•  19
    Mathematics and reality * by Mary Leng
    Analysis 71 (4): 798-799. 2011.
  •  11
    Models for the Logic of Possible Proofs
    Pacific Philosophical Quarterly 81 (1): 49-66. 2000.
  • Hellman, G., Mathematics without Numbers. Towards a Modal-Structural Interpretation (review)
    Tijdschrift Voor Filosofie 53 (4): 726. 1991.
  •  2
    Eindig, oneindig, meer dan oneindig. Grondslagen van de wiskundige wetenschappen
    Tijdschrift Voor Filosofie 67 (1): 175-177. 2005.
  •  117
    Revision Revisited
    with Graham E. Leigh, Hannes Leitgeb, and Philip Welch
    Review 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.
  •  13
    A Note Concerning The Notion Of Satisfiability
    Logique 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.
  •  130
    Computational Structuralism &dagger
    Philosophia 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
  •  3
    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 qu…Read more
  •  170
    Reflecting on Absolute Infinity
    with Philip Welch
    Journal 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.
  • Kessels, J., van der Dam, A., Tollenaar, J., De zaak Arlet. Inleiding in de kennistheorie (review)
    Tijdschrift Voor Filosofie 53 (1): 167. 1991.
  •  15
    Terugkeer van het subject? Verslag van de 23e Vlaams-Nederlandse filosofiedag, Kortrijk, 27 oktober 2001
    Algemeen Nederlands Tijdschrift voor Wijsbegeerte 94 (2): 155-158. 2002.
  •  34
    Godel's Disjunction: The Scope and Limits of Mathematical Knowledge (edited book)
    with Philip Welch
    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
  •  16
    The Logic of Intensional Predicates
    In Benedikt Löwe, Thoralf Räsch & Wolfgang Malzkorn (eds.), Foundations of the Formal Sciences II, Kluwer Academic Publishers. pp. 89--111. 2003.
  •  53
    Canonical naming systems
    Minds 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.
  •  5
    Reflecting in Epistemic Arithmetic
    Journal 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.
  •  130
    An argument concerning the unknowable
    Analysis 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
  •  2
    Preface
    In Volker Halbach & Leon Horsten (eds.), Principles of truth, Hänsel-hohenhausen. pp. 7-8. 2002.
  •  134
    Infinitesimal Probabilities
    with Vieri Benci and Sylvia Wenmackers
    British 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
  •  18
    Mathematical Philosophy?
    In Hanne Andersen, Dennis Dieks, Wenceslao González, Thomas Uebel & Gregory Wheeler (eds.), New Challenges to Philosophy of Science, Springer Verlag. pp. 73--86. 2013.
  •  2598
    Cantorian Infinity and Philosophical Concepts of God
    with Joanna Van der Veen
    European 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
  • Hughes, R.I.G., The Structure and Interpretation of Quantum Mechanics (review)
    Tijdschrift Voor Filosofie 54 (4): 735. 1992.
  •  60
    The work of mathematician and logician Alfred Tarski (1901--1983) marks the transition from substantial to deflationary views about truth.
  •  12
    Scope and rigidity
    Communication and Cognition: An Interdisciplinary Quarterly Journal 25 (4): 353-372. 1992.
  •  22
    Book Review: Stewart Shapiro. Vagueness in Context (review)
    Notre Dame Journal of Formal Logic 50 (2): 221-226. 2009.
  •  63
    Provability in principle and controversial constructivistic principles
    Journal 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
  •  42
    Principles 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
  •  110
    One Hundred Years of Semantic Paradox
    Journal 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