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Report on some ramified-type assignment systems and their model-theoretic semanticsIn Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica, Palgrave-macmillan. 2013.
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5Book reviews (review)History and Philosophy of Logic 14 (2): 221-263. 1993.Stewart Shapiro, Foundations without foundationalism: A case for second-order logic. Oxford: Clarendon Press, 1991. xvii + 277 pp. £35.00 A. Diaz, J, Echeverria and A. Ibarra, Structures in...
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386Where do sets come from?Journal of Symbolic Logic 56 (1): 150-175. 1991.A model-theoretic approach to the semantics of set-theoretic discourse.
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14Book Reviews (review)History and Philosophy of Logic 5 (2): 233-263. 1984.Albert Menne and Niels Öffenberger, Zur modernen Deutung der aristotelischen Logik. Band I:Über den Folgerungsbegriff in der aristotelischen Logik. Hildesheim and New York: Georg Olms Verlag, 1982. 220 pp. DM 48.Klaus Jacobi, Die Modalbegriffe in den logischen Schriften des Wilhelm von Shyreswood und in anderen Kompendien des 12. und 13. Jahrhunderts. Funktionsbestimmung und Gebrauch in der logischen Analyse. Leiden and KÖln: E.J. Brill, 1980. xiii + 528 pp. HFL 140.Nineteenth – Century Contrast…Read more
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240Cardinality logics. Part II: Definability in languages based on `exactly'Journal of Symbolic Logic 53 (3): 765-784. 1988.
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145Three Value Logics: An Introduction, A Comparison of Various Logical Lexica and Some Philosophical RemarksAnnals of Pure and Applied Logic 43 (2): 99-145. 1989.
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202Stewart Shapiro’s Philosophy of Mathematics (review)Philosophy and Phenomenological Research 65 (2). 2002.Two slogans define structuralism: contemporary mathematics studies structures; mathematical objects are places in those structures. Shapiro’s version of structuralism posits abstract objects of three sorts. A system is “a collection of objects with certain relations” between these objects. “An extended family is a system of people with blood and marital relationships.” A baseball defense, e.g., the Yankee’s defense in the first game of the 1999 World Series, is a also a system, “a collection of …Read more
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265Cut-conditions on sets of multiple-alternative inferencesMathematical Logic Quarterly 68 (1). 2022.I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation |- on Power(F), if |- is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey- Teichmüller Lemma. I then discuss relationships betwe…Read more
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421One-step Modal Logics, Intuitionistic and Classical, Part 1Journal of Philosophical Logic 50 (5): 837-872. 2021.This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section 2 prese…Read more
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339One-Step Modal Logics, Intuitionistic and Classical, Part 2Journal of Philosophical Logic 50 (5): 873-910. 2021.Part 1 [Hodes, 2021] “looked under the hood” of the familiar versions of the classical propositional modal logic K and its intuitionistic counterpart. This paper continues that project, addressing some familiar classical strengthenings of K and GL), and their intuitionistic counterparts. Section 9 associates two intuitionistic one-step proof-theoretic systems to each of the just mentioned intuitionistic logics, this by adding for each a new rule to those which generated IK in Part 1. For the sys…Read more
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51Jan von Plato and Sara Negri, Structural Proof Theory (review)Philosophical Review 115 (2): 255-258. 2006.
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36Book Review. Mechanism, Mentalism and Metamathematics. J Webb (review)Journal of Philosophy 81 (8): 456-64. 1984.
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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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238Where 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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27Review: Jaakko Hintikka, The Principles of Mathematics Revisited (review)Journal of Symbolic Logic 63 (4): 1615-1623. 1998.
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796Logicism and the ontological commitments of arithmeticJournal of Philosophy 81 (3): 123-149. 1984.
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22Book Review. Logic and Arithmetic, Volume I. D Bostock. (review)Journal of Philosophy 73 (6): 149-57. 1976.
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25Annual meeting of the Association for Symbolic Logic, New York City, December 1987Journal of Symbolic Logic 53 (4): 1287-1299. 1988.
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296Upper 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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271Individual-actualism and three-valued modal logics, part 2: Natural-deduction formalizationsJournal of Philosophical Logic 16 (1). 1987.
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10Book Review. Reflections. Kurt Godel. (review)THe Journal for Symbolic Logic 54 (3): 1095-98. 1989.
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19Wang 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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14Book Review. Abstract Objects. Bob Hale. (review)International Studies in Philosophy 24 (3): 146-48. 1992.
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370Some theorems on the expressive limitations of modal languagesJournal of Philosophical Logic 13 (1). 1984.
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512More 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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246Finite 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.