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Set Theory and its Philosophy: A Critical Introduction

Oxford, England: Oxford University Press (2004)

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  1. Great Philosophy: Discovery, Invention, and the Uses of Error.Christopher Norris - 2014 - International Journal of Philosophical Studies 22 (3):349-379.
    In this essay I consider what is meant by the description ‘great’ philosophy and then offer some broadly applicable criteria by which to assess candidate thinkers or works. On the one hand are philosophers in whose case the epithet, even if contested, is not grossly misconceived or merely the product of doctrinal adherence on the part of those who apply it. On the other are those – however gifted, acute, or technically adroit – to whom its application is inappropriate because (...)
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  • The Nature of Appearance in Kant’s Transcendentalism: A Seman- Tico-Cognitive Analysis.Sergey L. Katrechko - 2018 - Kantian Journal 37 (3):41-55.
  • Mathematics and Metaphilosophy.Justin Clarke-Doane - 2022 - Cambridge: Cambridge University Press.
    This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the challenge to explain the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the challenge to explain their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, (...)
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  • Reason, Causation and Compatibility with the Phenomena.Basil Evangelidis - 2020 - Wilmington, Delaware, USA: Vernon Press.
    'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...)
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  • Pluralism in Mathematics: A New Position in Philosophy of Mathematics.Michèle Friend - 2013 - Dordrecht, Netherland: Springer.
    The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...)
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  • Essential Properties - Analysis and Extension.Nathan Wildman - 2011 - Dissertation, Cambridge
  • Hilbert on Consistency as a Guide to Mathematical Reality.Fiona T. Doherty - 2017 - Logique Et Analyse 237:107-128.
  • Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2015 - Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  • The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Berlin: Springer. pp. 165-188.
    The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...)
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  • The Deflationary Theory of Ontological Dependence.David Mark Kovacs - 2018 - Philosophical Quarterly 68 (272):481-502.
    When an entity ontologically depends on another entity, the former ‘presupposes’ or ‘requires’ the latter in some metaphysical sense. This paper defends a novel view, Dependence Deflationism, according to which ontological dependence is what I call an aggregative cluster concept: a concept which can be understood, but not fully analysed, as a ‘weighted total’ of constructive and modal relations. The view has several benefits: it accounts for clear cases of ontological dependence as well as the source of disagreement in controversial (...)
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  • An Observation About Truth.David Kashtan - 2017 - Dissertation, University of Jerusalem
    Tarski's analysis of the concept of truth gives rise to a hierarchy of languages. Does this fragment the concept all the way to philosophical unacceptability? I argue it doesn't, drawing on a modification of Kaplan's theory of indexicals.
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  • Against Cumulative Type Theory.Tim Button & Robert Trueman - forthcoming - Review of Symbolic Logic:1-43.
    Standard Type Theory, ${\textrm {STT}}$, tells us that $b^n$ is well-formed iff $n=m+1$. However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$, which has more relaxed type-restrictions: according to $\textrm {CTT}$, $b^\beta $ is well-formed iff $\beta>\alpha $. In this paper, we set ourselves against $\textrm {CTT}$. We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$ (...)
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  • A Trivialist's Travails.Thomas Donaldson - 2014 - Philosophia Mathematica 22 (3):380-401.
    This paper is an exposition and evaluation of the Agustín Rayo's views about the epistemology and metaphysics of mathematics, as they are presented in his book The Construction of Logical Space.
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  • Sets, Logic, Computation: An Open Introduction to Metalogic.Richard Zach - 2021 - Open Logic Project.
    An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic.
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  • Mathematics and Its Applications, A Transcendental-Idealist Perspective.Jairo José da Silva - 2017 - Springer.
    This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...)
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  • Conceptual Metaphors and Mathematical Practice: On Cognitive Studies of Historical Developments in Mathematics.Dirk Schlimm - 2013 - Topics in Cognitive Science 5 (2):283-298.
    This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive analyses of (...)
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  • Sophist or Antiphilosopher? [REVIEW]Christopher Norris - 2012 - Journal of Critical Realism 11 (4):487-498.
    This essay takes Badiou’s recently published book as an opportunity to discuss not only his complex approach to Wittgenstein but also his evolving critical stance in relation to various other movements in present-day philosophical thought. In particular it examines his distinction between ‘sophistics’ and ‘anti-philosophy’, as developed very largely through his series of encounters with Wittgenstein. Beyond that, I offer some brief remarks about the role of set-theoretical concepts in Badiou’s thinking and the vexed question of their bearing on his (...)
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  • Grammar and Sets.B. H. Slater - 2006 - Australasian Journal of Philosophy 84 (1):59 – 73.
    'Philosophy arises through misconceptions of grammar', said Wittgenstein. Few people have believed him, and probably none, therefore, working in the area of the philosophy of mathematics. Yet his assertion is most evidently the case in the philosophy of Set Theory, as this paper demonstrates (see also Rodych 2000). The motivation for twentieth century Set Theory has rested on the belief that everything in Mathematics can be defined in terms of sets [Maddy 1994: 4]. But not only are there notable items (...)
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  • Modal Objectivity1.Clarke-Doane Justin - 2019 - Noûs:266-295.
    It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...)
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  • What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  • The Hidden Set-Theoretical Paradox of the Tractatus.Jing Li - 2018 - Philosophia 46 (1):159-164.
    We are familiar with various set-theoretical paradoxes such as Cantor's paradox, Burali-Forti's paradox, Russell's paradox, Russell-Myhill paradox and Kaplan's paradox. In fact, there is another new possible set-theoretical paradox hiding itself in Wittgenstein’s Tractatus. From the Tractatus’s Picture theory of language we can strictly infer the two contradictory propositions simultaneously: the world and the language are equinumerous; the world and the language are not equinumerous. I call this antinomy the world-language paradox. Based on a rigorous analysis of the Tractatus, with (...)
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  • Pluralism and “Bad” Mathematical Theories: Challenging Our Prejudices.Michèle Friend - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 277--307.
  • Wittgenstein on Incompleteness Makes Paraconsistent Sense.Francesco Berto - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 257--276.
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  • Proof Theory in Philosophy of Mathematics.Andrew Arana - 2010 - Philosophy Compass 5 (4):336-347.
    A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.
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  • Plurals.Agustín Rayo - 2007 - Philosophy Compass 2 (3):411–427.
    Forthcoming in Philosophical Compass. I explain why plural quantifiers and predicates have been thought to be philosophically significant.
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  • What Are Sets and What Are They For?Alex Oliver & Timothy Smiley - 2006 - Philosophical Perspectives 20 (1):123–155.
  • Foundations of Applied Mathematics I.Jeffrey Ketland - 2021 - Synthese 199 (1-2):4151-4193.
    This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: ZFCAσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathsf {ZFCA}_{\sigma }$$\end{document} with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents.
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  • Sets, Lies, and Analogy: A New Methodological Take.Giulia Terzian - 2020 - Philosophical Studies 178 (9):2759-2784.
    The starting point of this paper is a claim defended most famously by Graham Priest: that given certain observed similarities between the set-theoretic and the semantic paradoxes, we should be looking for a ‘uniform solution’ to the members of both families. Despite its indisputable surface attractiveness, I argue that this claim hinges on a problematic reasoning move. This is seen most clearly, I suggest, when the claim and its underlying assumptions are examined by the lights of a novel, quite general (...)
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  • Why Believe Infinite Sets Exist?Andrei Mărăşoiu - 2018 - Axiomathes 28 (4):447-460.
    The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s :481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of real sets, (...)
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  • Transfinite Recursion and Computation in the Iterative Conception of Set.Benjamin Rin - 2015 - Synthese 192 (8):2437-2462.
    Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative (...)
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  • Philosophy of Mathematics.Leon Horsten - 2008 - Stanford Encyclopedia of Philosophy.
    If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...)
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  • Understanding Programming Languages.Raymond Turner - 2007 - Minds and Machines 17 (2):203-216.
    We document the influence on programming language semantics of the Platonism/formalism divide in the philosophy of mathematics.
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  • How to Be a Minimalist About Sets.Luca Incurvati - 2012 - Philosophical Studies 159 (1):69-87.
    According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, (...)
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  • The Graph Conception of Set.Luca Incurvati - 2014 - Journal of Philosophical Logic 43 (1):181-208.
    The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA (...)
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  • Rigour, Proof and Soundness.Oliver M. W. Tatton-Brown - 2020 - Dissertation, University of Bristol
    The initial motivating question for this thesis is what the standard of rigour in modern mathematics amounts to: what makes a proof rigorous, or fail to be rigorous? How is this judged? A new account of rigour is put forward, aiming to go some way to answering these questions. Some benefits of the norm of rigour on this account are discussed. The account is contrasted with other remarks that have been made about mathematical proof and its workings, and is tested (...)
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  • The Limits of Classical Mereology: Mixed Fusions and the Failures of Mereological Hybridism.Joshua Kelleher - 2020 - Dissertation, The University of Queensland
    In this thesis I argue against unrestricted mereological hybridism, the view that there are absolutely no constraints on wholes having parts from many different logical or ontological categories, an exemplar of which I take to be ‘mixed fusions’. These are composite entities which have parts from at least two different categories – the membered (as in classes) and the non-membered (as in individuals). As a result, mixed fusions can also be understood to represent a variety of cross-category summation such as (...)
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  • Arithmetic, Set Theory, Reduction and Explanation.William D’Alessandro - 2018 - Synthese 195 (11):5059-5089.
    Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...)
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  • Quantification and Paradox.Edward Ferrier - 2018 - Dissertation, University of Massachusetts Amherst
    I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...)
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  • The Indifference Principle, its Paradoxes and Kolmogorov's Probability Space.Dan D. November - unknown
    In this paper I show that the validity of the Indifference Principle in light of its related paradoxes, is still an open question. I do so by offering an analysis of IP and its related paradoxes in the way they are manifested within the framework of Kolmogorov's probability theory. I describe the conditions that any mathematical formalization of IP must satisfy. Consequently, I show that IP's mathematical formalization has to be a set of constrains on probability spaces which mathematically describe (...)
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  • Antireductionism and Ordinals.Beau Madison Mount - 2019 - Philosophia Mathematica 27 (1):105-124.
    I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim (...)
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  • The Origin of the Theory of Types.Ryo Ito - 2018 - Annals of the Japan Association for Philosophy of Science 27:27-44.
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  • Bi-Modal Naive Set Theory.John Wigglesworth - 2018 - Australasian Journal of Logic 15 (2):139-150.
    This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators. We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some (...)
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  • Set-Theoretic Dependence.John Wigglesworth - 2015 - Australasian Journal of Logic 12 (3):159-176.
    In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible (...)
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  • Platitudes in Mathematics.Thomas Donaldson - 2015 - Synthese 192 (6):1799-1820.
    The term ‘continuous’ in real analysis wasn’t given an adequate formal definition until 1817. However, important theorems about continuity were proven long before that. How was this possible? In this paper, I introduce and refine a proposed answer to this question, derived from the work of Frank Jackson, David Lewis and other proponents of the ‘Canberra plan’. In brief, the proposal is that before 1817 the meaning of the term ‘continuous’ was determined by a number of ‘platitudes’ which had some (...)
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  • Against the Iterative Conception of Set.Edward Ferrier - 2019 - Philosophical Studies 176 (10):2681-2703.
    According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...)
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  • Composition and Identities.Manuel Lechthaler - 2017 - Dissertation, University of Otago
    Composition as Identity is the view that an object is identical to its parts taken collectively. I elaborate and defend a theory based on this idea: composition is a kind of identity. Since this claim is best presented within a plural logic, I develop a formal system of plural logic. The principles of this system differ from the standard views on plural logic because one of my central claims is that identity is a relation which comes in a variety of (...)
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  • Ontological Commitment.Daniel Durante Pereira Alves - 2018 - AL-Mukhatabat 1 (27):177-223.
    Disagreement over what exists is so fundamental that it tends to hinder or even to block dialogue among disputants. The various controversies between believers and atheists, or realists and nominalists, are only two kinds of examples. Interested in contributing to the intelligibility of the debate on ontology, in 1939 Willard van Orman Quine began a series of works which introduces the notion of ontological commitment and proposes an allegedly objective criterion to identify the exact conditions under which a theoretical discourse (...)
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  • Qual a motivação para se defender uma teoria causal da memória?César Schirmer Dos Santos - 2018 - In Juliano Santos do Carmo & Rogério F. Saucedo Corrêa (eds.), Linguagem e cognição. Pelotas: NEPFil. pp. 63-89.
    Este texto tem como objetivo apresentar a principal motivação filosófica para se defender uma teoria causal da memória, que é explicar como pode um evento que se deu no passado estar relacionado a uma experiência mnêmica que se dá no presente. Para tanto, iniciaremos apresentando a noção de memória de maneira informal e geral, para depois apresentar elementos mais detalhados. Finalizamos apresentando uma teoria causal da memória que se beneficia da noção de veritação (truthmaking).
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  • Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  • Abstraction and Additional Nature.Bob Hale & Crispin Wright - 2008 - Philosophia Mathematica 16 (2):182-208.
    What is wrong with abstraction’, Michael Potter and Peter Sullivan explain a further objection to the abstractionist programme in the foundations of mathematics which they first presented in their ‘Hale on Caesar’ and which they believe our discussion in The Reason's Proper Study misunderstood. The aims of the present note are: To get the character of this objection into sharper focus; To explore further certain of the assumptions—primarily, about reference-fixing in mathematics, about certain putative limitations of abstractionist set theory, and (...)
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