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355Jumping through the transfinite: The master code hierarchy of Turing degreesJournal of Symbolic Logic 45 (2): 204-220. 1980.Where $\underline{a}$ is a Turing degree and ξ is an ordinal $ , the result of performing ξ jumps on $\underline{a},\underline{a}^{(\xi)}$ , is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through $(\aleph_1)^{L^\underline{a}}$ of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operatio…Read more
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264Cardinality logics, part I: inclusions between languages based on ‘exactly’Annals of Pure and Applied Logic 39 (3): 199-238. 1988.
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35Book Review. Existence and Logic. Milton Munitz. (review)Philosophical Review 85 (3): 404-08. 1976.
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371Why Ramify?Notre Dame Journal of Formal Logic 56 (2): 379-415. 2015.This paper considers two reasons that might support Russell’s choice of a ramified-type theory over a simple-type theory. The first reason is the existence of purported paradoxes that can be formulated in any simple-type language, including an argument that Russell considered in 1903. These arguments depend on certain converse-compositional principles. When we take account of Russell’s doctrine that a propositional function is not a constituent of its values, these principles turn out to be too …Read more
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46Meeting of the association for symbolic logic: New York 1979Journal of Symbolic Logic 46 (2): 427-434. 1981.
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281The Modal Theory Of Pure Identity And Some Related Decision ProblemsMathematical Logic Quarterly 30 (26-29): 415-423. 1984.Relative to any reasonable frame, satisfiability of modal quantificational formulae in which “= ” is the sole predicate is undecidable; but if we restrict attention to satisfiability in structures with the expanding domain property, satisfiability relative to the familiar frames (K, K4, T, S4, B, S5) is decidable. Furthermore, relative to any reasonable frame, satisfiability for modal quantificational formulae with a single monadic predicate is undecidable ; this improves the result of Kripke co…Read more
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303Individual-actualism and three-valued modal logics, part 1: Model-theoretic semanticsJournal of Philosophical Logic 15 (4). 1986.
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22Book Review. Principles of Intuitionism. Michael Dummett (review)Philosophical Review 91 (2): 253-62. 1982.
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14A Minimal Upper Bound on a Sequence of Turing Degrees Which Represents that SequencePacific Journal of Mathematics 108 (1): 115-119. 1983.
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231Where Do the Cardinal Numbers Come From?Synthese 84 (3): 347-407. 1990.This paper presents a model-theoretic semantics for discourse "about" natural numbers, one that captures what I call "the mathematical-object picture", but avoids what I can "the mathematical-object theory".
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61Review: Jaakko Hintikka, The Principles of Mathematics Revisited (review)Journal of Symbolic Logic 63 (4): 1615-1623. 1998.
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752Logicism and the ontological commitments of arithmeticJournal of Philosophy 81 (3): 123-149. 1984.
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51Book Review. Logic and Arithmetic, Volume I. D Bostock. (review)Journal of Philosophy 73 (6): 149-57. 1976.
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36Annual meeting of the Association for Symbolic Logic, New York City, December 1987Journal of Symbolic Logic 53 (4): 1287-1299. 1988.
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275Upper bounds on locally countable admissible initial segments of a Turing degree hierarchyJournal of Symbolic Logic 46 (4): 753-760. 1981.Where AR is the set of arithmetic Turing degrees, 0 (ω ) is the least member of { $\mathbf{\alpha}^{(2)}|\mathbf{a}$ is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's O. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed …Read more
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254Individual-actualism and three-valued modal logics, part 2: Natural-deduction formalizationsJournal of Philosophical Logic 16 (1). 1987.
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19Book Review. Reflections. Kurt Godel. (review)THe Journal for Symbolic Logic 54 (3): 1095-98. 1989.
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48Book Review. Abstract Objects. Bob Hale. (review)International Studies in Philosophy 24 (3): 146-48. 1992.
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21Wang Hao. Reflections on Kurt Gödel. Bradford books. The MIT Press, Cambridge, Mass., and London, 1987, xxvi + 336 pp (review)Journal of Symbolic Logic 54 (3): 1095-1098. 1989.
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344Some theorems on the expressive limitations of modal languagesJournal of Philosophical Logic 13 (1). 1984.
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488More about uniform upper Bounds on ideals of Turing degreesJournal of Symbolic Logic 48 (2): 441-457. 1983.Let I be a countable jump ideal in $\mathscr{D} = \langle \text{The Turing degrees}, \leq\rangle$ . The central theorem of this paper is: a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump a (1) computes. We may replace "the join of an I-exact pair" in the above theorem by "a weak uniform upper bound on I". We also answer two minimality questions: the class of uniform upper bounds on I never has a minimal member; if ∪ I = L α [ A] ∩ ω ω for α admissible …Read more