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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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349Some theorems on the expressive limitations of modal languagesJournal of Philosophical Logic 13 (1). 1984.
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492More 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
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235Finite level borel games and a problem concerning the jump hierarchyJournal of Symbolic Logic 49 (4): 1301-1318. 1984.
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Book Review. Logic and Its Limits. P Shaw. (review)History and Philosophy of Logic 5 (2): 251. 1984.
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127An Exact Pair for the Arithmetic Degrees Whose Join is Not a Weak Uniform Upper BoundRecursive Function Theory-Newsletters 28. 1982.Proof uses forcing on perfect trees for 2-quantifier sentences in the language of arithmetic. The result extends to exact pairs for the hyperarithmetic degrees.
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327Uniform Upper Bounds on Ideals of Turing DegreesJournal of Symbolic Logic 43 (3): 601-612. 1978.
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96On some concepts associated with finite cardinal numbersBehavioral and Brain Sciences 31 (6): 657-658. 2008.I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians)
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25Jumping to a Uniform Upper BoundProceedings of the American Mathematical Society 85 (4): 600-602. 1982.A uniform upper bound on a class of Turing degrees is the Turing degree of a function which parametrizes the collection of all functions whose degree is in the given class. I prove that if a is a uniform upper bound on an ideal of degrees then a is the jump of a degree c with this additional property: there is a uniform bound b<a so that b V c < a.
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31Book Review. The Lambda-Calculus. H. P. Barendregt( (review)Philosophical Review 97 (1): 132-7. 1988.
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292Ontological Commitments, Thick and ThinIn George Boolos (ed.), Method, Reason and Language: Essays in Honor of Hilary Putnam, Cambridge University Press. pp. 235-260. 1990.Discourse carries thin commitment to objects of a certain sort iff it says or implies that there are such objects. It carries a thick commitment to such objects iff an account of what determines truth-values for its sentences say or implies that there are such objects. This paper presents two model-theoretic semantics for mathematical discourse, one reflecting thick commitment to mathematical objects, the other reflecting only a thin commitment to them. According to the latter view, for example,…Read more
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8An Exact Pair for the Arithmetic Degrees whose join is not a Weak Uniform Upper Bound, in the Recursive Function Theory-Newsletters, No. 28, August-September 1982.
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Book Review. Language and Philosophical Problems. Soren Stenland. (review)History and Philosophy of Logic 253-6. 1993.
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175
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581On The Sense and Reference of A Logical ConstantPhilosophical Quarterly 54 (214): 134-165. 2004.Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So the question “What is l…Read more
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361Jumping 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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274Cardinality logics, part I: inclusions between languages based on ‘exactly’Annals of Pure and Applied Logic 39 (3): 199-238. 1988.
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36Book Review. Existence and Logic. Milton Munitz. (review)Philosophical Review 85 (3): 404-08. 1976.
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380Why 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