017 1Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course. Contents 1. The Foundations: Logic and Proofs 2. Basic Structures: Sets, Functions, Sequences, Sum- s, and Matrices 3. Algorithms 4. Number Theory and Cryptography 5. Induction and Recursion 6. Counting 7. Discrete Probability 8. Advanced Counting Techniques 9. Relations 10. Graphs 11. Trees 12. Boolean Algebra 13. Modeling Computation 1 Chapter 1 The Foundations: Logic and Proofs Logic in Computer Science During the past fifty years there has been extensive, continuous, and growing interaction between logic and computer science. In many respects, logic provides computer science with both a u- nifying foundational framework and a tool for modeling compu- tational systems. In fact, logic has been called the calculus of computer science. The argument is that logic plays a fundamen- tal role in computer science, similar to that played by calculus in the physical sciences and traditional engineering disciplines. Indeed, logic plays an important role in areas of computer sci- ence as disparate as machine architecture, computer-aided de- sign, programming languages, databases, artificial intelligence, algorithms, and computability and complexity. Moshe Vardi 2 • The origins of logic can be dated back to Aristotle's time. • The birth of mathematical logic: { Leibnitz's idea { Russell paradox { Hilbert's plan { Three schools of modern logic: logicism (Frege, Russell, Whitehead) formalism (Hilbert) intuitionism (Brouwer) • One of the central problem for logicians is that: \why is this proof correct/incorrect?" • Boolean algebra owes to George Boole. -
Basic Concepts of Set Theory, Functions and Relations 1. Basic
Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6 -
Sets, Propositional Logic, Predicates, and Quantifiers
COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
Definitions and Nondefinability in Geometry 475 2
Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal. -
Principle of Acquaintance and Axiom of Reducibility
Principle of acquaintance and axiom of reducibility Brice Halimi Université Paris Nanterre 15 mars 2018 Russell-the-epistemologist is the founding father of the concept of acquaintance. In this talk, I would like to show that Russell’s theory of knowledge is not simply the next step following his logic, but that his logic (especially the system of Principia Mathematica) can also be understood as the formal underpinning of a theory of knowledge. So there is a concept of acquaintance for Russell-the-logician as well. Principia Mathematica’s logical types In Principia, Russell gives the following examples of first-order propositional functions of individuals: φx; (x; y); (y) (x; y);::: Then, introducing φ!zb as a variable first-order propositional function of one individual, he gives examples of second-order propositional functions: f (φ!zb); g(φ!zb; !zb); F(φ!zb; x); (x) F(φ!zb; x); (φ) g(φ!zb; !zb); (φ) F(φ!zb; x);::: Then f !(φb!zb) is introduced as a variable second-order propositional function of one first-order propositional function. And so on. A possible value of f !(φb!zb) is . φ!a. (Example given by Russell.) This has to do with the fact that Principia’s schematic letters are variables: It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. ] (Principia, p. 51) And variables are understood substitutionally (see Kevin Klement, “Russell on Ontological Fundamentality and Existence”, 2017). This explains that the language of Principia does not constitute an autonomous formal language. -
Taboo Versus Axiom
TABOO VERSUS AXIOM KATUZI ONO Some important formal systems are really developable from a finite number of axioms in the lower classical predicate logic LK or in the intuitionistic predicate logic LJ. Any system of this kind can be developed in LK (or in LJ) from the single conjunction of all the axioms of the system. I have been intending to develop usual formal systems starting from TABOOS and standing on the primitive logic LO at first, since the logic too could be brought up to the usual logics by means of TABOOS in LOυ. In developing any formal system from TABOOS, we are forced to assume unwillingly that all the TABOOS are mutually equivalent, as far as we adopt more than one TABOOS for the system. If we could develop formal systems from single- TABOO TABOO-systems, we could get rid of this unwilling assumption. It would be well expected that any formal theory developable from a finite number of axioms in LK or in LJ would be developable from a single TABOO.2) In the present paper, I will prove that this is really the case. Before stating the theorem, let us define the δ-TRANSFORM Spy] of any sentence ® with respect to an h-ary relation £y (λ>0). SΓ&3 is introduced by a structural recursive definition as follows: @C8Qs (r)((©->g(r))->#(r)) for any elementary sentence ©, Received January 20, 1966. *> As for this plan, see Ono [1]. LO is called PRIMITIVE SYSTEM OF POSITIVE LOGIC in Ono [1]. The terminology PRIMITIVE LOGIC together with its reference notation LO has been introduced in Ono [2], 2) In my work [1], I had to assume this to bring up logics by means of TABOO systems. -
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Gödel's Unpublished Papers on Foundations of Mathematics
G¨odel’s unpublished papers on foundations of mathematics W. W. Tait∗ Kurt G¨odel: Collected Works Volume III [G¨odel, 1995] contains a selec- tion from G¨odel’s Nachlass; it consists of texts of lectures, notes for lectures and manuscripts of papers that for one reason or another G¨odel chose not to publish. I will discuss those papers in it that are concerned with the foun- dations/philosophy of mathematics in relation to his well-known published papers on this topic.1 1 Cumulative Type Theory [*1933o]2 is the handwritten text of a lecture G¨odel delivered two years after the publication of his proof of the incompleteness theorems. The problem of giving a foundation for mathematics (i.e. for “the totality of methods actually used by mathematicians”), he said, falls into two parts. The first consists in reducing these proofs to a minimum number of axioms and primitive rules of inference; the second in justifying “in some sense” these axioms and rules. The first part he takes to have been solved “in a perfectly satisfactory way” ∗I thank Charles Parsons for an extensive list of comments on the penultimate version of this paper. Almost all of them have led to what I hope are improvements in the text. 1This paper is in response to an invitation by the editor of this journal to write a critical review of Volume III; but, with his indulgence and the author’s self-indulgence, something else entirely has been produced. 2I will adopt the device of the editors of the Collected Works of distinguishing the unpublished papers from those published by means of a preceding asterisk, as in “ [G¨odel, *1933o]”. -
Formal System Having Just One Primitive Notion
FORMAL SYSTEM HAVING JUST ONE PRIMITIVE NOTION KATUZI ONO Introduction In this paper, I would like to point out some characteristic features of formal systems having just one primitive notion. Most remarkable systems of this kind may be Zermelo's and FraenkeΓs set-theories, both having just one primitive notion e. In my former work [1], I have proved that J-series logics (intuitionistic predicate logic L J, Johansson's minimal predicate logic LM, and positive predicate logic LP) as well as K-series logics (fortified logics LK, LN, and LQ of LJ, LM, and LP by Peirce's rule that (A->B)->A implies Λy respectively; LK being the lower classical predicate logic) can be faithfully interpreted in the primitive logic LO. However, this does not mean that CONJUNCTION, DISJUNCTION, NEGATION (except for LP and LQ), and EXISTENTIAL QUANTIFICATION of these logics can be defined in LO in such ways that propositions behave with respect to the defined logical constants together with the original logical constants of LO just as propositions of J- and K-series logics concerning their provability. In reality, these logical constants are defined so only for a certain class of propositions in LO. If we restrict ourselves only to formal systems having just one primitive notion, however, we can really define CONJUNCTION, DISJUNCTION, and EXISTENTIAL QUANTIFICATION together with one NEGATION for LJ, a series of NEGATIONS for LM, and naturally no NEGATION for LP in such way that propositions behave with respect to the defined logical constants together with the original logical constants of LO just as propositions of these J-series logics, respectively. -
Neo-Logicism? an Ontological Reduction of Mathematics to Metaphysics
Edward N. Zalta 2 certain axiomatic, mathematics-free metaphysical theory. This thesis ap- pears to be a version of mathematical platonism, for if correct, it would make a certain simple and intuitive philosophical position about mathe- Neo-Logicism? An Ontological Reduction of matics much more rigorous, namely, that mathematics describes a realm of abstract objects. Nevertheless, there are two ways in which the present Mathematics to Metaphysics∗ view might constitute a kind of neo-logicism. The first is that the compre- hension principle for abstract objects that forms part of the metaphysical theory can be reformulated as a principle that `looks and sounds' like an Edward N. Zaltay analytic, if not logical, truth. Although we shall not argue here that the Center for the Study of Language and Information reformulated principle is analytic, other philosophers have argued that Stanford University principles analogous to it are. The second is that the abstract objects systematized by the metaphysical theory are, in some sense, logical ob- jects. By offering a reduction of mathematical objects to logical objects, the present view may thereby present us with a new kind of logicism. Die nat¨urlichen Zahlen hat der liebe Gott gemacht, der Rest ist Menschenwerk. To establish our thesis, we need two elements, the first of which is Leopold Kronecker already in place. The first element is the axiomatic, metaphysical theory of abstract objects. We shall employ the background ontology described It is now well accepted that logicism is false. The primitive notions by the axiomatic theory of abstract objects developed in Zalta [1983] and proper axioms of mathematical theories are not reducible to primi- and [1988].1 The axioms of this theory can be stated without appealing tive logical notions and logical axioms. -
Bertrand Russell – Principles of Mathematics
Principles of Mathematics “Unless we are very much mistaken, its lucid application and develop- ment of the great discoveries of Peano and Cantor mark the opening of a new epoch in both philosophical and mathematical thought” – The Spectator Bertrand Russell Principles of Mathematics London and New York First published in 1903 First published in the Routledge Classics in 2010 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2009. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. © 2010 The Bertrand Russell Peace Foundation Ltd Introduction © 1992 John G. Slater All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested ISBN 0-203-86476-X Master e-book ISBN ISBN 10: 0-415-48741-2 ISBN 10: 0-203-86476-X (ebk) ISBN 13: 978-0-415-48741-2 ISBN 13: 978-0-203-86476-0 (ebk) CONTENTS introduction to the 1992 edition xxv introduction to the second edition xxxi preface xliii PART I THE INDEFINABLES OF MATHEMATICS 1 1 Definition of Pure Mathematics 3 1.