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Notre Dame Journal of Logic
Notre Dame Journal of Logic
Boney, Will
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
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\[\sup_{\Vert M\Vert =\lambda}\vert S^{\kappa}(M)\vert =(\sup_{\Vert M\Vert =\lambda}\vert S^{1}(M)\vert )^{\kappa}.\] 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.
Vasey, Sebastien
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.
Teh, Wen Chean
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.
Griffiths, Owen
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 language of informal mathematics—and argue that it lends inferentialism no support.
Becker, Howard
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 (AD): 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.
Guzmán, Osvaldo; Hrušák, Michael; Martínez-Celis, Arturo
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 of $\mathsf{ZFC}$ there are $\mathsf{MAD}$ families whose Mathias forcing does not add a dominating real. We also...
Halimi, Brice
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 set theory can be compared to a background universe and shown to contain internal models. It then defines logical...
Bezhanishvili, Guram; Bezhanishvili, Nick
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 (si-logics for short).
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The $\vee$ -free reducts of Heyting algebras give rise to the $(\wedge,\to)$ -canonical formulas that we studied in an earlier work. Here we introduce the...
Field, Hartry
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 (“Two flavors of Curry’s paradox”) 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 solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction.
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Raftery, James G.
This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.
Kowalski, Tomasz; Humberstone, Lloyd
We show the admissibility for $\mathsf{BCI}$ of a rule form of the characteristic implicational axiom of abelian logic, this rule taking us from $(\alpha\to\beta)\to\beta$ to $\alpha$ . This is done in Section 8, with surrounding sections exploring the admissibility and derivability of various related rules in several extensions of $\mathsf{BCI}$ .
Dzik, Wojciech; Wojtylak, Piotr
We characterize all finitary consequence relations over $\mathbf{S4.3}$ , both syntactically, by exhibiting so-called (admissible) passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic $L$ extending $\mathbf{S4}$ has projective unification if and only if $L$ contains $\mathbf{S4.3}$ . In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known results by Bull and Fine, from logics, to consequence relations. We also show that the lattice...
Citkin, Alex
A term $\mathit{td}(p,q,r)$ is called a ternary deductive (TD) term for a variety of algebras $\mathcal{V}$ if the identity $\mathit{td}(p,p,r)\approxr$ holds in $\mathcal{V}$ and $(\mathsf{c},\mathsf{d})\in\theta(\mathsf{a},\mathsf{b})$ yields $\mathit{td}(\mathsf{a},\mathsf{b},\mathsf{c})\approx\mathit{td}(\mathsf{a},\mathsf{b},\mathsf{d})$ for any $\mathscr{A}\in\mathcal{V}$ and any principal congruence $\theta$ on $\mathscr{A}$ . A connective $f(p_{1},\dots,p_{n})$ is called $\mathit{td}$ -distributive if $\mathit{td}(p,q,f(r_{1},\dots,r_{n}))\approx$ $f(\mathit{td}(p,q,r_{1}),\dots,\mathit{td}(p,q,r_{n}))$ . If $\mathsf{L}$ is a propositional logic and $\mathcal{V}$ is a corresponding variety (algebraic semantic) that has a TD term $\mathit{td}$ , then any admissible in $\mathsf{L}$ rule, the premises of which contain only $\mathit{td}$ -distributive operations, is derivable, and the substitution $r\mapsto\mathit{td}(p,q,r)$ is a projective $\mathsf{L}$ -unifier for any formula containing...
Cabrer, Leonardo
In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive lattices.
Baader, Franz; Binh, Nguyen Thanh; Borgwardt, Stefan; Morawska, Barbara
Unification in description logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive description logic $\mathcal{E\!L}$ is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using $\mathcal{E\!L}$ . On the other hand, unification in $\mathcal{E\!L}$ has been shown to be NP-complete and, thus, of considerably lower complexity than unification in other description logics of similarly restricted expressive power.
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However, $\mathcal{E\!L}$ allows the use of the top concept ( $\top$ ), which represents the whole interpretation domain, whereas the large medical...
Iemhoff, Rosalie; Metcalfe, George
Hamkins, Joel David; Leahy, Cole
We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets; that is, $\mathrm{HOA}=\mathrm{HOD}$ . Moreover, we show that every (pointwise) algebraic model of $\mathrm{ZF}$ is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp—an algebraic analogue of the constructible universe—which is obtained by iteratively adding not only the sets that are definable over what has been built so far, but also those that are...
Suzuki, Tomoyuki
In this paper, we give a possible characterization of the distributivity on bi-approximation semantics. To this end, we introduce new notions of special elements on polarities and show that the distributivity is first-order definable on bi-approximation semantics. In addition, we investigate the dual representation of those structures and compare them with bi-approximation semantics for intuitionistic logic. We also discuss that two different methods to validate the distributivity—by the splitters and by the adjointness—can be explicated with the help of the axiom of choice as well.