•  526
    On The Sense and Reference of A Logical Constant
    Philosophical 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
  •  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
  •  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
  •  328
    Why 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
  •  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.
  •  318
    Jumping through the transfinite: The master code hierarchy of Turing degrees
    Journal 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
  •  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
  •  252
    The Modal Theory Of Pure Identity And Some Related Decision Problems
    Mathematical 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
  •  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
  •  234
  •  229
    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
  •  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
  •  210
    Well-behaved modal logics
    Journal of Symbolic Logic 49 (4): 1393-1402. 1984.
  •  199
    Cardinality logics. Part II: Definability in languages based on `exactly'
    Journal of Symbolic Logic 53 (3): 765-784. 1988.
  •  194
    Where 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".
  •  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
  •  127
    Jan von Plato and Sara Negri, Structural Proof Theory (review)
    Philosophical Review 115 (2): 255-258. 2006.
  •  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.