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Profile: Leon Horsten (Bristol University)
  1. Leon Horsten, A Note Concerning the Notion of Satisfiability.
    Tarski has shown how the argumentation of the liar paradox can be used to prove a theorem about truth in formalized languages. In this paper, it is shown how the paradox concerning the least undefinable ordinal can be used to prove a no go-theorem concerning the notion of satisfaction in formalized languages. Also, the connection of this theorem with the absolute notion of definability is discussed.
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  2. Leon Horsten, On Best Transitive Approximations to Simple Graphs.
    Given any finite graph, which transitive graphs approximate it most closely and how fast can we find them? The answer to this question depends on the concept of “closest approximation” involved. In [8,9] a qualitative concept of best approximation is formulated. Roughly, a qualitatively best transitive approximation of a graph is a transitive graph which cannot be “improved” without also going against the original graph. A quantitative concept of best approximation goes back at least to [10]. A quantitatively best transitive (...)
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  3. Vieri Benci, Leon Horsten & Sylvia Wenmackers (2013). Non-Archimedean Probability. Milan Journal of Mathematics 81 (1):121-151.
    We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by (...)
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  4. Leon Horsten (2013). Mathematical Philosophy?. In. In Hanne Andersen, Dennis Dieks, Wenceslao González, Thomas Uebel & Gregory Wheeler (eds.), New Challenges to Philosophy of Science. Springer Verlag. 73--86.
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  5. Sylvia Wenmackers & Leon Horsten (2013). Fair Infinite Lotteries. Synthese 190 (1):37-61.
    This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.
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  6. Vieri Benci, Leon Horsten & Sylvia Wenmackers (2012). Axioms for Non-Archimedean Probability (NAP). In De Vuyst J. & Demey L. (eds.), Future Directions for Logic; Proceedings of PhDs in Logic III - Vol. 2 of IfColog Proceedings. College Publications.
    In this contribution, we focus on probabilistic problems with a denumerably or non-denumerably infinite number of possible outcomes. Kolmogorov (1933) provided an axiomatic basis for probability theory, presented as a part of measure theory, which is a branch of standard analysis or calculus. Since standard analysis does not allow for non-Archimedean quantities (i.e. infinitesimals), we may call Kolmogorov's approach "Archimedean probability theory". We show that allowing non-Archimedean probability values may have considerable epistemological advantages in the infinite case. The current paper (...)
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  7. L. Horsten (2012). Vom Zahlen Zu den Zahlen: On the Relation Between Computation and Arithmetical Structuralism. Philosophia Mathematica 20 (3):275-288.
    This paper sketches an answer to the question how we, in our arithmetical practice, succeed in singling out the natural-number structure as our intended interpretation. It is argued that we bring this about by a combination of what we assert about the natural-number structure on the one hand, and our computational capacities on the other hand.
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  8. Leon Horsten, Graham E. Leigh, Hannes Leitgeb & Philip Welch (2012). Revision Revisited. Review of Symbolic Logic 5 (4):642-664.
    This article explores ways in which the Revision Theory of Truth can be expressed in the object language. In particular, we investigate the extent to which semantic deficiency, stable truth, and nearly stable truth can be so expressed, and we study different axiomatic systems for the Revision Theory of Truth.
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  9. L. Horsten (2011). Review of M. Leng, Mathematics and Reality. [REVIEW] Analysis 71 (4):798-799.
  10. Leon Horsten (2011). The Tarskian Turn. Deflationism and Axiomatic Truth. Mit Press.
    The work of mathematician and logician Alfred Tarski (1901--1983) marks the transition from substantial to deflationary views about truth.
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  11. Igor Douven, Leon Horsten & Jan-Willem Romeijn (2010). Probabilist antirealism. Pacific Philosophical Quarterly 91 (1):38-63.
    Until now, antirealists have offered sketches of a theory of truth, at best. In this paper, we present a probabilist account of antirealist truth in some formal detail, and we assess its ability to deal with the problems that are standardly taken to beset antirealism.
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  12. Leon Horsten (2010). Having an Interpretation. [REVIEW] Philosophical Studies 150 (3):449 - 459.
    I investigate what it means to have an interpretation of our language, how we manage to bestow a determinate interpretation to our utterances, and to which extent our interpretation of the world is determinate. All this is done in dialogue with van Fraassen's insightful discussion of Putnam's model-theoretic argument and of scientific structuralism.
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  13. Leon Horsten (2010). Impredicative Identity Criteria. Philosophy and Phenomenological Research 80 (2):411-439.
    In this paper, a general perspective on criteria of identity of kinds of objects is developed. The question of the admissibility of impredicative or circular identitycriteria is investigated in the light of the view that is articulated. It is argued that in and of itself impredicativity docs not constitute sufficient grounds for rejecting aputative identity criterion. The view that is presented is applied to Davidson's criterion of identity for events and to the structuralist criterion of identity of placesin a structure.
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  14. Leon Horsten (2010). Perceptual Indiscriminability and the Concept of a Color Shade. In Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and Clouds: Vaguenesss, its Nature and its Logic. Oup Oxford.
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  15. Leon Horsten & Irina Starikova (2010). Mathematical Knowledge: Intuition, Visualization, and Understanding. Topoi 29 (1):1-2.
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  16. Leon Horsten (2009). Levity. Mind 118 (471):555-581.
    In this article, the prospects of deflationism about the concept of truth are investigated. A new version of deflationism, called inferential deflationism, is articulated and defended. It is argued that it avoids the pitfalls of earlier deflationist views such as Horwich’s minimalist theory of truth and Field’s version of deflationism.
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  17. Leon Horsten (2009). An Argument Concerning the Unknowable. Analysis 69 (2):240-242.
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  18. Leon Horsten (2009). Book Review: Stewart Shapiro. Vagueness in Context. [REVIEW] Notre Dame Journal of Formal Logic 50 (2):221-226.
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  19. Leon Horsten (2009). Review of Jc Beall (Ed.), Revenge of the Liar: New Essays on the Paradox. [REVIEW] Notre Dame Philosophical Reviews 2009 (5).
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  20. Leon Horsten & Philip Welch (2009). Erratum: The Undecidability of Propositional Adaptive Logic. Synthese 169 (1):217 - 218.
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  21. Volker Halbach & Leon Horsten (2008). The Deflationists' Axioms for Truth. In J. C. Beall & Bradley Armour-Garb (eds.), Deflationism and Paradox. Oup Oxford.
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  22. Leon Horsten, Philosophy of Mathematics. 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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  23. Leon Horsten & Igor Douven (2008). Formal Methods in the Philosophy of Science. Studia Logica 89 (2):151 - 162.
    In this article, we reflect on the use of formal methods in the philosophy of science. These are taken to comprise not just methods from logic broadly conceived, but also from other formal disciplines such as probability theory, game theory, and graph theory. We explain how formal modelling in the philosophy of science can shed light on difficult problems in this domain.
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  24. Leon Horsten, Igor Douven & Erik Weber (2007). Wetenschapsfilosofie. Van Gorcum.
    Inleidend overzicht van thema's uit de wetenschapsfilosofie.
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  25. Leon Horsten & Philip Welch (2007). The Undecidability of Propositional Adaptive Logic. Synthese 158 (1):41 - 60.
    We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...)
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  26. Volker Halbach & Leon Horsten (2006). Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic 71 (2):677 - 712.
    We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in (...)
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  27. Jan Heylen & Leon Horsten (2006). Strict Conditionals: A Negative Result. Philosophical Quarterly 56 (225):536–549.
    Jonathan Lowe has argued that a particular variation on C.I. Lewis' notion of strict implication avoids the paradoxes of strict implication. We show that Lowe's notion of implication does not achieve this aim, and offer a general argument to demonstrate that no other variation on Lewis' notion of constantly strict implication describes the logical behaviour of natural-language conditionals in a satisfactory way.
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  28. Leon Horsten (2006). Axiomatizing Kripke's Theory of Truth. Journal of Symbolic Logic 71 (2):677 - 712.
    We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in (...)
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  29. Leon Horsten (2006). Formalizing Church's Thesis. In A. Olszewski, J. Wole'nski & R. Janusz (eds.), Church's Thesis After Seventy Years. Ontos Verlag. 1--253.
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  30. Rafael De Clercq & Leon Horsten (2005). Closer. Synthese 146 (3):371 - 393.
    Criteria of identity should mirror the identity relation in being reflexive, symmetrical, and transitive. However, this logical requirement is only rarely met by the criteria that we are most inclined to propose as candidates. The present paper addresses the question how such obvious candidates are best approximated by means of relations that have all of the aforementioned features, i.e., which are equivalence relations. This question divides into two more basic questions. First, what is to be considered a ‘best’ approximation. And (...)
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  31. Volker Halbach & Leon Horsten (2005). Computational Structuralism. Philosophia Mathematica 13 (2):174-186.
    According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by (nonstandard) models that do not instantiate this structure. Proponents of structuralism have put forward various accounts of how we succeed in fixing one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. (...)
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  32. Leon Horsten (2005). Canonical Naming Systems. Minds and Machines 15 (2):229-257.
    This paper outlines a framework for the abstract investigation of the concept of canonicity of names and of naming systems. Degrees of canonicity of names and of naming systems are distinguished. The structure of the degrees is investigated, and a notion of relative canonicity is defined. The notions of canonicity are formally expressed within a Carnapian system of second-order modal logic.
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  33. Leon Horsten (2005). Dennis E. Hesseling. Gnomes in the Fog: The Reception of Brouwer's Intuitionism in the 1920s. Basel, Boston, Berlin: Birkhäu-Ser Verlag, 2003. Pp. XXIII + 448. ISBN 3-7643-6536-. [REVIEW] Philosophia Mathematica 13 (1):111-113.
  34. Leon Horsten (2005). On the Quantitative Scalar or-Implicature. Synthese 146 (1-2):111 - 127.
    . Two simple generalized conversational implicatures are investigated :(1) the quantitative scalar implicature associated with ‘or’, and (2) the ‘not-and’-implicature, which is the dual to (1). It is argued that it is more fruitful to consider these implicatures as rules of interpretation and to model them in an algebraic fashion than to consider them as nonmonotonic rules of inference and to model them in a proof-theoretic way.
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  35. Leon Horsten (2005). Remarks on the Content and Extension of the Notion of Provability. Logique Et Analyse 48 (189-192):15-32.
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  36. Liza Verhoeven & Leon Horsten (2005). On the Exclusivity Implicature of 'Or' or on the Meaning of Eating Strawberries. Studia Logica 81 (1):19-24.
    This paper is a contribution to the program of constructing formal representations of pragmatic aspects of human reasoning. We propose a formalization within the framework of Adaptive Logics of the exclusivity implicature governing the connective ‘or’.Keywords: exclusivity implicature, Adaptive Logics.
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  37. Rafael De Clercq & Leon Horsten (2004). Perceptual Indiscriminability: In Defence of Wright's Proof. Philosophical Quarterly 54 (216):439 - 444.
    A series of unnoticeably small changes in an observable property may add up to a noticeable change. Crispin Wright has used this fact to prove that perceptual indiscriminability is a non-transitive relation. Delia Graff has recently argued that there is a 'tension' between Wright's assumptions. But Graff has misunderstood one of these, that 'phenomenal continua' are possible; and the other, that our powers of discrimination are finite, is sound. If the first assumption is properly understood, it is not in tension (...)
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  38. Rafael de Clercq & Leon Horsten (2004). Perceptual Indiscriminability: In Defence of Wright's Proof. Philosophical Quarterly 54 (216):439-444.
    A series of unnoticeably small changes in an observable property may add up to a noticeable change. Crispin Wright has used this fact to prove that perceptual indiscriminability is a non-transitive relation. Delia Graff has recently argued that there is a 'tension' between Wright's assumptions. But Graff has misunderstood one of these, that 'phenomenal continua' are possible; and the other, that our powers of discrimination are finite, is sound. If the first assumption is properly understood, it is not in tension (...)
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  39. Leon Horsten (2004). Bas C. Van Fraassen, The Empirical Stance. International Studies in the Philosophy of Science 18:95-97.
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  40. Leon Horsten (2004). Bespr. Van: The Empirical Stance (Bas C. Van Fraassen). International Studies in the Philosophy of Science 18 (1):95-97.
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  41. Leon Horsten (2003). The Logic of Intensional Predicates. In. In Benedikt Löwe, Thoralf Räsch & Wolfgang Malzkorn (eds.), Foundations of the Formal Sciences Ii. Kluwer. 89--111.
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  42. Volker Halbach & Leon Horsten (eds.) (2002). Principles of Truth. Hänsel-Hohenhausen.
  43. Leon Horsten (2002). Terugkeer van het subject? Verslag van de 23e Vlaams-Nederlandse filosofiedag, Kortrijk, 27 oktober 2001. Algemeen Nederlands Tijdschrift Voor Wijsbegeerte 94 (2):155-158.
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  44. Leon Horsten (2002). Review: Tomasz Placek, Mathematical Intuitionism and Intersubjectivity. A Critical Exposition of Arguments for Intuitionism. [REVIEW] Bulletin of Symbolic Logic 8 (4):518-520.
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  45. T. Placek & Leon Horsten (2002). REVIEWS-Mathematical Intuitionism and Intersubjectivity. Bulletin of Symbolic Logic 8 (4):518-519.
     
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  46. L. Horsten (2001). Platonistic Formalism. Erkenntnis 54 (2):173-194.
    The present paper discusses a proposal which says,roughly and with several qualifications, that thecollection of mathematical truths is identical withthe set of theorems of ZFC. It is argued that thisproposal is not as easily dismissed as outright falseor philosophically incoherent as one might think. Some morals of this are drawn for the concept ofmathematical knowledge.
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  47. Leon Horsten & Hannes Leitgeb (2001). No Future. Journal of Philosophical Logic 30 (3):259-265.
    The difficulties with formalizing the intensional notions necessity, knowability and omniscience, and rational belief are well-known. If these notions are formalized as predicates applying to (codes of) sentences, then from apparently weak and uncontroversial logical principles governing these notions, outright contradictions can be derived. Tense logic is one of the best understood and most extensively developed branches of intensional logic. In tense logic, the temporal notions future and past are formalized as sentential operators rather than as predicates. The question therefore (...)
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  48. Lieven Decock & Leon Horsten (eds.) (2000). Quine. Naturalized Epistemology, Perceptual Knowledge and Ontology. Poznan Studies in the Philosophy of the Sciences and the Humanities, Rodopi.
     
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  49. V. Halbach & L. Horsten (2000). Two Proof-Theoretic Remarks on EA + ECT. Mathematical Logic Quarterly 46 (4):461-466.
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  50. Leon Horsten (2000). Models for the Logic of Possible Proofs. Pacific Philosophical Quarterly 81 (1):49–66.
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