UNIVERSITY OF CALIFORNIA, IRVINE THREE ESSAYS ON THE FOUNDATIONS OF SCIENCE DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Social Science – Mathematical Behavioral Sciences by Rolf Henry Johansson Dissertation Committee: Professor Louis Narens, Chair Professor Donald Saari Professor Kent Johnson 2014 © 2014 Rolf Henry Johansson DEDICATION For my mother Lalla and my father Kurt for their continuing support. ii TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iv CURRICULUM VITAE v ABTRACT OF THE DISSERTATION viii ESSAY 1: THE UNIQUENESS PROBLEM FOR FINITE SEMIORDERS Abstract 1 1. Introduction 2 2. Preliminaries 8 3. Representation Problems for Semiorders 17 4. Theorems 24 5. Discussion 37 Appendix 40 References 45 ESSAY 2: TROUBLES WITH CONVENTION T Abstract 48 1. Introduction: Convention T and Natural Languages 50 2. Hintikka‟s Counterexample 55 3. Hintikka-Type Counterexamples 59 4. Proposed Solutions 63 4.1 On ambiguity versus context-sensitivity 64 4.2 Formalizing the metalanguage 70 4.3 Paraprasing the conditional 73 4.4 Dispensing with disquotation 76 5. Discussion 78 References 82 ESSAY 3: ELEMENTARY FORMULAS FOR THE nTH PRIME AND FOR THE NUMBER OF PRIMES UP TO A GIVEN LIMIT Abstract 84 1. Introduction 86 2. A Brief History of Prime Representing Functions 89 3. Formulas Based on Wilson‟s Theorem 93 4. Formulas Based on the Sieve of Eratosthenes 96 5. Elementary Formulas for Primes 100 References 107 iii ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my committee chair, Louis Narens, for many stimulating conversations over the years and for guidance throughout my graduate education. Louis‟ ability to weave together sophisticated mathematical, philosophical, and scientific knowledge is without compare, and I consider myself extremely fortunate to have worked so closely with such a creative and profound thinker. In addition, I would like to thank Duncan Luce, Jean-Claude Falmagne, and my other committee members Brian Skyrms and Don Saari. Both Duncan and Jean-Claude were of great help in shaping the research in ESSAY 1, and Duncan in particular gave the essay a very close reading and provided many helpful comments and criticisms. My other committee members Brian Skyrms and Don Saari also provided helpful comments on other parts of the research as it was developing. Financial support was provided by the University of California, Irvine, in the form of three Summer Research Fellowships and a Regents Dissertation Writing Fellowship. iv CURRICULUM VITAE ROLF JOHANSSON Education University of California, Irvine Ph.D. in Social Science – Mathematical Behavioral Sciences 2014 M.A. in Social Science – Mathematical Behavioral Sciences 2006 Columbia University Ph.D. Program in Philosophy 1994-95 M.I.T. Ph.D. Program in Linguistics and Philosophy 1986-88 University of Pennsylvania A.B. in Philosophy 1986 Areas of Specialization Philosophy of Language Philosophy of Religion Philosophy of Science Areas of Competence Mathematical Logic Decision Theory Metaphysics History of Analytic Philosophy Working Papers 1. “Troubles with Convention T” 2. “On Formulas for Primes” 3. “Measurement of Finite Semiorders” 4. “Ross‟ Principle of Possible Explanation” 5. “The Cosmological Argument and Russell‟s Paradox” 6. “Chomsky on the Indeterminacy of Translation” v 7. “Common Mistakes of Skepticism” 8. “Natural Languages are Not Vast” 9. “Sorites, Semiorders, and Individuation” 10. “Reduction, Ontology, Information” 11. “The Scotus-Ross Ontological Argument” Talks “Troubles with Deflationism,” Central Valley Philosophy Conference, University of California, Merced, (October, 2008) “Qualitative Semiorders and the Empirical Adequacy of Linear Representations.” INFORMS Conference, Session on Consumer‟s Preference Modeling, San Jose, CA (November 17, 2002) Courses Taught University of California, Merced Upper Division: PHIL 105: Philosophy of Language: Spring 2010, Fall 2011 PHIL 101: Metaphysics: Fall 2007, Spring 2009, Spring 2012 PHIL 160/MATH 171: Mathematical Logic: Spring 2008, Fall 2009, Spring 2013 PHIL 107: Philosophy of Religion: Spring 2007, Fall 2008, Spring 2011, Fall 2012 PHIL 195: Undergraduate Research: Spring 2009, Fall 2009, Fall 2011 PHIL 199: Individual Study: Fall 2007, Spring 2008, Spring 2010 Lower Division: PHIL 001: Introduction to Philosophy: Fall 2006, Fall 2007, Fall 2008, Fall 2009, Fall 2010, Fall 2011, Fall 2012 PHIL 005: Introduction to Logic: Spring 2007, Spring 2008, Spring 2009, Spring 2010 Spring 2011, Spring 2012, Spring 2013 Independent Studies Supervised: Teleology in Systems Theory (Fall 2011) Ontology and Ontological Commitment (Spring 2010) Theories of Immortality (Fall 2009) Philosophy of Quantum Mechanics (Spring 2009) vi Set Theory (Spring 2008) Gödel‟s Ontological Argument (Fall 2007) Other Experience Teaching Assistant, University of California, Irvine Problems of Philosophy, Inductive Logic, Decision Making, Probability and Statistics II, Probability and Statistics III, Basic Microeconomics, Intermediate Microeconomics, Fundamentals of Psychology, Social Psychology, Cultural Anthropology Teaching Assistant, Massachusetts Institute of Technology Introduction to Ethics (2 times) Research Assistant, University of Pennsylvania, Professor Paul Guyer Honors and Awards Distinguished Undergraduate Teaching Award, $1,000, UC Merced (2012-2013) Regents Dissertation Writing Fellowship, $4,600, U.C. Irvine Dean‟s Commendations for Excellence in Teaching, U.C. Irvine (3 times) Summer Research Fellowships, $3,400, U.C. Irvine (3 times) Institute Fellowship, $8,800, Massachusetts Institute of Technology Service to the Profession WASC Accreditation Reports (Spring 2010 and Spring 2011) Undergraduate Philosophy Club (Sponsor) Professional Affiliations American Philosophical Association Philosophy of Science Association vii ABSTRACT OF THE DISSERTATION THREE ESSAYS ON THE FOUNDATIONS OF SCIENCE By Rolf Henry Johansson Doctor of Philosophy in Social Science – Mathematical Behavioral Sciences University of California, Irvine, 2014 Professor Louis Narens, Chair The general uniqueness problem for finite semiorders is still unsolved, and this has impeded their wider use in decision modeling. In Essay 1, I show that for semiorders that satisfy some relatively weak constraints, unique representation (and hence interval scalability) may be obtained. In Essay 2, I discuss a type of counterexample to Tarski‟s Convention T that was originally discovered by Hintikka. I show that Hintikka‟s counterexample generalizes in quite unexpected ways, and that there are in fact a large number of unambiguous sentences that generate counterexamples of the same general type. I then show that various proposals for dealing with Hintikka‟s original counterexample are unsatisfactory, and that none of the proposed solutions can resolve all of the counterexamples presented in this essay. In Essay 3 I present elementary formulas for the nth prime and for the number of primes up to a given limit, both of which improve upon existing formulas by avoiding the computation of factorials and the exponential growth of terms. The formulas are based on the idea of “embedding” characteristic functions – a characteristic function for non-divisibility is used to construct a characteristic function for primality, and no use is made of either Wilson‟s theorem or the inclusion-exclusion process. viii ABSTRACT OF ESSAY 1 THE UNIQUENESS PROBLEM FOR FINITE SEMIORDERS By Rolf Henry Johansson Doctor of Philosophy in Social Science – Mathematical Behavioral Sciences University of California, Irvine, 2014 Professor Louis Narens, Chair Semiorders were introduced by Luce (1956) to account for the intransitivities found empirically in indifference judgments. In principle, they are superior to weak orders as descriptive models of choice behavior. However, the general uniqueness problem for finite semiorders is still unsolved, and this has impeded their wider use in decision modeling. First, we will discuss representational anomalies for semiorders in order to better understand the source of the difficulty in obtaining uniqueness. We will then show that for semiorders that satisfy some relatively weak constraints, unique representation (and hence interval scalability) may be obtained. This result follows by combining independent results of Suppes (1972) on equal- difference structures and Fishburn (1973b) on the construction of weak orders from fragmentary information. The finite semiorders for which unique representation may be obtained are “well- behaved” in the sense that they constitute partial information about an “underlying” equal- difference structure, and have a constant discrimination threshold. A very weak constraint on the size of the threshold enables the unique representation. Since most applications of utility models involve the comparison of alternatives within a limited range of utility values, over which discrimination thresholds are more or less constant, then well-behaved semiorders may have practical applications for qualitative modeling in such cases. 1 1. INTRODUCTION In the classical formulation of expected utility theory by von Neumann and Morgenstern (1944), strong assumptions were made about the structure of preference and indifference. It was assumed that preferences induce a linear ordering of a set of goods, and that transitivity holds for both preference and indifference. These assumptions were also made by Marschak (1950), Debreu (1954), and