Year:

  1.  4
    Strange Structures From Computable Model Theory.Howard Becker - 2017 - Notre Dame Journal of Formal Logic 58 (1):97-105.
    Let $L$ be a countable language, let ${\mathcal{I}}$ be an isomorphism-type of countable $L$-structures, and let $a\in2^{\omega}$. We say that ${\mathcal{I}}$ is $a$-strange if it contains a computable-from-$a$ structure and its Scott rank is exactly $\omega_{1}^{a}$. For all $a$, $a$-strange structures exist. Theorem : If $\mathcal{C}$ is a collection of $\aleph_{1}$ isomorphism-types of countable structures, then for a Turing cone of $a$’s, no member of $\mathcal{C}$ is $a$-strange.
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  2.  1
    Locally Finite Reducts of Heyting Algebras and Canonical Formulas.Guram Bezhanishvili & Nick Bezhanishvili - 2017 - Notre Dame Journal of Formal Logic 58 (1):21-45.
    The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the $\to$-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the $\vee$-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The $\vee$-free reducts of Heyting algebras give rise to the (...)
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  3. Computing the Number of Types of Infinite Length.Will Boney - 2017 - Notre Dame Journal of Formal Logic 58 (1):133-154.
    We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of $1$-types and the length of the sequences. Specifically, if $\kappa \leq \lambda$, then \[\sup_{\Vert M\Vert =\lambda}\vert S^{\kappa}\vert =\vert )^{\kappa}.\] We show that this holds for any abstract elementary class with $\lambda$-amalgamation. No such calculation is possible for nonalgebraic types. However, we introduce a subclass of nonalgebraic types for which the same upper bound holds.
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  4. Disarming a Paradox of Validity.Hartry Field - 2017 - Notre Dame Journal of Formal Logic 58 (1):1-19.
    Any theory of truth must find a way around Curry’s paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most “naive truth theorists.” To this end they recommend a radical (...)
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  5.  14
    Inferentialism and Quantification.Owen Griffiths - 2017 - Notre Dame Journal of Formal Logic 58 (1):107-113.
    Logical inferentialists contend that the meanings of the logical constants are given by their inference rules. Not just any rules are acceptable, however: inferentialists should demand that inference rules must reflect reasoning in natural language. By this standard, I argue, the inferentialist treatment of quantification fails. In particular, the inference rules for the universal quantifier contain free variables, which find no answer in natural language. I consider the most plausible natural language correlate to free variables—the use of variables in the (...)
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  6.  4
    Canjar Filters.Osvaldo Guzmán, Michael Hrušák & Arturo Martínez-Celis - 2017 - Notre Dame Journal of Formal Logic 58 (1):79-95.
    If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma}$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models (...)
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  7.  13
    Models as Universes.Brice Halimi - 2017 - Notre Dame Journal of Formal Logic 58 (1):47-78.
    Kreisel’s set-theoretic problem is the problem as to whether any logical consequence of ZFC is ensured to be true. Kreisel and Boolos both proposed an answer, taking truth to mean truth in the background set-theoretic universe. This article advocates another answer, which lies at the level of models of set theory, so that truth remains the usual semantic notion. The article is divided into three parts. It first analyzes Kreisel’s set-theoretic problem and proposes one way in which any model of (...)
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  8. Ramsey Algebras and Formal Orderly Terms.Wen Chean Teh - 2017 - Notre Dame Journal of Formal Logic 58 (1):115-125.
    Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman’s theorem. In this paper, we present the basic notions of Ramsey algebras by using terminology from mathematical logic. We also present some results regarding classification of Ramsey algebras.
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  9. Indiscernible Extraction and Morley Sequences.Sebastien Vasey - 2017 - Notre Dame Journal of Formal Logic 58 (1):127-132.
    We present a new proof of the existence of Morley sequences in simple theories. We avoid using the Erdős–Rado theorem and instead use only Ramsey’s theorem and compactness. The proof shows that the basic theory of forking in simple theories can be developed using only principles from “ordinary mathematics,” answering a question of Grossberg, Iovino, and Lessmann, as well as a question of Baldwin.
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