DOCSLIB.ORG

Approximating Formulas in the Frame of Basic Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Approximating Formulas in the Frame of Basic Logic UNIVERSITY OF OSTRAVA FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS APPROXIMATING FORMULAS IN THE FRAME OF BASIC LOGIC Ph.D. THESIS AUTHOR: Martina Daˇnkov´a SUPERVISOR: Irina Perfilieva 2005 OSTRAVSKA´ UNIVERZITA V OSTRAVEˇ PRˇ´IRODOVEDECKˇ A´ FAKULTA KATEDRA MATEMATIKY APROXIMACNˇ ´I FORMULE V RAMCI´ BASIC LOGIKY. DOKTORSKA´ DISERTACNˇ ´I PRACE´ AUTOR: Martina Daˇnkov´a VEDOUC´I PRACE:´ Irina Perfilieva 2005 Prohlaˇsuji, ˇze pˇredloˇzen´apr´ace je m´ym p˚uvodn´ım autorsk´ym d´ılem, kter´ejsem vypracoval samostatnˇe. Veˇskerou literaturu a dalˇs´ızdroje, z nichˇzjsem pˇri zpra- cov´an´ıˇcerpal, v pr´aci ˇr´adnˇecituji a jsou uvedeny v seznamu pouˇzit´eliteratury. Ostrava...................... ....................... .... (podpis) Beru na vˇedom´ı, ˇze tato doktorsk´adisertaˇcn´ıpr´ace je majetkem Ostravsk´euniverzity (autorsk´yz´akon C.ˇ 121/2000 Sb., §60 odst. 1), bez jej´ıho souhlasu nesm´ıb´yt nic z obsahu pr´ace publikov´ano. Souhlas´ım s prezenˇcn´ım zpˇr´ıstupnˇen´ım sv´epr´ace v Univerzitn´ıknihovnˇeOstravsk´e univerzity. Ostrava...................... ....................... .... (podpis) “It has long been an axiom of mine that the little things are infinitely the most important.” Sherlock Holmes 5 Summary An interesting class of decision making systems is that which is based on lin- guistic descriptions, i.e. finite sets of IF-THEN rules describing a process, situation, environment etc. These rules often include natural language expressions character- ized by a position on an ordered scale, usually on a real interval. As typical examples of such expressions, let us name light, loud, approximately 5, smaller then 7, warmer then 78 etc. The main task is to derive a conclusion from an observation on the basis of linguistic description by means of some feasible inference. This thesis proposes a methodology for dealing with the above-described situa- tion and studies its properties. The background for it is fuzzy predicate calculus. Considered fuzzy IF-THEN rules are understood in two different ways, i.e. as lin- guistically expressed logical conjunctions or implications. We strictly distinguish the syntactic from the semantic level of study. First, we introduce a generalized notion of normal forms aimed at formalizing linguistically expressed collection of fuzzy IF-THEN rules and prove their represen- tation abilities for a special class of formulas. Special inference rules determining conclusion from the observation are proposed as well. Moreover, the properties of such obtained conclusion are studied. Later on, we characterize this special class of formulas from the functional point of view. This leads to the general result relating to an approximations by functions representing normal forms. Finally, we apply all these result and present the complete automated learning method which conceives the rule-base composed of a set of IF-THEN rules from data assemble from a real process or environment. Keywords: Normal forms, Approximate inference, IF-THEN rules, Fuzzy logic, Fuzzy relations, Approximation, Fuzzy Systems, Genetic Algorithms. 6 Anotace V´yznamnou tˇr´ıdu rozhodovac´ıch syst´em˚utvoˇr´ısyst´emy zaloˇzen´ena jazykov´em popisu, tj. na koneˇcn´emnoˇzinˇepravidel typu JESTLIZE-PAKˇ popisuj´ıc´ıch proces, situaci, syst´emov´eprostˇred´ıap. Tato pravidla ˇcasto obsahuj´ıv´yrazy pˇrirozen´eho jazyka charakteristick´esvou pozic´ına uspoˇr´adan´eˇsk´ale, vˇetˇsinou na re´aln´em inter- valu. Typick´ym pˇr´ıkladem jsou v´yrazy jako lehk´y, hlasit´y, pˇribliˇznˇe5, m´enˇeneˇz7, teplejˇs´ıneˇz78 oC atd. Z´akladn´ım ´ukolem je odvodit z´avˇer z urˇcit´eho pozorov´an´ı na z´akladˇedan´eho jazykov´eho popisu prostˇrednictv´ım vhodn´einference. C´ılem t´eto disertace je navrhnout urˇcitou metodologii pro ˇreˇsen´ı situac´ı uve- den´ych v´yˇsea prozkoumat jej´ıvlastnosti. Tato metodologie je zaloˇzena na kalkulu predik´atov´efuzzy logiky. Uvaˇzovan´apravidla typu JESTLIZE-PAKˇ ch´apeme dvˇema odliˇsn´ymi zp˚usoby - bud’ jako jazykovˇevyj´adˇren´elogick´ekonjunkce, nebo implikace. V cel´epr´aci pˇr´ısnˇerozliˇsujeme syntaktickou a s´emantickou str´anku probl´emu. K tomuto ´uˇcelu nejprve zavedeme zobecnˇen´ypojem norm´aln´ıformy vhodn´ypro formalizaci jazykovˇevyj´adˇren´ych soubor˚upravidel typu JESTLIZE-PAKˇ a uk´aˇzeme, nakolik umoˇzˇnuje reprezentovat formule z urˇcit´etˇr´ıdy. Navrhneme rovnˇeˇzspeci´aln´ı odvozovac´ıpravidla pro stanoven´ız´avˇer˚uz pozorov´an´ıa prozkoum´ame vlastnosti takto z´ıskan´ych z´avˇer˚u. V dalˇs´ıˇc´asti pod´ame funkcion´aln´ıcharakterizaci uveden´etˇr´ıdy formul´ı, jeˇzvede k obecn´emu v´ysledku o aproximac´ıch funkcemi reprezentuj´ıc´ımi norm´aln´ıformy. Na z´avˇer provedeme aplikaci tˇechto v´ysledk˚ua pˇredvedeme metodu plnˇeauto- matick´eho uˇcen´ı, jeˇzvytv´aˇr´ıb´azi pravidel sloˇzenou z pravidel typu JESTLIZE-PAKˇ podle dat z´ıskan´ych z re´aln´ych proces˚uˇci prostˇred´ı. Kl´ıˇcov´aslova: Norm´aln´ı formy, Pˇribliˇzn´adedukce, JESTLIZE-PAKˇ pravidla, Fuzzy logika, Fuzzy relace, Aproximace, Fuzzy syst´emy, Genetick´ealgoritmy. 7 Preface Fuzzy logics are intend to handle information affected by inexactness or in the other words incorporate the vague phenomenon. Enormous number of applications in expert systems verify the role of fuzzy logic as an important tool compiling expert knowledge into a rule-base approximately describing a real system in a natural way. This is done using so called fuzzy IF-THEN rules. Formalizing these rules into the language of logic leads to formulas having special form commensurate with interpretation of available knowledge. The first type of fuzzy rules reflects a functional dependency, e.g. monotone dependency can be read as follows ”X is small” AND ”Y is small” OR ”X is medium” AND ”Y is medium” OR ”X is big” AND ”Y is big” Such interpretation can be naturally formalized as a disjunctive normal form. A different type of fuzzy rules is that which forms conjunctive normal form from the logical point of view. This type of fuzzy rules can be regarded as summarizing an expert knowledge. Having on mind the same dependency as above, we can read it as IF ”X is small” THEN ”Y is small” AND IF ”X is medium” THEN ”Y is medium” AND IF ”X is big” THEN ”Y is big” Generally, we may speak about the same system or process but we describe it in 8 two different ways with distinct functional realization. On the other side, we expect that both descriptions are at least partially equivalent. The aim of this thesis is to propose generalized normal forms suitable to formalize a collection of fuzzy IF-THEN rules, which satisfy the natural requirement for partial equivalence and further investigate their relationship. Furthermore, the significant task comprises of introducing a new automated learning mechanism able to find the normal forms on the basis of an available information. The results together with the whole methodology from this thesis will be used in the software system LFLC 2000 developed on Institute for Research and Applications of Fuzzy Modeling. This software system allows to design, test and apply linguistic descriptions, and additionally, it is endowed with automated learning procedures. It already proved itself to be useful in practical applications. I want to express my gratitude to my supervisor Prof. Irina Perfilieva for her support, valuable comments and permanent encouragement which made it possible to finish this thesis. Moreover, I would like to thank my boss Prof. Vil´em Nov´ak for working conditions he created in Institute for Research and Applications of Fuzzy Modeling. The warm thanks goes to Libor Bˇehounek and Anton´ın Dvoˇr´ak for the watchful reading throughout this thesis. I would also like to thank colleagues from our Institute making the friendly atmosphere. Last but not least, I thank my parents for warm atmosphere which gave me strength in difficult times. Ostrava, May 2005 Martina Daˇnkov´a 9 Table of Contents Summary 6 Anotace 7 Preface 8 Table of Contents 10 List of Figures 13 List of Symbols 15 1 Introduction 16 1.1 Fuzzy Logic in the modern scientific world . 16 1.2 NormalformsandfuzzyIF-THENrules . 17 1.3 GeneralizationofNormalforms . 18 1.4 Theoutlineofthiswork . 20 2 Preliminaries 22 2.1 Fuzzysetsandfuzzyrelations . 22 2.1.1 Basicnotions ........................... 22 2.1.2 T-norms and their residua as logical operations . 23 2.1.3 Extensional fuzzy relations . 28 2.2 Basicfuzzypredicatelogic . 34 2.2.1 Algebraic structures for BL and BL∀ .............. 35 2.2.2 Syntaxandsemantics. 37 2.2.3 Provable propositional and predicate tautologies . ..... 41 2.2.4 Normal Forms in BL∀ ...................... 43 10 3 Logical approximation 48 3.1 Normal Forms for Extensional Formulas . 50 3.1.1 Characterization of extensional formulas . 52 3.1.2 Logical Approximation by Finite Normal Forms . 56 3.2 Logical approximation in approximate inference . ...... 65 3.2.1 Approximate inferences based on Normal Forms . 67 3.2.2 Rules of inference and the theory of fuzzy control . 71 3.2.3 Properties of approximate inferences . 73 3.2.4 Consequences to fuzzy control . 76 4 Approximations over linearly ordered BL-algebras 83 4.1 Normal Forms for extensional fuzzy relations . 84 4.1.1 Approximating abilities of Normal Forms . 85 4.1.2 Representation of extensional fuzzy relations . 87 4.1.3 Illustrations ............................ 88 4.2 Functions associated with formulas of BL-logic . 88 4.2.1 Normal Forms for functions represented byLukasiewicz formulas 91 4.2.2 Functions associated with Product (Goguen) formulas . 94 4.2.3 Functions associated with G¨odel (Minimum) formulas . 100 4.2.4 Normal forms for functions associated with formulas of BL-logic106 4.3 Characterization of extensional fuzzy relations . .......108 4.3.1 Extensionality w.r.t. Reflexivity means Lipschitz Continuity . 110 4.3.2 From Lipschitz Continuity to Extensionality w.r.t. Reflexivity 114 4.3.3 Notes on ∧-extensionality . .124 4.3.4 Approximation Theorem . 125 5 Genetic Algorithm + Logical Approximation = Optimized Rule-Base 128 5.1 IntroductiontoGeneticAlgorithms .
Load more

Recommended publications
  • False Dilemma Wikipedia Contents
    False dilemma Wikipedia Contents 1 False dilemma 1 1.1 Examples ............................................... 1 1.1.1 Morton's fork ......................................... 1 1.1.2 False choice .......................................... 2 1.1.3 Black-and-white thinking ................................... 2 1.2 See also ................................................ 2 1.3 References ............................................... 3 1.4 External links ............................................. 3 2 Affirmative action 4 2.1 Origins ................................................. 4 2.2 Women ................................................ 4 2.3 Quotas ................................................. 5 2.4 National approaches .......................................... 5 2.4.1 Africa ............................................ 5 2.4.2 Asia .............................................. 7 2.4.3 Europe ............................................ 8 2.4.4 North America ........................................ 10 2.4.5 Oceania ............................................ 11 2.4.6 South America ........................................ 11 2.5 International organizations ...................................... 11 2.5.1 United Nations ........................................ 12 2.6 Support ................................................ 12 2.6.1 Polls .............................................. 12 2.7 Criticism ............................................... 12 2.7.1 Mismatching ......................................... 13 2.8 See also
    [Show full text]
  • Unexpected Hanging Paradox - Wikip… Unexpected Hanging Paradox from Wikipedia, the Free Encyclopedia
    2-12-2010 Unexpected hanging paradox - Wikip… Unexpected hanging paradox From Wikipedia, the free encyclopedia The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or prediction paradox is a paradox about a person's expectations about the timing of a future event (e.g. a prisoner's hanging, or a school test) which he is told will occur at an unexpected time. Despite significant academic interest, no consensus on its correct resolution has yet been established.[1] One approach, offered by the logical school of thought, suggests that the problem arises in a self-contradictory self- referencing statement at the heart of the judge's sentence. Another approach, offered by the epistemological school of thought, suggests the unexpected hanging paradox is an example of an epistemic paradox because it turns on our concept of knowledge.[2] Even though it is apparently simple, the paradox's underlying complexities have even led to it being called a "significant problem" for philosophy.[3] Contents 1 Description of the paradox 2 The logical school 2.1 Objections 2.2 Leaky inductive argument 2.3 Additivity of surprise 3 The epistemological school 4 See also 5 References 6 External links Description of the paradox The paradox has been described as follows:[4] A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
    [Show full text]
  • The Development of Mathematical Logic from Russell to Tarski: 1900–1935
    The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives .
    [Show full text]
  • Reflexivity and Self-Referentiality 0.8Em in Inverse Monoids and Categories
    Reflexivity and Self-Referentiality In Inverse Monoids and Categories Peter M. Hines York – Mathematics – 2019 www.peterhines.info Reflexivity and Self-Referentiality Some motivation ... This talk is about some inverse category theory closely associated with logic and theoretical computer science. The general topic is models of self-referentiality. We aim to: 1 Describe the historical context & importance. 2 Give concrete axioms & examples. 3 Do all this in the reversible (inverse monoid) setting. www.peterhines.info Reflexivity and Self-Referentiality Historical Context (I) — Foundations & Logic Scenes from the frog-mouse wars www.peterhines.info Reflexivity and Self-Referentiality The historical setting The late 19th and early to mid 20th century saw some- thing of a crisis in the foundations of mathematics. This can be compared to the controversy caused by the introduction of calculus that was resolved by rigorous no- tions of limit & convergence. However, it was more profound, and less easily resolved. Its aftermath is still relevant today. www.peterhines.info Reflexivity and Self-Referentiality The problems of infinity Georg Cantor lit the fuse, and stepped back to a safe distance ... His work was not always appreciated: A ”scientific charlatan”, a ”renegade” and a ”corrupter of youth” — Leopold Kroenecker Mathematics is ”ridden through and through with the per- nicious idioms of set theory”, which is ”utter nonsense” that is ”laughable” and ”wrong” — Ludwig Wittgenstein www.peterhines.info Reflexivity and Self-Referentiality A more balanced approach A very readable contemporaneous account: Mathematical Rigor, past and present – J. Pierpont (1928) The Mengenlehre of Cantor [Set Theory] has brought to light a number of paradoxes which have profoundly disturbed the mathematical community for a quarter of a century.
    [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]
  • Discontinuous Particle Dynamics Beyond Relativity
    From relativistic to quantum universe: Observation of a spatially- discontinuous particle dynamics beyond relativity Sergey A. Emelyanov Ioffe Institute, 194021 St. Petersburg, Russia Correspondence: [email protected] Abstract. We perform an experimental test where we directly observe light-induced electron transitions with a macroscopic spatial discontinuity. The effect is related to the fundamental indivisibility of macroscopic orbit-like quantum states reminiscent of so-called extended states in the integer quantum Hall system. The test has become realizable due to the discovering of a quantum phase with spontaneous pervasive quantum ordering reminiscent of that of a single atom. The observed transitions may be regarded as a peculiar quantum dynamics beyond relativity, which implies that the current relativistic model of universe should be replaced by a deeper quantum model. It is the Bohm’s model of undivided universe which now should involve a deeper-than-classical concept of absolute simultaneity and a deeper- than-relativistic concept of space and time. Ultimately, our test thus establishes a new hierarchy of fundamental physical theories where the de Broglie-Bohm realistic quantum theory is the deepest theory which does not contradict either classical physics or relativity but rather is beyond both. This is because the fact that quantum theory is dealing with a deeper reality where physical objects are not self-sufficient entities and therefore their discontinuous transitions are possible within an overall quantum system which may well be macroscopic. Keywords: atom-like macroscopic quantum ordering, spatially-discontinuous particle dynamics, Bohm’s undivided universe I. Introduction 1.1. The current view of the universe: why the Minkowski model of spacetime As it follows from the history of science, our view of universe is not unchangeable but determined by the physical theory which is regarded as being the most fundamental in a given historical period.
    [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]
  • INFORMATION– CONSCIOUSNESS– REALITY How a New Understanding of the Universe Can Help Answer Age-Old Questions of Existence the FRONTIERS COLLECTION
    THE FRONTIERS COLLECTION James B. Glattfelder INFORMATION– CONSCIOUSNESS– REALITY How a New Understanding of the Universe Can Help Answer Age-Old Questions of Existence THE FRONTIERS COLLECTION Series editors Avshalom C. Elitzur, Iyar, Israel Institute of Advanced Research, Rehovot, Israel Zeeya Merali, Foundational Questions Institute, Decatur, GA, USA Thanu Padmanabhan, Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India Maximilian Schlosshauer, Department of Physics, University of Portland, Portland, OR, USA Mark P. Silverman, Department of Physics, Trinity College, Hartford, CT, USA Jack A. Tuszynski, Department of Physics, University of Alberta, Edmonton, AB, Canada Rüdiger Vaas, Redaktion Astronomie, Physik, bild der wissenschaft, Leinfelden-Echterdingen, Germany THE FRONTIERS COLLECTION The books in this collection are devoted to challenging and open problems at the forefront of modern science and scholarship, including related philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science and research. Furthermore, it is intended to encourage active academics in all fields to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, conscious- ness, language and complex systems—the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge. More information about this series at http://www.springer.com/series/5342 For a full list of published titles, please see back of book or springer.com/series/5342 James B.
    [Show full text]
  • Reflexivity and Self-Referentiality 0.8Em in Inverse Monoids And
    Reflexivity and Self-Referentiality In Inverse Monoids and Categories Peter M. Hines York – Mathematics – 2019 www.peterhines.info Reflexivity and Self-Referentiality Some motivation ... This talk is about some inverse category theory closely associated with logic and theoretical computer science. The general topic is models of self-referentiality. We aim to: 1 Describe the historical context & importance. 2 Give concrete axioms & examples. 3 Do all this in the reversible (inverse monoid) setting. www.peterhines.info Reflexivity and Self-Referentiality Historical Context (I) — Foundations & Logic Scenes from the frog-mouse wars www.peterhines.info Reflexivity and Self-Referentiality The historical setting The late 19th and early to mid 20th century saw some- thing of a crisis in the foundations of mathematics. This can be compared to the controversy caused by the introduction of calculus that was resolved by rigorous no- tions of limit & convergence. However, it was more profound, and less easily resolved. Its aftermath is still relevant today. www.peterhines.info Reflexivity and Self-Referentiality The problems of infinity Georg Cantor lit the fuse, and stepped back to a safe distance ... His work was not always appreciated: A ”scientific charlatan”, a ”renegade” and a ”corrupter of youth” — Leopold Kroenecker Mathematics is ”ridden through and through with the per- nicious idioms of set theory”, which is ”utter nonsense” that is ”laughable” and ”wrong” — Ludwig Wittgenstein www.peterhines.info Reflexivity and Self-Referentiality A more balanced approach A very readable contemporaneous account: Mathematical Rigor, past and present – J. Pierpont (1928) The Mengenlehre of Cantor [Set Theory] has brought to light a number of paradoxes which have profoundly disturbed the mathematical community for a quarter of a century.
    [Show full text]
  • MMP-Elections and the Assembly Size
    MMP-elections and the assembly size BY Eivind Stensholt DISCUSSION PAPER Institutt for foretaksøkonomi Department of Business and Management Science FOR 15/2019 ISSN: 1500-4066 October 2019 29.10.2019 MMP-elections and the assembly size Eivind Stensholt, [email protected] mobile (047) 455 27 129 Norwegian School of Economics, Helleveien 30 5045 Bergen Norway Abstract MMP (Mixed Member Proportional) elections for legislatures have ballots with one vote in a local single seat tally and one vote for a party list in a multi-seat tally. In Germany, the multi-seat tally occasionally violated a Participation axiom. The federal Constitutional Court declared this unconstitutional in 2008. Rules were changed. In 2017, the result was a Bundestag with 709 members, 111 of them in extra-ordinary party seats. The paper considers two remedies against excessive assembly size. One is “faithful accounting” of ballot data in each local tally, another a change from Plurality to a Majority method. For this use, we consider IRV, i.e. Instant Runoff Voting, in combination with a 3- candidate Condorcet method. The mayoral IRV election in Burlington 2009 serves as an example, here in the special context of MMP. Violations of the Participation criterion occur also in the usual Majority methods for single seat elections. The legal adoption of a mathematical axiom from election theory have consequences seen in the context of established impossibility theorems. JEL classification D72 Key words: Mixed Member Proportional, Instant Runoff Voting, Participation criterion, legality, legitimacy, Burlington election 2009. Acknowledgement: Thanks to colleague Dirk Schindler for many and useful conversations! 1 MMP-elections and the assembly size Introduction Surveying elections of legislatures, "New Handbook" (Reynolds & al.
    [Show full text]
  • Software Studies: a Lexicon, Edited by Matthew Fuller, 2008
    fuller_jkt.qxd 4/11/08 7:13 AM Page 1 ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• S •••••••••••••••••••••••••••••••••••••new media/cultural studies ••••software studies •••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• O ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• F software studies\ a lexicon ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• T edited by matthew fuller Matthew Fuller is David Gee Reader in ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• This collection of short expository, critical, Digital Media at the Centre for Cultural ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• W and speculative texts offers a field guide Studies, Goldsmiths College, University of to the cultural, political, social, and aes- London. He is the author of Media ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• thetic impact of software. Computing and Ecologies: Materialist Energies in Art and A digital media are essential to the way we Technoculture (MIT Press, 2005) and ••••••••••••••••••••••••••••••••••••• •••• •••••••••••••••••••••••••••••••••• work and live, and much has been said Behind the Blip: Essays on the Culture of •••••••••••••••••••••••••••••••••••••
    [Show full text]
  • Copyrighted Material
    1 Foundations and Paradoxes In this chapter and the following, we shall learn lots of things in a short time.1 Initially, some of the things we will gain knowledge of may appear unrelated to each other, and their overall usefulness might not be clear either. However, it will turn out that they are all connected within Gödel’s symphony. Most of the work of these two chapters consists in preparing the instruments in order to play the music. We will begin by acquiring familiarity with the phenomenon of self-reference in logic – a phenomenon which, according to many, has to be grasped if one is to understand the deep meaning of Gödel’s result. Self-reference is closely connected to the famous logical paradoxes, whose understanding is also important to fully appreciate the Gödelian construction – a con- struction that, as we shall see, owes part of its timeless fascination to its getting quite close to a paradox without falling into it. But what is a paradox? A common first definition has it that a paradox is the absurd or blatantly counter-intuitive conclusion of an argument, which starts with intuitively plausible premises and advances via seem- ingly acceptable inferences. In The Ways of Paradox, Quine claims that “a paradox is just any conclusion that at first sounds absurd but that has an argument to sustain it.”2 We shall be particularly concerned not just with sentences that are paradoxical in the sense of being implausible, or contrary to common sense (“paradox” intended as something opposed to theCOPYRIGHTED δo¢ ξα, or to what is e’¢νδοξον MATERIAL, entrenched in pervasive 1 This chapter draws on Berto (2006a), (2007a), and (2007b) for an account of the basics of set theory and of logical paradoxes.
    [Show full text]