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 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 EDTTame Topology over dp-Minimal Structureshttps://projecteuclid.org/euclid.ndjfl/1547607904<strong>Pierre Simon</strong>, <strong>Erik Walsberg</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.
</p>projecteuclid.org/euclid.ndjfl/1547607904_20190115220514Tue, 15 Jan 2019 22:05 ESTTeachers, Learners, and Oracleshttps://projecteuclid.org/euclid.ndjfl/1547607905<strong>Achilles Beros</strong>, <strong>Colin de la Higuera</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We exhibit a family of computably enumerable sets which can be learned within polynomial resource bounds given access only to a teacher but which requires exponential resources to be learned given access only to a membership oracle. In general, we compare the families that can be learned with and without teachers and oracles for four measures of efficient learning.
</p>projecteuclid.org/euclid.ndjfl/1547607905_20190115220514Tue, 15 Jan 2019 22:05 ESTLevels of Uniformityhttps://projecteuclid.org/euclid.ndjfl/1547802297<strong>Rutger Kuyper</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of nonuniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses how uniform a reduction is. We study this notion for several well-known reductions from algorithmic randomness. Furthermore, since our new structures are Brouwer algebras, we study their propositional theories. Finally, we study if our new structures are elementarily equivalent to each other.
</p>projecteuclid.org/euclid.ndjfl/1547802297_20190118040504Fri, 18 Jan 2019 04:05 EST$\Pi ^{0}_{1}$ -Encodability and Omniscient Reductionshttps://projecteuclid.org/euclid.ndjfl/1547802298<strong>Benoit Monin</strong>, <strong>Ludovic Patey</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$ . By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this article, we extend this notion of computable encodability to subsets of the Baire space, and we characterize the $\Pi^{0}_{1}$ -encodable compact sets as those which admit a nonempty $\Sigma^{1}_{1}$ -subset. Thanks to this equivalence, we prove that weak weak König’s lemma is not strongly computably reducible to Ramsey’s theorem. This answers a question of Hirschfeldt and Jockusch.
</p>projecteuclid.org/euclid.ndjfl/1547802298_20190118040504Fri, 18 Jan 2019 04:05 ESTLayered Posets and Kunen’s Universal Collapsehttps://projecteuclid.org/euclid.ndjfl/1548385577<strong>Sean Cox</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 34 pages.</p><p><strong>Abstract:</strong><br/>
We develop the theory of layered posets and use the notion of layering to prove a new iteration theorem (Theorem 6: if $\kappa$ is weakly compact, then any universal Kunen iteration of $\kappa$ -cc posets (each possibly of size $\kappa$ ) is $\kappa$ -cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications:
1 For any $n\ge1$ , a wide variety of $\lt \omega_{n-1}$ -closed, $\omega_{n+1}$ -cc posets of size $\omega_{n+1}$ can consistently be absorbed (as regular suborders) by quotients of saturated ideals on $\omega_{n}$ (see Theorem 7 and Corollary 8).
2 For any $n\in\omega$ , the tree property at $\omega_{n+3}$ is consistent with Chang’s conjecture $(\omega_{n+3},\omega_{n+1})\twoheadrightarrow(\omega_{n+1},\omega_{n})$ (Theorem 9).
</p>projecteuclid.org/euclid.ndjfl/1548385577_20190124220651Thu, 24 Jan 2019 22:06 ESTAbstraction Principles and the Classification of Second-Order Equivalence Relationshttps://projecteuclid.org/euclid.ndjfl/1548385578<strong>Sean C. Ebels-Duggan</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 41 pages.</p><p><strong>Abstract:</strong><br/>
This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation $E$ is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem.
</p>projecteuclid.org/euclid.ndjfl/1548385578_20190124220651Thu, 24 Jan 2019 22:06 ESTDisjoint $n$ -Amalgamation and Pseudofinite Countably Categorical Theorieshttps://projecteuclid.org/euclid.ndjfl/1548730925<strong>Alex Kruckman</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
Disjoint $n$ -amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this article, we show that if a countably categorical theory $T$ admits an expansion with disjoint $n$ -amalgamation for all $n$ , then $T$ is pseudofinite. All theories which admit an expansion with disjoint $n$ -amalgamation for all $n$ are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain nonsimple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, $T^{*}_{\mathrm{feq}}$ and $T_{\mathrm{CPZ}}$ , and show that both are pseudofinite. The theories $T^{*}_{\mathrm{feq}}$ and $T_{\mathrm{CPZ}}$ are not simple, but they have NSOP $_{1}$ . This is established here for $T_{\mathrm{CPZ}}$ for the first time.
</p>projecteuclid.org/euclid.ndjfl/1548730925_20190128220230Mon, 28 Jan 2019 22:02 ESTThe Mereotopology of Timehttps://projecteuclid.org/euclid.ndjfl/1556179508<strong>Claudio Mazzola</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 38 pages.</p><p><strong>Abstract:</strong><br/>
Mereotopology is the discipline obtained from combining topology with the formal study of parts and their relation to wholes, or mereology. This article develops a mereotopological theory of time, illustrating how different temporal topologies can be effectively discriminated on this basis. Specifically, we demonstrate how the three principal types of temporal models—namely, the linear ones, the forking ones, and the circular ones—can be characterized by differently combining two sole mereotopological constraints: one to denote the absence of closed loops, and the other one to denote the absence of branches.
</p>projecteuclid.org/euclid.ndjfl/1556179508_20190425040516Thu, 25 Apr 2019 04:05 EDTTame Expansions of $\omega$ -Stable Theories and Definable Groupshttps://projecteuclid.org/euclid.ndjfl/1556179509<strong>Haydar Göral</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 34 pages.</p><p><strong>Abstract:</strong><br/>
We study groups definable in tame expansions of $\omega$ -stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair $(K,G)$ , where $K$ is an algebraically closed field and $G$ is a multiplicative subgroup of $K^{\times}$ with the Mann property, we show that the pair $(K,G)$ satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in $(K,G)$ in terms of algebraic groups in $K$ and interpretable groups in $G$ . Furthermore, we compute the Morley rank and the $\operatorname{U}$ -rank in $(K,G)$ and both ranks agree.
</p>projecteuclid.org/euclid.ndjfl/1556179509_20190425040516Thu, 25 Apr 2019 04:05 EDTA Modal Logic of Superveniencehttps://projecteuclid.org/euclid.ndjfl/1557129618<strong>Jie Fan</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
Inspired by the supervenience-determined consequence relation and the semantics of agreement operator, we introduce a modal logic of supervenience, which has a dyadic operator of supervenience as a sole modality. The semantics of supervenience modality very naturally correspond to the supervenience-determined consequence relation, in a quite similar way that the strict implication corresponds to the inference-determined consequence relation. We show that this new logic is more expressive than the modal logic of agreement, by proposing a notion of bisimulation for the latter. We provide a sound proof system for the new logic. We lift onto more general logics of supervenience. Related to this, we address an interesting open research direction listed in the literature, by comparing propositional logic of determinacy and noncontingency logic in expressive powers and axiomatizing propositional logic of determinacy over various classes of frames. We also obtain an alternative axiomatization for propositional logic of determinacy over universal models.
</p>projecteuclid.org/euclid.ndjfl/1557129618_20190506040033Mon, 06 May 2019 04:00 EDTOn $n$ -Dependencehttps://projecteuclid.org/euclid.ndjfl/1557129619<strong>Artem Chernikov</strong>, <strong>Daniel Palacin</strong>, <strong>Kota Takeuchi</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we develop and clarify some of the basic combinatorial properties of the new notion of $n$ -dependence (for $1\leq n\lt \omega$ ) recently introduced by Shelah. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, $n$ -dependence corresponds to the inability to encode a random $(n+1)$ -partite $(n+1)$ -hypergraph with a definable edge relation. We characterize $n$ -dependence by counting $\varphi $ -types over finite sets (generalizing the Sauer–Shelah lemma, answering a question of Shelah), and in terms of the collapse of random ordered $(n+1)$ -hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of $n$ -dependence is always witnessed by a formula in a single free variable).
</p>projecteuclid.org/euclid.ndjfl/1557129619_20190506040033Mon, 06 May 2019 04:00 EDTClosed Maximality Principles and Generalized Baire Spaceshttps://projecteuclid.org/euclid.ndjfl/1557281186<strong>Philipp Lücke</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 30 pages.</p><p><strong>Abstract:</strong><br/>
Given an uncountable regular cardinal $\kappa$ , we study the structural properties of the class of all sets of functions from $\kappa$ to $\kappa$ that are definable over the structure $\langle\mathrm{{H}}(\kappa^{+}),\in\rangle$ by a $\Sigma_{1}$ -formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of $\mathrm{{ZFC}}$ together with large cardinal axioms. In this paper, we present other canonical extensions of $\mathrm{{ZFC}}$ that provide a strong structure theory for these classes. These axioms are variations of the Maximality Principle introduced by Stavi and Väänänen and later rediscovered by Hamkins.
</p>projecteuclid.org/euclid.ndjfl/1557281186_20190507220646Tue, 07 May 2019 22:06 EDTA Note on Algebraic Semantics for $\mathsf{S5}$ with Propositional Quantifiershttps://projecteuclid.org/euclid.ndjfl/1557388819<strong>Wesley H. Holliday</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
In two of the earliest papers on extending modal logic with propositional quantifiers, R. A. Bull and K. Fine studied a modal logic S5 $\Pi $ extending S5 with axioms and rules for propositional quantification. Surprisingly, there seems to have been no proof in the literature of the completeness of S5 $\Pi $ with respect to its most natural algebraic semantics, with propositional quantifiers interpreted by meets and joins over all elements in a complete Boolean algebra. In this note, we give such a proof. This result raises the question: For which normal modal logics $\mathsf{L}$ can one axiomatize the quantified propositional modal logic determined by the complete modal algebras for $\mathsf{L}$ ?
</p>projecteuclid.org/euclid.ndjfl/1557388819_20190509040038Thu, 09 May 2019 04:00 EDTResidue Field Domination in Real Closed Valued Fieldshttps://projecteuclid.org/euclid.ndjfl/1562033115<strong>Clifton Ealy</strong>, <strong>Deirdre Haskell</strong>, <strong>Jana Maříková</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 333--351.</p><p><strong>Abstract:</strong><br/>
We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded $T$ -convex theory.
</p>projecteuclid.org/euclid.ndjfl/1562033115_20190725220358Thu, 25 Jul 2019 22:03 EDTAn Analytic Calculus for the Intuitionistic Logic of Proofshttps://projecteuclid.org/euclid.ndjfl/1562810593<strong>Brian Hill</strong>, <strong>Francesca Poggiolesi</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 353--393.</p><p><strong>Abstract:</strong><br/>
The goal of this article is to take a step toward the resolution of the problem of finding an analytic sequent calculus for the logic of proofs. For this, we focus on the system Ilp , the intuitionistic version of the logic of proofs. First we present the sequent calculus Gilp that is sound and complete with respect to the system Ilp ; we prove that Gilp is cut-free and contraction-free, but it still does not enjoy the subformula property. Then, we enrich the language of the logic of proofs and we formulate in this language a second Gentzen calculus $\mathbf{Gilp}^{*}$ . We show that $\mathbf{Gilp}^{*}$ is a conservative extension of Gilp , and that $\mathbf{Gilp}^{*}$ satisfies the subformula property.
</p>projecteuclid.org/euclid.ndjfl/1562810593_20190725220358Thu, 25 Jul 2019 22:03 EDTLD-Algebras Beyond I0https://projecteuclid.org/euclid.ndjfl/1560218425<strong>Vincenzo Dimonte</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 395--405.</p><p><strong>Abstract:</strong><br/>
The algebra of embeddings at the I3 level has been deeply analyzed, but nothing is known algebra-wise for embeddings above I3. In this article, we introduce an operation for embeddings at the level of I0 and above, and prove that they generate an LD-algebra that can be quite different from the one implied by I3.
</p>projecteuclid.org/euclid.ndjfl/1560218425_20190725220358Thu, 25 Jul 2019 22:03 EDTStrong Noncontingency: On the Modal Logics of an Operator Expressively Weaker Than Necessityhttps://projecteuclid.org/euclid.ndjfl/1562918419<strong>Jie Fan</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 407--435.</p><p><strong>Abstract:</strong><br/>
Operators can be compared in at least two respects: expressive strength and deductive strength. Inspired by Hintikka’s treatment of question embedding verbs, the variations of noncontingency operator, and also the various combinations of modal operators and Boolean connectives, we propose a logic with (deductively) strong noncontingency operator as the only primitive modality. The novel operator is deductively but not expressively stronger than both noncontingency operator and essence operator, and expressively but not deductively weaker than the necessity operator. The frame-definability power of this new logic is in between standard modal logic and noncontingency logic. A notion of bisimulation is proposed to characterize this logic within standard modal logic and first-order logic. Axiomatizations over various frame classes are presented, among which the minimal logic is related to the treatment of an alternative semantics of the agreement operator proposed by Lloyd Humberstone.
</p>projecteuclid.org/euclid.ndjfl/1562918419_20190725220358Thu, 25 Jul 2019 22:03 EDTMore Notions of Forcing Add a Souslin Treehttps://projecteuclid.org/euclid.ndjfl/1560218424<strong>Ari Meir Brodsky</strong>, <strong>Assaf Rinot</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 437--455.</p><p><strong>Abstract:</strong><br/>
An $\aleph _{1}$ -Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion— Cohen forcing —adds an $\aleph _{1}$ -Souslin tree.
In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a $\lambda ^{+}$ -Souslin tree. This class includes Prikry, Magidor, and Radin forcing.
</p>projecteuclid.org/euclid.ndjfl/1560218424_20190725220358Thu, 25 Jul 2019 22:03 EDTThe Eu Approach to Formalizing Euclid: A Response to “On the Inconsistency of Mumma’s Eu”https://projecteuclid.org/euclid.ndjfl/1562033114<strong>John Mumma</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 457--480.</p><p><strong>Abstract:</strong><br/>
In line with Ken Manders’s seminal account of Euclid’s diagrammatic method in the “The Euclidean Diagram,” two proof systems with a diagrammatic syntax have been advanced as formalizations of the method FG and Eu . In a paper examining Eu , Nathaniel Miller, the creator of FG , has identified a variety of technical problems with the formal details of Eu . This response shows how the problems are remedied.
</p>projecteuclid.org/euclid.ndjfl/1562033114_20190725220358Thu, 25 Jul 2019 22:03 EDTA Counterexample to Polynomially Bounded Realizability of Basic Arithmetichttps://projecteuclid.org/euclid.ndjfl/1562205626<strong>Mohammad Ardeshir</strong>, <strong>Erfan Khaniki</strong>, <strong>Mohsen Shahriari</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 481--489.</p><p><strong>Abstract:</strong><br/>
We give a counterexample to the claim that every provably total function of Basic Arithmetic is a polynomially bounded primitive recursive function.
</p>projecteuclid.org/euclid.ndjfl/1562205626_20190725220358Thu, 25 Jul 2019 22:03 EDTMartin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Setshttps://projecteuclid.org/euclid.ndjfl/1562810592<strong>Rodney G. Downey</strong>, <strong>Satyadev Nandakumar</strong>, <strong>André Nies</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 491--502.</p><p><strong>Abstract:</strong><br/>
This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space $\{0,1\}^{{\mathbb{N}}}$ with the uniform measure and the usual shift (erasing the first bit). We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets $\mathcal{P}$ of positive measure holds for iterations starting at the point. This means that for each $k\in{\mathbb{N}}$ there is an $n$ such that $n,2n,\ldots ,kn$ shifts of the point all end up in $\mathcal{P}$ . We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, $\Pi ^{0}_{1}$ with computable measure, and $\Pi ^{0}_{1}$ . The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the $k$ commuting shift operators on $\{0,1\}^{{\mathbb{N}}^{k}}$ .
</p>projecteuclid.org/euclid.ndjfl/1562810592_20190725220358Thu, 25 Jul 2019 22:03 EDTAdding a Nonreflecting Weakly Compact Sethttps://projecteuclid.org/euclid.ndjfl/1560218426<strong>Brent Cody</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 503--521.</p><p><strong>Abstract:</strong><br/>
For $n\lt \omega $ , we say that the $\Pi ^{1}_{n}$ -reflection principle holds at $\kappa $ and write $\operatorname{Refl}_{n}(\kappa )$ if and only if $\kappa $ is a $\Pi ^{1}_{n}$ -indescribable cardinal and every $\Pi ^{1}_{n}$ -indescribable subset of $\kappa $ has a $\Pi ^{1}_{n}$ -indescribable proper initial segment. The $\Pi ^{1}_{n}$ -reflection principle $\operatorname{Refl}_{n}(\kappa )$ generalizes a certain stationary reflection principle and implies that $\kappa $ is $\Pi ^{1}_{n}$ -indescribable of order $\omega $ . We define a forcing which shows that the converse of this implication can be false in the case $n=1$ ; that is, we show that $\kappa $ being $\Pi ^{1}_{1}$ -indescribable of order $\omega $ need not imply $\operatorname{Refl}_{1}(\kappa )$ . Moreover, we prove that if $\kappa $ is $(\alpha +1)$ -weakly compact where $\alpha \lt \kappa ^{+}$ , then there is a forcing extension in which there is a weakly compact set $W\subseteq\kappa $ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and $\kappa $ remains $(\alpha +1)$ -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down.
</p>projecteuclid.org/euclid.ndjfl/1560218426_20190725220358Thu, 25 Jul 2019 22:03 EDTOn Goodman Realizabilityhttps://projecteuclid.org/euclid.ndjfl/1562918420<strong>Emanuele Frittaion</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 523--550.</p><p><strong>Abstract:</strong><br/>
Goodman’s theorem states that $\mathsf{HA}^{\omega }+{\mathsf{AC}}+{\mathsf{RDC}}$ is conservative over ${\mathsf{HA}}$ . The same result applies to the extensional case, that is, ${\mathsf{E}}\text{-}{\mathsf{HA}}^{\omega}+{\mathsf{AC}}+{\mathsf{RDC}}$ is also conservative over ${\mathsf{HA}}$ . This is due to Beeson. In this article, we modify Goodman realizability and provide a new proof of the extensional case.
</p>projecteuclid.org/euclid.ndjfl/1562918420_20190725220358Thu, 25 Jul 2019 22:03 EDTConditionals and Conditional Probabilities without Trivialityhttps://projecteuclid.org/euclid.ndjfl/1562205627<strong>Alexander R. Pruss</strong>. <p><strong>Source: </strong>Notre Dame Journal of Formal Logic, Volume 60, Number 3, 551--558.</p><p><strong>Abstract:</strong><br/>
The Adams Thesis holds for a conditional $\rightarrow $ and a probability assignment $P$ if and only if $P(A\rightarrow B)=P(B\mid A)$ whenever $P(A)\gt 0$ . The restriction ensures that $P(B\mid A)$ is well defined by the classical formula $P(B\mid A)=P(B\cap A)/P(A)$ . Drawing on deep results of Maharam on measure algebras, it is shown that, notwithstanding well-known triviality results (Lewis, etc.), any probability space can be extended to a probability space with a new conditional satisfying the Adams Thesis and satisfying a number of axioms for conditionals. This puts significant limits on how far triviality results can go.
</p>projecteuclid.org/euclid.ndjfl/1562205627_20190725220358Thu, 25 Jul 2019 22:03 EDT