•  12
    Truth, Reflection, and Commitment
    In Carlo Nicolai & Johannes Stern (eds.), Modes of Truth: The Unified Approach to Truth, Modality, and Paradox, Routledge. pp. 69-87. 2021.
    Proof-theoretic reflection principles have been discussed in proof theory ever since Gödel’s discovery of the incompleteness theorems. But these reflection principles have not received much attention in the philosophical community. The present chapter aims to survey some of the principal meta-mathematical results on the iteration of proof-theoretic reflection principles and investigate these results from a logico-philosophical perspective; we will concentrate on the epistemological significance …Read more
  •  15
    Truth is Simple
    Mind. 2016.
  •  291
    Truth and Existence
    with Jan Heylen
    Thought: A Journal of Philosophy 6 (1): 106-114. 2017.
    Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing e…Read more
  •  608
    Strict conditionals: A negative result
    with Jan Heylen
    Philosophical Quarterly 56 (225). 2006.
    Jonathan Lowe has argued that a particular variation on C.I. Lewis' notion of strict implication avoids the paradoxes of strict implication. We show that Lowe's notion of implication does not achieve this aim, and offer a general argument to demonstrate that no other variation on Lewis' notion of constantly strict implication describes the logical behaviour of natural-language conditionals in a satisfactory way.
  •  7
  •  13
    Wetenschapsfilosofie
    Van Gorcum. 2007.
    Inleidend overzicht van thema's uit de wetenschapsfilosofie.
  •  78
    Unifying the Philosophy of Truth (edited book)
    with T. Achourioti, H. Galinon, J. Martínez Fernández, and K. Fujimoto
    Imprint: Springer. 2015.
    This anthology of the very latest research on truth features the work of recognized luminaries in the field, put together following a rigorous refereeing process. Along with an introduction outlining the central issues in the field, it provides a unique and unrivaled view of contemporary work on the nature of truth, with papers selected from key conferences in 2011 such as Truth Be Told (Amsterdam), Truth at Work (Paris), Paradoxes of Truth and Denotation (Barcelona) and Axiomatic Theories of Tr…Read more
  •  198
    Term Models for Abstraction Principles
    Journal of Philosophical Logic 45 (1): 1-23. 2016.
    Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the metho…Read more
  •  141
    Formal Methods in the Philosophy of Science
    Studia Logica 89 (2): 151-162. 2008.
    In this article, we reflect on the use of formal methods in the philosophy of science. These are taken to comprise not just methods from logic broadly conceived, but also from other formal disciplines such as probability theory, game theory, and graph theory. We explain how formal modelling in the philosophy of science can shed light on difficult problems in this domain.
  •  2
    Reflecting on incompleteness. The Journal of Symbolic Logic, vol. 56
    with S. Feferman, W. N. Reinhardt, V. Halbach, H. Friedman, and M. Sheard
    Bulletin of Symbolic Logic 16 (3): 424-428. 2010.
  •  3
    This book contains ten papers that were presented at the symposium about the realism debate, held at the Center for Logic, Philosophy of Science and Philosophy of Language of the Institute of Philosophy at the Katholieke Universiteit Leuven on 10 and 11 March 1995. The first group of papers are directly concerned with the realism/anti-realism debate in the general philosophy of science. This group includes the articles by Ernan McMullin, Diderik Batens/Joke Meheus, Igor Douven and Herman de Regt…Read more
  •  10
    Quine: Naturalized Epistemology, Perceptual Knowledge and Ontology (edited book)
    Poznan Studies in the Philosophy of the Sciences and the Humanities, Rodopi. 2000.
    Contents: Introduction. NATURALIZED EPISTEMOLOGY. Ton DERKSEN: Naturalistic Epistemology, Murder and Suicide? But what about the Promises! Christopher HOOKWAY: Naturalism and Rationality. Mia GOSSELIN: Quine's Hypothetical Theory of Language Learning. A Comparison of Different Conceptual Schemes of Their Logic. THE NATURE OF PERCEPTUAL KNOWLEDGE. Jaap van BRAKEL: Quine and Innate Similarity Spaces. Dirk KOPPELBERG: Quine and Davidson on the Structure of Empirical Knowledge. Eva PICARDI: Empathy …Read more
  •  192
    Earman (1993) distinguishes three notions of empirical indistinguishability and offers a rigorous framework to investigate how each of these notions relates to the problem of underdetermination of theory choice. He uses some of the results obtained in this framework to argue for a version of scientific anti- realism. In the present paper we first criticize Earman's arguments for that position. Secondly, we propose and motivate a modification of Earman's framework and establish several results co…Read more
  •  183
    Perceptual indiscriminability: In defence of Wright's proof
    Philosophical Quarterly 54 (216): 439-444. 2004.
    A series of unnoticeably small changes in an observable property may add up to a noticeable change. Crispin Wright has used this fact to prove that perceptual indiscriminability is a non-transitive relation. Delia Graff has recently argued that there is a 'tension' between Wright's assumptions. But Graff has misunderstood one of these, that 'phenomenal continua' are possible; and the other, that our powers of discrimination are finite, is sound. If the first assumption is properly understood, it…Read more
  •  101
    Probabilist antirealism
    Pacific Philosophical Quarterly 91 (1): 38-63. 2010.
    Until now, antirealists have offered sketches of a theory of truth, at best. In this paper, we present a probabilist account of antirealist truth in some formal detail, and we assess its ability to deal with the problems that are standardly taken to beset antirealism.
  •  167
    Closer
    Synthese 146 (3). 2005.
    Criteria of identity should mirror the identity relation in being reflexive, symmetrical, and transitive. However, this logical requirement is only rarely met by the criteria that we are most inclined to propose as candidates. The present paper addresses the question how such obvious candidates are best approximated by means of relations that have all of the aforementioned features, i.e., which are equivalence relations. This question divides into two more basic questions. First, what is to be c…Read more
  •  6
    The concept of truth is now a major research subject in analytic philosophy. At the same time, working in different areas, mathematical logicians have developed sophisticated theories of truth and its formal paradoxes. Recent developments of semantical paradoxes in logical theories 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, thi…Read more
  •  181
    Hypatia's silence
    Noûs 55 (1): 62-85. 2021.
    Hartry Field distinguished two concepts of type‐free truth: scientific truth and disquotational truth. We argue that scientific type‐free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non‐classical logical tre…Read more
  •  22
    Absolute Infinity in Class Theory and in Theology
    In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics, Springer International Publishing. 2016.
    In this article we investigate similarities between the role that ineffability of Absolute Infinity plays in class theory and in theology.
  •  15
    The Metaphysics and Mathematics of Arbitrary Objects
    Cambridge University Press. 2019.
    Building on the seminal work of Kit Fine in the 1980s, Leon Horsten here develops a new theory of arbitrary entities. He connects this theory to issues and debates in metaphysics, logic, and contemporary philosophy of mathematics, investigating the relation between specific and arbitrary objects and between specific and arbitrary systems of objects. His book shows how this innovative theory is highly applicable to problems in the philosophy of arithmetic, and explores in particular how arbitrary…Read more
  •  277
    Hartry Field distinguished two concepts of type-free truth: scientific truth and disquotational truth. We argue that scientific type-free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non-classical logical tre…Read more
  •  42
    Reasoning about Arbitrary Natural Numbers from a Carnapian Perspective
    Journal of Philosophical Logic 48 (4): 685-707. 2019.
    Inspired by Kit Fine’s theory of arbitrary objects, we explore some ways in which the generic structure of the natural numbers can be presented. Following a suggestion of Saul Kripke’s, we discuss how basic facts and questions about this generic structure can be expressed in the framework of Carnapian quantified modal logic.
  •  63
    Probability for the Revision Theory of Truth
    Journal of Philosophical Logic 48 (1): 87-112. 2019.
    We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap’s revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable prop…Read more
  •  44
    Triangulating non-archimedean probability
    with Hazel Brickhill
    Review of Symbolic Logic 11 (3): 519-546. 2018.
    We relate Popper functions to regular and perfectly additive such non-Archimedean probability functions by means of a representation theorem: every such non-Archimedean probability function is infinitesimally close to some Popper function, and vice versa. We also show that regular and perfectly additive non-Archimedean probability functions can be given a lexicographic representation. Thus Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability f…Read more
  •  82
    Generic Structures
    Philosophia Mathematica 27 (3): 362-380. 2019.
    In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regarding mathematical structuralism. I discuss how sui generis mathematical structures can be viewed as generic systems of mathematical objects, where mathematical objects are conceived of as arbitrary objects in Fine’s sense.
  •  58
    Human-Effective Computability†
    Philosophia Mathematica 27 (1): 61-87. 2018.
    We analyse Kreisel’s notion of human-effective computability. Like Kreisel, we relate this notion to a concept of informal provability, but we disagree with Kreisel about the precise way in which this is best done. The resulting two different ways of analysing human-effective computability give rise to two different variants of Church’s thesis. These are both investigated by relating them to transfinite progressions of formal theories in the sense of Feferman.
  •  21
    Iterated reflection over full disquotational truth
    with Fischer Martin and Nicolai Carlo
    Journal of Logic and Computation 27 (8): 2631-2651. 2017.
    Iterated reflection principles have been employed extensively to unfold epistemic commitments that are incurred by accepting a mathematical theory. Recently this has been applied to theories of truth. The idea is to start with a collection of Tarski-biconditionals and arrive by iterated reflection at strong compositional truth theories. In the context of classical logic, it is incoherent to adopt an initial truth theory in which A and ‘A is truen’ are inter-derivable. In this article, we show ho…Read more