Notre Dame Journal of Formal Logic Articles (Project Euclid)
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The latest articles from Notre Dame Journal of Formal Logic on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 28 Apr 2011 09:02 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Numerical Abstraction via the Frege Quantifier
http://projecteuclid.org/euclid.ndjfl/1276284780
<strong>G. Aldo Antonelli</strong><p><strong>Source: </strong>Notre Dame J. Formal Logic, Volume 51, Number 2, 161--179.</p><p><strong>Abstract:</strong><br/>
This paper presents a formalization of first-order arithmetic characterizing the
natural numbers as abstracta of the equinumerosity relation. The
formalization turns on the interaction of a nonstandard (but still first-order)
cardinality quantifier with an abstraction operator assigning objects to
predicates. The project draws its philosophical motivation from a
nonreductionist conception of logicism, a deflationary view of abstraction, and
an approach to formal arithmetic that emphasizes the cardinal properties
of the natural numbers over the structural ones.
</p>projecteuclid.org/euclid.ndjfl/1276284780_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTAdmissible Rules and the Leibniz Hierarchyhttp://projecteuclid.org/euclid.ndjfl/1472746139<strong>James G. Raftery</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 38 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ndjfl/1472746139_20160901120909Thu, 01 Sep 2016 12:09 EDTAn Abelian Rule for BCI—and Variationshttp://projecteuclid.org/euclid.ndjfl/1473686417<strong>Tomasz Kowalski</strong>, <strong>Lloyd Humberstone</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
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 $\mathsf{BCI}$ .
</p>projecteuclid.org/euclid.ndjfl/1473686417_20160912092028Mon, 12 Sep 2016 09:20 EDTUnification on Subvarieties of Pseudocomplemented Distributive Latticeshttp://projecteuclid.org/euclid.ndjfl/1473686418<strong>Leonardo Cabrer</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ndjfl/1473686418_20160912092028Mon, 12 Sep 2016 09:20 EDTSpecial Issue on Admissible Rules and Unificationhttp://projecteuclid.org/euclid.ndjfl/1477399541<strong>Rosalie Iemhoff</strong>, <strong>George Metcalfe</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 57, Number 4, 441--442.</p>projecteuclid.org/euclid.ndjfl/1477399541_20161025084553Tue, 25 Oct 2016 08:45 EDTLocally Finite Reducts of Heyting Algebras and Canonical Formulashttp://projecteuclid.org/euclid.ndjfl/1479351685<strong>Guram Bezhanishvili</strong>, <strong>Nick Bezhanishvili</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
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).
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 $(\wedge,\vee)$ -canonical formulas, which are obtained from the study of the $\to$ -free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by $(\wedge,\vee)$ -canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas.
One of the main ingredients of these formulas is a designated subset $D$ of pairs of elements of a finite subdirectly irreducible Heyting algebra $A$ . When $D=A^{2}$ , we show that the $(\wedge,\vee)$ -canonical formula of $A$ is equivalent to the Jankov formula of $A$ . On the other hand, when $D=\emptyset$ , the $(\wedge,\vee)$ -canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics.
</p>projecteuclid.org/euclid.ndjfl/1479351685_20161116220154Wed, 16 Nov 2016 22:01 ESTStrange Structures from Computable Model Theoryhttp://projecteuclid.org/euclid.ndjfl/1479351686<strong>Howard Becker</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ndjfl/1479351686_20161116220154Wed, 16 Nov 2016 22:01 ESTDisarming a Paradox of Validityhttp://projecteuclid.org/euclid.ndjfl/1479351687<strong>Hartry Field</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
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.
In this paper I argue that substructuralism is unnecessary. Diagnosing the “v-Curry” is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem.
</p>projecteuclid.org/euclid.ndjfl/1479351687_20161116220154Wed, 16 Nov 2016 22:01 ESTInferentialism and Quantificationhttp://projecteuclid.org/euclid.ndjfl/1480042819<strong>Owen Griffiths</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 7 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ndjfl/1480042819_20161124220039Thu, 24 Nov 2016 22:00 ESTComputing the Number of Types of Infinite Lengthhttp://projecteuclid.org/euclid.ndjfl/1480042820<strong>Will Boney</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
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
\[\sup_{\Vert M\Vert =\lambda}\vert S^{\kappa}(M)\vert =(\sup_{\Vert M\Vert =\lambda}\vert S^{1}(M)\vert )^{\kappa}.\] 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.
</p>projecteuclid.org/euclid.ndjfl/1480042820_20161124220039Thu, 24 Nov 2016 22:00 ESTRamsey Algebras and Formal Orderly Termshttp://projecteuclid.org/euclid.ndjfl/1480647803<strong>Wen Chean Teh</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ndjfl/1480647803_20161201220349Thu, 01 Dec 2016 22:03 ESTIndiscernible Extraction and Morley Sequenceshttp://projecteuclid.org/euclid.ndjfl/1480647804<strong>Sebastien Vasey</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 6 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ndjfl/1480647804_20161201220349Thu, 01 Dec 2016 22:03 ESTModels as Universeshttp://projecteuclid.org/euclid.ndjfl/1481684566<strong>Brice Halimi</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 32 pages.</p><p><strong>Abstract:</strong><br/>
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 consequence with respect to a model of ZFC, solves the model-scaled version of Kreisel’s set-theoretic problem, and presents various further results bearing on internal models. Finally, internal models are presented as accessible worlds, leading to an internal modal logic in which internal reflection corresponds to modal reflexivity, and resplendency corresponds to modal axiom 4 .
</p>projecteuclid.org/euclid.ndjfl/1481684566_20161213220303Tue, 13 Dec 2016 22:03 ESTEditorial Noticehttp://projecteuclid.org/euclid.ndjfl/1484125216<p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 58, Number 1, 155--155.</p>projecteuclid.org/euclid.ndjfl/1484125216_20170111040129Wed, 11 Jan 2017 04:01 ESTErratumhttp://projecteuclid.org/euclid.ndjfl/1484125217<p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 58, Number 1, 157--157.</p>projecteuclid.org/euclid.ndjfl/1484125217_20170111040129Wed, 11 Jan 2017 04:01 ESTInfinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Fieldhttp://projecteuclid.org/euclid.ndjfl/1484902818<strong>Patrick Reeder</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
The primary purpose of this paper is to analyze the relationship between the familiar non-Archimedean field of hyperreals from Abraham Robinson’s nonstandard analysis and Paolo Giordano’s ring extension of the real numbers containing nilpotents. There is an interesting nontrivial homomorphism from the limited hyperreals into the Giordano ring, whereas the only nontrivial homomorphism from the Giordano ring to the hyperreals is the standard part function, namely, the function that maps a value to its real part. We interpret this asymmetry to mean that the nilpotent infinitesimal values of Giordano’s ring are “smaller” than the hyperreal infinitesimals. By viewing things from the “point of view” of the hyperreals, all nilpotents are zero, whereas by viewing things from the “point of view” of Giordano’s ring, nonnilpotent, nonzero infinitesimals register as nonzero infinitesimals. This suggests that Giordano’s infinitesimals are more fine-grained.
</p>projecteuclid.org/euclid.ndjfl/1484902818_20170120040045Fri, 20 Jan 2017 04:00 ESTUniversal Structureshttp://projecteuclid.org/euclid.ndjfl/1485572517<strong>Saharon Shelah</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
We deal with the existence of universal members in a given cardinality for several classes. First, we deal with classes of abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality or $\lambda=\lambda^{\aleph_{0}}$ . We use versions of being reduced—replacing $\mathbb{Q}$ by a subring (defined by a sequence $\bar{t}$ )—and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with (variants of) the oak property (from a work of Džamonja and the author), a property of complete first-order theories sufficient for the nonexistence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for quantifier-free formulas) and deals more with the existence of universals.
</p>projecteuclid.org/euclid.ndjfl/1485572517_20170127220220Fri, 27 Jan 2017 22:02 ESTWhy Intuitionistic Relevant Logic Cannot Be a Core Logichttp://projecteuclid.org/euclid.ndjfl/1486177446<strong>Joseph Vidal-Rosset</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 8 pages.</p><p><strong>Abstract:</strong><br/>
At the end of the 1980s, Tennant invented a logical system that he called “intuitionistic relevant logic” ( $\mathbf{IR}$ , for short). Now he calls this same system “Core logic.” In Section 1, by reference to the rules of natural deduction for $\mathbf{IR}$ , I explain why $\mathbf{IR}$ is a relevant logic in a subtle way. Sections 2, 3, and 4 give three reasons to assert that $\mathbf{IR}$ cannot be a core logic.
</p>projecteuclid.org/euclid.ndjfl/1486177446_20170203220420Fri, 03 Feb 2017 22:04 ESTIndependence of the Dual Axiom in Modal $\mathbf{K}$ with Primitive $\lozenge$http://projecteuclid.org/euclid.ndjfl/1486630844<strong>Richmond Thomason</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 5 pages.</p><p><strong>Abstract:</strong><br/>
Explicit axioms relating $\lozenge \phi$ and $\Box \phi$ appear to be needed if $\lozenge$ is taken to be primitive. We prove that such axioms are in fact indispensable.
</p>projecteuclid.org/euclid.ndjfl/1486630844_20170209040058Thu, 09 Feb 2017 04:00 ESTInfinite Computations with Random Oracleshttp://projecteuclid.org/euclid.ndjfl/1487646410<strong>Merlin Carl</strong>, <strong>Philipp Schlicht</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of $\mathrm{ZFC}$ for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and resetting infinite-time register machines, and $\alpha$ -Turing machines for countable admissible ordinals $\alpha$ .
</p>projecteuclid.org/euclid.ndjfl/1487646410_20170220220709Mon, 20 Feb 2017 22:07 ESTDunn–Priest Quotients of Many-Valued Structureshttp://projecteuclid.org/euclid.ndjfl/1487646411<strong>Thomas Macaulay Ferguson</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
J. Michael Dunn’s Theorem in 3-Valued Model Theory and Graham Priest’s Collapsing Lemma provide the means of constructing first-order, three-valued structures from classical models while preserving some control over the theories of the ensuing models. The present article introduces a general construction that we call a Dunn–Priest quotient , providing a more general means of constructing models for arbitrary many-valued, first-order logical systems from models of any second system. This technique not only counts Dunn’s and Priest’s techniques as special cases, but also provides a generalized Collapsing Lemma for Priest’s more recent plurivalent semantics in general. We examine when and how much control may be exerted over the resulting theories in particular cases. Finally, we expand the utility of the construction by showing that taking Dunn–Priest quotients of a family of structures commutes with taking an ultraproduct of that family, increasing the versatility of the tool.
</p>projecteuclid.org/euclid.ndjfl/1487646411_20170220220709Mon, 20 Feb 2017 22:07 ESTConcrete Fibrationshttp://projecteuclid.org/euclid.ndjfl/1487646412<strong>Ruggero Pagnan</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
As far as we know, no notion of concrete fibration is available. We provide one such notion in adherence to the foundational attitude that characterizes the adoption of the fibrational perspective in approaching fundamental subjects in category theory and discuss it in connection with the notion of concrete category and the notions of locally small and small fibrations. We also discuss the appropriateness of our notion of concrete fibration for fibrations of small maps, which is relevant to algebraic set theory.
</p>projecteuclid.org/euclid.ndjfl/1487646412_20170220220709Mon, 20 Feb 2017 22:07 ESTOn Polynomial-Time Relation Reducibilityhttp://projecteuclid.org/euclid.ndjfl/1488510091<strong>Su Gao</strong>, <strong>Caleb Ziegler</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
We study the notion of polynomial-time relation reducibility among computable equivalence relations. We identify some benchmark equivalence relations and show that the reducibility hierarchy has a rich structure. Specifically, we embed the partial order of all polynomial-time computable sets into the polynomial-time relation reducibility hierarchy between two benchmark equivalence relations $\mathsf{E}_{\lambda}$ and $\mathsf{id}$ . In addition, we consider equivalence relations with finitely many nontrivial equivalence classes and those whose equivalence classes are all finite.
</p>projecteuclid.org/euclid.ndjfl/1488510091_20170302220204Thu, 02 Mar 2017 22:02 ESTBimodal Logics with a “Weakly Connected” Component without the Finite Model Propertyhttp://projecteuclid.org/euclid.ndjfl/1489028416<strong>Agi Kurucz</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
There are two known general results on the finite model property (fmp) of commutators $[L_{0},L_{1}]$ (bimodal logics with commuting and confluent modalities). If $L$ is finitely axiomatizable by modal formulas having universal Horn first-order correspondents, then both $[L,{\mathbf{K}}]$ and $[L,{\mathbf{S5}}]$ are determined by classes of frames that admit filtration, and so they have the fmp. On the negative side, if both $L_{0}$ and $L_{1}$ are determined by transitive frames and have frames of arbitrarily large depth, then $[L_{0},L_{1}]$ does not have the fmp. In this paper we show that commutators with a “weakly connected” component often lack the fmp. Our results imply that the above positive result does not generalize to universally axiomatizable component logics, and even commutators without “transitive” components such as $[{\mathbf{K3}},{\mathbf{K}}]$ can lack the fmp. We also generalize the above negative result to cases where one of the component logics has frames of depth one only, such as $[{\mathbf{S4.3}},{\mathbf{S5}}]$ and the decidable product logic ${\mathbf{S4.3}}\!\times\!{\mathbf{S5}}$ . We also show cases when already half of commutativity is enough to force infinite frames.
</p>projecteuclid.org/euclid.ndjfl/1489028416_20170308220053Wed, 08 Mar 2017 22:00 ESTRandomness and Semimeasureshttp://projecteuclid.org/euclid.ndjfl/1489543214<strong>Laurent Bienvenu</strong>, <strong>Rupert Hölzl</strong>, <strong>Christopher P. Porter</strong>, <strong>Paul Shafer</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 28 pages.</p><p><strong>Abstract:</strong><br/>
A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak $2$ -randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Löf randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a positive answer to which would also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases.
</p>projecteuclid.org/euclid.ndjfl/1489543214_20170314220031Tue, 14 Mar 2017 22:00 EDTNormal Numbers and Limit Computable Cantor Serieshttp://projecteuclid.org/euclid.ndjfl/1490148081<strong>Achilles Beros</strong>, <strong>Konstantinos Beros</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 6 pages.</p><p><strong>Abstract:</strong><br/>
Given any oracle, $A$ , we construct a basic sequence $Q$ , computable in the jump of $A$ , such that no $A$ -computable real is $Q$ -distribution-normal. A corollary to this is that there is a $\Delta^{0}_{n+1}$ basic sequence with respect to which no $\Delta^{0}_{n}$ real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.
</p>projecteuclid.org/euclid.ndjfl/1490148081_20170321220151Tue, 21 Mar 2017 22:01 EDTA Completed System for Robin Smith’s Incomplete Ecthetic Syllogistichttp://projecteuclid.org/euclid.ndjfl/1490234420<strong>Pierre Joray</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we first show that Robin Smith’s ecthetic system $\mathit{SE}$ for Aristotle’s assertoric syllogistic is not complete, despite what is claimed by Smith. $\mathit{SE}$ is then not adequate to establish that ecthesis allows one to dispense with indirect or per impossibile deductions in Aristotle’s assertoric logic. As an alternative to $\mathit{SE}$ , we then present a stronger system $\mathit{EC}$ which is adequate for this purpose. $\mathit{EC}$ is a nonexplosive ecthetic system which is shown to be sound and complete with respect to all valid syllogistic arguments with a consistent set of premises.
</p>projecteuclid.org/euclid.ndjfl/1490234420_20170322220112Wed, 22 Mar 2017 22:01 EDTTwo Upper Bounds on Consistency Strength of $\neg\square_{\aleph_{\omega}}$ and Stationary Set Reflection at Two Successive $\aleph_{n}$http://projecteuclid.org/euclid.ndjfl/1491012044<strong>Martin Zeman</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 24 pages.</p><p><strong>Abstract:</strong><br/>
We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a $\kappa^{+}$ -supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into $\aleph_{\omega}$ and make the principle $\square_{\aleph_{\omega},\lt \omega}$ fail in the generic extension. We also show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into $\aleph_{2}$ and arrange in the generic extension that simultaneous reflection holds at $\aleph_{2}$ , and at the same time, every stationary subset of $\aleph_{3}$ concentrating on points of cofinality $\omega$ has a reflection point of cofinality $\omega_{1}$ .
</p>projecteuclid.org/euclid.ndjfl/1491012044_20170331220111Fri, 31 Mar 2017 22:01 EDTNonstandard Functional Interpretations and Categorical Modelshttp://projecteuclid.org/euclid.ndjfl/1492567509<strong>Amar Hadzihasanovic</strong>, <strong>Benno van den Berg</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 38 pages.</p><p><strong>Abstract:</strong><br/>
Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica , a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization . We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models.
</p>projecteuclid.org/euclid.ndjfl/1492567509_20170418220520Tue, 18 Apr 2017 22:05 EDTClub-Isomorphisms of Aronszajn Trees in the Extension with a Suslin Treehttp://projecteuclid.org/euclid.ndjfl/1492567510<strong>Teruyuki Yorioka</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
We show that, under $\mathsf{PFA}(S)$ , a coherent Suslin tree forces that every two Aronszajn trees are club-isomorphic.
</p>projecteuclid.org/euclid.ndjfl/1492567510_20170418220520Tue, 18 Apr 2017 22:05 EDTSelective and Ramsey Ultrafilters on $G$ -spaceshttp://projecteuclid.org/euclid.ndjfl/1492567511<strong>Oleksandr Petrenko</strong>, <strong>Igor Protasov</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 7 pages.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a group, and let $X$ be an infinite transitive $G$ -space. A free ultrafilter $\mathcal{U}$ on $X$ is called $G$ -selective if, for any $G$ -invariant partition $\mathcal{P}$ of $X$ , either one cell of $\mathcal{P}$ is a member of $\mathcal{U}$ , or there is a member of $\mathcal{U}$ which meets each cell of $\mathcal{P}$ in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a $G$ -selective ultrafilter on $X$ . We describe all $G$ -spaces $X$ such that each free ultrafilter on $X$ is $G$ -selective, and we prove that a free ultrafilter $\mathcal{U}$ on $\omega$ is selective if and only if $\mathcal{U}$ is $G$ -selective with respect to the action of any countable group $G$ of permutations of $\omega$ .
A free ultrafilter $\mathcal{U}$ on $X$ is called $G$ -Ramsey if, for any $G$ -invariant coloring $\chi:[X]^{2}\to\{0,1\}$ , there is $U\in\mathcal{U}$ such that $[U]^{2}$ is $\chi$ -monochromatic. We show that each $G$ -Ramsey ultrafilter on $X$ is $G$ -selective. Additional theorems give a lot of examples of ultrafilters on $\mathbb{Z}$ that are $\mathbb{Z}$ -selective but not $\mathbb{Z}$ -Ramsey.
</p>projecteuclid.org/euclid.ndjfl/1492567511_20170418220520Tue, 18 Apr 2017 22:05 EDTA Diamond Principle Consistent with ADhttp://projecteuclid.org/euclid.ndjfl/1492761611<strong>Daniel Cunningham</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
We present a diamond principle $\lozenge_{\mathbb{R}}$ concerning all subsets of $\Theta$ , the supremum of the ordinals that are the surjective image of $\mathbb{R}$ . We prove that $\lozenge_{\mathbb{R}}$ holds in Steel’s core model $\mathbf{K}(\mathbb{R})$ , a canonical inner model for determinacy.
</p>projecteuclid.org/euclid.ndjfl/1492761611_20170421040040Fri, 21 Apr 2017 04:00 EDTDecidable Fragments of the Simple Theory of Types with Infinity and $\mathrm{NF}$http://projecteuclid.org/euclid.ndjfl/1492761612<strong>Anuj Dawar</strong>, <strong>Thomas Forster</strong>, <strong>Zachiri McKenzie</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
We identify complete fragments of the simple theory of types with infinity ( $\mathrm{TSTI}$ ) and Quine’s new foundations ( $\mathrm{NF}$ ) set theory. We show that $\mathrm{TSTI}$ decides every sentence $\phi$ in the language of type theory that is in one of the following forms:
(A) $\phi=\forall x_{1}^{r_{1}}\cdots \forall x_{k}^{r_{k}}\exists y_{1}^{s_{1}}\cdots \exists y_{l}^{s_{l}}\theta$ where the superscripts denote the types of the variables, $s_{1}\gt \cdots \gt s_{l}$ , and $\theta$ is quantifier-free,
(B) $\phi=\forall x_{1}^{r_{1}}\cdots \forall x_{k}^{r_{k}}\exists y_{1}^{s}\cdots \exists y_{l}^{s}\theta$ where the superscripts denote the types of the variables and $\theta$ is quantifier-free.
This shows that $\mathrm{NF}$ decides every stratified sentence $\phi$ in the language of set theory that is in one of the following forms:
(A $ ) $\phi=\forall x_{1}\cdots \forall x_{k}\exists y_{1}\cdots \exists y_{l}\theta$ where $\theta$ is quantifier-free and $\phi$ admits a stratification that assigns distinct values to all of the variables $y_{1},\ldots,y_{l}$ ,
(B $ ) $\phi=\forall x_{1}\cdots \forall x_{k}\exists y_{1}\cdots \exists y_{l}\theta$ where $\theta$ is quantifier-free and $\phi$ admits a stratification that assigns the same value to all of the variables $y_{1},\ldots,y_{l}$ .
</p>projecteuclid.org/euclid.ndjfl/1492761612_20170421040040Fri, 21 Apr 2017 04:00 EDTGrades of Discrimination: Indiscernibility, Symmetry, and Relativityhttp://projecteuclid.org/euclid.ndjfl/1493085740<strong>Tim Button</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas and draws connections with definability theory.
</p>projecteuclid.org/euclid.ndjfl/1493085740_20170424220234Mon, 24 Apr 2017 22:02 EDTNew Degree Spectra of Abelian Groupshttp://projecteuclid.org/euclid.ndjfl/1494640857<strong>Alexander G. Melnikov</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
We show that for every computable ordinal of the form $\beta=\delta+2n+1\gt 1$ , where $\delta$ is zero or a limit ordinal and $n\in\omega$ , there exists a torsion-free abelian group having an $X$ -computable copy if and only if $X$ is nonlow $_{\beta}$ .
</p>projecteuclid.org/euclid.ndjfl/1494640857_20170512220117Fri, 12 May 2017 22:01 EDTProspects for a Naive Theory of Classeshttp://projecteuclid.org/euclid.ndjfl/1496736029<strong>Hartry Field</strong>, <strong>Harvey Lederman</strong>, <strong>Tore Fjetland Øgaard</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 46 pages.</p><p><strong>Abstract:</strong><br/>
The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak.
</p>projecteuclid.org/euclid.ndjfl/1496736029_20170606040111Tue, 06 Jun 2017 04:01 EDTForking and Dividing in Henson Graphshttp://projecteuclid.org/euclid.ndjfl/1496736030<strong>Gabriel Conant</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
For $n\geq3$ , define $T_{n}$ to be the theory of the generic $K_{n}$ -free graph, where $K_{n}$ is the complete graph on $n$ vertices. We prove a graph-theoretic characterization of dividing in $T_{n}$ and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, $T_{n}$ provides a counterexample to a question of Chernikov and Kaplan.
</p>projecteuclid.org/euclid.ndjfl/1496736030_20170606040111Tue, 06 Jun 2017 04:01 EDTClassifications of Computable Structureshttp://projecteuclid.org/euclid.ndjfl/1498788255<strong>Karen Lange</strong>, <strong>Russell Miller</strong>, <strong>Rebecca M. Steiner</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{K}$ be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from $\mathcal{K}$ such that every structure in $\mathcal{K}$ is isomorphic to exactly one structure on the list. Such a list is called a computable classification of $\mathcal{K}$ , up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a $\mathbf{0}'$ -oracle, we can obtain similar classifications of the families of computable equivalence structures and of computable finite-branching trees. However, there is no computable classification of the latter, nor of the family of computable torsion-free abelian groups of rank $1$ , even though these families are both closely allied with computable algebraic fields.
</p>projecteuclid.org/euclid.ndjfl/1498788255_20170629220431Thu, 29 Jun 2017 22:04 EDTThe Logical Strength of Compositional Principleshttp://projecteuclid.org/euclid.ndjfl/1499241609<strong>Richard G. Heck Jr.</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 33 pages.</p><p><strong>Abstract:</strong><br/>
This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism . Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken together, have substantial logical strength, amounting to a kind of abstract consistency statement.
</p>projecteuclid.org/euclid.ndjfl/1499241609_20170705040018Wed, 05 Jul 2017 04:00 EDTEkman’s Paradoxhttp://projecteuclid.org/euclid.ndjfl/1500364943<strong>Peter Schroeder-Heister</strong>, <strong>Luca Tranchini</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
Prawitz observed that Russell’s paradox in naive set theory yields a derivation of absurdity whose reduction sequence loops. Building on this observation, and based on numerous examples, Tennant claimed that this looping feature, or more generally, the fact that derivations of absurdity do not normalize, is characteristic of the paradoxes. Striking results by Ekman show that looping reduction sequences are already obtained in minimal propositional logic, when certain reduction steps, which are prima facie plausible, are considered in addition to the standard ones. This shows that the notion of reduction is in need of clarification. Referring to the notion of identity of proofs in general proof theory, we argue that reduction steps should not merely remove redundancies, but must respect the identity of proofs. Consequentially, we propose to modify Tennant’s paradoxicality test by basing it on this refined notion of reduction.
</p>projecteuclid.org/euclid.ndjfl/1500364943_20170718040302Tue, 18 Jul 2017 04:03 EDTOn the Jumps of the Degrees Below a Recursively Enumerable Degreehttp://projecteuclid.org/euclid.ndjfl/1500537625<strong>David R. Belanger</strong>, <strong>Richard A. Shore</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
We consider the set of jumps below a Turing degree, given by $\mathsf{JB}(\mathbf{a})=\{\mathbf{x}':\mathbf{x}\leq\mathbf{a}\}$ , with a focus on the problem: Which recursively enumerable (r.e.) degrees $\mathbf{a}$ are uniquely determined by $\mathsf{JB}(\mathbf{a})$ ? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order $\mathcal{R}$ of r.e. degrees. Namely, we show that if every high ${}_{2}$ r.e. degree $\mathbf{a}$ is determined by $\mathsf{JB}(\mathbf{a})$ , then $\mathcal{R}$ cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing pairs $\mathbf{a}_{0}$ , $\mathbf{a}_{1}$ of distinct r.e. degrees such that $\mathsf{JB}(\mathbf{a}_{0})=\mathsf{JB}(\mathbf{a}_{1})$ within any possible jump class $\{\mathbf{x}:\mathbf{x}'=\mathbf{c}\}$ . We give some extensions of the construction and suggest ways to salvage the attack on rigidity.
</p>projecteuclid.org/euclid.ndjfl/1500537625_20170720040101Thu, 20 Jul 2017 04:01 EDTThe Admissible Rules of ${{\mathsf{BD}_{2}}}$ and ${\mathsf{GSc}}$http://projecteuclid.org/euclid.ndjfl/1501639384<strong>Jeroen P. Goudsmit</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 29 pages.</p><p><strong>Abstract:</strong><br/>
The Visser rules form a basis of admissibility for the intuitionistic propositional calculus. We show how one can characterize the existence of covers in certain models by means of formulae. Through this characterization, we provide a new proof of the admissibility of a weak form of the Visser rules. Finally, we use this observation, coupled with a description of a generalization of the disjunction property, to provide a basis of admissibility for the intermediate logics ${{\mathsf{BD}_{2}}}$ and ${\mathsf{GSc}}$ .
</p>projecteuclid.org/euclid.ndjfl/1501639384_20170801220326Tue, 01 Aug 2017 22:03 EDTInvariance and Definability, with and without Equalityhttps://projecteuclid.org/euclid.ndjfl/1503626493<strong>Denis Bonnay</strong>, <strong>Fredrik Engström</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 24 pages.</p><p><strong>Abstract:</strong><br/>
The dual character of invariance under transformations and definability by some operations has been used in classical works by, for example, Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this article, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in $\mathscr{L}_{\infty\infty}$ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee’s theorem about quantifiers invariant under all permutations and definability in pure $\mathscr{L}_{\infty\infty}$ as a particular case. We also prove some optimality results along the way, regarding the kinds of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.
</p>projecteuclid.org/euclid.ndjfl/1503626493_20170824220205Thu, 24 Aug 2017 22:02 EDTNegation-Free and Contradiction-Free Proof of the Steiner–Lehmus Theoremhttps://projecteuclid.org/euclid.ndjfl/1504252824<strong>Victor Pambuccian</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.
</p>projecteuclid.org/euclid.ndjfl/1504252824_20170901040102Fri, 01 Sep 2017 04:01 EDTCardinality and Acceptable Abstractionhttps://projecteuclid.org/euclid.ndjfl/1510802482<strong>Roy T. Cook</strong>, <strong>Øystein Linnebo</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but we analyze the interesting idea on which it is based, namely that an acceptable abstraction has to “generate” the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.
</p>projecteuclid.org/euclid.ndjfl/1510802482_20171115222139Wed, 15 Nov 2017 22:21 ESTActualism, Serious Actualism, and Quantified Modal Logichttps://projecteuclid.org/euclid.ndjfl/1510888080<strong>William H. Hanson</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 52 pages.</p><p><strong>Abstract:</strong><br/>
This article studies seriously actualistic quantified modal logics. A key component of the language is an abstraction operator by means of which predicates can be created out of complex formulas. This facilitates proof of a uniform substitution theorem: if a sentence is logically true, then any sentence that results from substituting a (perhaps complex) predicate abstract for each occurrence of a simple predicate abstract is also logically true. This solves a problem identified by Kripke early in the modern semantic study of quantified modal logic. A tableau proof system is presented and proved sound and complete with respect to logical truth. The main focus is on seriously actualistic T (SAT), an extension of T, but the results established hold also for systems based on other propositional modal logics (e.g., K, B, S4, and S5). Following Menzel it is shown that the formal language studied also supports an actualistic account of truth simpliciter.
</p>projecteuclid.org/euclid.ndjfl/1510888080_20171116220824Thu, 16 Nov 2017 22:08 ESTOstrowski Numeration Systems, Addition, and Finite Automatahttps://projecteuclid.org/euclid.ndjfl/1513998207<strong>Philipp Hieronymi</strong>, <strong>Alonza Terry Jr.</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When $a$ is quadratic, addition in the Ostrowski numeration system based on $a$ is recognizable by a finite automaton. We deduce that a subset of $X\subseteq\mathbb{N}^{n}$ is definable in $(\mathbb{N},+,V_{a})$ , where $V_{a}$ is the function that maps a natural number $x$ to the smallest denominator of a convergent of $a$ that appears in the Ostrowski representation based on $a$ of $x$ with a nonzero coefficient if and only if the set of Ostrowski representations of elements of $X$ is recognizable by a finite automaton. The decidability of the theory of $(\mathbb{N},+,V_{a})$ follows.
</p>projecteuclid.org/euclid.ndjfl/1513998207_20171222220351Fri, 22 Dec 2017 22:03 ESTErratahttps://projecteuclid.org/euclid.ndjfl/1515121303<p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 59, Number 1, 135--138.</p>projecteuclid.org/euclid.ndjfl/1515121303_20180104220202Thu, 04 Jan 2018 22:02 ESTNonreduction of Relations in the Gromov Space to Polish Actionshttps://projecteuclid.org/euclid.ndjfl/1515402015<strong>Jesús A. Álvarez López</strong>, <strong>Alberto Candel</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
We show that in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov–Hausdorff distance cannot be reduced to the equivalence relation defined by any Polish action.
</p>projecteuclid.org/euclid.ndjfl/1515402015_20180108040031Mon, 08 Jan 2018 04:00 ESTA Problem in Pythagorean Arithmetichttps://projecteuclid.org/euclid.ndjfl/1515467280<strong>Victor Pambuccian</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 8 pages.</p><p><strong>Abstract:</strong><br/>
Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples $(a,b,c)$ of positive integers for which $ab-c$ , $bc-a$ , and $ca-b$ are all powers of $2$ . We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.
</p>projecteuclid.org/euclid.ndjfl/1515467280_20180108220821Mon, 08 Jan 2018 22:08 EST