Plurals and Mereology
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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”. -
Lecture Notes on the Lambda Calculus 2.4 Introduction to Β-Reduction
2.2 Free and bound variables, α-equivalence. 8 2.3 Substitution ............................. 10 Lecture Notes on the Lambda Calculus 2.4 Introduction to β-reduction..................... 12 2.5 Formal definitions of β-reduction and β-equivalence . 13 Peter Selinger 3 Programming in the untyped lambda calculus 14 3.1 Booleans .............................. 14 Department of Mathematics and Statistics 3.2 Naturalnumbers........................... 15 Dalhousie University, Halifax, Canada 3.3 Fixedpointsandrecursivefunctions . 17 3.4 Other data types: pairs, tuples, lists, trees, etc. ....... 20 Abstract This is a set of lecture notes that developed out of courses on the lambda 4 The Church-Rosser Theorem 22 calculus that I taught at the University of Ottawa in 2001 and at Dalhousie 4.1 Extensionality, η-equivalence, and η-reduction. 22 University in 2007 and 2013. Topics covered in these notes include the un- typed lambda calculus, the Church-Rosser theorem, combinatory algebras, 4.2 Statement of the Church-Rosser Theorem, and some consequences 23 the simply-typed lambda calculus, the Curry-Howard isomorphism, weak 4.3 Preliminary remarks on the proof of the Church-Rosser Theorem . 25 and strong normalization, polymorphism, type inference, denotational se- mantics, complete partial orders, and the language PCF. 4.4 ProofoftheChurch-RosserTheorem. 27 4.5 Exercises .............................. 32 Contents 5 Combinatory algebras 34 5.1 Applicativestructures. 34 1 Introduction 1 5.2 Combinatorycompleteness . 35 1.1 Extensionalvs. intensionalviewoffunctions . ... 1 5.3 Combinatoryalgebras. 37 1.2 Thelambdacalculus ........................ 2 5.4 The failure of soundnessforcombinatoryalgebras . .... 38 1.3 Untypedvs.typedlambda-calculi . 3 5.5 Lambdaalgebras .......................... 40 1.4 Lambdacalculusandcomputability . 4 5.6 Extensionalcombinatoryalgebras . 44 1.5 Connectionstocomputerscience . -
On the Boundary Between Mereology and Topology
On the Boundary Between Mereology and Topology Achille C. Varzi Istituto per la Ricerca Scientifica e Tecnologica, I-38100 Trento, Italy [email protected] (Final version published in Roberto Casati, Barry Smith, and Graham White (eds.), Philosophy and the Cognitive Sciences. Proceedings of the 16th International Wittgenstein Symposium, Vienna, Hölder-Pichler-Tempsky, 1994, pp. 423–442.) 1. Introduction Much recent work aimed at providing a formal ontology for the common-sense world has emphasized the need for a mereological account to be supplemented with topological concepts and principles. There are at least two reasons under- lying this view. The first is truly metaphysical and relates to the task of charac- terizing individual integrity or organic unity: since the notion of connectedness runs afoul of plain mereology, a theory of parts and wholes really needs to in- corporate a topological machinery of some sort. The second reason has been stressed mainly in connection with applications to certain areas of artificial in- telligence, most notably naive physics and qualitative reasoning about space and time: here mereology proves useful to account for certain basic relation- ships among things or events; but one needs topology to account for the fact that, say, two events can be continuous with each other, or that something can be inside, outside, abutting, or surrounding something else. These motivations (at times combined with others, e.g., semantic transpar- ency or computational efficiency) have led to the development of theories in which both mereological and topological notions play a pivotal role. How ex- actly these notions are related, however, and how the underlying principles should interact with one another, is still a rather unexplored issue. -
The Joint Philosophical Program of Russell and Wittgenstein and Its Demise
The Joint Philosophical Program of Russell and Wittgenstein and Its Demise Abstract Between April and November 1912, Bertrand Russell and Ludwig Wittgenstein were en- gaged in a joint philosophical program. Wittgenstein‘s meeting with Gottlob Frege in De- cember 1912 led, however, to its dissolution – the joint program was abandoned. Section 2 of this paper outlines the key points of that program, identifying what Russell and Wittgenstein each contributed to it. The third section determines precisely those features of their collabora- tive work that Frege criticized. Finally, building upon the evidence developed in the preced- ing two sections, section 4 recasts along previously undeveloped lines Wittgenstein‘s logical– philosophical discoveries in the two years following his encounter with Frege in 1912. The paper concludes, in section 5, with an overview of the dramatic consequences the Frege- Wittgenstein critique had for Russell‘s philosophical development. 1. Introductory Remarks The subject of our investigation is the interaction between the three founding fathers of analytic philosophy, Russell, Wittgenstein and Frege, in a most important period of their philosophical development. We discuss the joint program that Russell and Wittgenstein worked on from April till November 1912, as well as its collapse after Wittgenstein visited Frege in Jena in December the same year. Frege‘s criticism of elements of the program underpinned and motivated Wittgenstein‘s attack on Russell‘s approach in philosophy that culminated in May and June 1913 when, facing Wittgenstein‘s criticism, Russell stopped writing the Theory of Knowledge book project. Frege‘s arguments also urged Wittgenstein to rethink and reformulate his own philosophical ideas. -
Abstract 1. Russell As a Mereologist
THE 1900 TURN IN BERTRAND RUSSELL’S LOGIC, THE EMERGENCE OF HIS PARADOX, AND THE WAY OUT Prof. Dr. Nikolay Milkov, Universität Paderborn, [email protected] Abstract Russell‘s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell‘s logic was nothing but mereology. First, his acquaintance with Peano‘s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‗denoting phrase‘. Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell‘s new con- ception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell‘s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and sim- ples. (ii) The elimination of classes, with the help of the ‗substitutional theory‘,1 and of prop- ositions, by means of the Multiple Relation Theory of Judgment,2 completed this process. 1. Russell as a Mereologist In 1898, Russell abandoned his short period of adherence to the Neo-Hegelian position in the philosophy of mathematics and replaced it with what can be called the ‗analytic philoso- phy of mathematics‘, substantiated by the logic of relations. -
Category Theory and Lambda Calculus
Category Theory and Lambda Calculus Mario Román García Trabajo Fin de Grado Doble Grado en Ingeniería Informática y Matemáticas Tutores Pedro A. García-Sánchez Manuel Bullejos Lorenzo Facultad de Ciencias E.T.S. Ingenierías Informática y de Telecomunicación Granada, a 18 de junio de 2018 Contents 1 Lambda calculus 13 1.1 Untyped λ-calculus . 13 1.1.1 Untyped λ-calculus . 14 1.1.2 Free and bound variables, substitution . 14 1.1.3 Alpha equivalence . 16 1.1.4 Beta reduction . 16 1.1.5 Eta reduction . 17 1.1.6 Confluence . 17 1.1.7 The Church-Rosser theorem . 18 1.1.8 Normalization . 20 1.1.9 Standardization and evaluation strategies . 21 1.1.10 SKI combinators . 22 1.1.11 Turing completeness . 24 1.2 Simply typed λ-calculus . 24 1.2.1 Simple types . 25 1.2.2 Typing rules for simply typed λ-calculus . 25 1.2.3 Curry-style types . 26 1.2.4 Unification and type inference . 27 1.2.5 Subject reduction and normalization . 29 1.3 The Curry-Howard correspondence . 31 1.3.1 Extending the simply typed λ-calculus . 31 1.3.2 Natural deduction . 32 1.3.3 Propositions as types . 34 1.4 Other type systems . 35 1.4.1 λ-cube..................................... 35 2 Mikrokosmos 38 2.1 Implementation of λ-expressions . 38 2.1.1 The Haskell programming language . 38 2.1.2 De Bruijn indexes . 40 2.1.3 Substitution . 41 2.1.4 De Bruijn-terms and λ-terms . 42 2.1.5 Evaluation . -
Mereology and the Sciences SYNTHESE LIBRARY
Mereology and the Sciences SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE Editor-in-Chief LUCIANO FLORIDI, University of Oxford, Oxford Internet Institute, United Kingdom Editors THEO A.F. KUIPERS, University of Groningen Fac. Philosophy, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University Dept. Philosophy, USA PATRICK SUPPES, Stanford University Ctr. Study of Language & Information, USA JAN WOLEÑSKI, Jagiellonian University of Kraków Institute of Philosophy, Poland DIRK VAN DALEN, Utrecht University Department of Philosophy, The Netherlands VOLUME 371 For further volumes: http://www.springer.com/series/6607 Claudio Calosi Pierluigi Graziani Editors Mereology and the Sciences Parts and Wholes in the Contemporary Scientific Context 123 Editors Claudio Calosi Pierluigi Graziani Department of Basic Sciences and Department of Basic Sciences and Foundations Foundations University of Urbino University of Urbino Urbino Urbino Italy Italy ISBN 978-3-319-05355-4 ISBN 978-3-319-05356-1 (eBook) DOI 10.1007/978-3-319-05356-1 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014939928 c Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. -
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. -
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. -
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/. -
Legal Theory Workshop UCLA School of Law Asya Passinsky
Legal Theory Workshop UCLA School of Law Asya Passinsky Visiting Assistant Professor, University of North Carolina at Chapel Hill “NORM AND OBJECT: A NORMATIVE HYLOMORPHIC THEORY OF SOCIAL OBJECTS” Thursday, March 4, 2021, 5:00-6:00 pm Via Zoom Draft, February 19, 2021. For UCLA Workshop. Please Don’t Cite Or Quote Without Permission. Norm and Object: A Normative Hylomorphic Theory of Social Objects Asya Passinsky ABSTRACT: This paper is an investigation into the metaphysics of social objects such as political borders, states, and organizations. I articulate a metaphysical puzzle concerning such objects and then propose a novel account of social objects that provides a solution to the puzzle. The basic idea behind the puzzle is that under appropriate circumstances, seemingly concrete social objects can apparently be created by acts of agreement, decree, declaration, or the like. Yet there is reason to believe that no concrete object can be created in this way. The central idea of my positive account is that social objects have a normative component to them, and seemingly concrete social objects have both normative and material components. I develop this idea more rigorously using resources from the Aristotelian hylomorphic tradition. The resulting normative hylomorphic account, I argue, solves the puzzle by providing a satisfying explanation of creation-by-agreement and the like, while at the same time avoiding the difficulties facing extant accounts of social objects. 1. Introduction This paper is an investigation into the metaphysics of social objects such as political borders, states, and organizations. Roughly speaking, social objects are things that can be created through the performance of social acts such as agreement, decree, declaration, or the like. -
Computational Lambda-Calculus and Monads
Computational lambda-calculus and monads Eugenio Moggi∗ LFCS Dept. of Comp. Sci. University of Edinburgh EH9 3JZ Edinburgh, UK [email protected] October 1988 Abstract The λ-calculus is considered an useful mathematical tool in the study of programming languages, since programs can be identified with λ-terms. However, if one goes further and uses βη-conversion to prove equivalence of programs, then a gross simplification1 is introduced, that may jeopardise the applicability of theoretical results to real situations. In this paper we introduce a new calculus based on a categorical semantics for computations. This calculus provides a correct basis for proving equivalence of programs, independent from any specific computational model. 1 Introduction This paper is about logics for reasoning about programs, in particular for proving equivalence of programs. Following a consolidated tradition in theoretical computer science we identify programs with the closed λ-terms, possibly containing extra constants, corresponding to some features of the programming language under con- sideration. There are three approaches to proving equivalence of programs: • The operational approach starts from an operational semantics, e.g. a par- tial function mapping every program (i.e. closed term) to its resulting value (if any), which induces a congruence relation on open terms called operational equivalence (see e.g. [Plo75]). Then the problem is to prove that two terms are operationally equivalent. ∗On leave from Universit`a di Pisa. Research partially supported by the Joint Collaboration Contract # ST2J-0374-C(EDB) of the EEC 1programs are identified with total functions from values to values 1 • The denotational approach gives an interpretation of the (programming) lan- guage in a mathematical structure, the intended model.