DOCSLIB.ORG

Mathematics, Philosophy of 11/4/10 10:42 PM Oxford Bibliographies Online the World’S Largest University Press: Authority and Innovation for Research

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematics, Philosophy of 11/4/10 10:42 PM Oxford Bibliographies Online the World’S Largest University Press: Authority and Innovation for Research Oxford Bibliographies Online - Mathematics, Philosophy of 11/4/10 10:42 PM Oxford Bibliographies Online The World’s Largest University Press: Authority and Innovation for Research Philosophy of Mathematics by Otávio Bueno Introduction Philosophy of mathematics is arguably one of the oldest branches of philosophy, and one that bears significant connections with core philosophical areas, particularly metaphysics, epistemology, and (more recently) the philosophy of science. This entry focuses on contemporary developments, which have yielded novel approaches (such as new forms of Platonism and nominalism, structuralism, neo-Fregeanism, empiricism, and naturalism) as well as several new issues (such as the significance of the application of mathematics, the role of visualization in mathematical reasoning, particular attention to mathematical practice and to the nature of mathematical explanation). Excellent work has also been done on particular philosophical issues that arise in the context of specific branches of mathematics, such as algebra, analysis, and geometry, as well as particular mathematical theories, such as set theory and category theory. Due to limitations of space, this work goes beyond the scope of the present entry. General Overviews There are several general overviews of the philosophy of mathematics, varying in how detailed or up-to-date they are. Horsten 2008 and Detlefsen 1996 are very readable and thoughtful surveys of the field. The former is up-to-date and freely available online; the latter offers more detailed coverage of the issues it addresses. Longer treatments of particular topics in the philosophy of mathematics may be found in the papers collected in Shapiro 2005, Irvine 2009, Bueno and Linnebo 2009, and Schirn 1998. An excellent and up-to-date survey of Platonism in the philosophy of mathematics, which is also freely available online, is given in Linnebo 2009. Burgess and Rosen 1997 offers a critical survey of some nominalist views, but the work is no longer up-to-date. Bueno, Otávio, and Øystein Linnebo, eds. New Waves in Philosophy of Mathematics. Basingstoke, UK: Palgrave Macmillan, 2009. Find this resource: A collection of thirteen essays by promising young researchers that offers an up-to-date picture of contemporary philosophy of mathematics, including a reassessment of orthodoxy in the field, the question of realism in mathematics, relations between mathematical practice and the methodology of mathematics, and connections between philosophical logic and the philosophy of mathematics. Burgess, John P., and Gideon A. Rosen. A Subject with No Objects: Strategies for Nominalistic Interpretation of Mathematics. Oxford: Clarendon, 1997. Find this resource: A critical examination of major nominalist interpretations of mathematics by two authors who do not defend nominalism. Detlefsen, Michael. “Philosophy of Mathematics in the Twentieth Century.” In Philosophy of Science, Logic and Mathematics in the Twentieth Century. Routledge History of Philosophy 9. Edited by Stuart G. Shanker, 50–123. New York: Routledge, 1996. Find this resource: A careful survey of the philosophy of mathematics focusing on some of the central proposals in the 20th century. http://www.oxfordbibliographiesonline.com/display/id/obo-9780195396577-0069 Page 1 of 16 Oxford Bibliographies Online - Mathematics, Philosophy of 11/4/10 10:42 PM Horsten, Leon. “Philosophy of Mathematics.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2008. A useful and up-to-date survey of the philosophy of mathematics, including a discussion of four classic approaches (logicism, intuitionism, formalism, and predicativism) as well as more recent proposals (Platonism, structuralism, and nominalism) and some special topics (philosophy of set theory, categoricity, and computation and proof). Irvine, Andrew D., ed. Philosophy of Mathematics. Handbook of the Philosophy of Science series. Amsterdam: North Holland, 2009. Find this resource: An up-to-date survey of the philosophy of mathematics composed by fifteen specially commissioned essays that cover central issues and conceptions in the field, with emphasis on realism and antirealism, empiricism, Kantianism, as well as logicism, formalism, and constructivism. Philosophical issues that emerge in set theory, probability theory, computability theory in addition to inconsistent and applied mathematics are also examined. Linnebo, Øystein. “Platonism in the Philosophy of Mathematics.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2009. An up-to-date survey of the main forms of mathematical Platonism as well as the central arguments for this conception and the main objections that have been raised against it. Schirn, Matthias, ed. The Philosophy of Mathematics Today. Oxford: Clarendon, 1998. Find this resource: A comprehensive panorama of the philosophy of mathematics given by twenty specially commissioned essays by leading philosophers of mathematics, which examine a range of issues from the nature of mathematical knowledge and the existence of mathematical objects through the characterization of the concepts of set and natural number to logical consequence and abstraction. Shapiro, Stewart, ed. The Oxford Handbook of Philosophy of Mathematics and Logic. New York: Oxford University Press, 2005. Find this resource: A comprehensive and up-to-date overview of the philosophy of mathematics, written by some of the main contributors to the field, and that also explores significant connections with the philosophy of logic. Balanced treatments of logicism, formalism, intuitionism, empiricism, naturalism, nominalism, structuralism, as well as the application of mathematics, concepts of logical consequence and relevance, and higher-order logic, are offered. Textbooks There are excellent textbooks in the philosophy of mathematics. The most accessible for undergraduate students with a limited background in mathematics is Friend 2007. Brown 2008 is also very accessible, and it covers topics that usually are not addressed in elementary introductions to the philosophy of mathematics (such as the epistemological significance of pictures in mathematical proofs and the importance of mathematical notation). George and Velleman 2002 offers an excellent examination of how philosophical programs in the philosophy of mathematics (such as logicism, intuitionism, and finitism) combine a philosophical vision about mathematics with particular mathematical developments. A primarily historical approach to the philosophy of mathematics is carefully presented in Bostock 2009, whereas Shapiro 2000 combines a historical and conceptual discussion of major philosophical views about mathematics (both classical and contemporary). Bostock, David. Philosophy of Mathematics: An Introduction. Oxford: Wiley-Blackwell, 2009. Find this resource: A historical introduction to the philosophy of mathematics, starting with a discussion of the works of Plato and Aristotle, going through Kant, Frege, Russell, Hilbert, and Brouwer, all the way to Gödel, Quine, Putnam, and the contemporary scene. Brown, James Robert. Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and http://www.oxfordbibliographiesonline.com/display/id/obo-9780195396577-0069 Page 2 of 16 Oxford Bibliographies Online - Mathematics, Philosophy of 11/4/10 10:42 PM Pictures. 2d ed. New York: Routledge, 2008. Find this resource: A very readable and accessible introduction to the philosophy of mathematics, covering some topics not typically discussed in elementary introductions to the field, such as the role of pictures in proofs, the significance of mathematical notation, and a refutation of the continuum hypothesis. Friend, Michèle. Introducing Philosophy of Mathematics. Montreal: McGill-Queen’s University Press, 2007. Find this resource: An extremely accessible introduction to the philosophy of mathematics for undergraduate students with a limited background in mathematics. It assesses the advantages and limitations of Platonism, logicism, formalism, constructivism, and structuralism as well as psychologism, fictionalism, and Meinongianism. George, Alexander, and David J. Velleman. Philosophies of Mathematics. Malden, MA: Blackwell, 2002. Find this resource: A careful and very readable discussion of three classical programs in the philosophy of mathematics—logicism, intuitionism, and finitism—examining how they combine a certain philosophical vision of mathematics with particular mathematical developments. Shapiro, Stewart Thinking about Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press, 2000. Find this resource: A balanced combination of historical and conceptual discussion of central proposals in the philosophy of mathematics, including an examination of the three main classical conceptions—logicism, formalism, and intuitionism—as well as central proposals in the contemporary scene—Platonism, nominalism, and structuralism. Anthologies Different anthologies in the philosophy of mathematics focus on different aspects of the field. Benacerraf and Putnam 1983 is a classic; it brings together seminal papers by preeminent philosophers and mathematicians working on the foundations of mathematics. An excellent anthology on the significant connections between history and philosophy of mathematics is Asprey and Kitcher 1988. Hart 1996 collects some of the most influential papers in the philosophy of mathematics in the second half of the 20th century, whereas Jacquette 2002 brings together essays
Load more

Recommended publications
  • Finitism, Divisibility, and the Beginning of the Universe: Replies to Loke and Dumsday
    This is a preprint of an article whose final and definitive form will be published in the Australasian Journal of Philosophy. The Australasian Journal of Philosophy is available online at: http://www.tandf.co.uk/journals/. FINITISM, DIVISIBILITY, AND THE BEGINNING OF THE UNIVERSE: REPLIES TO LOKE AND DUMSDAY Stephen Puryear North Carolina State University Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies. Keywords: Space, time, continuity, infinity, creation 1. Introduction Some philosophers argue that the universe must have had a beginning, because otherwise reaching the present would have required something impossible, namely, the occurrence or ‘traversal’ of an actually infinite sequence of events. But this argument faces the objection that actual infinities are traversed all the time: for example, whenever a finite period of time elapses [Morriston 2002: 162; Sinnott-Armstrong 2004: 42–3]. Let us call these the finitist argument and the Zeno objection. In response to the Zeno objection, the most ardent proponent of the finitist argument in our day, William Lane Craig [Craig and Sinclair 2009: 112–13, 119; cf. Craig and Smith 1993: 27–30], has taken the position that ‘time is logically prior to the divisions we make within it’. On this view, time divides into parts only in so far as we divide it, and thus divides into only a finite number of parts.
    [Show full text]
  • Aristotelian Finitism
    Synthese DOI 10.1007/s11229-015-0827-9 S.I. : INFINITY Aristotelian finitism Tamer Nawar1 Received: 12 January 2014 / Accepted: 25 June 2015 © Springer Science+Business Media Dordrecht 2015 Abstract It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments con- sidered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logi- cal’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed. Keywords Aristotle · Aristotelian commentators · Infinity · Mathematics · Metaphysics 1 Introduction It is widely known that Aristotle embraced some sort of finitism and denied the exis- tence of so-called ‘actual infinities’ while allowing for the existence of ‘potential infinities’. It is difficult to overestimate the influence of Aristotle’s views on this score and the denial of the (actual) existence of infinities became a commonplace among philosophers for over two thousand years. However, the precise grounds for Aristo- tle’s finitism have not been discussed in much detail and, insofar as they have received attention, his reasons for ruling out the existence of (actual) infinities have often been B Tamer Nawar [email protected] 1 University of Oxford, 21 Millway Close, Oxford OX2 8BJ, UK 123 Synthese deemed obscure or ad hoc (e.g.
    [Show full text]
  • NSP4 Pragmatist Kant
    Nordic NSP Studies in Pragmatism Helsinki — 2019 Giovanni Maddalena “Anti-Kantianism as a Necessary Characteristic of Pragmatism” In: Krzysztof Piotr Skowronski´ and Sami Pihlstrom¨ (Eds.) (2019). Pragmatist Kant—Pragmatism, Kant, and Kantianism in the Twenty-first Century (pp. 43–59). Nordic Studies in Pragmatism 4. Helsinki: Nordic Pragmatism Network. issn-l 1799-3954 issn 1799-3954 isbn 978-952-67497-3-0 Copyright c 2019 The Authors and the Nordic Pragmatism Network. This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License. CC BY NC For more information, see http://creativecommons.org/licenses/by-nc/3.0/ Nordic Pragmatism Network, NPN Helsinki 2019 www.nordprag.org Anti-Kantianism as a Necessary Characteristic of Pragmatism Giovanni Maddalena Universit`adel Molise 1. Introduction Pragmatists declared their anti-Cartesianism at the first appearance of the movement, in Peirce’s series on cognition written for the Journal of Specu- lative Philosophy (1867–8). As is well known, the brilliant young scientist characterized Cartesian doubt as a “paper doubt”, by opposing it to sci- entists’ true “living doubt” (Peirce 1998 [1868], 115).1 Some readers have not understood the powerful novelty that his opposition to Cartesianism implies. According to Peirce, research does not proceed from skeptical, “paper” doubt. For Peirce, doubt is possible because of a previous cer- tainty, a position which is similar to the one held by Augustine (Augustine 1970). Research moves from one certainty to another; the abandonment of an initial certainty is only reasonable in the presence of a real and surprising phenomenon that alters one of the pillars on which it stands.
    [Show full text]
  • Constructivity in Homotopy Type Theory
    Ludwig Maximilian University of Munich Munich Center for Mathematical Philosophy Constructivity in Homotopy Type Theory Author: Supervisors: Maximilian Doré Prof. Dr. Dr. Hannes Leitgeb Prof. Steve Awodey, PhD Munich, August 2019 Thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Logic and Philosophy of Science contents Contents 1 Introduction1 1.1 Outline................................ 3 1.2 Open Problems ........................... 4 2 Judgements and Propositions6 2.1 Judgements ............................. 7 2.2 Propositions............................. 9 2.2.1 Dependent types...................... 10 2.2.2 The logical constants in HoTT .............. 11 2.3 Natural Numbers.......................... 13 2.4 Propositional Equality....................... 14 2.5 Equality, Revisited ......................... 17 2.6 Mere Propositions and Propositional Truncation . 18 2.7 Universes and Univalence..................... 19 3 Constructive Logic 22 3.1 Brouwer and the Advent of Intuitionism ............ 22 3.2 Heyting and Kolmogorov, and the Formalization of Intuitionism 23 3.3 The Lambda Calculus and Propositions-as-types . 26 3.4 Bishop’s Constructive Mathematics................ 27 4 Computational Content 29 4.1 BHK in Homotopy Type Theory ................. 30 4.2 Martin-Löf’s Meaning Explanations ............... 31 4.2.1 The meaning of the judgments.............. 32 4.2.2 The theory of expressions................. 34 4.2.3 Canonical forms ...................... 35 4.2.4 The validity of the types.................. 37 4.3 Breaking Canonicity and Propositional Canonicity . 38 4.3.1 Breaking canonicity .................... 39 4.3.2 Propositional canonicity.................. 40 4.4 Proof-theoretic Semantics and the Meaning Explanations . 40 5 Constructive Identity 44 5.1 Identity in Martin-Löf’s Meaning Explanations......... 45 ii contents 5.1.1 Intensional type theory and the meaning explanations 46 5.1.2 Extensional type theory and the meaning explanations 47 5.2 Homotopical Interpretation of Identity ............
    [Show full text]
  • The Proper Explanation of Intuitionistic Logic: on Brouwer's Demonstration
    The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem Göran Sundholm, Mark van Atten To cite this version: Göran Sundholm, Mark van Atten. The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem. van Atten, Mark Heinzmann, Gerhard Boldini, Pascal Bourdeau, Michel. One Hundred Years of Intuitionism (1907-2007). The Cerisy Conference, Birkhäuser, pp.60- 77, 2008, 978-3-7643-8652-8. halshs-00791550 HAL Id: halshs-00791550 https://halshs.archives-ouvertes.fr/halshs-00791550 Submitted on 24 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem Göran Sundholm Philosophical Institute, Leiden University, P.O. Box 2315, 2300 RA Leiden, The Netherlands. [email protected] Mark van Atten SND (CNRS / Paris IV), 1 rue Victor Cousin, 75005 Paris, France. [email protected] Der … geführte Beweis scheint mir aber trotzdem . Basel: Birkhäuser, 2008, 60–77. wegen der in seinem Gedankengange enthaltenen Aussagen Interesse zu besitzen. (Brouwer 1927B, n. 7)1 Brouwer’s demonstration of his Bar Theorem gives rise to provocative ques- tions regarding the proper explanation of the logical connectives within intu- itionistic and constructivist frameworks, respectively, and, more generally, re- garding the role of logic within intuitionism.
    [Show full text]
  • Gödel on Finitism, Constructivity and Hilbert's Program
    Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert’s program Solomon Feferman 1. Gödel, Bernays, and Hilbert. The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel’s collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post- Kantian philosophy, and contains Gödel’s thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their life-long warm personal relationship. I have given a detailed synopsis of the Bernays Gödel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Gödel’s Collected Works, pp. 41- 79.1 My purpose here is to focus on only one group of interrelated topics from these exchanges, namely the light that ittogether with assorted published and unpublished articles and lectures by Gödelthrows on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert’s program.2 In that connection, this piece has an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career. 1 The five volumes of Gödel’s Collected Works (1986-2003) are referred to below, respectively, as CW I, II, III, IV and V. CW I consists of the publications 1929-1936, CW II of the publications 1938-1974, CW III of unpublished essays and letters, CW IV of correspondence A-G, and CW V of correspondence H-Z.
    [Show full text]
  • THE LOOICAL ATOMISM J. D. MACKENZIE Date Submitted Thesis
    THE LOOICAL ATOMISM OF F. P. RAMSEY J. D. MACKENZIE Date submitted Thesis submitted for the Degree of Master of Arts in the School of Philosophy University of New South Wales (i) SYNOPSIS The first Chapter sets Ramsey in histor:iealperspective as a Logical Atomist. Chapter Two is concerned with the impasse in which Russell found himself ,d.th general propositions, Wittgenstein's putative solution in terms of his Doctrine of Showing, and Ramsey's "Wittgensteinian" solution, which is not satisfactory. An attempt is then ma.de to describe a Ramseian solution on the basis of what he says about the Axiom of Infi- nity, and to criticize this solution. In Chapter Three Ramsay's objections to the Pl4 definition of identity are considered, and consequences of his rejection of that definition for the Theory of Classes and the Axiom of Choice are drawn. In Chapter Four, Ramsey•s modifications to Russell's Theory of Types are discussed. His division of the Paradoxes into two groups is defended, but his redefinition of 'predicative' is rejected. Chapter Five deals with Ra.msey's analysis of propositional attitudes and negative propositions, and Chapter Six considers the dispute between Russell and Ramsey over the nature and status of universals. In Chapter Seven, the conclusions are summarized, and Ramsay's contribution to Logical Atom.ism are assessed. His main fail­ ing is found to be his lack of understanding of impossibility, especially with regard to the concept of infinity. (ii) PREFACE The thesis is divided into chapters, which are in turn divided into sections.
    [Show full text]
  • Understanding Intuitionism
    Understanding Intuitionism by Edward Nelson Department of Mathematics Princeton University http:==www.math.princeton.edu= nelson=papers.html ∼ Intuitionism was the creation of L. E. J. Brouwer [Br], and I like to think that classical mathematics was the creation of Pythagoras. Imag- ine a conversation between a classical mathematician and an intuitionist, say a woman and a man respectively. She speaks first: I have just proved xA. Congratulations! What9 is it? I don't know. I assumed x A and derived a contradiction. Oh. You proved x A.8 : That's what I said.:8 : Or another: I have proved A B. Good. Which did_ you prove? What? You said you proved A or B; which did you prove? Neither; I assumed A & B and derived a contradiction. Oh, you proved [ :A & :B]. That's right. It's: another: : way of putting the same thing. But he does not agree with her last statement; they have a different semantics and a different notion of proof. This paper is an attempt to understand the differences between them. I am grateful to Mitsuru Yasuhara for stimulating discussions of this material and for pinpointing errors and obscurities in earlier drafts. I also wish to thank Simon Kochen and Per Martin-L¨offor helpful comments. 1. Incomplete communications For a classical mathematician, a closed formula, true in a given structure, is a complete communication. It expresses an objective state of affairs in the universe of discourse; it is an ontological assertion. But 1 2 1. INCOMPLETE COMMUNICATIONS to an intuitionist, a closed true formula may be an incomplete communi- cation.
    [Show full text]
  • Intuition As Evidence in Philosophical Analysis: Taking Connectionism Seriously
    Intuition as Evidence in Philosophical Analysis: Taking Connectionism Seriously by Tom Rand A thesis submitted in conformity with the requirements for the degree of Ph.D. Graduate Department of Philosophy University of Toronto (c) Copyright by Tom Rand 2008 Intuition as Evidence in Philosophical Analysis: Taking Connectionism Seriously Ph.D. Degree, 2008, by Tom Rand, Graduate Department of Philosophy, University of Toronto ABSTRACT 1. Intuitions are often treated in philosophy as a basic evidential source to confirm/discredit a proposed definition or theory; e.g. intuitions about Gettier cases are taken to deny a justified-true-belief analysis of ‘knowledge’. Recently, Weinberg, Nichols & Stitch (WN&S) provided evidence that epistemic intuitions vary across persons and cultures. In- so-far as philosophy of this type (Standard Philosophical Methodology – SPM) is committed to provide conceptual analyses, the use of intuition is suspect – it does not exhibit the requisite normativity. I provide an analysis of intuition, with an emphasis on its neural – or connectionist – cognitive backbone; the analysis provides insight into its epistemic status and proper role within SPM. Intuition is initially characterized as the recognition of a pattern. 2. The metaphysics of ‘pattern’ is analyzed for the purpose of denying that traditional symbolic computation is capable of differentiating the patterns of interest. 3. The epistemology of ‘recognition’ is analyzed, again, to deny that traditional computation is capable of capturing human acts of recognition. 4. Fodor’s informational semantics, his Language of Thought and his Representational Theory of Mind are analyzed and his arguments denied. Again, the purpose is to deny traditional computational theories of mind.
    [Show full text]
  • Constructive Logic for All
    Constructive Logic for All Greg Restall∗ Philosophy Department Macquarie University June 14, 2000 Abstract It is a commonplace in recent metaphysics that one's logical commit- ments go hand in hand with one's metaphysics. Brouwer, Heyting and Dummett have each championed the move to constructive (intuitionistic) reasoning on the grounds of anti-realism. I hope to break this close connec- tion, to explain why a realist ought to reason constructively. 1 Introduction Let me start by explaining the terms of the discussion. In this area of philo- sophical logic there seems to be some confusion in the use of terms such as “intuitionistic” and “constructive.” It will help to get the use of these terms somewhat fixed before I start to argue my case. 1.1 Logic First, logic. The subject Logic is the study of logical consequence. Logical con- sequence is a matter of the validity and invalidity of arguments. An argument is valid just when in any case in which the premises of the argument are true, so is the conclusion. It is often helpful, in discussing logical consequence, to have a formal language in which to express the premises and conclusions of ar- guments. One such language is the language of first-order logic, with primitive expressions for the connectives and quantifiers ^ conjunction _ disjunction conditional ⊃ ∼ negation x universal quantifier 8 (with x a variable) x existential quantifier 9 (with x a variable) For some, logic is at heart the study of the behaviour of these connectives. For others, the singling out of particular parts of the language is an incidental mat- ter.
    [Show full text]
  • Paraconsistency and Duality
    Paraconsistency and duality: between ontological and 1 epistemological views 1 2 WALTER CARNIELLI AND ABILIO RODRIGUES 2 Abstract: The aim of this paper is to show how an epistemic approach to paraconsistency may be philosophically justified based on the duality be- tween paraconsistency and paracompleteness. The invalidity of excluded middle in intuitionistic logic may be understood as expressing that no con- 3 structive proof of a pair of propositions A and :A is available. Analogously, in order to explain the invalidity of the principle of explosion in paraconsis- tent logics, it is not necessary to consider that A and :A are true, but rather that there are conflicting and non-conclusive evidence for both. Keywords: Paraconsistent logic, Philosophy of paraconsistency, Evidence, Intuitionistic logic 4 1 Introduction 5 Paraconsistent logics have been assuming an increasingly important place 6 in contemporary philosophical debate. Although from the strictly techni- 7 cal point of view paraconsistent formal systems have reached a point of 8 remarkable development, there are still some aspects of their philosophi- 9 cal significance that have not been fully explored yet. The distinctive fea- 10 ture of paraconsistent logics is that the principle of explosion, according to 11 which anything follows from a contradiction, does not hold, thus allowing 12 for the presence of contradictions without deductive triviality. Dialetheism 13 is the view according to which there are true contradictions (cf. e.g. Priest 14 1The first author acknowledges support from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo, thematic project LogCons) and from a CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) research grant.
    [Show full text]
  • Scott Soames, Philosophical Analysis in the Twentieth Century: Volume 1: the Dawn of Analysis Princeton University Press, 2003
    Philos Stud (2006) 129:637–643 DOI 10.1007/s11098-006-0015-3 DISCUSSION Scott Soames, Philosophical Analysis in the Twentieth Century: Volume 1: The Dawn of Analysis Princeton University Press, 2003 R.M. Sainsbury Accepted: 27 March 2006 Ó Springer Science+Business Media B.V. 2006 The jacket of this first volume of the two volume work tells us that this is a history of analytic philosophy from 1900 to mid-century. Accordingly, we can properly eval- uate it under three heads: (1) the care and accuracy of the scholarship; (2) the light it throws on the relationships among the various activities it describes; and (3) the quality of the philosophical argumentation. Evaluated under the third, the book deserves high praise. It provides serious philosophy: arguments are carefully set out and taken to pieces, objections and possible responses are developed in an orderly way. Most readers will learn something of philosophical value, and even on the issues about which they disagree with Soames, they will find him a worthy and helpful opponent. In this respect the text provides a model to which students should aspire. Every page demonstrates one way in which philosophy can be done excel- lently, as will be no surprise to those familiar with Soames’s other work. Evaluated under the other two heads, however, the book in my judgement falls short. Not only are there historical inaccuracies, the approach to reading, evaluating and interpreting texts is one I find uncongenial; and the attempt to impose some kind of overall developmental themes on the material under discussion strikes me as unsuccessful.
    [Show full text]