Logical Foundations of Physics. Part I. Zeno's Dichotomy Paradox
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Beyond Good and Evil—1
Nietzsche & Asian Philosophy Beyond Good and Evil—1 Beyond good and Evil Preface supposing truth is a woman philosophers like love-sick suitors who don’t understand the woman-truth central problem of philosophy is Plato’s error: denying perspective, the basic condition of all life On the Prejudices of Philosophers 1) questioning the will to truth who is it that really wants truth? What in us wants truth? Why not untruth? 2) origin of the will to truth out of the will to untruth, deception can anything arise out of its opposite? A dangerous questioning? Nietzsche sees new philosophers coming up who have the strength for the dangerous “maybe.” Note in general Nietzsche’s preference for the conditional tense, his penchant for beginning his questioning with “perhaps” or “suppose” or “maybe.” In many of the passages throughout this book Nietzsche takes up a perspective which perhaps none had dared take up before, a perspective to question what had seemed previously to be unquestionable. He seems to constantly be tempting the reader with a dangerous thought experiment. This begins with the questioning of the will to truth and the supposition that, perhaps, the will to truth may have arisen out of its opposite, the will to untruth, ignorance, deception. 3) the supposition that the greater part of conscious thinking must be included among instinctive activities Nietzsche emphasizes that consciousness is a surface phenomenon conscious thinking is directed by what goes on beneath the surface contrary to Plato’s notion of pure reason, the conscious -
Epistemological Consequences of the Incompleteness Theorems
Epistemological Consequences of the Incompleteness Theorems Giuseppe Raguní UCAM - Universidad Católica de Murcia, Avenida Jerónimos 135, Guadalupe 30107, Murcia, Spain - [email protected] After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main episte- mological consequences of the first incompleteness Theorem for the two funda- mental arithmetical theories are shown: the non-mechanizability for the truths of the first-order arithmetic and the peculiarities for the model of the second- order arithmetic. Finally, the common epistemological interpretation of the second incompleteness Theorem is corrected, proposing the new Metatheorem of undemonstrability of internal consistency. KEYWORDS: semantics, languages, epistemology, paradoxes, arithmetic, in- completeness, first-order, second-order, consistency. 1 Semantics in the Languages Consider an arbitrary language that, as normally, makes use of a countable1 number of characters. Combining these characters in certain ways, are formed some fundamental strings that we call terms of the language: those collected in a dictionary. When the terms are semantically interpreted, i. e. a certain meaning is assigned to them, we have their distinction in adjectives, nouns, verbs, etc. Then, a proper grammar establishes the rules arXiv:1602.03390v1 [math.GM] 13 Jan 2016 of formation of sentences. While the terms are finite, the combinations of grammatically allowed terms form an infinite-countable amount of possible sentences. In a non-trivial language, the meaning associated to each term, and thus to each ex- pression that contains it, is not always unique. The same sentence can enunciate different things, so representing different propositions. -
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”. -
The Strength of Mac Lane Set Theory
The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret -
Derridean Deconstruction and Feminism
DERRIDEAN DECONSTRUCTION AND FEMINISM: Exploring Aporias in Feminist Theory and Practice Pam Papadelos Thesis Submitted for the Degree of Doctor of Philosophy in the Discipline of Gender, Work and Social Inquiry Adelaide University December 2006 Contents ABSTRACT..............................................................................................................III DECLARATION .....................................................................................................IV ACKNOWLEDGEMENTS ......................................................................................V INTRODUCTION ..................................................................................................... 1 THESIS STRUCTURE AND OVERVIEW......................................................................... 5 CHAPTER 1: LAYING THE FOUNDATIONS – FEMINISM AND DECONSTRUCTION ............................................................................................... 8 INTRODUCTION ......................................................................................................... 8 FEMINIST CRITIQUES OF PHILOSOPHY..................................................................... 10 Is Philosophy Inherently Masculine? ................................................................ 11 The Discipline of Philosophy Does Not Acknowledge Feminist Theories......... 13 The Concept of a Feminist Philosopher is Contradictory Given the Basic Premises of Philosophy..................................................................................... -
Type Theory and Applications
Type Theory and Applications Harley Eades [email protected] 1 Introduction There are two major problems growing in two areas. The first is in Computer Science, in particular software engineering. Software is becoming more and more complex, and hence more susceptible to software defects. Software bugs have two critical repercussions: they cost companies lots of money and time to fix, and they have the potential to cause harm. The National Institute of Standards and Technology estimated that software errors cost the United State's economy approximately sixty billion dollars annually, while the Federal Bureau of Investigations estimated in a 2005 report that software bugs cost U.S. companies approximately sixty-seven billion a year [90, 108]. Software bugs have the potential to cause harm. In 2010 there were a approximately a hundred reports made to the National Highway Traffic Safety Administration of potential problems with the braking system of the 2010 Toyota Prius [17]. The problem was that the anti-lock braking system would experience a \short delay" when the brakes where pressed by the driver of the vehicle [106]. This actually caused some crashes. Toyota found that this short delay was the result of a software bug, and was able to repair the the vehicles using a software update [91]. Another incident where substantial harm was caused was in 2002 where two planes collided over Uberlingen¨ in Germany. A cargo plane operated by DHL collided with a passenger flight holding fifty-one passengers. Air-traffic control did not notice the intersecting traffic until less than a minute before the collision occurred. -
METALOGIC METALOGIC an Introduction to the Metatheory of Standard First Order Logic
METALOGIC METALOGIC An Introduction to the Metatheory of Standard First Order Logic Geoffrey Hunter Senior Lecturer in the Department of Logic and Metaphysics University of St Andrews PALGRA VE MACMILLAN © Geoffrey Hunter 1971 Softcover reprint of the hardcover 1st edition 1971 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First published 1971 by MACMILLAN AND CO LTD London and Basingstoke Associated companies in New York Toronto Dublin Melbourne Johannesburg and Madras SBN 333 11589 9 (hard cover) 333 11590 2 (paper cover) ISBN 978-0-333-11590-9 ISBN 978-1-349-15428-9 (eBook) DOI 10.1007/978-1-349-15428-9 The Papermac edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent, in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. To my mother and to the memory of my father, Joseph Walter Hunter Contents Preface xi Part One: Introduction: General Notions 1 Formal languages 4 2 Interpretations of formal languages. Model theory 6 3 Deductive apparatuses. Formal systems. Proof theory 7 4 'Syntactic', 'Semantic' 9 5 Metatheory. The metatheory of logic 10 6 Using and mentioning. Object language and metalang- uage. Proofs in a formal system and proofs about a formal system. Theorem and metatheorem 10 7 The notion of effective method in logic and mathematics 13 8 Decidable sets 16 9 1-1 correspondence. -
Computable Diagonalizations and Turing's Cardinality Paradox
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library J Gen Philos Sci (2014) 45:239–262 DOI 10.1007/s10838-014-9244-x ARTICLE Computable Diagonalizations and Turing’s Cardinality Paradox Dale Jacquette Published online: 26 February 2014 Ó Springer Science+Business Media Dordrecht 2014 Abstract A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimen- sional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two- dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. -
Axiomatic Set Theory As a Basis for the Construction of Mathematics"
"Axiomatic set theory as a basis for the construction of mathematics"# A diosertatien submitted in support of candidature for the Master of Science degree of the University of London (Faculty of Science) by BRIAN ROTMAN March, 1962% ir • ProQuest Number: 10097259 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest. ProQuest 10097259 Published by ProQuest LLC(2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ABSTRACT It is widely known that one of the major tasks of •Foundations* is to construct a formal system which can he said to contain the whole of mathematics# For various reasons axiomatic set theory is a very suitable • choice for such a system and it is one which has proved acceptable to both logicians and mathematicians# The particular demands of mathematicians and logicians, however, are not the same# As a result there exist at the moment two different formulations of set theory idiich can be rou^ily said to cater for logicians and mathematicians respectively*# It is these systems which are the subject of this dissertation# The -
Hume's Law, Moore's Open Question and Aquinas' Human Intellect
Hume’s Law, Moore’s Open Question and Aquinas’ Human Intellect Augusto Trujillo Werner University of Malaga Abstract This article concerns Aquinas’ practical doctrine on two philosophical difficulties underlying much contemporary ethical debate. One is Hume’s Is-ought thesis and the other is its radical consequence, Moore’s Open-question argument. These ethical paradoxes appear to have their roots in epistemological scepticism and in a deficient anthropology. A possible response to them can be found in that a) Aquinas defends the substantial unity and rationality of the human being; b) Thomistic natural law is a natural consequence of the rational being; c) Thomistic human intellect is essentially theoretical and practical at the same time; d) Aquinas’ human reason naturally performs three main operations. First, to apprehend the intellecta and universal notions ens, verum and bonum. Second, to formulate the first theoretical and practical principles. Third, to order that the intellectum and universal good be done and the opposite avoided. For these reasons, Thomistic philosophical response to both predicaments will not be exclusively ethical, but will embrace ontology, anthropology and epistemology. Aquinas’ moral philosophy is fundamentally different from ethics that qualifies actions as good either by mere social consensus (contractualism) or just by calculating its consequences (consequentialism). Keywords: ontology, anthropology, epistemology, ethics. 1. Hume’s Law and Moore’s Open Question The first part of this article will study Aquinas’ possible response to Hume’s law. According to shared interpretation, David Hume sought to reform philosophy (Mackie 1980) and this paper will focus on his moral philosophy, by arguing against his famous Is-ought thesis or Hume’s Law.1 It may be briefly defined as being unlawful to derive ought (what ought to be) from is (what is). -
Structural Dichotomy Theorems in Descriptive Set Theory
Structural dichotomy theorems in descriptive set theory Benjamin Miller Universit¨atWien Winter Semester, 2019 Introduction The goal of these notes is to provide a succinct introduction to the primary structural dichotomy theorems of descriptive set theory. The only prerequisites are a rudimentary knowledge of point-set topology and set theory. Working in the base theory ZF + DC, we first discuss trees, the corresponding representations of closed, Borel, and Souslin sets, and Baire category. We then consider consequences of the open dihypergraph dichotomy and variants of the G0 dichotomy. While pri- marily focused upon Borel structures, we also note that minimal modi- fications of our arguments can be combined with well-known structural consequences of determinacy (which we take as a black box) to yield generalizations into the projective hierarchy and beyond. iii Contents Introduction iii Chapter 1. Preliminaries 1 1. Closed sets 1 2. Ranks 2 3. Borel sets 3 4. Souslin sets 4 5. Baire category 9 6. Canonical objects 13 Chapter 2. The box-open dihypergraph dichotomy 27 1. Colorings of box-open dihypergraphs 27 2. Partial compactifications 31 3. Separation by unions of closed hyperrectangles 34 Chapter 3. The G0 dichotomy, I: Abstract colorings 39 1. Colorings within cliques 39 2. Discrete perfect sets within cliques 42 3. Scrambled sets 45 Chapter 4. The G0 dichotomy, II: Borel colorings 49 1. Borel colorings 49 2. Index two subequivalence relations 52 3. Perfect antichains 54 4. Parametrization and uniformization 56 Chapter 5. The (G0; H0) dichotomy 61 1. Borel local colorings 61 2. Linearizability of quasi-orders 67 Bibliography 71 Index 73 v CHAPTER 1 Preliminaries 1. -
Philosophy, Ἱερατική, and the Damascian Dichotomy: Pursuing the Bacchic Ideal in the Sixth-Century Academy Todd Krulak
Philosophy, Ἱερατική, and the Damascian Dichotomy: Pursuing the Bacchic Ideal in the Sixth-Century Academy Todd Krulak HE BEGINNING of the sixth century CE found the venerable Athenian Academy unexpectedly struggling T to maintain its very existence. The institution enjoyed both curricular and political stability for much of the fifth century under the capable leadership of Proclus of Lycia, but after his death in 485, the school struggled to find a capable successor. Edward Watts has suggested that the failing fortunes of the Academy in this period were due to a dearth of candi- dates who possessed the right combination of the intellectual heft required to carry on the grand tradition of the Academy and the political deftness needed to protect it against encroach- ing Christian authorities. As a result, the school was perched on the razor’s edge. In particular, Watts postulates that an im- politic emphasis on traditional forms of religious ritual in the Academy in the post-Proclan era, especially by the indiscrete Hegias, the head of the school around the beginning of the sixth century, intensified Christian scrutiny.1 Thus, the assump- tion of the position of scholarch, the head of the school, by the 1 See E. J. Watts, City and School in Late Antique Athens and Alexandria (Berke- ley 2006) 118–142; and “Athens between East and West: Elite Self-Presen- tation and the Durability of Traditional Cult,” GRBS 57 (2017) 191–213, which contends that, with regard to religion, Athens retained a decidedly traditional character well into the fifth century. Even after the temples were shuttered in the second quarter of that century, elites supportive of ancestral practices extended financial support to the Academy, for it was known to be sympathetic to traditional deities and rites.