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40On Play by Means of Computing Machines .A Theory of Higher Order Probabilities.Knowledge and Efficient Computation.Realizability Semantics for Error-Tolerant Logics (review)Journal of Symbolic Logic 53 (2): 669. 1988.
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22An Extension of a Theorem of Gaifman-Hales-SolovayJournal of Symbolic Logic 34 (1): 131-132. 1969.
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26Saul A. Kripke. An extension of a theorem of Gaifman-Hales-Solovay. Fundamenta mathematicae, vol. 61 , pp. 29–32Journal of Symbolic Logic 34 (1): 131-132. 1969.
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519Context-dependent UtilitiesIn Wiebe Van Der Hoek, Wesley H. Holliday & Wen Fang Wang (eds.), Logic, Rationality, and Interaction, Springer. pp. 90-101. 2015.Savage's framework of subjective preference among acts provides a paradigmatic derivation of rational subjective probabilities within a more general theory of rational decisions. The system is based on a set of possible states of the world, and on acts, which are functions that assign to each state a consequence. The representation theorem states that the given preference between acts is determined by their expected utilities, based on uniquely determined probabilities (assigned to sets of sta…Read more
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746A Simpler and More Realistic Subjective Decision TheorySynthese 195 (10): 4205--4241. 2018.In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that satisf…Read more
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138Is the "bottom-up" approach from the theory of meaning to metaphysics possible?Journal of Philosophy 93 (8): 373-407. 1996.Dummett’s The Logical Foundations of Metaphysics (LFM) outlines an ambitious project that has been at the core of his work during the last forty years. The project is built around a particular conception of the theory of meaning (or philosophy of language), according to which such a theory should constitute the corner stone of philosophy and, in particular, provide answers to various metaphysical questions. The present paper is intended as a critical evaluation of some of the main features of th…Read more
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98Naming and Diagonalization, from Cantor to Gödel to KleeneLogic Journal of the IGPL 14 (5): 709-728. 2006.We trace self-reference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form ,{ }), where D is a non-empty set; for every a∈ D, which is a name of a k-ary function, {a}: Dk → D is the function named by a, and type is the type of a, which tells us if a is a name and, if it is, the arity of the named function. Under quite general conditions we get a fixed point theorem, whose special cas…Read more
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80Deceptive updating and minimal information methodsSynthese 187 (1): 147-178. 2012.The technique of minimizing information (infomin) has been commonly employed as a general method for both choosing and updating a subjective probability function. We argue that, in a wide class of cases, the use of infomin methods fails to cohere with our standard conception of rational degrees of belief. We introduce the notion of a deceptive updating method and argue that non-deceptiveness is a necessary condition for rational coherence. Infomin has been criticized on the grounds that there ar…Read more
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333Vagueness, tolerance and contextual logicSynthese 174 (1). 2010.The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as…Read more
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23Is the "Bottom-Up" Approach from the Theory of Meaning to Metaphysics Possible?Journal of Philosophy 93 (8): 373-407. 1996.
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104A reason for theoretical termsErkenntnis 32 (2). 1990.The presence of nonobservational vocabulary is shown to be necessary for wide application of a conservative principle of theory revision.
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73The paper outlines a project in the philosophy of mathematics based on a proposed view of the nature of mathematical reasoning. It also contains a brief evaluative overview of the discipline and some historical observations; here it points out and illustrates the division between the philosophical dimension, where questions of realism and the status of mathematics are treated, and the more descriptive and looser dimension of epistemic efficiency, which has to do with ways of organizing the mathe…Read more
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144Probabilities over rich languages, testing and randomnessJournal of Symbolic Logic 47 (3): 495-548. 1982.
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68Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the…Read more
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72This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observations on circularity and draw brief comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motives for it are made obvious. We s…Read more
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99What Godel's Incompleteness Result Does and Does Not ShowJournal of Philosophy 97 (8): 462. 2000.In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer.1 McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made without following McCall’s route; it is then straighter and more forceful. Yet the argument fails in an i…Read more
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164Reasoning with limited resources and assigning probabilities to arithmetical statementsSynthese 140 (1-2). 2004.There are three sections in this paper. The first is a philosophical discussion of the general problem of reasoning under limited deductive capacity. The second sketches a rigorous way of assigning probabilities to statements in pure arithmetic; motivated by the preceding discussion, it can nonetheless be read separately. The third is a philosophical discussion that highlights the shifting contextual character of subjective probabilities and beliefs.
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218On ontology and realism in mathematicsReview of Symbolic Logic 5 (3): 480-512. 2012.The paper is concerned with the way in which “ontology” and “realism” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sectio…Read more
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74This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders the proof of the so–called fixed point theorem transparent. We also point out various historical details and make some observations on circularity and some comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motive for arithmetization and what it should accomplish are made obvious. We s…Read more
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61The semantic paradoxes, whose paradigm is the Liar, played a crucial role at a crucial juncture in the development of modern logic. In his 1908 seminal paper, Russell outlined a system, soon to become that of the Principia Mathematicae, whose main goal was the solution of the logical paradoxes, both semantic and settheoretic. Russell did not distinguish between the two and his theory of types was designed to solve both kinds in the same uniform way. Set theoreticians, however, were content to tr…Read more
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Areas of Specialization
Philosophy of Language |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Probability |