•  71
    Classical Logic
    In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University. 2018.
    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.
  •  66
    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
  •  65
    Predicativism as a Form of Potentialism
    Review of Symbolic Logic 16 (1): 1-32. 2023.
    In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.
  •  65
    Friedrich Waismann: The Open Texture of Analytic Philosophy (edited book)
    Palgrave Macmillan. 2019.
    This edited collection covers Friedrich Waismann's most influential contributions to twentieth-century philosophy of language: his concepts of open texture and language strata, his early criticism of verificationism and the analytic-synthetic distinction, as well as their significance for experimental and legal philosophy. In addition, Waismann's original papers in ethics, metaphysics, epistemology and the philosophy of mathematics are here evaluated. They introduce Waismann's theory of action a…Read more
  •  65
    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
  •  65
    Logical pluralism and normativity
    Inquiry: An Interdisciplinary Journal of Philosophy 63 (3-4): 389-410. 2020.
    We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which po…Read more
  •  64
    Vagueness and Context
    Inquiry: An Interdisciplinary Journal of Philosophy 59 (4): 343-381. 2016.
    A number of recent accounts for vague terms postulate a kind of context-sensitivity, one that kicks in after the usual ‘external’ contextual factors like comparison class are established and held fixed. In a recent paper, ‘Vagueness without Context Change’: 275–92), Rosanna Keefe criticizes all such accounts. The arguments are variations on considerations that have been brought against context-sensitive accounts of knowledge, predicates of personal taste, epistemic modals, and the like. The issu…Read more
  •  63
    Structure and Ontology
    Philosophical Topics 17 (2): 145-171. 1989.
  •  62
    A Note on Choice Principles in Second-Order Logic
    with Benjamin Siskind and Paolo Mancosu
    Review of Symbolic Logic 16 (2): 339-350. 2023.
    Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We als…Read more
  •  61
    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.
  •  59
    The status of logic
    In Paul Artin Boghossian & Christopher Peacocke (eds.), New Essays on the A Priori, Oxford University Press. pp. 333--366. 2000.
  •  58
    Effectiveness
    In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics, Springer. pp. 37--49. 2006.
  •  58
    Proof and Truth
    Journal of Philosophy 95 (10): 493-521. 1998.
  •  58
    Book reviews (review)
    with Ben-Ami Scharfstein, Gary Jason, John Blackmore, R. A. Naulty, and F. Bradford Wallack
    Philosophia 17 (4): 551-570. 1987.
  •  56
    Intentional mathematics (edited book)
    Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co.. 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.
  •  55
    Cardinals, Ordinals, and the Prospects for a Fregean Foundation
    Royal Institute of Philosophy Supplement 82 77-107. 2018.
    There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fund…Read more
  •  54
    Review of Michael P. Lynch, Truth as One and Many (review)
    Notre Dame Philosophical Reviews 2009 (9). 2009.
  •  52
    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
  •  52
    Turing projectability
    Notre Dame Journal of Formal Logic 28 (4): 520-535. 1987.
  •  49
    Life on the Ship of Neurath: Mathematics in the Philosophy of Mathematics
    Croatian Journal of Philosophy 26 (2): 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
  •  49
    Possibilities, models, and intuitionistic logic: Ian Rumfitt’s The boundary stones of thought
    Inquiry: An Interdisciplinary Journal of Philosophy 62 (7): 812-825. 2019.
    ABSTRACTAIan Rumfitt's new book presents a distinctive and intriguing philosophy of logic, one that ultimately settles on classical logic as the uniquely correct one–or at least rebuts some prominent arguments against classical logic. The purpose of this note is to evaluate Rumfitt's perspective by focusing on some themes that have occupied me for some time: the role and importance of model theory and, in particular, the place of counter-arguments in establishing invalidity, higher-order logic, …Read more
  •  48
    Vagueness in Context
    Oxford University Press UK. 2006.
    Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professional basketball players. The main feature of S…Read more
  •  48
    Book reviews (review)
    with Rudolf Haller, L. Nathan Oaklander, George N. Schlesinger, Richard Shusterman, and L. E. Goodman
    Philosophia 14 (1-2): 225-250. 1984.
  •  47
    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
  •  46
    Vagueness in Context (review)
    Philosophy and Phenomenological Research 76 (2): 471-483. 2008.