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 EDTConcrete 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 ESTBlurring: An Approach to Conflationhttps://projecteuclid.org/euclid.ndjfl/1516849225<strong>David Ripley</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
I consider the phenomenon of conflation —treating distinct things as one—and develop logical tools for modeling it. These tools involve a purely consequence-theoretic treatment, independent of any proof or model theory, as well as a four-valued valuational treatment.
</p>projecteuclid.org/euclid.ndjfl/1516849225_20180124220048Wed, 24 Jan 2018 22:00 ESTOn Superstable Expansions of Free Abelian Groupshttps://projecteuclid.org/euclid.ndjfl/1517216524<strong>Daniel Palacín</strong>, <strong>Rizos Sklinos</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We prove that $(\mathbb{Z},+,0)$ has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank $\omega$ . Additionally, our methods yield other superstable expansions such as $(\mathbb{Z},+,0)$ equipped with the set of factorial elements.
</p>projecteuclid.org/euclid.ndjfl/1517216524_20180129040225Mon, 29 Jan 2018 04:02 ESTTwo More Characterizations of K -Trivialityhttps://projecteuclid.org/euclid.ndjfl/1517540521<strong>Noam Greenberg</strong>, <strong>Joseph S. Miller</strong>, <strong>Benoit Monin</strong>, <strong>Daniel Turetsky</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 7 pages.</p><p><strong>Abstract:</strong><br/> We give two new characterizations of $K$ -triviality. We show that if for all $Y$ such that $\Omega$ is $Y$ -random, $\Omega$ is $(Y\oplusA)$ -random, then $A$ is $K$ -trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of $K$ -triviality and answering a question of Yu. We also prove that if $A$ is $K$ -trivial, then for all $Y$ such that $\Omega$ is $Y$ -random, $(Y\oplus A)\equiv_{\textup{LR}}Y$ . This answers a question of Merkle and Yu. The other direction is immediate, so we have the second characterization of $K$ -triviality. The proof of the first characterization uses a new cupping result. We prove that if $A\nleq_{\textup{LR}}B$ , then for every set $X$ there is a $B$ -random set $Y$ such that $X$ is computable from $Y\oplus A$ . </p>projecteuclid.org/euclid.ndjfl/1517540521_20180201220215Thu, 01 Feb 2018 22:02 ESTThe Complexity of Primes in Computable Unique Factorization Domainshttps://projecteuclid.org/euclid.ndjfl/1519722286<strong>Damir D. Dzhafarov</strong>, <strong>Joseph R. Mileti</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
In many simple integral domains, such as $\mathbb{Z}$ or $\mathbb{Z}[i]$ , there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact that such a naive approach does not immediately translate to integral domains like $\mathbb{Z}[x]$ or the ring of integers in an algebraic number field, there still exist computational procedures that work to determine the prime elements in these cases. In contrast, we will show how to computably extend $\mathbb{Z}$ in such a way that we can control the ordinary integer primes in any $\Pi_{2}^{0}$ way, all while maintaining unique factorization. As a corollary, we establish the existence of a computable unique factorization domain (UFD) such that the set of primes is $\Pi_{2}^{0}$ -complete in every computable presentation.
</p>projecteuclid.org/euclid.ndjfl/1519722286_20180227040455Tue, 27 Feb 2018 04:04 ESTCoding and Definability in Computable Structureshttps://projecteuclid.org/euclid.ndjfl/1525140052<strong>Antonio Montalbán</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
These are the lecture notes from a 10-hour course that the author gave at the University of Notre Dame in September 2010. The objective of the course was to introduce some basic concepts in computable structure theory and develop the background needed to understand the author’s research on back-and-forth relations.
</p>projecteuclid.org/euclid.ndjfl/1525140052_20180430220056Mon, 30 Apr 2018 22:00 EDTStable Formulas in Intuitionistic Logichttps://projecteuclid.org/euclid.ndjfl/1525420860<strong>Nick Bezhanishvili</strong>, <strong>Dick de Jongh</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are (up to provable equivalence) exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of (descriptive and Kripke) frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that these are (up to frame equivalence) the formulas preserved in monotonic images of (descriptive and Kripke) frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013.
</p>projecteuclid.org/euclid.ndjfl/1525420860_20180504040107Fri, 04 May 2018 04:01 EDTOn the Spectrum of Characters of Ultrafiltershttps://projecteuclid.org/euclid.ndjfl/1529373704<strong>Shimon Garti</strong>, <strong>Menachem Magidor</strong>, <strong>Saharon Shelah</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
We show that the character spectrum $\operatorname{Sp}_{\chi}(\lambda)$ (for a singular cardinal $\lambda$ of countable cofinality) may include any prescribed set of regular cardinals between $\lambda$ and $2^{\lambda}$ .
</p>projecteuclid.org/euclid.ndjfl/1529373704_20180618220153Mon, 18 Jun 2018 22:01 EDTEhrenfeucht’s Lemma in Set Theoryhttps://projecteuclid.org/euclid.ndjfl/1529460366<strong>Gunter Fuchs</strong>, <strong>Victoria Gitman</strong>, <strong>Joel David Hamkins</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying $V=\mathsf{HOD}$ . We show that the lemma fails in the forcing extension of the universe by adding a Cohen real. We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht’s lemma, namely, the principles of the form $\mathsf{EL}(A,P,Q)$ , which asserts that $P$ -definability from $A$ implies $Q$ -discernibility. We also consider various analogues of Ehrenfeucht’s lemma obtained by using algebraicity in place of definability, where a set $b$ is algebraic in $a$ if it is a member of a finite set definable from $a$ . Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using a similar analysis, we answer two open questions posed earlier by the third author and C. Leahy, showing that (i) algebraicity and definability need not coincide in models of set theory and (ii) the internal and external notions of being ordinal algebraic need not coincide.
</p>projecteuclid.org/euclid.ndjfl/1529460366_20180619220625Tue, 19 Jun 2018 22:06 EDTA Note on Gabriel Uzquiano’s “Varieties of Indefinite Extensibility”https://projecteuclid.org/euclid.ndjfl/1529481616<strong>Simon Hewitt</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 5 pages.</p><p><strong>Abstract:</strong><br/>
Gabriel Uzquiano has offered an account of indefinite extensibility for sets in the context of a modal logic. The modal operators are interpreted in terms of linguistic extensibility. After reviewing the proposal, I argue that the view should be understood as a version of in rebus structuralism about set theory. As such it is subject to the usual problems for in rebus structuralism. In particular, there is no good extra set-theoretic reason to assent to an ontology of sufficient cardinality to make true the theorems of ZFC.
</p>projecteuclid.org/euclid.ndjfl/1529481616_20180620040031Wed, 20 Jun 2018 04:00 EDTSemigroups in Stable Structureshttps://projecteuclid.org/euclid.ndjfl/1529481617<strong>Yatir Halevi</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
Assume that $G$ is a definable group in a stable structure $M$ . Newelski showed that the semigroup $S_{G}(M)$ of complete types concentrated on $G$ is an inverse limit of the $\infty$ -definable (in $M^{\mathrm{eq}}$ ) semigroups $S_{G,\Delta}(M)$ . He also showed that it is strongly $\pi$ -regular: for every $p\inS_{G,\Delta}(M)$ , there exists $n\in\mathbb{N}$ such that $p^{n}$ is in a subgroup of $S_{G,\Delta}(M)$ . We show that $S_{G,\Delta}(M)$ is in fact an intersection of definable semigroups, so $S_{G}(M)$ is an inverse limit of definable semigroups, and that the latter property is enjoyed by all $\infty$ -definable semigroups in stable structures.
</p>projecteuclid.org/euclid.ndjfl/1529481617_20180620040031Wed, 20 Jun 2018 04:00 EDTOn the Status of Reflection and Conservativity in Replacement Theories of Truthhttps://projecteuclid.org/euclid.ndjfl/1529978580<strong>Jeffrey R. Schatz</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
This article examines Kevin Scharp’s formal solution to the alethic paradoxes, ADT, which stands for ascending and descending truth. One of the main supposed benefits of ADT over its competitors is that it alone can validate the uses of truth concepts in theoretical contexts, such as truth-theoretic semantics. The appendixes contain a new consistency proof for ADT, and additionally show that it is conservative. As a result of its conservativity, the article argues that ADT faces a problem in accounting for certain mathematical uses of truth. Thus, Scharp’s theory needs to be amended in order to fulfill its aim of replicating all substantive uses of truth.
</p>projecteuclid.org/euclid.ndjfl/1529978580_20180625220320Mon, 25 Jun 2018 22:03 EDTA Partition Theorem of $\omega^{\omega^{\alpha}}$https://projecteuclid.org/euclid.ndjfl/1529978581<strong>Claribet Piña</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
We consider finite partitions of the closure $\overline{\mathcal{F}}$ of an $\omega^{\alpha}$ -uniform barrier $\mathcal{F}$ . For each partition, we get a homogeneous set having both the same combinatorial and topological structure as $\overline{\mathcal{F}}$ , seen as a subspace of the Cantor space $2^{\mathbb{N}}$ .
</p>projecteuclid.org/euclid.ndjfl/1529978581_20180625220320Mon, 25 Jun 2018 22:03 EDTSet Mappings on $4$ -Tupleshttps://projecteuclid.org/euclid.ndjfl/1529978582<strong>Shahram Mohsenipour</strong>, <strong>Saharon Shelah</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we study set mappings on 4-tuples. We continue a previous work of Komjath and Shelah by getting new finite bounds on the size of free sets in a generic extension. This is obtained by an entirely different forcing construction. Moreover, we prove a ZFC result for set mappings on 4-tuples. Also, as another application of our forcing construction, we give a consistency result for set mappings on triples.
</p>projecteuclid.org/euclid.ndjfl/1529978582_20180625220320Mon, 25 Jun 2018 22:03 EDTStable Forking and Imaginarieshttps://projecteuclid.org/euclid.ndjfl/1531792823<strong>Enrique Casanovas</strong>, <strong>Joris Potier</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 6 pages.</p><p><strong>Abstract:</strong><br/>
We prove that a theory $T$ has stable forking if and only if $T^{\mathrm{eq}}$ has stable forking.
</p>projecteuclid.org/euclid.ndjfl/1531792823_20180716220047Mon, 16 Jul 2018 22:00 EDTSecond-Order Logic of Paradoxhttps://projecteuclid.org/euclid.ndjfl/1536653099<strong>Allen P. Hazen</strong>, <strong>Francis Jeffry Pelletier</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
The logic of paradox, $\mathit{LP}$ , is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order $\mathit{LP}$ is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely difficult to appeal to second-order $\mathit{LP}$ for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made.
</p>projecteuclid.org/euclid.ndjfl/1536653099_20180911040523Tue, 11 Sep 2018 04:05 EDTMore Automorphism Groups of Countable, Arithmetically Saturated Models of Peano Arithmetichttps://projecteuclid.org/euclid.ndjfl/1538445762<strong>James H. Schmerl</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 6 pages.</p><p><strong>Abstract:</strong><br/>
There is an infinite set ${\mathcal{T}}$ of Turing-equivalent completions of Peano Arithmetic ( $\mathsf{PA}$ ) such that whenever ${\mathcal{M}}$ and ${\mathcal{N}}$ are nonisomorphic countable, arithmetically saturated models of $\mathsf{PA}$ and $\operatorname{Th}({\mathcal{M}})$ , $\operatorname{Th}({\mathcal{N}})\in{\mathcal{T}}$ , then $\operatorname{Aut}({\mathcal{M}})\ncong\operatorname{Aut}({\mathcal{N}})$ .
</p>projecteuclid.org/euclid.ndjfl/1538445762_20181001220259Mon, 01 Oct 2018 22:02 EDTOn Some Mistaken Beliefs About Core Logic and Some Mistaken Core Beliefs About Logichttps://projecteuclid.org/euclid.ndjfl/1539137244<strong>Neil Tennant</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
This is in part a reply to a recent work of Vidal-Rosset, which expresses various mistaken beliefs about Core Logic. Rebutting these leads us further to identify, and argue against, some mistaken core beliefs about logic.
</p>projecteuclid.org/euclid.ndjfl/1539137244_20181009220740Tue, 09 Oct 2018 22:07 EDTRefining the Taming of the Reverse Mathematics Zoohttps://projecteuclid.org/euclid.ndjfl/1539309631<strong>Sam Sanders</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
Reverse mathematics is a program in the foundations of mathematics. It provides an elegant classification in which the majority of theorems of ordinary mathematics fall into only five categories, based on the “big five” logical systems. Recently, a lot of effort has been directed toward finding exceptional theorems, that is, those which fall outside the big five. The so-called reverse mathematics zoo is a collection of such exceptional theorems (and their relations). It was previously shown that a number of uniform versions of the zoo theorems, that is, where a functional computes the objects stated to exist, fall in the third big five category, arithmetical comprehension , inside Kohlenbach’s higher-order reverse mathematics. In this paper, we extend and refine these previous results. In particular, we establish analogous results for recent additions to the reverse mathematics zoo, thus establishing that the latter disappear at the uniform level. Furthermore, we show that the aforementioned equivalences can be proved using only intuitionistic logic. Perhaps most surprisingly, these explicit equivalences are extracted from nonstandard equivalences in Nelson’s internal set theory , and we show that the nonstandard equivalence can be recovered from the explicit ones. Finally, the following zoo theorems are studied in this paper: $\Pi^{0}_{1}\textsf{G}$ (existence of uniformly $\Pi^{0}_{1}$ -generics), $\textsf{FIP}$ (finite intersection principle), 1-GEN (existence of $1$ -generics), OPT (omitting partial types principle), AMT (atomic model theorem), SADS (stable ascending or descending sequence), AST (atomic model theorem with subenumerable types), NCS (existence of noncomputable sets), and KPT (Kleene–Post theorem that there exist Turing incomparable sets).
</p>projecteuclid.org/euclid.ndjfl/1539309631_20181011220110Thu, 11 Oct 2018 22:01 EDTA Long Pseudo-Comparison of Premice in $L[x]$https://projecteuclid.org/euclid.ndjfl/1539309632<strong>Farmer Schlutzenberg</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 6 pages.</p><p><strong>Abstract:</strong><br/>
A significant open problem in inner model theory is the analysis of $\mathrm{HOD}^{L[x]}$ as a strategy premouse, for a Turing cone of reals $x$ . We describe here an obstacle to such an analysis. Assuming sufficient large cardinals, for a Turing cone of reals $x$ there are proper class $1$ -small premice $M,N$ , with Woodin cardinals $\delta,\varepsilon$ , respectively, such that $M|\delta$ and $N|\varepsilon$ are in $L[x]$ , $(\delta^{+})^{M}$ and $(\varepsilon^{+})^{N}$ are countable in $L[x]$ , and the pseudo-comparison of $M$ with $N$ succeeds, is in $L[x]$ , and lasts exactly $\omega_{1}^{L[x]}$ stages. Moreover, we can take $M=M_{1}$ , the minimal iterable proper class inner model with a Woodin cardinal, and take $N$ to be $M_{1}$ -like and short-tree-iterable.
</p>projecteuclid.org/euclid.ndjfl/1539309632_20181011220110Thu, 11 Oct 2018 22:01 EDTOn the Uniform Computational Content of the Baire Category Theoremhttps://projecteuclid.org/euclid.ndjfl/1539396028<strong>Vasco Brattka</strong>, <strong>Matthew Hendtlass</strong>, <strong>Alexander P. Kreuzer</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 32 pages.</p><p><strong>Abstract:</strong><br/>
We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete (i.e., “large”) metric space cannot be decomposed into countably many nowhere dense (i.e., small) pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets.
</p>projecteuclid.org/euclid.ndjfl/1539396028_20181012220112Fri, 12 Oct 2018 22:01 EDTEnumeration $1$ -Genericity in the Local Enumeration Degreeshttps://projecteuclid.org/euclid.ndjfl/1539396032<strong>Liliana Badillo</strong>, <strong>Charles M. Harris</strong>, <strong>Mariya I. Soskova</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 29 pages.</p><p><strong>Abstract:</strong><br/>
We discuss a notion of forcing that characterizes enumeration $1$ -genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration $1$ -generic sets and their degrees. We construct an enumeration operator $\Delta$ such that, for any $A$ , the set $\Delta^{A}$ is enumeration $1$ -generic and has the same jump complexity as $A$ . We deduce from this and other recent results from the literature that not only does every degree $a$ bound an enumeration $1$ -generic degree $b$ such that $a'=b'$ , but also that, if $a$ is nonzero, then we can find such $b$ satisfying $0_{e}\lt b\lt a$ . We conclude by proving the existence of both a nonzero low and a properly $\Sigma_{2}^{0}$ nonsplittable enumeration $1$ -generic degree, hence proving that the class of $1$ -generic degrees is properly subsumed by the class of enumeration $1$ -generic degrees.
</p>projecteuclid.org/euclid.ndjfl/1539396032_20181012220112Fri, 12 Oct 2018 22:01 EDTA Propositional Theory of Truthhttps://projecteuclid.org/euclid.ndjfl/1539396036<strong>Yannis Stephanou</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 43 pages.</p><p><strong>Abstract:</strong><br/>
The liar and kindred paradoxes show that we can derive contradictions if our language possesses sentences lending themselves to paradox and we reason classically from schema (T) about truth: \[\mathbf{S}\mbox{ is true iff }p,\vspace*{12pt}\] where the letter $p$ is to be replaced with a sentence and the letter $\mathbf{S}$ with a name of that sentence. This article presents a theory of truth that keeps (T) at the expense of classical logic. The theory is couched in a language that possesses paradoxical sentences. It incorporates all the instances of the analogue of (T) for that language and also includes other platitudes about truth. The theory avoids contradiction because its logical framework is an appropriately constructed nonclassical propositional logic. The logic and the theory are different from others that have been proposed for keeping (T), and the methods used in the main proofs are novel.
</p>projecteuclid.org/euclid.ndjfl/1539396036_20181012220112Fri, 12 Oct 2018 22:01 EDT