Is Dedekind a Logicist? Why Does Such a Question Arise?
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Nominalism, Trivialism, Logicism
Nominalism, Trivialism, Logicism Agustín Rayo∗ May 1, 2014 This paper is an effort to extract some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase-functions for mathematical languages, and sug- gest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases. Although much of the material in this paper is drawn from the book—and from an earlier paper (Rayo 2008)—I hope the present discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. 1 Nominalism Mathematical Nominalism is the view that there are no mathematical objets. A standard problem for nominalists is that it is not obvious that they can explain what the point of a mathematical assertion would be. For it is natural to think that mathematical sentences like ‘the number of the dinosaurs is zero’ or ‘1 + 1 = 2’ can only be true if mathematical objects exist. But if this is right, the nominalist is committed to the view that such sentences are untrue. And if the sentences are untrue, it not immediately obvious why they would be worth asserting. ∗For their many helpful comments, I am indebted to Vann McGee, Kevin Richardson, Bernhard Salow and two anonymous referees for Philosophia Mathematica. I would also like to thank audiences at Smith College, the Università Vita-Salute San Raffaele, and MIT’s Logic, Langauge, Metaphysics and Mind Reading Group. Most of all, I would like to thank Steve Yablo. -
Biography Paper – Georg Cantor
Mike Garkie Math 4010 – History of Math UCD Denver 4/1/08 Biography Paper – Georg Cantor Few mathematicians are house-hold names; perhaps only Newton and Euclid would qualify. But there is a second tier of mathematicians, those whose names might not be familiar, but whose discoveries are part of everyday math. Examples here are Napier with logarithms, Cauchy with limits and Georg Cantor (1845 – 1918) with sets. In fact, those who superficially familier with Georg Cantor probably have two impressions of the man: First, as a consequence of thinking about sets, Cantor developed a theory of the actual infinite. And second, that Cantor was a troubled genius, crippled by Freudian conflict and mental illness. The first impression is fundamentally true. Cantor almost single-handedly overturned the Aristotle’s concept of the potential infinite by developing the concept of transfinite numbers. And, even though Bolzano and Frege made significant contributions, “Set theory … is the creation of one person, Georg Cantor.” [4] The second impression is mostly false. Cantor certainly did suffer from mental illness later in his life, but the other emotional baggage assigned to him is mostly due his early biographers, particularly the infamous E.T. Bell in Men Of Mathematics [7]. In the racially charged atmosphere of 1930’s Europe, the sensational story mathematician who turned the idea of infinity on its head and went crazy in the process, probably make for good reading. The drama of the controversy over Cantor’s ideas only added spice. 1 Fortunately, modern scholars have corrected the errors and biases in older biographies. -
Our Conceptual Understanding of a Phenomenon, While the Logic of Induction Adds Quantitative Details to the Conceptual Knowledge
DOCUMENT RESUME ED 376 173 TM 021 982 AUTHOR Ho, Yu Chong TITLE Abduction? Deduction? Induction? Is There a Logic of Exploratory Data Analysis? PUB DATE Apr 94 NOTE 28p.; Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 4-8, 1994). PUB TYPE Reports Descriptive (141) Speeches/Conference Papers (150) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS *Comprehension; *Deduction; Hypothesis Testing; *Induction; *Logic IDENTIFIERS *Abductive Reasoning; *Exploratory Data Analysis; Peirce (Charles S) ABSTRACT The philosophical notions introduced by Charles Sanders Peirce (1839-1914) are helpfu: for researchers in understanding the nature of knowledge and reality. In the Peircean logical system, the logic of abduction and deduction contribute to our conceptual understanding of a phenomenon, while the logic of induction adds quantitative details to the conceptual knowledge. Although Peirce justified the validity of induction as a self-corrective process, he asserted that neither induction nor deduction can help us to unveil the internal structure of meaning. As exploratory data analysis performs the function of a model builder for confirmatory data analysis, abduction plays the role of explorer of viable paths to further inquiry. Thus, the logic of abduction fits well into exploratory data analysis. At the stage of abduction, the goal is to explore the data, find out a pattern, and suggest a plausible hypothesis; deduction is to refine the hypothesis based upon other plausible premises; and induction is the empirical substantiation. (Contains 55 references.) (Author) *********************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. is *********************************************************************** Abduction? Deduction? Induction? Is there a Logic of Exploratory Data Analysis? Yu Chong Ho University of Oklahoma Internet: [email protected] April 4, 1994 U S. -
Explanations
© Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. CHAPTER 1 Explanations 1.1 SALLIES Language is an instrument of Logic, but not an indispensable instrument. Boole 1847a, 118 We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. De Morgan 1868a,71 That which is provable, ought not to be believed in science without proof. Dedekind 1888a, preface If I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems whilst the root drives into the depthswx . Frege 1893a, xiii Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. Russell 1903a, 451 1.2 SCOPE AND LIMITS OF THE BOOK 1.2.1 An outline history. The story told here from §3 onwards is re- garded as well known. It begins with the emergence of set theory in the 1870s under the inspiration of Georg Cantor, and the contemporary development of mathematical logic by Gottlob Frege andŽ. especially Giuseppe Peano. A cumulation of these and some related movements was achieved in the 1900s with the philosophy of mathematics proposed by Alfred North Whitehead and Bertrand Russell. -
Infinite Time Computable Model Theory
INFINITE TIME COMPUTABLE MODEL THEORY JOEL DAVID HAMKINS, RUSSELL MILLER, DANIEL SEABOLD, AND STEVE WARNER Abstract. We introduce infinite time computable model theory, the com- putable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time theory generalizes to the infinite time context, but several fundamental questions, including the infinite time computable analogue of the Completeness Theorem, turn out to be independent of ZFC. 1. Introduction Computable model theory is model theory with a view to the computability of the structures and theories that arise (for a standard reference, see [EGNR98]). Infinite time computable model theory, which we introduce here, carries out this program with the infinitary notions of computability provided by infinite time Turing ma- chines. The motivation for a broader context is that, while finite time computable model theory is necessarily limited to countable models and theories, the infinitary context naturally allows for uncountable models and theories, while retaining the computational nature of the undertaking. Many constructions generalize from finite time computable model theory, with structures built on N, to the infinitary theory, with structures built on R. In this article, we introduce the basic theory and con- sider the infinitary analogues of the completeness theorem, the L¨owenheim-Skolem Theorem, Myhill’s theorem and others. It turns out that, when stated in their fully general infinitary forms, several of these fundamental questions are independent of ZFC. The analysis makes use of techniques both from computability theory and set theory. This article follows up [Ham05]. -
The Constituents of the Propositions of Logic Kevin C
10 The Constituents of the Propositions of Logic Kevin C. Klement 1 Introduction Many founders of modern logic—Frege and Russell among them—bemoan- ed the tendency, still found in most textbook treatments, to define the subject matter of logic as “the laws of thought” or “the principles of inference”. Such descriptions fail to capture logic’s objective nature; they make it too dependent on human psychology or linguistic practices. It is one thing to identify what logic is not about. It is another to say what it is about. I tell my students that logic studies relationships between the truth-values of propositions that hold in virtue of their form. But even this characterization leaves me uneasy. I don’t really know what a “form” is, and even worse perhaps, I don’t really know what these “propositions” are that have these forms. If propositions are considered merely as sentences or linguistic assertions, the definition does not seem like much of an improvement over the psychological definitions. Language is a human invention, but logic is more than that, or so it seems. It is perhaps forgiveable then that at certain times Russell would not have been prepared to give a very good answer to the question “What is Logic?”, such as when he attempted, but failed, to compose a paper with that title in October 1912. Given that Russell had recently completed Principia Mathematica, a work alleging to establish the reducibility of mathematics to logic, one might think this overly generous. What does the claim that mathematics reduces to logic come to if we cannot independently speci- Published in Acquaintance, Knowledge and Logic: New Essays on Bertrand Russell’s Problems of Philosophy, edited by Donovan Wishon and Bernard Linsky. -
Vasiliev and the Foundations of Logic
Chapter 4 Vasiliev and the Foundations of Logic Otávio Bueno Abstract Nikolai Vasiliev offered a systematic approach to the development of a class of non-classical logics, which he called “Imaginary Logics”. In this paper, IexaminecriticallysomeofthecentralfeaturesofVasiliev’sapproachtological theory, suggesting its relevance to contemporary debates in the philosophy of logic. IarguethatthereismuchofsignificantvalueinVasiliev’swork,whichdeserves close philosophical engagement. Keywords Vasiliev • logical pluralism • Revisability • a priori • negation 4.1 Introduction: Six Central Features of Vasiliev’s Approach to Logical Theory Nikolai Vasiliev’s approach to logical theory has a number of features. Six of them, in particular, are worth highlighting: (a) logical pluralism (there is a plurality of logics, depending on the subject matter under consideration); (b) logical revisability (certain logical laws can be revised depending on the subject matter); (c) logical non-a priorism (certain logical laws are empirically based); (d) logical contingency (given the empirical nature of some logical laws, they are ultimately contingent; in this context, issues regarding the scope of logic are also examined, with the accompanying distinction between laws of objects and laws of thought); (e) the nature of negation (negation is characterized via incompatibility; it is not just difference, nor is it grounded on absence, and it is inferred rather than perceived); and (f) logical commitment (why Vasiliev is not a dialetheist, after all). In this paper, I will examine each of these features, and suggest the relevance of many of Vasiliev’s proposals to contemporary philosophical reflection about the foundations of logic. Although the terms I use to describe some of the views O. Bueno (!) Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] ©SpringerInternationalPublishingAG2017 43 V. -
Abstract Consequence and Logics
Abstract Consequence and Logics Essays in honor of Edelcio G. de Souza edited by Alexandre Costa-Leite Contents Introduction Alexandre Costa-Leite On Edelcio G. de Souza PART 1 Abstraction, unity and logic 3 Jean-Yves Beziau Logical structures from a model-theoretical viewpoint 17 Gerhard Schurz Universal translatability: optimality-based justification of (not necessarily) classical logic 37 Roderick Batchelor Abstract logic with vocables 67 Juliano Maranh~ao An abstract definition of normative system 79 Newton C. A. da Costa and Decio Krause Suppes predicate for classes of structures and the notion of transportability 99 Patr´ıciaDel Nero Velasco On a reconstruction of the valuation concept PART 2 Categories, logics and arithmetic 115 Vladimir L. Vasyukov Internal logic of the H − B topos 135 Marcelo E. Coniglio On categorial combination of logics 173 Walter Carnielli and David Fuenmayor Godel's¨ incompleteness theorems from a paraconsistent perspective 199 Edgar L. B. Almeida and Rodrigo A. Freire On existence in arithmetic PART 3 Non-classical inferences 221 Arnon Avron A note on semi-implication with negation 227 Diana Costa and Manuel A. Martins A roadmap of paraconsistent hybrid logics 243 H´erculesde Araujo Feitosa, Angela Pereira Rodrigues Moreira and Marcelo Reicher Soares A relational model for the logic of deduction 251 Andrew Schumann From pragmatic truths to emotional truths 263 Hilan Bensusan and Gregory Carneiro Paraconsistentization through antimonotonicity: towards a logic of supplement PART 4 Philosophy and history of logic 277 Diogo H. B. Dias Hans Hahn and the foundations of mathematics 289 Cassiano Terra Rodrigues A first survey of Charles S. -
The History of Logic
c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). -
Paradox and Foundation Zach Weber Submitted in Total Fulfilment of The
Paradox and Foundation Zach Weber Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy May 2009 School of Philosophy, Anthropology and Social Inquiry The University of Melbourne This is to certify that - the thesis comprises only my original work towards the PhD, - due acknowledgement has been made in the text to all other material used, - the thesis is less than 100,000 words in length. Preface Dialethic paraconsistency is an approach to formal and philosophical theories in which some but not all contradictions are true. Advancing that program, this thesis is about paradoxes and the foundations of mathematics, and is divided accordingly into two main parts. The first part concerns the history and philosophy of set theory from Cantor through the independence proofs, focusing on the set concept. A set is any col- lection of objects that is itself an object, with identity completely determined by membership. The set concept is called naive because it is inconsistent. I argue that the set concept is inherently and rightly paradoxical, because sets are both intensional and extensional objects: Sets are predicates in extension. All consistent characterizations of sets are either not explanatory or not coherent. To understand sets, we need to reason about them with an appropriate logic; paraconsistent naive set theory is situated as a continuation of the original foundational project. The second part produces a set theory deduced from an unrestricted compre- hension principle using the weak relevant logic DLQ, dialethic logic with quantifiers. I discuss some of the problems involved with embedding in DLQ, especially related to identity and substitution. -
Notes on Incompleteness Theorems
The Incompleteness Theorems ● Here are some fundamental philosophical questions with mathematical answers: ○ (1) Is there a (recursive) algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is true? ○ (2) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of Peano or Robinson Arithmetic? ○ (3) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of pure (first-order) logic? ○ (4) Is there a complete (even if not recursive) recursively axiomatizable theory in the language of first-order arithmetic? ○ (5) Is there a recursively axiomatizable sub-theory of Peano Arithmetic that proves the consistency of Peano Arithmetic (even if it leaves other questions undecided)? ○ (6) Is there a formula of arithmetic that defines arithmetic truth in the standard model, N (even if it does not represent it)? ○ (7) Is the (non-recursively enumerable) set of truths in the language of first-order arithmetic categorical? If not, is it ω-categorical (i.e., categorical in models of cardinality ω)? ● Questions (1) -- (7) turn out to be linked. Their philosophical interest depends partly on the following philosophical thesis, of which we will make frequent, but inessential, use. ○ Church-Turing Thesis:A function is (intuitively) computable if/f it is recursive. ■ Church: “[T]he notion of an effectively calculable function of positive integers should be identified with that of recursive function (quoted in Epstein & Carnielli, 223).” ○ Note: Since a function is recursive if/f it is Turing computable, the Church-Turing Thesis also implies that a function is computable if/f it is Turing computable. -
How Peircean Was the “'Fregean' Revolution” in Logic?
HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0.