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63The Expressive Power of TruthReview of Symbolic Logic 8 (2): 345-369. 2015.There are two perspectives from which formal theories can be viewed. On the one hand, one can take a theory to be about some privileged models. On the other hand, one can take all models of a theory to be on a par. In contrast with what is usually done in philosophical debates, we adopt the latter viewpoint. Suppose that from this perspective we want to add an adequate truth predicate to a background theory. Then on the one hand the truth theory ought to be semantically conservative over the bac…Read more
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Norms for Theories of Reflexive TruthIn T. Achourioti, H. Galinon, J. Martínez Fernández & K. Fujimoto (eds.), Unifying the Philosophy of Truth, Imprint: Springer. 2015.
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368Axiomatizing Kripke’s Theory of TruthJournal of Symbolic Logic 71 (2). 2006.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 K…Read more
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163Impredicative Identity CriteriaPhilosophy and Phenomenological Research 80 (2): 411-439. 2010.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 identity criteria is investigated in the light of the view that is articulated. It is argued that in and of itself impredicativity does not constitute sufficient grounds for rejecting a putative identity criterion. The view that is presented is applied to Davidson’s criterion of identity for events and to the structuralist criterion of iden…Read more
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12Gödels disjunctieTijdschrift Voor Filosofie 60 (1). 1998.In his Gibbs lecture, Gödel argued for the thesis that either the human mind is not a Turing machine, or there exist absolutely undecidable mathematical propositions. He believed that this disjunction can be deduced with mathematical certainty from certain results in mathematical logic. He thought that his disjunctive thesis is of great philosophical importance. First, Gödel's argument for his disjunctive thesis is discussed. It is argued that thisargument contains an ambiguity. But when it is m…Read more
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116Truth is SimpleMind 126 (501): 195-232. 2017.Even though disquotationalism is not correct as it is usually formulated, a deep insight lies behind it. Specifically, it can be argued that, modulo implicit commitment to reflection principles, all there is to the notion of truth is given by a simple, natural collection of truth-biconditionals.
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10`Contemporary Methods for Investigating the Concept of Truth – An Introduction'In Volker Halbach & Leon Horsten (eds.), Principles of truth, Hänsel-hohenhausen. pp. 11-36. 2002.
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239Reflecting in epistemic arithmeticJournal of Symbolic Logic 61 (3): 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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26Two Proof-Theoretic Remarks on EA + ECTMathematical Logic Quarterly 46 (4): 461-466. 2000.In this note two propositions about the epistemic formalization of Church's Thesis are proved. First it is shown that all arithmetical sentences deducible in Shapiro's system EA of Epistemic Arithmetic from ECT are derivable from Peano Arithmetic PA + uniform reflection for PA. Second it is shown that the system EA + ECT has the epistemic disjunction property and the epistemic numerical existence property for arithmetical formulas
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4PrefaceIn Volker Halbach & Leon Horsten (eds.), Principles of truth, Hänsel-hohenhausen. pp. 7-8. 2002.
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55Axioms 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. 2012.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 probabi…Read more
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136Mathematical knowledge: Intuition, visualization, and understandingTopoi 29 (1): 1-2. 2010.This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initi…Read more
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39Modal-Epistemic Variants of Shapiro’s System of Epistemic ArithmeticNotre Dame Journal of Formal Logic 35 (2): 284-291. 1994.
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Hellman, G., Mathematics without Numbers. Towards a Modal-Structural Interpretation (review)Tijdschrift Voor Filosofie 53 (4): 726. 1991.
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The Semantical Paradoxes, the Neutrality of Truth and the Neutrality of the Minimalist Theory of TruthIn P. Cartois (ed.), The Many Problems of Realism (Studies in the General Philosophy of Science: Volume 3), Tilberg University Press. 1995.
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2Eindig, oneindig, meer dan oneindig. Grondslagen van de wiskundige wetenschappenTijdschrift Voor Filosofie 67 (1): 175-177. 2005.
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117Revision RevisitedReview of Symbolic Logic 5 (4): 642-664. 2012.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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13A Note Concerning The Notion Of SatisfiabilityLogique Et Analyse 47. 2004.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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Perceptual Indiscriminability and the Concept of a Color ShadeIn Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and Clouds: Vaguenesss, its Nature and its Logic, Oxford University Press. 2010.
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130Computational Structuralism &daggerPhilosophia Mathematica 13 (2): 174-186. 2005.According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by 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…Read more
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3Given 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 qu…Read more
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170Reflecting on Absolute InfinityJournal of Philosophy 113 (2): 89-111. 2016.This article is concerned with reflection principles in the context of Cantor’s conception of the set-theoretic universe. We argue that within such a conception reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity.
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Kessels, J., van der Dam, A., Tollenaar, J., De zaak Arlet. Inleiding in de kennistheorie (review)Tijdschrift Voor Filosofie 53 (1): 167. 1991.
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15Terugkeer van het subject? Verslag van de 23e Vlaams-Nederlandse filosofiedag, Kortrijk, 27 oktober 2001Algemeen Nederlands Tijdschrift voor Wijsbegeerte 94 (2): 155-158. 2002.
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34Godel's Disjunction: The Scope and Limits of Mathematical Knowledge (edited book)Oxford University Press UK. 2016.The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that in…Read more
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16The Logic of Intensional PredicatesIn Benedikt Löwe, Thoralf Räsch & Wolfgang Malzkorn (eds.), Foundations of the Formal Sciences II, Kluwer Academic Publishers. pp. 89--111. 2003.
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53Canonical naming systemsMinds and Machines 15 (2): 229-257. 2004.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.