DOCSLIB.ORG

Quantification and Paradox

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantification and Paradox University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses March 2018 Quantification and arP adox Edward Ferrier University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Logic and Foundations of Mathematics Commons Recommended Citation Ferrier, Edward, "Quantification and arP adox" (2018). Doctoral Dissertations. 1165. https://doi.org/10.7275/11246028.0 https://scholarworks.umass.edu/dissertations_2/1165 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. QUANTIFICATION AND PARADOX A Dissertation Presented by EDWARD FERRIER Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY February 2018 Philosophy c Copyright by Edward Ferrier 2018 All Rights Reserved QUANTIFICATION AND PARADOX A Dissertation Presented by EDWARD FERRIER Approved as to style and content by: Kevin Klement, Chair Phillip Bricker, Member Christopher Meacham, Member Robert Kusner, Member Joseph Levine, Department Head Philosophy DEDICATION To all my wonderful colleagues and teachers who were so patient with me. ABSTRACT QUANTIFICATION AND PARADOX FEBRUARY 2018 EDWARD FERRIER B.A., THOMAS AQUINAS COLLEGE B.A., UNIVERSITY OF NORTH CAROLINA GREENSBORO M.A., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Kevin Klement I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and vari- ants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantifi- cation, but on the \logical" conception of set which motivates naive set theory. The accepted solution is to replace this with the \iterative" conception of set. I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2{3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set v theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. vi TABLE OF CONTENTS Page ABSTRACT .......................................................... v CHAPTER 1. ABSOLUTISM AND RESTRICTIVISM ........................... 1 1.1 Introduction . 1 1.2 Absolutism . 7 1.2.1 Defining absolutism . 7 1.2.2 Motivating absolutism . 9 1.3 Non-restrictivist arguments against absolutism . 11 1.4 Restrictivism. 14 1.4.1 Defining restrictivism . 14 1.4.2 A quick argument for restrictivism? . 15 1.4.3 Indefinite extensibility . 22 1.4.4 Generality relativism and limitivism . 27 2. THREE SET-THEORETIC PARADOXES ........................ 29 2.1 Basic terminology and definitions . 30 2.2 Cantor's paradox . 33 2.2.1 Proof of Cantor's theorem . 33 2.2.2 Derivations of Cantor's paradox . 34 2.2.2.1 Cantor I . 36 2.2.2.2 Cantor II . 39 2.2.2.3 Cantor III . 41 2.3 Burali-Forti's paradox . 42 2.3.1 Derivations of Burali-Forti's paradox . 46 vii 2.3.1.1 Burali-Forti I . 47 2.3.1.2 Burali-Forti II . 49 2.3.1.3 Burali-Forti III . 51 2.4 Russell's Paradox . 51 2.4.1 Derivation of Russell's Paradox . 52 2.5 Genuine paradoxes or mere reductios? . 52 2.5.1 Cantor, Burali-Forti and Russell . 56 2.5.2 Paradoxes of Logic and Paradoxes of Set Theory . 60 2.6 Appendix . 62 3. RUSSELL'S SCHEMA ............................................ 69 3.1 Russell's schema . 69 3.1.1 Self-reproductive properties . 71 3.2 Putting the paradoxes into the form of Russell's schema . 73 3.2.1 Russell's schema: Russell's paradox . 73 3.2.2 Russell's schema: Cantor's paradox . 74 3.2.2.1 First translation . 75 3.2.2.2 Second translation . 76 3.2.3 Russell's schema: Burali-Forti's paradox . 77 3.3 The vicious circle principle . 78 3.4 Logical relations between No Function and No Set . 81 3.4.1 f1 and f2: No Function is equivalent to No Set . 81 3.4.2 f3 and f4: No Function is independent from No Set . 82 3.4.3 A uniform solution . 85 4. THE LOGICAL CONCEPTION OF SET ......................... 86 4.1 Concepts . 87 4.2 Extensions . 91 4.2.1 Involvement and membership . 94 4.2.2 Identity . 94 4.3 The priority of concepts to extensions . 96 viii 4.3.1 Extensions as pluralities . 99 4.4 Inconsistency of naive set theory . 104 4.5 Limitation of Size . 107 4.5.1 Failure to respect the intuitive notion of an extension . 109 4.5.2 Circular explanations . 113 4.6 First-order set restrictivism . 117 4.6.1 A modalized variant of FC . 122 4.6.2 The All-in-One principle . 125 4.6.3 Explanation and Expressibility . 127 5. EXPRESSING RESTRICTIVISM ............................... 130 5.1 Semantic ascent . 131 5.1.1 Williamson . 132 5.1.2 Fine . 134 5.2 Ambiguous assertion: any and every . 141 5.3 The generality of any-statements . 143 5.3.1 An alternative account . 147 5.4 Expressing restrictivism . 149 5.5 Concluding Remark . 153 6. THE ITERATIVE CONCEPTION OF SET ..................... 156 6.1 The process of set formation . 160 6.1.1 Priority and Explanation . 167 6.2 Iterative set theories . 171 6.2.1 How Class Comprehension blocks the paradoxes . 174 6.2.2 Is NBG set theory restrictivist? . 176 6.2.3 How Separation blocks the paradoxes . 178 6.2.4 Expressing the Iterative Conception: ZF vs. NBG . 180 6.3 Motivating the set existence axioms . 181 6.4 Accounts of Priority . 200 6.4.1 The constructivist account of priority . 201 6.4.2 The modal account of priority . 203 ix 6.4.2.1 Modal set theory . 209 6.4.2.2 Understanding the modal operators . 211 6.4.3 The dependence account of priority . 218 6.5 Conclusion . 219 7. SETLESS VARIANTS OF THE PARADOXES .................. 221 7.1 Russell's Paradox: Semantic Version . ..
Load more

Recommended publications
  • Redalyc.Sets and Pluralities
    Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Sistema de Información Científica Gustavo Fernández Díez Sets and Pluralities Revista Colombiana de Filosofía de la Ciencia, vol. IX, núm. 19, 2009, pp. 5-22, Universidad El Bosque Colombia Available in: http://www.redalyc.org/articulo.oa?id=41418349001 Revista Colombiana de Filosofía de la Ciencia, ISSN (Printed Version): 0124-4620 [email protected] Universidad El Bosque Colombia How to cite Complete issue More information about this article Journal's homepage www.redalyc.org Non-Profit Academic Project, developed under the Open Acces Initiative Sets and Pluralities1 Gustavo Fernández Díez2 Resumen En este artículo estudio el trasfondo filosófico del sistema de lógica conocido como “lógica plural”, o “lógica de cuantificadores plurales”, de aparición relativamente reciente (y en alza notable en los últimos años). En particular, comparo la noción de “conjunto” emanada de la teoría axiomática de conjuntos, con la noción de “plura- lidad” que se encuentra detrás de este nuevo sistema. Mi conclusión es que los dos son completamente diferentes en su alcance y sus límites, y que la diferencia proviene de las diferentes motivaciones que han dado lugar a cada uno. Mientras que la teoría de conjuntos es una teoría genuinamente matemática, que tiene el interés matemático como ingrediente principal, la lógica plural ha aparecido como respuesta a considera- ciones lingüísticas, relacionadas con la estructura lógica de los enunciados plurales del inglés y el resto de los lenguajes naturales. Palabras clave: conjunto, teoría de conjuntos, pluralidad, cuantificación plural, lógica plural. Abstract In this paper I study the philosophical background of the relatively recent (and in the last few years increasingly flourishing) system of logic called “plural logic”, or “logic of plural quantifiers”.
    [Show full text]
  • 1 Lecture 13: Quantifier Laws1 1. Laws of Quantifier Negation There
    Ling 409 lecture notes, Lecture 13 Ling 409 lecture notes, Lecture 13 October 19, 2005 October 19, 2005 Lecture 13: Quantifier Laws1 true or there is at least one individual that makes φ(x) true (or both). Those are also equivalent statements. 1. Laws of Quantifier Negation Laws 2 and 3 suggest a fundamental connection between universal quantification and There are a number of equivalences based on the set-theoretic semantics of predicate logic. conjunction and existential quantification and disjunction: Take for instance a statement of the form (∃x)~ φ(x)2. It asserts that there is at least one individual d which makes ~φ(x) true. That means that d makes ϕ(x) false. That, in turn, (∀x) ψ(x) is true iff ψ(a) and ψ(b) and ψ(c) … (where a, b, c … are the names of the means that (∀x) φ(x) is false, which means that ~(∀x) φ(x) is true. The same reasoning individuals in the domain.) can be applied in the reverse direction and the result is the first quantifier law: ∃ ψ ψ ψ ψ ( x) (x) is true iff (a) or (b) or c) … (where a, b, c … are the names of the (1) Laws of Quantifier Negation individuals in the domain.) Law 1: ~(∀x) φ(x) ⇐⇒ (∃x)~ φ(x) From that point of view, Law 1 resembles a generalized DeMorgan’s Law: (Possible English correspondents: Not everyone passed the test and Someone did not pass (4) ~(ϕ(a) & ϕ(b) & ϕ(c) … ) ⇐⇒ ~ϕ(a) v ~ϕ(b) v ϕ(c) … the test.) (5) Law 4: (∀x) ψ(x) v (∀x)φ(x) ⇒ (∀x)( ψ(x) v φ(x)) ϕ ⇐ ϕ Because of the Law of Double Negation (~~ (x) ⇒ (x)), Law 1 could also be written (6) Law 5: (∃x)( ψ(x) & φ(x)) ⇒ (∃x) ψ(x) & (∃x)φ(x) as follows: These two laws are one-way entailments, NOT equivalences.
    [Show full text]
  • Arxiv:1804.02439V1
    DATHEMATICS: A META-ISOMORPHIC VERSION OF ‘STANDARD’ MATHEMATICS BASED ON PROPER CLASSES DANNY ARLEN DE JESUS´ GOMEZ-RAM´ ´IREZ ABSTRACT. We show that the (typical) quantitative considerations about proper (as too big) and small classes are just tangential facts regarding the consistency of Zermelo-Fraenkel Set Theory with Choice. Effectively, we will construct a first-order logic theory D-ZFC (Dual theory of ZFC) strictly based on (a particular sub-collection of) proper classes with a corresponding spe- cial membership relation, such that ZFC and D-ZFC are meta-isomorphic frameworks (together with a more general dualization theorem). More specifically, for any standard formal definition, axiom and theorem that can be described and deduced in ZFC, there exists a corresponding ‘dual’ ver- sion in D-ZFC and vice versa. Finally, we prove the meta-fact that (classic) mathematics (i.e. theories grounded on ZFC) and dathematics (i.e. dual theories grounded on D-ZFC) are meta-isomorphic. This shows that proper classes are as suitable (primitive notions) as sets for building a foundational framework for mathematics. Mathematical Subject Classification (2010): 03B10, 03E99 Keywords: proper classes, NBG Set Theory, equiconsistency, meta-isomorphism. INTRODUCTION At the beginning of the twentieth century there was a particular interest among mathematicians and logicians in finding a general, coherent and con- sistent formal framework for mathematics. One of the main reasons for this was the discovery of paradoxes in Cantor’s Naive Set Theory and related sys- tems, e.g., Russell’s, Cantor’s, Burati-Forti’s, Richard’s, Berry’s and Grelling’s paradoxes [12], [4], [14], [3], [6] and [11].
    [Show full text]
  • Lexical Scoping As Universal Quantification
    University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science March 1989 Lexical Scoping as Universal Quantification Dale Miller University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Dale Miller, "Lexical Scoping as Universal Quantification", . March 1989. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-89-23. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/785 For more information, please contact [email protected]. Lexical Scoping as Universal Quantification Abstract A universally quantified goal can be interpreted intensionally, that is, the goal ∀x.G(x) succeeds if for some new constant c, the goal G(c) succeeds. The constant c is, in a sense, given a scope: it is introduced to solve this goal and is "discharged" after the goal succeeds or fails. This interpretation is similar to the interpretation of implicational goals: the goal D ⊃ G should succeed if when D is assumed, the goal G succeeds. The assumption D is discharged after G succeeds or fails. An interpreter for a logic programming language containing both universal quantifiers and implications in goals and the body of clauses is described. In its non-deterministic form, this interpreter is sound and complete for intuitionistic logic. Universal quantification can provide lexical scoping of individual, function, and predicate constants. Several examples are presented to show how such scoping can be used to provide a Prolog-like language with facilities for local definition of programs, local declarations in modules, abstract data types, and encapsulation of state.
    [Show full text]
  • 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.
    [Show full text]
  • Collapse, Plurals and Sets 421
    doi: 10.5007/1808-1711.2014v18n3p419 COLLAPSE,PLURALS AND SETS EDUARDO ALEJANDRO BARRIO Abstract. This paper raises the question under what circumstances a plurality forms a set. My main point is that not always all things form sets. A provocative way of presenting my position is that, as a result of my approach, there are more pluralities than sets. Another way of presenting the same thesis claims that there are ways of talking about objects that do not always collapse into sets. My argument is related to expressive powers of formal languages. Assuming classical logic, I show that if all plurality form a set and the quantifiers are absolutely general, then one gets a trivial theory. So, by reductio, one has to abandon one of the premiss. Then, I argue against the collapse of the pluralities into sets. What I am advocating is that the thesis of collapse limits important applications of the plural logic in model theory, when it is assumed that the quantifiers are absolutely general. Keywords: Pluralities; absolute generality; sets; hierarchies. We often say that some things form a set. For instance, every house in Beacon Hill may form a set. Also, all antimatter particles in the universe, all even numbers, all odd numbers, and in general all natural numbers do so. Naturally, following this line of thought, one might think that the plurality of all things constitutes a set. And al- though natural language allows us, by means of its plural constructions, to talk about objects without grouping them in one entity, there are also nominalization devices to turn constructions involving high order expressive resources into others that only make use of first order ones.
    [Show full text]
  • CHAPTER 1 the Main Subject of Mathematical Logic Is Mathematical
    CHAPTER 1 Logic The main subject of Mathematical Logic is mathematical proof. In this chapter we deal with the basics of formalizing such proofs and analysing their structure. The system we pick for the representation of proofs is natu- ral deduction as in Gentzen (1935). Our reasons for this choice are twofold. First, as the name says this is a natural notion of formal proof, which means that the way proofs are represented corresponds very much to the way a careful mathematician writing out all details of an argument would go any- way. Second, formal proofs in natural deduction are closely related (via the Curry-Howard correspondence) to terms in typed lambda calculus. This provides us not only with a compact notation for logical derivations (which otherwise tend to become somewhat unmanageable tree-like structures), but also opens up a route to applying the computational techniques which un- derpin lambda calculus. An essential point for Mathematical Logic is to fix the formal language to be used. We take implication ! and the universal quantifier 8 as basic. Then the logic rules correspond precisely to lambda calculus. The additional connectives (i.e., the existential quantifier 9, disjunction _ and conjunction ^) will be added via axioms. Later we will see that these axioms are deter- mined by particular inductive definitions. In addition to the use of inductive definitions as a unifying concept, another reason for that change of empha- sis will be that it fits more readily with the more computational viewpoint adopted there. This chapter does not simply introduce basic proof theory, but in addi- tion there is an underlying theme: to bring out the constructive content of logic, particularly in regard to the relationship between minimal and classical logic.
    [Show full text]
  • Plurals and Mereology
    Journal of Philosophical Logic (2021) 50:415–445 https://doi.org/10.1007/s10992-020-09570-9 Plurals and Mereology Salvatore Florio1 · David Nicolas2 Received: 2 August 2019 / Accepted: 5 August 2020 / Published online: 26 October 2020 © The Author(s) 2020 Abstract In linguistics, the dominant approach to the semantics of plurals appeals to mere- ology. However, this approach has received strong criticisms from philosophical logicians who subscribe to an alternative framework based on plural logic. In the first part of the article, we offer a precise characterization of the mereological approach and the semantic background in which the debate can be meaningfully reconstructed. In the second part, we deal with the criticisms and assess their logical, linguistic, and philosophical significance. We identify four main objections and show how each can be addressed. Finally, we compare the strengths and shortcomings of the mereologi- cal approach and plural logic. Our conclusion is that the former remains a viable and well-motivated framework for the analysis of plurals. Keywords Mass nouns · Mereology · Model theory · Natural language semantics · Ontological commitment · Plural logic · Plurals · Russell’s paradox · Truth theory 1 Introduction A prominent tradition in linguistic semantics analyzes plurals by appealing to mere- ology (e.g. Link [40, 41], Landman [32, 34], Gillon [20], Moltmann [50], Krifka [30], Bale and Barner [2], Chierchia [12], Sutton and Filip [76], and Champollion [9]).1 1The historical roots of this tradition include Leonard and Goodman [38], Goodman and Quine [22], Massey [46], and Sharvy [74]. Salvatore Florio [email protected] David Nicolas [email protected] 1 Department of Philosophy, University of Birmingham, Birmingham, United Kingdom 2 Institut Jean Nicod, Departement´ d’etudes´ cognitives, ENS, EHESS, CNRS, PSL University, Paris, France 416 S.
    [Show full text]
  • Beyond Plurals
    Beyond Plurals Agust´ınRayo philosophy.ucsd.edu/arayo July 10, 2008 I have two main objectives. The first is to get a better understanding of what is at issue between friends and foes of higher-order quantification, and of what it would mean to extend a Boolos-style treatment of second-order quantification to third- and higher- order quantification. The second objective is to argue that in the presence of absolutely general quantification, proper semantic theorizing is essentially unstable: it is impossible to provide a suitably general semantics for a given language in a language of the same logical type. I claim that this leads to a trilemma: one must choose between giving up absolutely general quantification, settling for the view that adequate semantic theorizing about certain languages is essentially beyond our reach, and countenancing an open-ended hierarchy of languages of ever ascending logical type. I conclude by suggesting that the hierarchy may be the least unattractive of the options on the table. 1 Preliminaries 1.1 Categorial Semantics Throughout this paper I shall assume the following: Categorial Semantics Every meaningful sentence has a semantic structure,1 which may be represented 1 as a certain kind of tree.2 Each node in the tree falls under a particular se- mantic category (e.g. `sentence', `quantifier’, `sentential connective'), and has an intension that is appropriate for that category. The semantic category and intension of each non-terminal node in the tree is determined by the semantic categories and intensions of nodes below it. Although I won't attempt to defend Categorial Semantics here,3 two points are worth emphasizing.
    [Show full text]
  • Plural Reference.Pdf
    OUP UNCORRECTED PROOF – FIRST PROOF, 11/12/2015, SPi Plurality and Unity Dictionary: NOSD 0002624321.INDD 1 11/12/2015 3:07:20 PM OUP UNCORRECTED PROOF – FIRST PROOF, 11/12/2015, SPi Dictionary: NOSD 0002624321.INDD 2 11/12/2015 3:07:20 PM OUP UNCORRECTED PROOF – FIRST PROOF, 11/12/2015, SPi Plurality and Unity Logic, Philosophy, and Linguistics !"#$!" %& Massimiliano Carrara, Alexandra Arapinis, and Friederike Moltmann Dictionary: NOSD 0002624321.INDD 3 11/12/2015 3:07:20 PM OUP UNCORRECTED PROOF – FIRST PROOF, 11/12/2015, SPi Great Clarendon Street, Oxford, OX' (DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors ')*( +e moral rights of the authors have been asserted First Edition published in ')*( Impression: * All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press *,- Madison Avenue, New York, NY *))*(, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: ')*.,//0/.
    [Show full text]
  • Precomplete Negation and Universal Quantification
    PRECOMPLETE NEGATION AND UNIVERSAL QUANTIFICATION by Paul J. Vada Technical Report 86-9 April 1986 Precomplete Negation And Universal Quantification. Paul J. Voda Department of Computer Science, The University of British Columbia, Vancouver, B.C. V6T IW5, Canada. ABSTRACT This paper is concerned with negation in logic programs. We propose to extend negation as failure by a stronger form or negation called precomplete negation. In contrast to negation as failure, precomplete negation has a simple semantic charaterization given in terms of computational theories which deli­ berately abandon the law of the excluded middle (and thus classical negation) in order to attain computational efficiency. The computation with precomplete negation proceeds with the direct computation of negated formulas even in the presence of free variables. Negated formulas are computed in a mode which is dual to the standard positive mode of logic computations. With uegation as failure the formulas with free variables must be delayed until the latter obtain values. Consequently, in situations where delayed formulas are never sufficiently instantiated, precomplete negation can find solutions unattainable with negation as failure. As a consequence of delaying, negation as failure cannot compute unbounded universal quantifiers whereas precomplete negation can. Instead of concentrating on the model-theoretical side of precomplete negation this paper deals with questions of complete computations and efficient implementations. April 1986 Precomplete Negation And Universal Quanti.flcation. Paul J. Voda Department or Computer Science, The University or British Columbia, Vancouver, B.C. V6T 1W5, Canada. 1. Introduction. Logic programming languages ba.5ed on Prolog experience significant difficulties with negation. There are many Prolog implementations with unsound negation where the computation succeeds with wrong results.
    [Show full text]
  • Set Theory for Computer Science by Prof Glynn Winskel
    Notes on Set Theory for Computer Science by Prof Glynn Winskel c Glynn Winskel 2 i A brief history of sets A set is an unordered collection of objects, and as such a set is determined by the objects it contains. Before the 19th century it was uncommon to think of sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural number, or being irrational, but it was rare to think of say the collection of rational numbers as itself an object. (There were exceptions. From Euclid mathe- maticians were used to thinking of geometric objects such as lines and planes and spheres which we might today identify with their sets of points.) In the mid 19th century there was a renaissance in Logic. For thousands of years, since the time of Aristotle and before, learned individuals had been familiar with syllogisms as patterns of legitimate reasoning, for example: All men are mortal. Socrates is a man. Therefore Socrates is mortal. But syllogisms involved descriptions of properties. The idea of pioneers such as Boole was to assign a meaning as a set to these descriptions. For example, the two descriptions \is a man" and \is a male homo sapiens" both describe the same set, viz. the set of all men. It was this objectification of meaning, understanding properties as sets, that led to a rebirth of Logic and Mathematics in the 19th century. Cantor took the idea of set to a revolutionary level, unveiling its true power. By inventing a notion of size of set he was able compare different forms of infinity and, almost incidentally, to shortcut several traditional mathematical arguments.
    [Show full text]