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5Reflecting in Epistemic ArithmeticJournal of Symbolic Logic 61 (2): 788-801. 1996.An epistemic formalization of arithmetic is constructed in which certain non-trivial metatheoretical inferences about the system itself can be made. These inferences involve the notion of provability in principle, and cannot be made in any consistent extensions of Stewart Shapiro's system of epistemic arithmetic. The system constructed in the paper can be given a modal-structural interpretation.
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130An argument concerning the unknowableAnalysis 69 (2): 240-242. 2009.Williamson has forcefully argued that Fitch's argument shows that the domain of the unknowable is non-empty. And he exhorts us to make more inroads into the land of the unknowable. Concluding his discussion of Fitch's argument, he writes: " Once we acknowledge that [the domain of the unknowable] is non-empty, we can explore more effectively its extent. … We are only beginning to understand the deeper limits of our knowledge. " I shall formulate and evaluate a new argument concerning the domain o…Read more
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2PrefaceIn Volker Halbach & Leon Horsten (eds.), Principles of truth, Hänsel-hohenhausen. pp. 7-8. 2002.
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2617Cantorian Infinity and Philosophical Concepts of GodEuropean Journal for Philosophy of Religion 5 (3): 117--138. 2013.It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be considered are fundamentally unclear. In this article we argue that Cantor’s views on this subject, at least up until around 1896, are relatively clear, coherent, and interesting. We then go on to argue that Cantor’s views about the set theoretic universe as a whole have implications for theology that have hitherto not been sufficiently recognised. However, the theological implications in question, …Read more
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133Infinitesimal ProbabilitiesBritish Journal for the Philosophy of Science 69 (2): 509-552. 2016.Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. _1_ Introduction _2_ The Limits of Classical Probability Theory _2.1_ Classical probability functions _2.2_ Limitations _2.3_ Infinitesimals to the rescue? _3_ NAP Theory _3.1_ Fir…Read more
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18Mathematical Philosophy?In Hanne Andersen, Dennis Dieks, Wenceslao J. Gonzalez, Thomas Uebel & Gregory Wheeler (eds.), New Challenges to Philosophy of Science, Springer Verlag. pp. 73--86. 2013.
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60The Tarskian Turn: Deflationism and Axiomatic TruthMIT Press. 2011.The work of mathematician and logician Alfred Tarski (1901--1983) marks the transition from substantial to deflationary views about truth.
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Hughes, R.I.G., The Structure and Interpretation of Quantum Mechanics (review)Tijdschrift Voor Filosofie 54 (4): 735. 1992.
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12Scope and rigidityCommunication and Cognition: An Interdisciplinary Quarterly Journal 25 (4): 353-372. 1992.
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22Book Review: Stewart Shapiro. Vagueness in Context (review)Notre Dame Journal of Formal Logic 50 (2): 221-226. 2009.
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63Provability in principle and controversial constructivistic principlesJournal of Philosophical Logic 26 (6): 635-660. 1997.New epistemic principles are formulated in the language of Shapiro's system of Epistemic Arithmetic. It is argued that some plausibility can be attributed to these principles. The relations between these principles and variants of controversial constructivistic principles are investigated. Special attention is given to variants of the intuitionistic version of Church's thesis and to variants of Markov's principle
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1453Fair infinite lotteriesSynthese 190 (1): 37-61. 2013.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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42Principles of truth (edited book)Hänsel-Hohenhausen. 2002.On the one hand, the concept of truth is a major research subject in analytic philosophy. On the other hand, mathematical logicians have developed sophisticated logical theories of truth and the paradoxes. Recent developments in logical theories of the semantical paradoxes are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume …Read more
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110One Hundred Years of Semantic ParadoxJournal of Philosophical Logic (6): 1-15. 2015.This article contains an overview of the main problems, themes and theories relating to the semantic paradoxes in the twentieth century. From this historical overview I tentatively draw some lessons about the way in which the field may evolve in the next decade
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159Vom Zahlen zu den Zahlen: On the Relation Between Computation and Arithmetical StructuralismPhilosophia Mathematica 20 (3): 275-288. 2012.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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The layered structure of physics according to Peter GalisonTijdschrift Voor Filosofie 61 (4): 747-778. 1999.
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Gomperts, M.C., Neeltje komt dinsdag in evakostuum (review)Tijdschrift Voor Filosofie 55 (3): 571. 1993.
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33Dennis 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. 2005.
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23Review of jc Beall (ed.), Revenge of the Liar: New Essays on the Paradox (review)Notre Dame Philosophical Reviews 2009 (5). 2009.
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22An Axiomatic Investigation of Provability as a Primitive PredicateIn Volker Halbach & Leon Horsten (eds.), Principles of truth, Hänsel-hohenhausen. pp. 203-220. 2002.
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46Platonistic formalismErkenntnis 54 (2): 173-194. 2001.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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58On the Exclusivity Implicature of ‘Or’ or on the Meaning of Eating StrawberriesStudia Logica 81 (1): 19-24. 2005.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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203Non-Archimedean ProbabilityMilan Journal of Mathematics 81 (1): 121-151. 2013.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 Kolm…Read more
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88No futureJournal of Philosophical Logic 30 (3): 259-265. 2001.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…Read more
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59The Undecidability of Propositional Adaptive LogicSynthese 158 (1): 41-60. 2007.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 decid…Read more
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83In defense of epistemic arithmeticSynthese 116 (1): 1-25. 1998.This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.
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23Formalizing Church's ThesisIn A. Olszewski, J. Wole'nski & R. Janusz (eds.), Church's Thesis After Seventy Years, Ontos Verlag. pp. 1--253. 2006.
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88The church-Turing thesis and effective mundane proceduresMinds and Machines 5 (1): 1-8. 1995.We critically discuss Cleland''s analysis of effective procedures as mundane effective procedures. She argues that Turing machines cannot carry out mundane procedures, since Turing machines are abstract entities and therefore cannot generate the causal processes that are generated by mundane procedures. We argue that if Turing machines cannot enter the physical world, then it is hard to see how Cleland''s mundane procedures can enter the world of numbers. Hence her arguments against versions o…Read more