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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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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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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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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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300Ontological 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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551On 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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405Jumping 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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282Cardinality logics, part I: inclusions between languages based on ‘exactly’Annals of Pure and Applied Logic 39 (3): 199-238. 1988.
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389Why 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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22Book Review. Existence and Logic. Milton Munitz. (review)Philosophical Review 85 (3): 404-08. 1976.
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258The 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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46Meeting of the association for symbolic logic: New York 1979Journal of Symbolic Logic 46 (2): 427-434. 1981.