•  61
    Vagueness, Open-Texture, and Retrievability
    Inquiry: An Interdisciplinary Journal of Philosophy 56 (2-3): 307-326. 2013.
    Just about every theorist holds that vague terms are context-sensitive to some extent. What counts as ?tall?, ?rich?, and ?bald? depends on the ambient comparison class, paradigm cases, and/or the like. To take a stock example, a given person might be tall with respect to European entrepreneurs and downright short with respect to professional basketball players. It is also generally agreed that vagueness remains even after comparison class, paradigm cases, etc. are fixed, and so this context sen…Read more
  •  60
    Space, number and structure: A tale of two debates
    Philosophia Mathematica 4 (2): 148-173. 1996.
    Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates ill…Read more
  •  59
    Frege meets dedekind: A neologicist treatment of real analysis
    Notre Dame Journal of Formal Logic 41 (4): 335--364. 2000.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of ratio…Read more
  •  56
    Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language.
  •  53
    Effectiveness
    In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 37--49. 2006.
  •  49
    Reasoning, logic and computation
    Philosophia Mathematica 3 (1): 31-51. 1995.
    The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism…Read more
  •  46
    The paradox of the Unexpected Hanging, related prediction paradoxes, and the Sorites paradoxes all involve reasoning about ordered collections of entities: days ordered by date in the case of the Unexpected Hanging; men ordered by the number of hairs on their heads the case of the bald man version of the Sorites. The reasoning then assigns each entity a value that depends on the previously assigned value of one of the neighboring entities. The final result is paradoxical because it conflicts wit…Read more
  •  46
    Structure and ontology
    Philosophical Topics 17 (2): 145-171. 1989.
  •  45
    Translating Logical Terms
    Topoi 38 (2): 291-303. 2019.
    The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the not…Read more
  •  45
    Intentional Mathematics (edited book)
    Elsevier. 1985.
    Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.
  •  43
    Life on the Ship of Neurath: Mathematics in the Philosophy of Mathematics
    In Majda Trobok Nenad Miščević & Berislav Žarnić (eds.), Croatian Journal of Philosophy, Springer. pp. 11--27. 2012.
    Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics
  •  43
    Turing projectability
    with Timothy McCarthy
    Notre Dame Journal of Formal Logic 28 (4): 520-535. 1987.
  •  42
    Review of Michael P. Lynch, Truth as One and Many (review)
    Notre Dame Philosophical Reviews 2009 (9). 2009.
  •  41
    Remarks on the development of computability
    History and Philosophy of Logic 4 (1-2): 203-220. 1983.
    The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized …Read more
  •  40
    Book reviews (review)
    with Ben-Ami Scharfstein, Gary Jason, John Blackmore, R. A. Naulty, and F. Bradford Wallack
    Philosophia 17 (4): 551-570. 1987.
  •  39
    This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.
  •  37
    The status of logic
    In Paul Boghossian & Christopher Peacocke (eds.), New Essays on the a Priori, Oxford University Press. pp. 333--338. 2000.
  •  37
    ABSTRACT Michael Rescorla has argued that it makes sense to compute directly with numbers, and he faulted Turing for not giving an analysis of number-theoretic computability. However, in line with a later paper of his, it only makes sense to compute directly with syntactic entities, such as strings on a given alphabet. Computing with numbers goes via notation. This raises broader issues involving de re propositional attitudes towards numbers and other non-syntactic abstract entities.
  •  36
    Review of T. Franzen, Godel's theorem: An incomplete guide to its use and abuse (review)
    Philosophia Mathematica 14 (2): 262-264. 2006.
    This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience with the rigor of mathemat…Read more
  •  36
    Varieties of Logic
    Oxford University Press. 2014.
    Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.
  •  36
    The Company Kept by Cut Abstraction (and its Relatives)
    Philosophia Mathematica 19 (2): 107-138. 2011.
    This article concerns the ongoing neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections . The overall plan is to develop branches of established mathematics using abstraction principles in the form: Formula where a and b are variables of a g…Read more
  •  33
    The Classical Continuum without Points – CORRIGENDUM
    with G. Hellman
    Review of Symbolic Logic 6 (3): 571-571. 2013.
  •  31
    Life on the Ship of Neurath
    Croatian Journal of Philosophy 9 (2): 149-166. 2009.
    Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics
  •  31
    On the notion of effectiveness
    History and Philosophy of Logic 1 (1-2): 209-230. 1980.
    This paper focuses on two notions of effectiveness which are not treated in detail elsewhere. Unlike the standard computability notion, which is a property of functions themselves, both notions of effectiveness are properties of interpreted linguistic presentations of functions. It is shown that effectiveness is epistemically at least as basic as computability in the sense that decisions about computability normally involve judgments concerning effectiveness. There are many occurrences of the pr…Read more
  •  29
    Understanding the Infinite
    with Mary Ellen Smircich and Shaughan Lavine
    Philosophical Review 105 (2): 256. 1996.
    Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.