•  46
    Platonistic formalism
    Erkenntnis 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.
  •  203
    Non-Archimedean Probability
    with Vieri Benci and Sylvia Wenmackers
    Milan 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
  •  88
    No future
    Journal 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
  •  91
    On the Exclusivity Implicature of ‘Or’ or on the Meaning of Eating Strawberries
    with Liza Verhoeven
    Studia 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.
  •  83
    In defense of epistemic arithmetic
    Synthese 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.
  •  118
    The Undecidability of Propositional Adaptive Logic
    with Philip Welch
    Synthese 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
  •  23
    Formalizing Church's Thesis
    In A. Olszewski, J. Wole'nski & R. Janusz (eds.), Church's Thesis After Seventy Years, Ontos Verlag. pp. 1--253. 2006.
  •  88
    The church-Turing thesis and effective mundane procedures
    Minds 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
  •  54
    Bas C. van Fraassen, The Empirical Stance (review)
    International Studies in the Philosophy of Science 18 (1): 95-97. 2004.
  •  326
    Philosophy of mathematics
    Stanford 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
  • The deflationists' axioms for truth
    with Volker Halbach
    In J. C. Beall & Bradley Armour-Garb (eds.), Deflationism and Paradox, Oxford University Press. 2005.
  •  85
    On the Quantitative Scalar or-Implicature
    Synthese 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.