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The Foundations of Arithmetic

Evanston: Ill., Northwestern University Press (1884/1950)

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  1. What Frege Asked Alex the Parrot: Inferentialism, Number Concepts, and Animal Cognition.Erik Nelson - 2019 - Philosophical Psychology 33 (2):206-227.
    While there has been significant philosophical debate on whether nonlinguistic animals can possess conceptual capabilities, less time has been devoted to considering 'talking' animals, such as parrots. When they are discussed, their capabilities are often downplayed as mere mimicry. The most explicit philosophical example of this can be seen in Brandom's frequent comparisons of parrots and thermostats. Brandom argues that because parrots (like thermostats) cannot grasp the implicit inferential connections between concepts, their vocal articulations do not actually have any conceptual (...)
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  • La Fundamentación Filosófica de Los Principios No-Empíricos de Investigación.Sergio H. Menna - 2004 - Principia: An International Journal of Epistemology 8 (1):55-83.
    Non-empirical principles have always been a subject of interest of philoso-phers. Authors from different times and traditions agree that principles such as analogy or simplicity are present in the scientific practice. The disagreement comes out when these authors affirm that these principles have an epistemic function, and when they try to present reasons in order to found this state-ment. The first goal of this paper is to describe these principles and to point out their methodological importance. The second goal is (...)
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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.
  • The Serpent's Trail: William James, Object‐Oriented Programming, and Critical Realism.Larry J. Crockett - 2012 - Zygon 47 (2):388-414.
    Pragmatism has played only a small role in the half century and more of the science‐and‐religion dialogue, in part because pragmatism was at a low ebb in the 1950s. Even though Jamesean pragmatism in particular is experiencing a resurgence, owing partly to the work of Rorty and Putnam, it remains inconspicuous in the dialogue. Excepting artificial intelligence and artificial life, computer science also has not played a large role in the dialogue. Recent research into the foundations of object‐oriented programming, however, (...)
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  • Interpreting the Personal: Expression and the Formation of Feelings.Naomi Scheman - 2000 - Philosophical Review 109 (1):118.
    One of Adrian Piper’s “reactive guerrilla performances” dealing with issues of race and racism was a calling card that she handed out to individuals who made racist remarks that they would not have made if they had taken themselves to be in the presence of a person of color. The card reads.
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  • Mathematics as a Science of Patterns. [REVIEW]Mark Steiner - 2000 - Philosophical Review 109 (1):115-118.
    For the past hundred years, mathematics, for its own reasons, has been shifting away from the study of “mathematical objects” and towards the study of “structures”. One would have expected philosophers to jump onto the bandwagon, as in many other cases, to proclaim that this shift is no accident, since mathematics is “essentially” about structures, not objects. In fact, structuralism has not been a very popular philosophy of mathematics, probably because of the hostility of Frege and other influential logicists, and (...)
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  • El Tractatus al rescate de Principia Mathematica: Ramsey y los fundamentos logicistas de las matemáticas.Emilio Méndez Pinto - 2022 - Critica 54 (161):43-69.
    Mi objetivo es discutir las principales dificultades que Frank P. Ramsey encontró en Principia Mathematica y la solución que, vía el Tractatus Logico-Philosophicus, propuso al respecto. Sostengo que las principales dificultades que Ramsey encontró en Principia Mathematica están, todas, relacionadas con que Russell y Whitehead desatendieron la forma lógica de las proposiciones matemáticas, las cuales, según Ramsey, deben ser tautológicas.
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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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  • Pragmatic a Priori Knowledge: A Pragmatic Approach to the Nature and Object of What Can Be Known Independently of Experience.Lauri Järvilehto - 2011 - Jyväskylä University Printing House.
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  • Logical Indefinites.Jack Woods - 2014 - Logique Et Analyse -- Special Issue Edited by Julien Murzi and Massimiliano Carrara 227: 277-307.
    I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.
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  • Neo-Fregean Ontology.Matti Eklund - 2006 - Philosophical Perspectives 20 (1):95-121.
    Neo-Fregeanism in the philosophy of mathematics consists of two main parts: the logicist thesis, that mathematics (or at least branches thereof, like arithmetic) all but reduce to logic, and the platonist thesis, that there are abstract, mathematical objects. I will here focus on the ontological thesis, platonism. Neo-Fregeanism has been widely discussed in recent years. Mostly the discussion has focused on issues specific to mathematics. I will here single out for special attention the view on ontology which underlies the neo-Fregeans’ (...)
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  • Essential Properties - Analysis and Extension.Nathan Wildman - 2011 - Dissertation, Cambridge
  • Anti-exceptionalism about logic as tradition rejection.Ben Martin & Ole Thomassen Hjortland - 2022 - Synthese 200 (2):1-33.
    While anti-exceptionalism about logic is now a popular topic within the philosophy of logic, there’s still a lack of clarity over what the proposal amounts to. currently, it is most common to conceive of AEL as the proposal that logic is continuous with the sciences. Yet, as we show here, this conception of AEL is unhelpful due to both its lack of precision, and its distortion of the current debates. Rather, AEL is better understood as the rejection of certain traditional (...)
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  • Evidence in Logic.Ben Martin & Ole Thomassen Hjortland - forthcoming - In Maria Lasonen-Aarnio & Clayton M. Littlejohn (eds.), Routledge Handbook of the Philosophy of Evidence. Routledge.
    The historical consensus is that logical evidence is special. Whereas empirical evidence is used to support theories within both the natural and social sciences, logic answers solely to a priori evidence. Further, unlike other areas of research that rely upon a priori evidence, such as mathematics, logical evidence is basic. While we can assume the validity of certain inferences in order to establish truths within mathematics and test scientifi c theories, logicians cannot use results from mathematics or the empirical sciences (...)
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  • Neo-Fregeanism Naturalized: The Role of One-to-One Correspondence in Numerical Cognition.Lieven Decock - 2008 - Behavioral and Brain Sciences 31 (6):648-649.
    Rips et al. argue that the construction of math schemas roughly similar to the Dedekind/Peano axioms may be necessary for arriving at arithmetical skills. However, they neglect the neo-Fregean alternative axiomatization of arithmetic, based on Hume's principle. Frege arithmetic is arguably a more plausible start for a top-down approach in the psychological study of mathematical cognition than Peano arithmetic.
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  • The Entanglement of Logic and Set Theory, Constructively.Laura Crosilla - 2022 - Inquiry: An Interdisciplinary Journal of Philosophy 65 (6).
    ABSTRACT Theories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. In (...)
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  • Abstraction Relations Need Not Be Reflexive.Jonathan Payne - 2013 - Thought: A Journal of Philosophy 2 (2):137-147.
    Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this article I argue that abstraction relations need not be reflexive. I furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles.
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  • A Refutation of Frege’s Context Principle?Aaron Barth - 2012 - Thought: A Journal of Philosophy 1 (1):26-35.
    This paper explores the limitations of current empirical approaches to the philosophy of language in light of a recent criticism of Frege's context principle. According to this criticism, the context principle is in conflict with certain features of natural language use and this is held to undermine its application in Foundations of Arithmetic. I argue that this view is mistaken because the features with which the context principle is alleged to be in conflict are irrelevant to the principle's methodological significance (...)
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  • Meaning and Metaphysical Necessity.Tristan Grøtvedt Haze - 2022 - New York: Routledge.
    This book is about the idea that some true statements would have been true no matter how the world had turned out, while others could have been false. It develops and defends a version of the idea that we tell the difference between these two types of truths in part by reflecting on the meanings of words. It has often been thought that modal issues—issues about possibility and necessity—are related to issues about meaning. In this book, the author defends the (...)
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  • Calculus as Method or Calculus as Rules? Boole and Frege on the Aims of a Logical Calculus.Dirk Schlimm & David Waszek - 2021 - Synthese 199 (5-6):11913-11943.
    By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the other hand, (...)
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  • Philosophy of Language in the Twentieth Century.Jason Stanley - 2008 - In Dermot Moran (ed.), The Routledge Companion to Twentieth Century Philosophy. Routledge. pp. 382-437.
    In the Twentieth Century, Logic and Philosophy of Language are two of the few areas of philosophy in which philosophers made indisputable progress. For example, even now many of the foremost living ethicists present their theories as somewhat more explicit versions of the ideas of Kant, Mill, or Aristotle. In contrast, it would be patently absurd for a contemporary philosopher of language or logician to think of herself as working in the shadow of any figure who died before the Twentieth (...)
     
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  • Intentional Psychologism.David Pitt - 2009 - Philosophical Studies 146 (1):117-138.
    In the past few years, a number of philosophers ; Horgan and Tienson 2002; Pitt 2004) have maintained the following three theses: there is a distinctive sort of phenomenology characteristic of conscious thought, as opposed to other sorts of conscious mental states; different conscious thoughts have different phenomenologies; and thoughts with the same phenomenology have the same intentional content. The last of these three claims is open to at least two different interpretations. It might mean that the phenomenology of a (...)
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  • Identifying Logical Evidence.Ben Martin - 2020 - Synthese 198 (10):9069-9095.
    Given the plethora of competing logical theories of validity available, it’s understandable that there has been a marked increase in interest in logical epistemology within the literature. If we are to choose between these logical theories, we require a good understanding of the suitable criteria we ought to judge according to. However, so far there’s been a lack of appreciation of how logical practice could support an epistemology of logic. This paper aims to correct that error, by arguing for a (...)
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  • Logicism Revisited.Otávio Bueno - 2001 - Principia 5 (1-2):99-124.
    In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a cruciai principie in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist approach beyond arithmetic. Finally, I argue (...)
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  • Representationalism and Sensory Modalities: An Argument for Intermodal Representationalism.David Bourget - 2017 - American Philosophical Quarterly 54 (3):251-268.
    Intermodal representationalists hold that the phenomenal characters of experiences are fully determined by their contents. In contrast, intramodal representationalists hold that the phenomenal characters of experiences are determined by their contents together with their intentional modes or manners of representation, which are nonrepresentational features corresponding roughly to the sensory modalities. This paper discusses a kind of experience that provides evidence for an intermodal representationalist view: intermodal experiences, experiences that unify experiences in different modalities. I argue that such experiences are much (...)
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  • Arguing on the Toulmin Model: New Essays in Argument Analysis and Evaluation.David Hitchcock & Bart Verheij (eds.) - 2006 - Dordrecht, Netherland: Springer.
    In The Uses of Argument, Stephen Toulmin proposed a model for the layout of arguments: claim, data, warrant, qualifier, rebuttal, backing. Since then, Toulmin’s model has been appropriated, adapted and extended by researchers in speech communications, philosophy and artificial intelligence. This book assembles the best contemporary reflection in these fields, extending or challenging Toulmin’s ideas in ways that make fresh contributions to the theory of analysing and evaluating arguments.
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  • The Bounds of Logic: A Generalized Viewpoint.Gila Sher - 1991 - MIT Press.
    The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...)
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  • A concept and its structures. Methodological analysis.Vladimir Kuznetsov (ed.) - 1997 - Institute of philosophy.
    The triplet model treats a concept as complex structure that expresses three kinds of information. The first is about entities subsumed under a concept,their properties and relations. The second is about means and ways of representing the first information in intelligent systems. The third is about linkage between the first and second ones and methods of its constructing. The application of triplet models to generalization and development of concept models in philosophy, logic, cognitive psychology, cognitive science, linguistics, artificial intelligence has (...)
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  • Ser, existencia y facticidad.David Torrijos Castrillejo - 2021 - In Manuel Alejandro Serra Pérez (ed.), La cualidad metafísica del ser respecto a la forma: Estudio de la crítica de Lawrence Dewan a Étienne Gilson. Pamplona: EUNSA. pp. 93-108.
    This paper tries to understand existence as identified with Aquinas' esse in a right way. The author distinguishes among facticity and existence, because facticity it a mere negative understanding of existence as the negation of nothing. But existence is not only a negative thing, but a positive one, therefore it is not still described by this negative definition. Existence includes facticity but is not only facticity. Existence is the actuality of essence and that description does not diminish Aquinas' understanding of (...)
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  • Understanding, Justification and the a Priori.David Hunter - 1997 - Philosophical Studies 87 (2):119-141.
    What I wish to consider here is how understanding something is related to the justification of beliefs about what it means. Suppose, for instance, that S understands the name “Clinton” and has a justified belief that it names Clinton. How is S’s understanding related to that belief’s justification? Or suppose that S understands the sentence “Clinton is President”, or Jones’ assertive utterance of it, and has a justified belief that that sentence expresses the proposition that Clinton is President, or that (...)
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  • The Three Circles of Consciousness.Uriah Kriegel - forthcoming - In M. Guillot & M. Garcia-Carpintero (eds.), Self-Experience: Essays on Inner Awareness. Oxford University Press.
    A widespread assumption in current philosophy of mind is that a conscious state’s phenomenal properties vary with its representational contents. In this paper, I present (rather dogmatically) an alternative picture that recognizes two kinds of phenomenal properties that do not vary concomitantly with content. First, it admits phenomenal properties that vary rather with attitude: what it is like for me to see rain is phenomenally different from what it is like for me to remember (indistinguishable) rain, which is different again (...)
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  • Anti-exceptionalism and the justification of basic logical principles.Matthew Carlson - 2022 - Synthese 200 (3):1-19.
    Anti-exceptionalism about logic is the thesis that logic is not special. In this paper, I consider, and reject, a challenge to this thesis. According to this challenge, there are basic logical principles, and part of what makes such principles basic is that they are epistemically exceptional. Thus, according to this challenge, the existence of basic logical principles provides reason to reject anti-exceptionalism about logic. I argue that this challenge fails, and that the exceptionalist positions motivated by it are thus unfounded. (...)
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  • Mental Representation.David Pitt - 2020 - Stanford Encyclopedia of Philosophy.
    The notion of a "mental representation" is, arguably, in the first instance a theoretical construct of cognitive science. As such, it is a basic concept of the Computational Theory of Mind, according to which cognitive states and processes are constituted by the occurrence, transformation and storage (in the mind/brain) of information-bearing structures (representations) of one kind or another.
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  • Questions in Action.Daniel Hoek - 2022 - Journal of Philosophy 119 (3):113-143.
    Choices confront us with questions. How we act depends on our answers to those questions. So the way our beliefs guide our choices is not just a function of their informational content, but also depends systematically on the questions those beliefs address. This paper gives a precise account of the interplay between choices, questions and beliefs, and harnesses this account to obtain a principled approach to the problem of deduction. The result is a novel theory of belief-guided action that explains (...)
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  • Frege on Existence and Non‐Existence.Karen Green - 2015 - Theoria 81 (4):293-310.
    Despite its importance for early analytic philosophy, Gottlob Frege's account of existence statements, according to which they classify concepts, has been thought to succumb to a number of well-worn criticisms. This article does two things. First, it argues that, by remaining faithful to the letter of Frege's claim that concepts are functions, the Fregean account can be saved from many of the standard criticisms. Second, it examines the problem that Frege's account fails to generalize to cases which involve definite descriptions (...)
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  • Logic, Logicism, and Intuitions in Mathematics.Besim Karakadılar - 2001 - Dissertation, Middle East Technical University
    In this work I study the main tenets of the logicist philosophy of mathematics. I deal, basically, with two problems: (1) To what extent can one dispense with intuition in mathematics? (2) What is the appropriate logic for the purposes of logicism? By means of my considerations I try to determine the pros and cons of logicism. My standpoint favors the logicist line of thought. -/- .
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  • Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method.Carlo Cellucci - 2013 - Dordrecht, Netherland: Springer.
    This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...)
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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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  • Dedekind and Wolffian Deductive Method.José Ferreirós & Abel Lassalle-Casanave - forthcoming - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie:1-21.
    Dedekind’s methodology, in his classic booklet on the foundations of arithmetic, has been the topic of some debate. While some authors make it closely analogue to Hilbert’s early axiomatics, others emphasize its idiosyncratic features, most importantly the fact that no axioms are stated and its careful deductive structure apparently rests on definitions alone. In particular, the so-called Dedekind “axioms” of arithmetic are presented by him as “characteristic conditions” in the definition of the complex concept of a simply infinite system. Making (...)
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  • Epistemology Versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf.P. Dybjer, Sten Lindström, Erik Palmgren & G. Sundholm (eds.) - 2012 - Dordrecht, Netherland: Springer.
    This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...)
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  • The Conceptual Structure of Reality.Gal Yehezkel - 2014 - Springer.
    This book describes a novel conception of reality, one that uniquely incorporates an idealistic view of existence with an account of objectivity. It introduces a general model of conceptual analysis and demonstrates its effectiveness in exposing and establishing the existence of conceptual ties. The book begins by introducing the tools and principles needed for the conceptual analysis undertaken in chapters that follow. Next, it presents a detailed examination into existence, contingency, idealism, self-consciousness and natural laws. In the process, the author (...)
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  • The Province of Conceptual Reason: Hegel's Post-Kantian Rationalism.William Clark Wolf - unknown
    In this dissertation, I seek to explain G.W.F. Hegel’s view that human accessible conceptual content can provide knowledge about the nature or essence of things. I call this view “Conceptual Transparency.” It finds its historical antecedent in the views of eighteenth century German rationalists, which were strongly criticized by Immanuel Kant. I argue that Hegel explains Conceptual Transparency in such a way that preserves many implications of German rationalism, but in a form that is largely compatible with Kant’s criticisms of (...)
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  • Approaching Infinity.Michael Huemer - 2016 - New York: Palgrave Macmillan.
    Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters. The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It (...)
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  • Elme És Evolúció.Bence Nanay - 2000 - Kávé..
  • The Nature of Truth.M. J. Frapolli - 2013 - Springer.
    The book offers a proposal on how to define truth in all its complexity, without reductionism, showing at the same time which questions a theory of truth has to answer and which questions, although related to truth, do not belong within the scope of such a theory. Just like any other theory, a theory of truth has its structure and limits. The semantic core of the position is that truth-ascriptions are pro-forms, i.e. natural language propositional variables. The book also offers (...)
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  • Before the Law.Mark Eli Kalderon - 2011 - Philosophical Issues 21 (1):219-244.
    Before the law sits a gatekeeper. To this gatekeeper comes a man from the country who asks to gain entry into the law. But the gatekeeper says that he cannot grant him entry at the moment. The man thinks about it and then asks if he will be allowed to come in sometime later on. “It is possible,” says the gatekeeper, “but not now.”—Franz Kafka..
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  • Linguistic Aspects, Meaninglessness and Paradox: A Rejoinder to John David Stone. [REVIEW]Laurence Goldstein - 1980 - Linguistics and Philosophy 4 (4):579 - 592.
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  • On Naturalizing the Epistemology of Mathematics.Jeffrey W. Roland - 2009 - Pacific Philosophical Quarterly 90 (1):63-97.
    In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.
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  • To Exist and to Count: A Note on the Minimalist View.Francesco Berto & Massimiliano Carrara - 2009 - Dialectica 63 (3):343-356.
    Sometimes mereologists have problems with counting. We often don't want to count the parts of maximally connected objects as full-fledged objects themselves, and we don't want to count discontinuous objects as parts of further, full-fledged objects. But whatever one takes "full-fledged object" to mean, the axioms and theorems of classical, extensional mereology commit us to the existence both of parts and of wholes – all on a par, included in the domain of quantification – and this makes mereology look counterintuitive (...)
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  • The Second-Order Property View of Existence.Joel Katzav - 2008 - Pacific Philosophical Quarterly 89 (4):486-496.
    Abstract: In this paper, I examine the current case against the second-order property view of existence through a discussion of Colin McGinn's up to date statement of this case. I conclude that the second-order property view of existence remains viable.
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