DOCSLIB.ORG

Self-Referential Paradoxes Three Motivating Examples Philosophical Noson S

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Self-Referential Paradoxes Three Motivating Examples Philosophical Noson S Self- Referential Paradoxes Noson S. Yanofsky Preliminaries Self-Referential Paradoxes Three Motivating Examples Philosophical Noson S. Yanofsky Interlude Cantor's Inequalities Brooklyn College, CUNY Main Theorem February 17, 2021 Turing's Halting Problem Fixed Points in Logic Two Other Paradoxes Further Reading My Agenda Self- Some of the most profound and famous theorems in Referential Paradoxes mathematics and computer science of the past 150 years are Noson S. instances of self-referential paradoxes. Yanofsky Georg Cantor's theorem that shows there are dierent Preliminaries levels of innity; Three Bertrand Russell's paradox which proves that simple set Motivating Examples theory is inconsistent; Philosophical Interlude Kurt Gödel's famous incompleteness theorems that Cantor's demonstrates a limitation of the notion of proof; Inequalities Alan Turing's realization that some problems can never be Main Theorem solved by a computer; Turing's and much more. Halting Problem Amazingly, all these diverse theorems can be seen as instances Fixed Points of a single simple theorem of basic category theory. We describe in Logic this theorem and show some of the instances. No category Two Other Paradoxes theory is needed for this talk. Further Reading My Hidden Agenda Self- Referential Paradoxes Noson S. Yanofsky This talk is a single section (of 58) of an introductory category Preliminaries theory textbook I am writing called Three Motivating MONOIDAL CATEGORIES Examples A Unifying Concept in Mathematics, Physics, and Computing Philosophical Interlude Cantor's The goal is to show the power of category theory (and to get Inequalities you to buy the book!!!) The point is that with a little category Main Theorem theory, one can know a hell-of-a-lot of mathematics, physics, Turing's and computers. Halting Problem Fixed Points in Logic Two Other Paradoxes Further Reading Outline of the Talk Self- Referential Paradoxes 1 Some Preliminaries Noson S. Yanofsky 2 Three Motivating Examples Preliminaries 3 Philosophical Interlude Three Motivating 4 Examples Cantor's Inequalities Philosophical Interlude 5 The Main Theorem About Self-Referential Paradoxes Cantor's Inequalities 6 Turing's Halting Problem Main Theorem 7 Fixed Points in Logic Turing's Halting Problem 8 Two Other Paradoxes Fixed Points in Logic 9 Further Reading Two Other Paradoxes Further Reading Some Preliminaries Self- Referential Paradoxes Before we leap into all the examples, there are some technical Noson S. Yanofsky ideas about sets. Preliminaries Let 2 = f0; 1g be a set with two values which correspond Three to true and false. Motivating Examples Let S be a set. A function g : S −! 2 is a characteristic Philosophical function and describes a subset of . Interlude S Cantor's Consider a set function f : S × S −! 2. The f accepts two Inequalities elements of S and outputs either 0 or 1. Think of Main Theorem Turing's f (a; b) = 1 Halting Problem meaning is a part of or is described by . Fixed Points a b a b in Logic Two Other Paradoxes Further Reading Some Preliminaries Self- Referential Paradoxes For any element s0 of S, consider the function f where the Noson S. Yanofsky second input is always s0. We say that s0 is hardwired into the function. This gives us a function Preliminaries Three 2 Motivating f ( ; s0): S −! Examples Philosophical with only one input. Since this function goes from S to 2, Interlude it also is a characteristic function and describes a subset of Cantor's Inequalities S. The subset is Main Theorem fs 2 S : f (s; s0) = 1g: Turing's Halting Problem It is all the elements that are part of s0 or all the Fixed Points in Logic elements that are described by s0. Two Other Paradoxes Further Reading Some Preliminaries Self- Referential Paradoxes Noson S. Yanofsky For dierent f 's and various elements of S, there are Preliminaries dierent characteristic functions which describe various Three subsets. Motivating Examples We now ask a simple question: given g : S −! 2 and Philosophical 2, is there an in such that Interlude f : S × S −! s0 S g Cantor's characterizes the same subset as f ( ; s0)? Inequalities To restate, for a given g and f , does there exist an s0 2 S Main Theorem such that g( ) = f ( ; s0)? If such an s0 exists, then we Turing's say g can be represented by f . Halting Problem Fixed Points in Logic Two Other Paradoxes Further Reading Some Preliminaries Self- Referential For every set S there is a set map ∆: S −! S × S called Paradoxes the diagonal map that takes an element t to the ordered Noson S. Yanofsky pair (t; t). This is the core of self reference. Preliminaries If f : S × S −! 2 is a function that evaluates the Three relationship of S elements to S elements, then Motivating Examples f Philosophical S × S / 2 Interlude < ∆ Cantor's Inequalities Main S Theorem Turing's takes every element as follows Halting t 2 S Problem Fixed Points t 7−! (t; t) 7−! f (t; t): in Logic Two Other Paradoxes This evaluates t with itself. Further Reading The Barber Paradox Self- Bertrand Russell was a great expositor. The barber paradox is Referential Paradoxes attributed to Russell and is used to explain some of the central Noson S. ideas of self-referential systems. Imagine an isolated village in Yanofsky the Austrian alps where it is dicult for villagers to leave and Preliminaries for itinerant barbers to come to the village. This village has Three exactly one barber and there is a strict rule that is enforced: Motivating Examples Philosophical A villager cuts his own hair i he does not go to the barber. Interlude Cantor's If the villager will cut his own hair, why should he go to the Inequalities barber? On the other hand, if the villager goes to the barber, Main Theorem he will not need to cut his own hair. This works out very well Turing's for the villagers except for one: the barber. Who cuts the Halting Problem barber's hair? If the barber cuts his own hair, then he is Fixed Points violating the village ordinance by cutting his own hair and in Logic having his hair cut by the barber. If he goes to the barber, then Two Other Paradoxes he is also cutting his own hair. This is illegal! Further Reading The Barber Paradox Self- Let us formalize the problem. Let the set Vill consist of all the Referential Paradoxes villagers in the village. The function Noson S. Yanofsky f : Vill × Vill −! 2 Preliminaries describes who cuts whose hair in the village. It is dened for 0 Three villagers v and v as Motivating Examples 8 1 : if the hair of v is cut by v 0 Philosophical 0 < Interlude f (v; v ) = 0 Cantor's : 0 : if the hair of v is not cut by v : Inequalities Main We can now express the village ordinance as saying that for all v Theorem Turing's f (v; v) = 1 if and only if f (v; barber) = 0: Halting Problem This is true for all v including v = barber. In this case we get: Fixed Points in Logic f (barber; barber) = 1 if and only if f (barber; barber) = 0: Two Other Paradoxes This is a contradiction! Further Reading The Barber Paradox Self- Referential Let us be more categorical. There is Paradoxes The diagonal set function that is Noson S. ∆: Vill −! Vill × Vill Yanofsky dened as ∆(v) = (v; v). Preliminaries There is also a negation function NOT : 2 −! 2 dened as Three NOT (0) = 1 and NOT (1) = 0. Motivating Examples Composing f with ∆ and NOT gives us g : Vill −! 2 as Philosophical Interlude in the following commutative diagram: Cantor's Inequalities f Vill × Vill / 2 Main 9 Theorem ∆ NOT Turing's Halting 2 Problem Vill g / Fixed Points in Logic g = NOT ◦ f ◦ ∆: Two Other Paradoxes Further Reading The Barber Paradox Self- For a villager v, Referential Paradoxes Noson S. g(v) == NOT (f (∆(v))) = NOT (f (v; v)): Yanofsky So Preliminaries g(v) = 1 Three Motivating if and only if Examples Philosophical NOT (f (v; v)) = 1 Interlude if and only if Cantor's Inequalities f (v; v) = 0 Main Theorem if and only if Turing's the hair of is not cut by . Halting v v Problem In other words g(v) = 1 if and only if v does not cut his own Fixed Points hair. in Logic Two Other g is the characteristic function of the subset of villagers who do Paradoxes not cut their own hair. Further Reading The Barber Paradox Self- We now ask the simple question: can g be represented by f ? In Referential Paradoxes other words, is there a villager v0 such that g( ) = f ( ; v0)? It Noson S. stands to reason that the barber is the villager who can Yanofsky represent g. After all, f ( ; barber) describes all the villagers Preliminaries who get their hair cut by the barber. Three Motivating Examples g(v) = f (v; barber) Philosophical Interlude What about v = barber. Cantor's Inequalities barber barber barber Main g( ) = f ( ; ): Theorem Turing's But the denition of g is given as Halting Problem g(barber) = NOT (f (barber; barber)). We conclude that g is Fixed Points not represented by f ( ; barber) That is, the set of villagers in Logic who do not cut their own hair can not be the same as the set of Two Other Paradoxes villagers who get their hair cut by anyone. Further Reading The Barber Paradox Matrix Form Self- Referential It is helpful to describe this problem in matrix form. Let us Paradoxes consider the set Vill as fv1; v2; v3;:::; v g.
Load more

Recommended publications
  • Philosophy and the Foreigner in Plato's Dialogues
    Philosophy and the Foreigner in Plato’s Dialogues By Rebecca LeMoine A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Political Science) at the UNIVERSITY OF WISCONSIN-MADISON 2014 Date of final oral examination: 06/20/2014 The dissertation is approved by the following members of the Final Oral Committee: Richard Avramenko, Associate Professor, Political Science Alex Dressler, Assistant Professor, Classics Daniel Kapust, Associate Professor, Political Science Helen Kinsella, Associate Professor, Political Science John Zumbrunnen, Professor, Political Science i ABSTRACT The place of foreigners in Plato’s thought remains understudied despite the prevalence of foreign characters, myths, and practices throughout his dialogues. Attending to this gap in the scholarly literature, this dissertation challenges conventional depictions of Plato as hostile to diversity by showing that Plato makes a compelling case for why we should engage with foreigners: the epistemological benefits of cross-cultural engagement. Through exegetical readings of the Republic, Laws, Phaedrus, and Menexenus, I argue that Plato finds cross-cultural dialogue epistemologically beneficial owing to its ability to provoke us to philosophize together, an activity at once conducive to the quest for wisdom and generative of friendship. Put simply, conversations with foreigners perform the same role as the Socratic gadfly of stinging us into consciousness. This finding has major implications for the field of political theory and, specifically, for the role of the new subfield commonly referred to as comparative political theory. By demonstrating the centrality of cross-cultural dialogue to Plato’s conception of political theory, this dissertation suggests that comparative political theory is not a deviation from the tradition of Western political theory, but a restoration of it.
    [Show full text]
  • Barbers & Big Ideas: Paradox in Math and Poetry
    Barbers & Big Ideas: Paradox in Math and Poetry by Alice Major Published in the Journal of Mathematics and the Arts Volume 8, Issue 1-2, 2014 Special Issue: Poetry and Mathematics The final version of this paper is Available at Taylor & Francis: http://dx.doi.org/10.1080/17513472.2014.943999 Abstract Paradox intrigues both mathematicians and artists of all kinds. Throughout the recorded history of human thought, paradox has been a signal that we have to look hard for explanations, whether in natural language or the symbols of math and logic. The particular resonance between math and poetry is related to the fact that paradoxical concepts can translate from one form of expression to another surprisingly well. The author examines paradoxes that have intrigued her and presents five of her poems that have been inspired as a result. Keywords: mathematical poetry; paradox; Zeno; Eubulides; Russell’s Paradox; Twin Paradox; Liar’s Paradox AMS Subject Classification: 00A99; 03A10; 03B99; 97E20 1. Russell and Zeno Paradox has long been a mental sandbox in which mathematicians, logicians and poets like to play around. A paradox is something that is both true and not true, simultaneously logical and illogical. Its contradictions can seem trivial, nonsensical, and yet lead on to powerful insight. And it serves as a hinge between two modes of thought—mathematical and artistic—that are often considered different. Perhaps this is the case because, at heart, mathematics and art are deeply paradoxical activities. “The paradox of the arts is that they are all made up and yet they allow us to get at truths about who and what we are or might be…” wrote poet Seamus Heaney [10, p 69].
    [Show full text]
  • The Liar Paradox As a Reductio Ad Absurdum Argument
    University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 3 May 15th, 9:00 AM - May 17th, 5:00 PM The Liar Paradox as a reductio ad absurdum argument Menashe Schwed Ashkelon Academic College Follow this and additional works at: https://scholar.uwindsor.ca/ossaarchive Part of the Philosophy Commons Schwed, Menashe, "The Liar Paradox as a reductio ad absurdum argument" (1999). OSSA Conference Archive. 48. https://scholar.uwindsor.ca/ossaarchive/OSSA3/papersandcommentaries/48 This Paper is brought to you for free and open access by the Conferences and Conference Proceedings at Scholarship at UWindsor. It has been accepted for inclusion in OSSA Conference Archive by an authorized conference organizer of Scholarship at UWindsor. For more information, please contact [email protected]. Title: The Liar Paradox as a Reductio ad Absurdum Author: Menashe Schwed Response to this paper by: Lawrence Powers (c)2000 Menashe Schwed 1. Introduction The paper discusses two seemingly separated topics: the origin and function of the Liar Paradox in ancient Greek philosophy and the Reduction ad absurdum mode of argumentation. Its goal is to show how the two topics fit together and why they are closely connected. The accepted tradition is that Eubulides of Miletos was the first to formulate the Liar Paradox correctly and that the paradox was part of the philosophical discussion of the Megarian School. Which version of the paradox was formulated by Eubulides is unknown, but according to some hints given by Aristotle and an incorrect version given by Cicero1, the version was probably as follows: The paradox is created from the Liar sentence ‘I am lying’.
    [Show full text]
  • Self-Referential Problems in Mathematics
    Michigan Law Review Volume 90 Issue 5 1992 Some Lesson About the Law From Self-Referential Problems in Mathematics John M. Rogers University of Kentucky Robert E. Molzon University of Kentucky Follow this and additional works at: https://repository.law.umich.edu/mlr Part of the Mathematics Commons, and the Public Law and Legal Theory Commons Recommended Citation John M. Rogers & Robert E. Molzon, Some Lesson About the Law From Self-Referential Problems in Mathematics, 90 MICH. L. REV. 992 (1992). Available at: https://repository.law.umich.edu/mlr/vol90/iss5/3 This Essay is brought to you for free and open access by the Michigan Law Review at University of Michigan Law School Scholarship Repository. It has been accepted for inclusion in Michigan Law Review by an authorized editor of University of Michigan Law School Scholarship Repository. For more information, please contact [email protected]. ESSAY SOME LESSONS ABOUT THE LAW FROM SELF-REFERENTIAL PROBLEMS IN MATHEMATICS John M. Rogers* and Robert E. Molzon ** If systems of law can be analogized to axiomatic systems of mathe­ matical logic in a tight enough fashion, then some conclusions of twen­ tieth-century mathematicians regarding number theory systems may usefully be applied to legal systems as well. In particular, legal theo­ rists must become comfortable with the incompleteness of legal sys­ tems, no matter how carefully constructed, in the same way that mathematicians and philosophers have become comfortable with the incompleteness of axiomatic systems of number theory. The analogy sheds light on some currently perceived problems with our constitu­ tional system.
    [Show full text]
  • Philosophy Sunday, July 8, 2018 12:01 PM
    Philosophy Sunday, July 8, 2018 12:01 PM Western Pre-Socratics Fanon Heraclitus- Greek 535-475 Bayle Panta rhei Marshall Mcluhan • "Everything flows" Roman Jakobson • "No man ever steps in the same river twice" Saussure • Doctrine of flux Butler Logos Harris • "Reason" or "Argument" • "All entities come to be in accordance with the Logos" Dike eris • "Strife is justice" • Oppositional process of dissolving and generating known as strife "The Obscure" and "The Weeping Philosopher" "The path up and down are one and the same" • Theory about unity of opposites • Bow and lyre Native of Ephesus "Follow the common" "Character is fate" "Lighting steers the universe" Neitzshce said he was "eternally right" for "declaring that Being was an empty illusion" and embracing "becoming" Subject of Heideggar and Eugen Fink's lecture Fire was the origin of everything Influenced the Stoics Protagoras- Greek 490-420 BCE Most influential of the Sophists • Derided by Plato and Socrates for being mere rhetoricians "Man is the measure of all things" • Found many things to be unknowable • What is true for one person is not for another Could "make the worse case better" • Focused on persuasiveness of an argument Names a Socratic dialogue about whether virtue can be taught Pythagoras of Samos- Greek 570-495 BCE Metempsychosis • "Transmigration of souls" • Every soul is immortal and upon death enters a new body Pythagorean Theorem Pythagorean Tuning • System of musical tuning where frequency rations are on intervals based on ration 3:2 • "Pure" perfect fifth • Inspired
    [Show full text]
  • Paradoxes Situations That Seems to Defy Intuition
    Paradoxes Situations that seems to defy intuition PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 08 Jul 2014 07:26:17 UTC Contents Articles Introduction 1 Paradox 1 List of paradoxes 4 Paradoxical laughter 16 Decision theory 17 Abilene paradox 17 Chainstore paradox 19 Exchange paradox 22 Kavka's toxin puzzle 34 Necktie paradox 36 Economy 38 Allais paradox 38 Arrow's impossibility theorem 41 Bertrand paradox 52 Demographic-economic paradox 53 Dollar auction 56 Downs–Thomson paradox 57 Easterlin paradox 58 Ellsberg paradox 59 Green paradox 62 Icarus paradox 65 Jevons paradox 65 Leontief paradox 70 Lucas paradox 71 Metzler paradox 72 Paradox of thrift 73 Paradox of value 77 Productivity paradox 80 St. Petersburg paradox 85 Logic 92 All horses are the same color 92 Barbershop paradox 93 Carroll's paradox 96 Crocodile Dilemma 97 Drinker paradox 98 Infinite regress 101 Lottery paradox 102 Paradoxes of material implication 104 Raven paradox 107 Unexpected hanging paradox 119 What the Tortoise Said to Achilles 123 Mathematics 127 Accuracy paradox 127 Apportionment paradox 129 Banach–Tarski paradox 131 Berkson's paradox 139 Bertrand's box paradox 141 Bertrand paradox 146 Birthday problem 149 Borel–Kolmogorov paradox 163 Boy or Girl paradox 166 Burali-Forti paradox 172 Cantor's paradox 173 Coastline paradox 174 Cramer's paradox 178 Elevator paradox 179 False positive paradox 181 Gabriel's Horn 184 Galileo's paradox 187 Gambler's fallacy 188 Gödel's incompleteness theorems
    [Show full text]
  • In the History of Philosophy, Logical Empiricism Is Often Referred to As Neo-Positivism
    In the history of philosophy, Logical Empiricism is often referred to as neo-positivism. Since the positivism debate of the 1960s at the very latest, positivism has been considered the very embodiment of an affirmative philosophy that, in contrast to critical theory, is completely devoid of a social criticism component and lacks even the capacity to develop a transcendental perspective with respect to the existing social and economic order. Ever since, generations of students have been intellectually socialized with this negative image of positivism, which was also transferred to the Vienna Circle and Logical Empiricism of the interwar years. In the context of the historical centers of continental European Logical Empiricism, the research project analyzed two organizations that were dedicated to the dissemination of this approach in philosophy of science: The Ernst Mach Association and the Berlin Society of Empirical Philosophy (renamed Berlin Society of Scientific Philosophy in 1931). The associations were founded independently of one another in 1927 and 1928; however, a comparative analysis of their origins demonstrates remarkable parallels. Many representatives of the ‘scientific world-conception’ (which was the title of the Vienna Circle’s manifesto) came from an ideologically informed late-Enlightenment milieu that was anything but apolitical. Associations such as the Monists, the Freethinkers and even representatives of non-religious humanitarian ethics played key roles in the establishment of both societies. Although the activities of both associations became increasingly ‘scientific’ in the following years, they were never apolitical. Red Vienna and Red Berlin were opposed by Black Vienna and Black Berlin, both of which forcefully attacked their respective counterparts, as analysis of contemporary media coverage shows.
    [Show full text]
  • The Berlin Group and the Philosophy of Logical Empiricism BOSTON STUDIES in the PHILOSOPHY and HISTORY of SCIENCE
    The Berlin Group and the Philosophy of Logical Empiricism BOSTON STUDIES IN THE PHILOSOPHY AND HISTORY OF SCIENCE Editors ROBERT S. COHEN, Boston University JURGEN¨ RENN, Max Planck Institute for the History of Science KOSTAS GAVROGLU, University of Athens Managing Editor LINDY DIVARCI, Max Planck Institute for the History of Science Editorial Board THEODORE ARABATZIS, University of Athens ALISA BOKULICH, Boston University HEATHER E. DOUGLAS, University of Pittsburgh JEAN GAYON, Universit´eParis1 THOMAS F. GLICK, Boston University HUBERT GOENNER, University of Goettingen JOHN HEILBRON, University of California, Berkeley DIANA KORMOS-BUCHWALD, California Institute of Technology CHRISTOPH LEHNER, Max Planck Institute for the History of Science PETER MCLAUGHLIN, Universit¨at Heidelberg AGUSTI´ NIETO-GALAN, Universitat Aut`onoma de Barcelona NUCCIO ORDINE, Universit´a della Calabria ANA SIMOES,˜ Universidade de Lisboa JOHN J. STACHEL, Boston University SYLVAN S. SCHWEBER, Harvard University BAICHUN ZHANG, Chinese Academy of Science VOLUME 273 For further volumes: http://www.springer.com/series/5710 Nikolay Milkov • Volker Peckhaus Editors The Berlin Group and the Philosophy of Logical Empiricism 123 Editors Nikolay Milkov Volker Peckhaus Department of Philosophy Department of Philosophy University of Paderborn University of Paderborn 33098 Paderborn 33098 Paderborn Germany Germany ISSN 0068-0346 ISBN 978-94-007-5484-3 ISBN 978-94-007-5485-0 (eBook) DOI 10.1007/978-94-007-5485-0 Springer Dordrecht Heidelberg New York London Library of Congress
    [Show full text]
  • Grelling on Dependence
    Grelling on dependence Jouko Va¨an¨ anen¨ ∗ Abstract Kurt Grelling wrote a paper in 1939 presenting various concepts of de- pendence. The paper remained unpublished, but deserves to be read today. Many of the ideas of the paper have been subsequently reinvented but one concept, which we call G-dependence, is still genuinely new, and that is the main topic of this paper. We isolate some basic properties of G-dependence and pose the question of finding simple axioms for it. 1 Introduction In his 1939 [11] the mathematician and logician Kurt Grelling2 developed, in co- operation with Paul Oppenheim3, a completely abstract theory of dependence and independence with apparently no connection to algebra or probability theory, al- though a study of dependence was emerging in these fields too. His starting point seems to have been so-called Gestalt Theory [23], but his prime example was the earthly way in which commercial price depends on supply and demand. This paper will present an overview of Grelling’s theory. Department of Mathematics and Statistics, University of Helsinki, Finland · Institute for Logic, Language and Computation, University of Amsterdam, The Netherlands ∗ Research partially supported by grant 40734 of the Academy of Finland. 2 Kurt Grelling was born in Berlin in 1886. He studied in Gottingen¨ with David Hilbert as his supervisor, graduating in 1910 with a thesis on set theory. Already in 1908 he published a paper with L. Nelson on Russell’s paradox, in which they introduced what is today known as the Grelling- Nelson paradox. He was subsequently influential in philosophy in the so-called Berlin Circle.
    [Show full text]
  • Passmore, J. (1967). Logical Positivism. in P. Edwards (Ed.). the Encyclopedia of Philosophy (Vol. 5, 52- 57). New York: Macmillan
    Passmore, J. (1967). Logical Positivism. In P. Edwards (Ed.). The Encyclopedia of Philosophy (Vol. 5, 52- 57). New York: Macmillan. LOGICAL POSITIVISM is the name given in 1931 by A. E. Blumberg and Herbert Feigl to a set of philosophical ideas put forward by the Vienna circle. Synonymous expressions include "consistent empiricism," "logical empiricism," "scientific empiricism," and "logical neo-positivism." The name logical positivism is often, but misleadingly, used more broadly to include the "analytical" or "ordinary language philosophies developed at Cambridge and Oxford. HISTORICAL BACKGROUND The logical positivists thought of themselves as continuing a nineteenth-century Viennese empirical tradition, closely linked with British empiricism and culminating in the antimetaphysical, scientifically oriented teaching of Ernst Mach. In 1907 the mathematician Hans Hahn, the economist Otto Neurath, and the physicist Philipp Frank, all of whom were later to be prominent members of the Vienna circle, came together as an informal group to discuss the philosophy of science. They hoped to give an account of science which would do justice -as, they thought, Mach did not- to the central importance of mathematics, logic, and theoretical physics, without abandoning Mach's general doctrine that science is, fundamentally, the description of experience. As a solution to their problems, they looked to the "new positivism" of Poincare; in attempting to reconcile Mach and Poincare; they anticipated the main themes of logical positivism. In 1922, at the instigation of members of the "Vienna group," Moritz Schlick was invited to Vienna as professor, like Mach before him (1895-1901), in the philosophy of the inductive sciences. Schlick had been trained as a scientist under Max Planck and had won a name for himself as an interpreter of Einstein's theory of relativity.
    [Show full text]
  • List of Paradoxes 1 List of Paradoxes
    List of paradoxes 1 List of paradoxes This is a list of paradoxes, grouped thematically. The grouping is approximate: Paradoxes may fit into more than one category. Because of varying definitions of the term paradox, some of the following are not considered to be paradoxes by everyone. This list collects only those instances that have been termed paradox by at least one source and which have their own article. Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis. Logic • Barbershop paradox: The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. • What the Tortoise Said to Achilles "Whatever Logic is good enough to tell me is worth writing down...," also known as Carroll's paradox, not to be confused with the physical paradox of the same name. • Crocodile Dilemma: If a crocodile steals a child and promises its return if the father can correctly guess what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned? • Catch-22 (logic): In need of something which can only be had by not being in need of it. • Drinker paradox: In any pub there is a customer such that, if he or she drinks, everybody in the pub drinks. • Paradox of entailment: Inconsistent premises always make an argument valid. • Horse paradox: All horses are the same color. • Lottery paradox: There is one winning ticket in a large lottery. It is reasonable to believe of a particular lottery ticket that it is not the winning ticket, since the probability that it is the winner is so very small, but it is not reasonable to believe that no lottery ticket will win.
    [Show full text]
  • The Barber Paradox: on Its Paradoxicality and Its Relationship to Russell’S Paradox
    Prolegomena 13 (2) 2014: 269–278 The Barber Paradox: On its Paradoxicality and its Relationship to Russell’s Paradox JIRI RACLAVSKY Department of Philosophy, Masaryk University, Arne Novaka 1, 602 00 Brno, Czech Republic [email protected] ORIGINAL SCIENTIFIC ARTICLE / RECEIVED: 210914 ACCEPTED: 261014 ABSTRACT: The Barber Paradox is often introduced as a popular version of Russell’s paradox, though some philosophers and logicians (e.g. Church) have denied their similarity, even calling the Barber paradox a pseudoparadox. In the first part of the paper, we demonstrate mainly that in the standard (Quinean) definition of a paradox the Barber paradox is a clear-cut example of a non-paradox. In the second part of the paper, we examine a probable source of the paradoxicality of the Barber Paradox, which is found in a certain ambivalence in terms of meaning. The two different read- ings of the crucial phrase yield distinct existential assumptions which produce the paradoxical conclusion. KEYWORDS: Degrees of paradoxicality, pseudoparadoxicality, Russell’s paradox, the Barber paradox. 1. Introduction One of the paradoxical aspects of the Barber paradox (BP) is that it is not a paradox, though many people still think it is.1 It is also paradoxical that its authorship is often attributed to Russell, 2 even though he did not invent it,3 and even warned against it as a false analogy to Russell’s paradox (RP).4 1 Including some philosophers that address paradoxes, e.g., Rescher (2001: 143–147). 2 Cf., e.g., the popular text Joyce (2002). 3 The probable author of the BP is the Austrian philosopher Mally.
    [Show full text]