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323Philosophy of mathematicsStanford Encyclopedia of Philosophy. 2008.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 with respect to th…Read more
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The deflationists' axioms for truthIn J. C. Beall & Bradley Armour-Garb (eds.), Deflationism and Paradox, Oxford University Press. 2005.
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81On the Quantitative Scalar or-ImplicatureSynthese 146 (1-2): 111-127. 2005.. 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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32A Kripkean Approach to Unknowability and TruthNotre Dame Journal of Formal Logic 39 (3): 389-405. 1998.We consider a language containing partial predicates for subjective knowability and truth. For this language, inductive hierarchy rules are proposed which build up the extension and anti-extension of these partial predicates in stages. The logical interaction between the extension of the truth predicate and the anti-extension of the knowability predicate is investigated
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120Having an interpretation (review)Philosophical Studies 150 (3). 2010.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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29Two problems concerning Frege's distinction between concepts and objectsLogique Et Analyse 127 (27): 267-284. 1989.
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18De gelaagde structuur Van de natuurkunde volgens Peter GalisonTijdschrift Voor Filosofie 61 (4). 1999.This article discusses Peter Galison's views on the structure and evolution of experimental and instrumental cultures in 20th century particle physics, which are unfolded in his recent book Image and Logic. A Material Culture of Microphysics. First a description is given of the uncomfortable predicament in which the Kuhnian tradition finds itself in the past two decades. It is then explained how Galison distinguishes a layered structure in the practice of modern particle physics. Physics as a pr…Read more
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31Remarks on the content and extension of the notion of provabilityLogique Et Analyse 48 (189-192): 15-32. 2005.
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Peelen, G.J. , Het voordeel van de twijfel. In gesprek met de wetenschap (review)Tijdschrift Voor Filosofie 53 (4): 737. 1991.
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61The 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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363Axiomatizing 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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157Impredicative 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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106Truth 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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8`Contemporary Methods for Investigating the Concept of Truth – An Introduction'In Leon Horsten & Volker Halbach (eds.), Principles of Truth, De Gruyter. pp. 11-36. 2003.
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237Reflecting 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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25Two 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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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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38Modal-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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109Revision 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.