•  5
    Book reviews (review)
    with Michael Resnik, John Bigelow, Albert Lewis, Massimo Galuzzi, M. Franchella, Gabriel Nuchelmans, Alan Perreiah, Besprechung Von Christoph Demmerling, I. Grattan-Guinness, Michele Di Francesco, Thomas Oberdan, Wolfe Mays, John Martin, H. A. Ide, E. J. Lowe, J. Wolenski, Liliana Albertazzi, C. W. Kilmister, Christoph Demmerling, S. B. Russ, and Geregory Moore
    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...
  •  324
    Where do sets come from?
    Journal of Symbolic Logic 56 (1): 150-175. 1991.
    A model-theoretic approach to the semantics of set-theoretic discourse.
  •  9
    Book Reviews (review)
    with Robin Smith, N. J. Green-Pedersen, David Holdcroft, Rezensiert von Peter Schroeder-Heister, Peter Loptson, Recensione di Corrado Mangione, P. M. Simons, and G. J. Tee
    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
  •  199
    Cardinality logics. Part II: Definability in languages based on `exactly'
    Journal of Symbolic Logic 53 (3): 765-784. 1988.
  •  177
    Stewart 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
  •  221
    Cut-conditions on sets of multiple-alternative inferences
    Mathematical 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
  •  363
    One-step Modal Logics, Intuitionistic and Classical, Part 1
    Journal 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
  •  272
    One-Step Modal Logics, Intuitionistic and Classical, Part 2
    Journal 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
  •  127
    Jan von Plato and Sara Negri, Structural Proof Theory (review)
    Philosophical Review 115 (2): 255-258. 2006.
  •  59
    Intensional Mathematics. Stewart Shapiro (review)
    Philosophy of Science 56 (1): 177-178. 1989.
  •  461
    More about uniform upper Bounds on ideals of Turing degrees
    Journal 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
  •  91
    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.
  •  87
    On some concepts associated with finite cardinal numbers
    Behavioral 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)
  •  21
    Jumping to a Uniform Upper Bound
    Proceedings 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.
  •  28
    Book Review. The Lambda-Calculus. H. P. Barendregt( (review)
    Philosophical Review 97 (1): 132-7. 1988.
  •  72
    Book Review. Basic Set Theory. Azriel Levy (review)
    Philosophical Review 90 (2): 298-300. 1981.
  •  210
    Well-behaved modal logics
    Journal of Symbolic Logic 49 (4): 1393-1402. 1984.
  • Association for Symbolic Logic
    with Jon Barwise, Howard S. Becker, Chi Tat Chong, Herbert B. Enderton, Michael Hallett, C. Ward Henson, Neil Immerman, Phokion Kolaitis, and Alistair Lachlan
    Bulletin of Symbolic Logic 4 (4): 465-510. 1998.
  •  238
    Ontological Commitments, Thick and Thin
    In 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
  •  8
    An 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.