39 found
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  1. Logicism and the ontological commitments of arithmetic.Harold T. Hodes - 1984 - Journal of Philosophy 81 (3):123-149.
  2. Why Ramify?Harold T. Hodes - 2015 - Notre Dame Journal of Formal Logic 56 (2):379-415.
    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 implausible to make these arguments troubling. (...)
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  3. On modal logics which enrich first-order S5.Harold T. Hodes - 1984 - Journal of Philosophical Logic 13 (4):423 - 454.
  4. Some theorems on the expressive limitations of modal languages.Harold T. Hodes - 1984 - Journal of Philosophical Logic 13 (1):13 - 26.
  5. Axioms for actuality.Harold T. Hodes - 1984 - Journal of Philosophical Logic 13 (1):27 - 34.
  6. Where do the natural numbers come from?Harold T. Hodes - 1990 - Synthese 84 (3):347-407.
  7. The composition of Fregean thoughts.Harold T. Hodes - 1982 - Philosophical Studies 41 (2):161 - 178.
  8. Ontological Commitments, Thick and Thin.Harold T. Hodes - 1990 - In George Boolos (ed.), Method, Reason and Language: Essays in Honor of Hilary Putnam. Cambridge University Press. pp. 235-260.
    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, the semantic role (...)
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  9. One-step Modal Logics, Intuitionistic and Classical, Part 1.Harold T. Hodes - 2021 - Journal of Philosophical Logic 50 (5):837-872.
    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 presents (...)
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  10. Where do sets come from?Harold T. Hodes - 1991 - Journal of Symbolic Logic 56 (1):150-175.
    A model-theoretic approach to the semantics of set-theoretic discourse.
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  11. Individual-actualism and three-valued modal logics, part 1: Model-theoretic semantics.Harold T. Hodes - 1986 - Journal of Philosophical Logic 15 (4):369 - 401.
  12. Jumping through the transfinite: The master code hierarchy of Turing degrees.Harold T. Hodes - 1980 - Journal of Symbolic Logic 45 (2):204-220.
    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 operation.
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  13. Uniform Upper Bounds on Ideals of Turing Degrees.Harold T. Hodes - 1978 - Journal of Symbolic Logic 43 (3):601-612.
  14. Cut-conditions on sets of multiple-alternative inferences.Harold T. Hodes - 2022 - Mathematical Logic Quarterly 68 (1):95 - 106.
    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 between various (...)
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  15. One-Step Modal Logics, Intuitionistic and Classical, Part 2.Harold T. Hodes - 2021 - Journal of Philosophical Logic 50 (5):873-910.
    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 systems associated with the intuitionistic counterparts (...)
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  16. An Exact Pair for the Arithmetic Degrees Whose Join is Not a Weak Uniform Upper Bound.Harold T. Hodes - 1982 - Recursive Function Theory-Newsletters 28.
    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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  17.  53
    Jan von Plato and Sara Negri, Structural Proof Theory. [REVIEW]Harold T. Hodes - 2006 - Philosophical Review 115 (2):255-258.
  18. Individual-actualism and three-valued modal logics, part 2: Natural-deduction formalizations.Harold T. Hodes - 1987 - Journal of Philosophical Logic 16 (1):17 - 63.
  19. More about uniform upper Bounds on ideals of Turing degrees.Harold T. Hodes - 1983 - Journal of Symbolic Logic 48 (2):441-457.
    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 (...)
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  20. The Modal Theory Of Pure Identity And Some Related Decision Problems.Harold T. Hodes - 1984 - Mathematical Logic Quarterly 30 (26-29):415-423.
    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 concerning formulae with two monadic predicates.
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  21. Upper bounds on locally countable admissible initial segments of a Turing degree hierarchy.Harold T. Hodes - 1981 - Journal of Symbolic Logic 46 (4):753-760.
    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 (...)
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  22. Corrections to "where do sets come from?".Harold T. Hodes - 1991 - Journal of Symbolic Logic 56 (4):1486.
  23. Where Do the Cardinal Numbers Come From?Harold T. Hodes - 1990 - Synthese 84 (3):347-407.
    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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  24. Finite level borel games and a problem concerning the jump hierarchy.Harold T. Hodes - 1984 - Journal of Symbolic Logic 49 (4):1301-1318.
  25. Well-behaved modal logics.Harold T. Hodes - 1984 - Journal of Symbolic Logic 49 (4):1393-1402.
  26.  25
    Annual meeting of the Association for Symbolic Logic, New York City, December 1987.Nicholas Goodman, Harold T. Hodes, Carl G. Jockusch & Kenneth McAloon - 1988 - Journal of Symbolic Logic 53 (4):1287-1299.
  27.  15
    A Minimal Upper Bound on a Sequence of Turing Degrees Which Represents that Sequence.Harold T. Hodes - 1983 - Pacific Journal of Mathematics 108 (1):115-119.
  28. Harold Hodes: Bibliography.Harold T. Hodes - unknown
    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.
     
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  29.  25
    Jumping to a Uniform Upper Bound.Harold T. Hodes - 1982 - Proceedings of the American Mathematical Society 85 (4):600-602.
    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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  30.  97
    On some concepts associated with finite cardinal numbers.Harold T. Hodes - 2008 - Behavioral and Brain Sciences 31 (6):657-658.
    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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  31.  44
    Book Review. Basic Set Theory. Azriel Levy. [REVIEW]Harold T. Hodes - 1981 - Philosophical Review 90 (2):298-300.
  32.  37
    Book Review. Mechanism, Mentalism and Metamathematics. J Webb. [REVIEW]Harold T. Hodes - 1984 - Journal of Philosophy 81 (8):456-64.
  33.  16
    Book Review. Abstract Objects. Bob Hale. [REVIEW]Harold T. Hodes - 1992 - International Studies in Philosophy 24 (3):146-48.
  34. Book Review. Logic and Its Limits. P Shaw. [REVIEW]Harold T. Hodes - 1984 - History and Philosophy of Logic 5 (2):251.
  35. Book Review. Language and Philosophical Problems. Soren Stenland. [REVIEW]Harold T. Hodes - 1993 - History and Philosophy of Logic:253-6.
  36.  28
    Book Review. Principles of Intuitionism. Michael Dummett. [REVIEW]Harold T. Hodes - 1982 - Philosophical Review 91 (2):253-62.
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  37.  35
    Intensional Mathematics. Stewart Shapiro. [REVIEW]Harold T. Hodes - 1989 - Philosophy of Science 56 (1):177-178.
  38.  20
    Wang Hao. Reflections on Kurt Gödel. Bradford books. The MIT Press, Cambridge, Mass., and London, 1987, xxvi + 336 pp. [REVIEW]Harold T. Hodes - 1989 - Journal of Symbolic Logic 54 (3):1095-1098.
  39.  11
    Book Review. Reflections. Kurt Godel. [REVIEW]Harold T. Hodes - 1989 - THe Journal for Symbolic Logic 54 (3):1095-98.