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The Principles of Mathematics

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  1. Russell and Jin Yuelin on Truth: A Comparative Study.Chen Bo - 2021 - Contemporary Chinese Thought 52 (1-2):43-78.
    Jin Yuelin’s logical and philosophical thought was deeply influenced by the philosophy of Bertrand Russell. The same influence existed also in the case of his view on truth, which was considerably close to the views maintained by Russell in his phase of logical atomism. In their investigations, Russell and Jin not only focused on similar topics, but also occupied similar philosophical positions, such as realism in the domain of ontology, empiricism in epistemology, and the correspondence theory in the domain of (...)
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  • Reversing Logical Nihilism.Tristan Grøtvedt Haze - forthcoming - Synthese.
    Gillian Russell has recently proposed counterexamples to such elementary argument forms as Conjunction Introduction (e.g. ‘Snow is white. Grass is green. Therefore, snow is white and grass is green’) and Identity (e.g. ‘Snow is white. Therefore, snow is white’). These purported counterexamples involve expressions that are sensitive to linguistic context—for example, a sentence which is true when it appears alone but false when embedded in a larger sentence. If they are genuine counterexamples, it looks as though logical nihilism—the view that (...)
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  • Chalmers and Semantics.Panu Raatikainen - 2021 - Theoria 87 (5):1193-1221.
    David Chalmers’ two-dimensionalism is an ambitious philosophical program that aims to “ground” or “construct” Fregean meanings and restore “the golden triangle” of apriority, necessity, and meaning that Kripke seemingly broke. This paper aims to examine critically what Chalmers’ theory can in reality achieve. It is argued that the theory faces severe challenges. There are some gaps in the overall arguments, and the reasoning is in some places somewhat circular. Chalmers’ theory is effectively founded on certain strong philosophical assumptions. It is (...)
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  • Epistemic Modality, Mind, and Mathematics.Hasen Khudairi - 2021 - Dissertation, University of St Andrews
    This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and (...)
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  • CRITIQUE OF IMPURE REASON: Horizons of Possibility and Meaning.Steven James Bartlett - 2021 - Salem, USA: Studies in Theory and Behavior.
    PLEASE NOTE: This is the corrected 2nd eBook edition, 2021.¶¶ _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly, technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost.¶¶ The printed edition was released on September 1, 2021 and is now available through all booksellers, (...)
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  • Modal Cognitivism and Modal Expressivism.Hasen Khudairi - manuscript
    This paper aims to provide a mathematically tractable background against which to model both modal cognitivism and modal expressivism. I argue that epistemic modal algebras, endowed with a hyperintensional truthmaker semantics, comprise a materially adequate fragment of the language of thought. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are dual. I examine, in particular, the virtues unique to the modal expressivist approach here proffered in the setting of (...)
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  • Closed Structure.Peter Fritz, Harvey Lederman & Gabriel Uzquiano - forthcoming - Journal of Philosophical Logic:1-43.
    According to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that (...)
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  • Duration Enough for Presentism.Robert E. Pezet - 2020 - Axiomathes 30 (4):391-421.
    This paper considers a problem for dynamic presentism that has received little attention: its apparent inability to accommodate the duration of events. After outlining the problem, I defend presentism from it. This defence proceeds in two stages. First, I argue the objection rests on a faulty assumption: that duration is temporal extension. The paper challenges that assumption on several different ways of conceiving of temporal extension. This is the negative case and forms the bulk of the paper. Second, after diagnosing (...)
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  • Structure by Proxy, with an Application to Grounding.Peter Fritz - 2019 - Synthese 198 (7):6045-6063.
    An argument going back to Russell shows that the view that propositions are structured is inconsistent in standard type theories. Here, it is shown that such type theories may nevertheless provide entities which can serve as proxies for structured propositions. As an illustration, such proxies are applied to the case of grounding, as standard views of grounding require a degree of propositional structure which suffices for a version of Russell’s argument. While this application solves some of the problems grounding faces, (...)
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  • Is Objectual Identity Really Dispensable?Eric Updike - 2019 - Erkenntnis 84 (4):761-782.
    Kai Wehmeier’s Wittgensteinian Predicate Logic is a formulation of first-order logic under the exclusive interpretation of the quantifiers. W-logic has a distinguished relation constant for co-reference but no sign for objectual identity. Wehmeier denies that objectual identity exists on the grounds that it cannot be a genuine binary relation. Fortunately W-logic is equi-expressive with standard first-order logic with identity and it appears that objectual identity is dispensable across the broader logical enterprise. This paper challenges the latter claim as objectual identity (...)
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  • Constitutive principles versus comprehensibility conditions in post-Kantian physics.Olivier Darrigol - 2020 - Synthese 197 (10):4571-4616.
    The relativistic revolution led to varieties of neo-Kantianism in which constitutive principles define the object of scientific knowledge in a domain-dependent and historically mutable manner. These principles are a priori insofar as they are necessary premises for the formulation of empirical laws in a given domain, but they lack the self-evidence of Kant’s a priori and they cannot be identified without prior knowledge of the theory they purport to frame. In contrast, the rationalist endeavors of a few masters of theoretical (...)
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  • Paradoxes and the Limits of Theorizing About Propositional Attitudes.Dustin Tucker - 2018 - Synthese 198 (Suppl 5):1075-1094.
    Propositions are central to at least most theorizing about the connection between our mental lives and the world: we use them in our theories of an array of attitudes including belief, desire, hope, fear, knowledge, and understanding. Unfortunately, when we press on these theories, we encounter a relatively neglected family of paradoxes first studied by Arthur Prior. I argue that these paradoxes present a fatal problem for most familiar resolutions of paradoxes. In particular, I argue that truth-value gap, contextualist, situation (...)
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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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  • The Singularity of Experiences and Thoughts.Alberto Voltolini - 2020 - Topoi 39 (2):459-473.
    Recently, various people have maintained that one must revise either the externalistically-based notion of singular thought or the naïve realism-inspired notion of relational particularity, as respectively applied to some thoughts and to some perceptual experiences. In order to do so, one must either provide a broader notion of singular thought or flank the notion of relational particularity with a broader notion of phenomenal particularity. I want to hold that there is no need of that revision. For the original notions can (...)
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  • Paradoks znatljivosti iz raslovske perspektive.Pierdaniele Giaretta - 2009 - Prolegomena 8 (2):141-158.
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  • On Dedekind's Logicism.José Ferreirós - unknown
    The place of Richard Dedekind in the history of logicism is a controversial matter. The conception of logic incorporated in his work is certainly old-fashioned, in spite of innovative elements that would play an important role in late 19th and early 20th century discussions. Yet his understanding of logic and logicism remains of interest for the light it throws upon the development of modern logic in general, and logicist views of the foundations of mathematics in particular. The paper clarifies Dedekind's (...)
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  • Ontological Motivation in Obtaining Certain Quantum Equations: A Case for Panexperientialism.Villacrés Juan - unknown
    In this work I argue for the existence of an ontological state in which no entity in it can be more basic than the others in such a state. This is used to provide conceptual justification for a method that is applied to obtain the Schr\"{o}dinger equation, the Klein-Gordon equation, and the Klein-Gordon equation for a particle in an electromagnetic field. Additionally, it is argued that the existence of such state is incompatible with indirect realism; and the discussion suggests that (...)
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  • If Structured Propositions Are Logical Procedures Then How Are Procedures Individuated?Marie Duží - 2019 - Synthese 196 (4):1249-1283.
    This paper deals with two issues. First, it identifies structured propositions with logical procedures. Second, it considers various rigorous definitions of the granularity of procedures, hence also of structured propositions, and comes out in favour of one of them. As for the first point, structured propositions are explicated as algorithmically structured procedures. I show that these procedures are structured wholes that are assigned to expressions as their meanings, and their constituents are sub-procedures occurring in executed mode. Moreover, procedures are not (...)
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  • The Prenective View of Propositional Content.Robert Trueman - 2018 - Synthese 195 (4):1799-1825.
    Beliefs have what I will call ‘propositional content’. A belief is always a belief that so-and-so: a belief that grass is green, or a belief that snow is white, or whatever. Other things have propositional content too, such as sentences, judgments and assertions. The Standard View amongst philosophers is that what it is to have a propositional content is to stand in an appropriate relation to a proposition. Moreover, on this view, propositions are objects, i.e. the kind of thing you (...)
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  • Ontology of Time and Hyperdynamism.Erwin Tegtmeier - 2009 - Metaphysica 10 (2):185-198.
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  • The Criterion or Criteria of Change.Xiaoqiang Han - 2009 - Metaphysica 10 (2):149-156.
    In this paper, I offer an examination of the two existing criteria of change, one indicated, implicitly, by Aristotle and the other proposed, quite formally, by Russell. Both criteria engender problems. While the Aristotelian criterion is both too narrow and too broad, as it includes bogus changes and excludes subjectless changes, the Russellian criterion avoids the distinction between genuine changes and bogus changes completely. The aim of the paper is to address these problems and to show how these two existing (...)
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  • Russell and His Sources for Non-Classical Logics.Irving H. Anellis - 2009 - Logica Universalis 3 (2):153-218.
    My purpose here is purely historical. It is not an attempt to resolve the question as to whether Russell did or did not countenance nonclassical logics, and if so, which nonclassical logics, and still less to demonstrate whether he himself contributed, in any manner, to the development of nonclassical logic. Rather, I want merely to explore and insofar as possible document, whether, and to what extent, if any, Russell interacted with the various, either the various candidates or their, ideas that (...)
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  • John Cook Wilson.Mathieu Marion - 2010 - Stanford Encyclopedia of Philosophy.
    John Cook Wilson (1849–1915) was Wykeham Professor of Logic at New College, Oxford and the founder of ‘Oxford Realism’, a philosophical movement that flourished at Oxford during the first decades of the 20th century. Although trained as a classicist and a mathematician, his most important contribution was to the theory of knowledge, where he argued that knowledge is factive and not definable in terms of belief, and he criticized ‘hybrid’ and ‘externalist’ accounts. He also argued for direct realism in perception, (...)
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  • The Number of Senses.Kevin C. Klement - 2003 - Erkenntnis 58 (3):303 - 323.
    Many philosophers still countenance senses or meanings in the broadly Fregean vein.However, it is difficult to posit the existence of senses without positing quite a lot ofthem, including at least one presenting every entity in existence. I discuss a number ofCantorian paradoxes that seem to result from an overly large metaphysics of senses, and various possible solutions. Certain more deflationary and non-traditional understandings of senses, and to what extent they fare better in solving the problems, are also discussed. In the (...)
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  • Brandom, Hegel and Inferentialism.Tom Rockmore - 2002 - International Journal of Philosophical Studies 10 (4):429 – 447.
    In the course of developing a semantics with epistemological intent, Brandom claims that his inferentialism is Hegelian. This paper argues that, even on a charitable reading, Brandom is an anti-Hegelian.
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  • If It Ain’T Moving It Shall Not Be Moved.Emiliano Boccardi - 2015 - Topoi 34 (1):171-185.
    There are two no-change objections that can be raised against the B-theory of time. One stems from the observation that in a B-theoretic scenario changes of determinations can only be represented by propositions which have eternal truth values. The other derives from the principle that nothing can vary over a period of time if it doesn’t instantiate a state of change at all the instants of time which compose it. Here I argue that both objections apply to all comparative conceptions (...)
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  • What is the Problem of Non-Existence?Tim Crane - 2012 - Philosophia 40 (3):417-434.
    It is widely held that there is a problem of talking about or otherwise representing things that not exist. But what exactly is this problem? This paper presents a formulation of the problem in terms of the conflict between the fact that there are truths about non-existent things and the fact that truths must be answerable to reality, how things are. Given this, the problem of singular negative existential statements is no longer the central or most difficult aspect of the (...)
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  • Sorites.Bertil Rolf - 1984 - Synthese 58 (2):219 - 250.
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  • To Be is to Be the Object of a Possible Act of Choice.Massimiliano Carrara & Enrico Martino - 2010 - Studia Logica 96 (2):289-313.
    Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference . Plural quantification will be then explained in terms of plural reference. As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite (...)
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  • Causal Slingshots.Michael Baumgartner - 2010 - Erkenntnis 72 (1):111-133.
    Causal slingshots are formal arguments advanced by proponents of an event ontology of token-level causation which, in the end, are intended to show two things: (i) The logical form of statements expressing causal dependencies on token level features a binary predicate ‘‘... causes ...’’ and (ii) that predicate takes events as arguments. Even though formalisms are only revealing with respect to the logical form of natural language statements, if the latter are shown to be adequately captured within a corresponding formalism, (...)
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  • Farewell to McTaggart’s Argument?Michael Tooley - 2010 - Philosophia 38 (2):243-255.
    Philosophers have responded to McTaggart’s famous argument for the unreality of time in a variety of ways. Some of those responses are not easy to evaluate, since they involve, for example, sometimes murky questions concerning whether a certain infinite regress is or is not vicious. In this paper I set out a response that has not, I think, been advanced by any other author, and which, if successful, is absolutely clear-cut. The basic idea is simply that a tensed approach to (...)
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  • A Repair of Frege’s Theory of Thoughts.Mark Textor - 2009 - Synthese 167 (1):105 - 123.
    Frege’s writings contain arguments for the thesis (i) that a thought expressed by a sentence S is a structured object whose composition pictures the composition of S, and for the thesis (ii) that a thought is an unstructured object. I will argue that Frege’s reasons for both (i) and (ii) are strong. Frege’s explanation of the difference in sense between logically equivalent sentences rests on assumption (i), while Frege’s claim that the same thought can be decomposed differently makes (ii) plausible. (...)
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  • Hilbert, Logicism, and Mathematical Existence.José Ferreirós - 2009 - Synthese 170 (1):33 - 70.
    David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...)
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  • The Foundations of Kaplan's Direct Reference Theory for Demonstratives.Lawrence D. Roberts - 1994 - Philosophia 23 (1-4):91-116.
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  • Chains of Life: Turing, Lebensform, and the Emergence of Wittgenstein’s Later Style.Juliet Floyd - 2016 - Nordic Wittgenstein Review 5 (2):7-89.
    This essay accounts for the notion of _Lebensform_ by assigning it a _logical _role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the _PI_ occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise _The Brown Book_. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later (...)
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  • How Wittgenstein Defeated Russell’s Multiple Relation Theory of Judgment.Peter W. Hanks - 2007 - Synthese 154 (1):121 - 146.
    In 1913 Wittgenstein raised an objection to Russell’s multiple relation theory of judgment that eventually led Russell to abandon his theory. As he put it in the Tractatus, the objection was that “the correct explanation of the form of the proposition, ‘A makes the judgement p’, must show that it is impossible for a judgement to be a piece of nonsense. (Russell’s theory does not satisfy this requirement,” (5.5422). This objection has been widely interpreted to concern type restrictions on the (...)
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  • Literature Survey: Recent Publications in the History and Philosophy of Mathematics From the Renaissance to Berkeley. [REVIEW]Paolo Mancosu - 1999 - Metascience 8 (1):102-124.
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  • Structure and Transition: Towards an Accretivist Theory of Time.David Preston Taylor - unknown
    This dissertation is a defense of a particular theory of the metaphysics of time which I call "accretivism", but which is popularly known in a form usually called the "Growing Block Theory". The goal of a metaphysics of time is to incorporate the various aspects of our temporal experience into a single, comprehensive whole. To this end I delineate five aspects of our ordinary experience of time: 1) The Tensed Aspect, in virtue of which objects are presented to us as (...)
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  • The Ontology of Intentionality II: Dependence Ontology as Prolegomenon to Noetic Modal Semantics.Gilbert T. Null - 2007 - Husserl Studies 23 (2):119-159.
    This is the second in a sequence of three essays which axiomatize and apply Edmund Husserl's dependence ontology of parts and wholes as a non-Diodorean, non-Kantian temporal semantics for first-order predicate modal languages. The Ontology of Intentionality I introduced enough of Husserl's dependence-ontology of parts and wholes to formulate his account of order as effected by relating moments of unity, and The Ontology of Intentionality II extends that axiomatic dependence-ontology far enough to enable its semantic application. Formalizing the compatibility [Vereinbarkeit] (...)
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  • Hertz and Wittgenstein's Philosophy of Science.Peter C. Kjaergaard - 2002 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 33 (1):121-149.
    The German physicist Heinrich Hertz played a decisive role for Wittgenstein's use of a unique philosophical method. Wittgenstein applied this method successfully to critical problems in logic and mathematics throughout his life. Logical paradoxes and foundational problems including those of mathematics were seen as pseudo-problems requiring clarity instead of solution. In effect, Wittgenstein's controversial response to David Hilbert and Kurt Gödel was deeply influenced by Hertz and can only be fully understood when seen in this context. To comprehend the arguments (...)
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  • Jean van Heijenoort’s Conception of Modern Logic, in Historical Perspective.Irving H. Anellis - 2012 - Logica Universalis 6 (3-4):339-409.
    I use van Heijenoort’s published writings and manuscript materials to provide a comprehensive overview of his conception of modern logic as a first-order functional calculus and of the historical developments which led to this conception of mathematical logic, its defining characteristics, and in particular to provide an integral account, from his most important publications as well as his unpublished notes and scattered shorter historico-philosophical articles, of how and why the mathematical logic, whose he traced to Frege and the culmination of (...)
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  • Toward A Visual Proof System: Lewis Carroll’s Method of Trees.Francine F. Abeles - 2012 - Logica Universalis 6 (3-4):521-534.
    In the period 1893–1897 Charles Dodgson, writing as Lewis Carroll, published two books and two articles on logic topics. Manuscript material first published in 1977 together with letters and diary entries provide evidence that he was working toward a visual proof system for complex syllogistic propositional logic based on a mechanical tree method that he devised.
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  • Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament.Anssi Korhonen - 2012 - Logica Universalis 6 (3-4):597-613.
    Since its publication in 1967, van Heijenoort’s paper, “Logic as Calculus and Logic as Language” has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege’s Begriffsschrift (1879) to the work of Herbrand, Gödel and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on (...)
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  • Russell on Spinoza’s Substance Monism.Pierfrancesco Basile - 2012 - Metaphysica 13 (1):27-41.
    Russell’s critique of substance monism is an ideal starting point from which to understand some main concepts in Spinoza’s difficult metaphysics. This paper provides an in-depth examination of Spinoza’s proof that only one substance exists. On this basis, it rejects Russell’s interpretation of Spinoza’s theory of reality as founded upon the logical doctrine that all propositions consist of a predicate and a subject. An alternative interpretation is offered: Spinoza’s substance is not a bearer of properties, as Russell implied, but an (...)
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  • The Bearable Lightness of Being (Vol 20, Pg 399, 2010).Bob Hale - 2011 - Axiomathes 21 (4):597 - 597.
    How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories—such as object , property , and relation —are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical (...)
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  • On Fundamental Differences Between Dependent and Independent Meanings.Claire Ortiz Hill - 2010 - Axiomathes 20 (2-3):313-332.
    In “Function and Concept” and “On Concept and Object”, Frege argued that certain differences between dependent and independent meanings were inviolable and “founded deep in the nature of things” but, in those articles, he was not explicit about the actual consequences of violating such differences. However, since by creating a law that permitted one to pass from a concept to its extension, he himself mixed dependent and independent meanings, we are in a position to study some of the actual consequences (...)
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  • What is a Line?D. F. M. Strauss - 2014 - Axiomathes 24 (2):181-205.
    Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...)
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  • What Are Groups?Katherine Ritchie - 2013 - Philosophical Studies 166 (2):257-272.
    In this paper I argue for a view of groups, things like teams, committees, clubs and courts. I begin by examining features all groups seem to share. I formulate a list of six features of groups that serve as criteria any adequate theory of groups must capture. Next, I examine four of the most prominent views of groups currently on offer—that groups are non-singular pluralities, fusions, aggregates and sets. I argue that each fails to capture one or more of the (...)
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  • Frege’s Logicism and the Neo-Fregean Project.Matthias Schirn - 2014 - Axiomathes 24 (2):207-243.
    Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...)
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  • On the Exhaustion of Mathematical Entities by Structures.Adrian Heathcote - 2014 - Axiomathes 24 (2):167-180.
    There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.
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