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Paradoxes (Trends in Logic Trends in Logic Volume 31 For further volumes: http://www.springer.com/series/6645 TRENDS IN LOGIC Studia Logica Library VOLUME 31 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Wieslaw Dziobiak, University of Puerto Rico at Mayagüez, USA Melvin Fitting, City University of New York, USA Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics ‘‘Ulisse Dini’’, University of Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany SCOPE OF THE SERIES Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica – that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time. Volume Editor Heinrich Wansing Piotr Łukowski Paradoxes 123 Piotr Łukowski Department of Cognitive Science University of Łódz´ Smugowa 10/12 91-433 Łódz´ Poland e-mail: [email protected] Translated by Marek Gensler ISBN 978-94-007-1475-5 e-ISBN 978-94-007-1476-2 DOI 10.1007/978-94-007-1476-2 Springer Dordrecht Heidelberg London New York Ó Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Cover design: eStudio Calamar, Berlin/Figueres Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my Parents, Maria and Zdzisław Contents 1 Introduction ........................................ 1 References . 3 2 Paradoxes of Wrong Intuition ........................... 5 2.1 Bottle Imp Paradox (Stevenson’s Bottle), or the Unintuitive Character of a Conclusion Following a Sufficient Multiplication of a Simple Reasoning . 5 2.2 Newcomb’s Paradox, or the Unintuitive Character of a Conclusion Drawn from Premises with Non-intuitive Assumption . 7 2.3 Paradox of Common Birthday, or the Unintuitive Character of Some Results in Probability Calculus. 11 2.4 Paradox of Approximation and Paradox of the Equator, or the Unintuitive Character of Some Results in Euclidean Geometry. 13 2.5 Horses’ Paradox, or the Intuitive Character of an Erroneous Application of Mathematical Induction. 15 2.6 Hempel’s Paradox (Raven, Confirmation), or the Unintuitive Character of Some Inductive Reasoning Results . 16 2.7 Paradoxes of Infinity, or Unintuitive Character of Some Results Obtained in Set Theory . 19 2.7.1 Aristotelian Circles Paradox, or the Definition of an Infinite Set. 19 2.7.2 Holy Trinity Paradox, or Application of the Definition of Infinite Set in Theology. 27 2.8 Fitch’s Paradox, or the Conflict of Two Intuitions. 32 References . 35 3 Paradoxes of Ambiguity ................................ 37 3.1 Protagoras’ (Law Teacher’s) Paradox. 37 3.2 Electra’s (the Veiled One’s) Paradox and Other Equivocations. 48 vii viii Contents 3.3 The Horny One’s Paradox . 51 3.4 Nameless Club Paradox . 52 3.5 Paradox of a Stone, or an Attempt at Disproving God’s Omnipotence . 54 References . 73 4 Paradoxes of Self-Reference ............................. 75 4.1 Möbius Ribbon and Klein Bottle, or Self-Reference in Mathematics . 75 4.2 Great Semantical Paradoxes . 80 4.2.1 Liar Antinomy . 80 4.2.2 Buridan’s Paradox. 106 4.2.3 Generalized Form of Liar Antinomy . 108 4.2.4 Curry’s Paradox . 109 4.3 Other Semantic Paradoxes . 110 4.3.1 Barber’s Antinomy . 111 4.3.2 Richard’s and Berry’s Antinomies. 113 4.3.3 Grelling’s Antinomy (Vox Non Appellans Se) . 118 4.4 Unexpected Examination (Hangman’s) Paradox, or Self-Reflexive Reasoning . 119 4.5 Crocodile’s Paradox, or Baron Münchhausen Fallacy. 124 References . 128 5 Ontological Paradoxes ................................. 131 5.1 Paradoxes of Difference - Paradox of a Heap . 131 5.1.1 What is Vagueness? . 141 5.1.2 Proposals Substituting Preciseness for Vagueness . 151 5.1.3 Vagueness Respecting Proposals. 166 5.2 Paradoxes of Change . 171 5.2.1 Paradox of the Moment of Death (Change of State Paradox) . 171 5.2.2 Paradoxes of Identity. 172 5.2.3 The Paradoxes of Motion. 175 5.3 Diagnosis of Paradoxes of Vagueness and Change . 181 References . 185 Names Index ........................................... 189 Subject Index .......................................... 193 Chapter 1 Introduction Because of their attractive intellectual form, it is tempting to treat paradoxes lightly, as if they were only an intellectual entertainment. Admittedly, when presented in interesting and non-trivial way, they can offer an attractive diversion at a party. Reducing them to entertainment only, however, is disrespectful not only to their authors but also to ourselves, since paradoxes should be treated as alarm signals informing of danger resulting from some errors in our thinking. This absolutely serious function which they play ought to be recognized especially by people who are interested in logic and philosophy. It is for this reason that para- doxes deserve not only attention but also serious consideration of the conclusions following from them. Regrettably, it is not so. One can still encounter logical and, especially, philosophical ideas, which seem to ignore the existence of paradoxical arguments, or at best question the sense of dealing with them. Many philosophical ideas can survive only because they disregard completely such paradoxes as sorites, paradoxes of change, motion or identity. If treated seriously, the conclu- sions following from those paradoxes could be death blows to a number of metaphysical or ontological ideas. Paradox is a thought construction, which leads to an unexpected contradiction.1 The essence of this definition lies in the word ‘‘unexpected’’. We often reach contradictions, but not every such situation is seen as paradoxical. True, a contra- diction understood as a conjunction of two propositions/sentences/statements/ claims, of which one is the negation of the other, is something that we should reject. But every such conjunction is paradoxical for us: if I consciously accept some P together with non-P, I should not be surprised that I have to accept a contradiction P and non-P. A paradox appears only when we get a contradiction in spite of: • Using logically correct rules of inference • Being convinced that we formulated our thoughts precisely • Certitude that the opinions we have so far are rational. 1 For stylistic reasons, the word ‘‘paradox’’ will be sometimes replaced by the word ‘‘dilemma’’ in order to avoid repetitions. P. Łukowski, Paradoxes, Trends in Logic, 31, 1 DOI: 10.1007/978-94-007-1476-2_1, Ó Springer Science+Business Media B.V. 2011 2 1 Introduction Then, indeed, the contradiction which appears is unexpected, since the procedure applied in our reasoning was devised specially to prevent it. Logical checks were functioning and yet we obtained a contradiction, a very serious error. By tradition, a paradox (gr. paqadonof, paradoxos) is an opinion, conclusion etc., which is contrary to common belief. According to Aristotle (384–322 BC), a statement paradoxical when it is contrary to the belief of a group of people. One and the same opinion can be a paradox for one group of people but not for another. The word ‘‘infinity’’ is often used in its every day sense of something without an end, something inexhaustible. For a common man, the mathematical understanding of infinity and its properties would be a surprise or even something unbelievable. Yet that knowledge is nothing strange for a person with mathematical education. In a narrower sense, a paradox is an antinomy (Greek amsimolia, antinomia; anti— against, molof, nomos—law). We shall call a reasoning antinomic when, despite its logically correct form, it leads to a contradiction: P and non-P. In a still narrower sense, an antinomy is a logically correct reasoning, which justifies an equivalence: P if and only if non-P. It is in this sense that antinomy is understood most often. A special kind of paradoxes are sophisms (Greek: rouirla, sophisma—false conclusion). Aristotle uses his name for every argument, which seems to be correct but is not. Usually, the word ‘‘sophism’’ is understood to include an intention to deceive the listener. For this reason, one distinguishes a paralogism (Greek: paqakocirlof, paralogismos—false conclusion), which is an incorrect reasoning without an intention to deceive. A person, who presents such a reasonings, is either unaware that it is erroneous or knows that it is wrong but treats the presentation as a joke or a didactic trick to be explained soon. Classification of paradoxes is rather difficult, since it is not clear according to which criteria it should be done. In this book, we have accepted the essence of a paradox as the fundament of classification. If the essence of a paradox is an error of ambiguity, it is classified as a paradox ambiguity, regardless of the character of its narration. This can be manifestly seen in the case of such paradoxes as Newcomb’s and Protagoras’s. Accordingly, the book has four chapters. The first one analyzes those paradoxes, which result from a clash between a logically correct reasoning and our previously accepted opinions. Here we can place e.g., Georg Cantor’s set theory paradoxes, but because of their strictly mathematical character they will not be discussed in this book.
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