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  1.  6
    Projective Games on the Reals.Juan P. Aguilera & Sandra Müller - 2020 - Notre Dame Journal of Formal Logic 61 (4):573-589.
    Let Mn♯ denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn the class-sized model obtained by iterating the topmost measure of Mn class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn, under the assumption that projective games on reals are determined:1. for even n, Σ1Mn=⅁RΠn+11;2. for odd n, Σ1Mn=⅁RΣn+11.This generalizes a theorem of Martin and Steel for L, (...)
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  2.  10
    Mirroring Theorems in Free Logic.Ethan Brauer - 2020 - Notre Dame Journal of Formal Logic 61 (4):561-572.
    Linnebo and Shapiro have recently given an analysis of potential infinity using modal logic. A key technical component of their account is to show that under a suitable translation ◊ of nonmodal language into modal language, nonmodal sentences ϕ 1, …, ϕ n entail ψ just in case ϕ 1 ◊, …, ϕ n ◊ entail ψ ◊ in the modal logic S4.2. Linnebo and Shapiro establish this result in nonfree logic. In this note I argue that their analysis of (...)
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  3.  3
    Term-Space Semantics of Typed Lambda Calculus.Ryo Kashima, Naosuke Matsuda & Takao Yuyama - 2020 - Notre Dame Journal of Formal Logic 61 (4):591-600.
    Barendregt gave a sound semantics of the simple type assignment system λ → by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.
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  4.  1
    Weak Density and Nondensity Among Transfinite Levels of the Ershov Hierarchy.Yong Liu & Cheng Peng - 2020 - Notre Dame Journal of Formal Logic 61 (4):521-536.
    We show that for any ω-r.e. degree d and n-r.e. degree b with d
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  5.  6
    Four-Valued Logics of Truth, Nonfalsity, Exact Truth, and Material Equivalence.Adam Přenosil - 2020 - Notre Dame Journal of Formal Logic 61 (4):601-621.
    The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth. Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, (...)
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  6.  4
    Reverse Mathematics of Topology: Dimension, Paracompactness, and Splittings.Sam Sanders - 2020 - Notre Dame Journal of Formal Logic 61 (4):537-559.
    Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A, B, C such that A (...)
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  7.  2
    Existence of Certain Finite Relation Algebras Implies Failure of Omitting Types for L N.Tarek Sayed Ahmed - 2020 - Notre Dame Journal of Formal Logic 61 (4):503-519.
    Fix 2 < n < ω. Let CA n denote the class of cylindric algebras of dimension n, and let RCA n denote the variety of representable CA n ’s. Let L n denote first-order logic restricted to the first n variables. Roughly, CA n, an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of L n, namely, its proof theory, while RCA n algebraically and geometrically represents the Tarskian semantics of L n. Unlike Boolean (...)
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  8.  1
    Existentially Closed Closure Algebras.Philip Scowcroft - 2020 - Notre Dame Journal of Formal Logic 61 (4):623-661.
    The study of existentially closed closure algebras begins with Lipparini’s 1982 paper. After presenting new nonelementary axioms for algebraically closed and existentially closed closure algebras and showing that these nonelementary classes are different, this paper shows that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extends part of Hausdorff’s theory of reducible sets to existentially closed closure algebras, and shows that finitely generic and infinitely generic closure algebras are elementarily (...)
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  9.  25
    Begriffsschrift’s Logic.Calixto Badesa & Joan Bertran-San Millán - 2020 - Notre Dame Journal of Formal Logic 61 (3):409-440.
    In Begriffsschrift, Frege presented a formal system and used it to formulate logical definitions of arithmetical notions and to deduce some noteworthy theorems by means of logical axioms and inference rules. From a contemporary perspective, Begriffsschrift’s deductions are, in general, straightforward; it is assumed that all of them can be reproduced in a second-order formal system. Some deductions in this work present—according to this perspective—oddities that have led many scholars to consider it to be Frege’s inaccuracies which should be amended. (...)
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  10.  3
    Ideals and Their Generic Ultrafilters.David Chodounský & Jindřich Zapletal - 2020 - Notre Dame Journal of Formal Logic 61 (3):403-408.
    Let I be an F σ -ideal on natural numbers. We characterize the ultrafilters which are generic over the model L for the poset of I -positive sets of natural numbers ordered by inclusion.
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  11.  10
    Connexive Restricted Quantification.Nissim Francez - 2020 - Notre Dame Journal of Formal Logic 61 (3):383-402.
    This paper investigates the meaning of restricted quantification when the embedded conditional is taken as the conditional of some first-order connexive logics. The study is carried out by checking the suitability of RQ for defining a connexive class theory, in analogy to the definition of Boolean class theory by using RQ in classical logic. Negative results are obtained for Wansing’s first-order connexive logic QC and one variant of Priest’s first-order connexive logic QP. A positive result is obtained for another variant (...)
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  12.  1
    Definable Functions and Stratifications in Power-Bounded T -Convex Fields.Erick García Ramírez - 2020 - Notre Dame Journal of Formal Logic 61 (3):441-465.
    We study properties of definable sets and functions in power-bounded T -convex fields, proving that the latter have the multidimensional Jacobian property and that the theory of T -convex fields is b -minimal with centers. Through these results and work of I. Halupczok we ensure that a certain kind of geometrical stratifications exist for definable objects in said fields. We then discuss a number of applications of those stratifications, including applications to Archimedean o-minimal geometry.
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  13.  4
    On the Proof of Elimination of Imaginaries in Algebraically Closed Valued Fields.Will Johnson - 2020 - Notre Dame Journal of Formal Logic 61 (3):363-381.
    We give a simplified proof of elimination of imaginaries in ACVF, based on ideas of Hrushovski. This proof manages to avoid many of the technical issues which arose in the original proof by Haskell, Hrushovski, and Macpherson.
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  14.  39
    Prospects for a Theory of Decycling.Jon Erling Litland - 2020 - Notre Dame Journal of Formal Logic 61 (3):467-499.
    Seemingly natural principles about the logic of ground generate cycles of ground; how can this be if ground is asymmetric? The goal of the theory of decycling is to find systematic and principled ways of getting rid of such cycles of ground. In this paper—drawing on graph-theoretic and topological ideas—I develop a general framework in which various theories of decycling can be compared. This allows us to improve on proposals made earlier by Fine and Litland. However, it turns out that (...)
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  15.  10
    Erratum for “Conditionals and Conditional Probabilities Without Triviality”.Alexander R. Pruss - 2020 - Notre Dame Journal of Formal Logic 61 (3):501-501.
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  16.  21
    A Note on FDE “All the Way Up”.Jc Beall & Caleb Camrud - 2020 - Notre Dame Journal of Formal Logic 61 (2):283-296.
    A very natural and philosophically important subclassical logic is FDE. This account of logical consequence can be seen as going beyond the standard two-valued account to a four-valued account. A natural question arises: What account of logical consequence arises from considering further combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different context. In this note we generalize Priest’s result to show (...)
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  17.  4
    Isometry Groups of Borel Randomizations.Alexander Berenstein & Rafael Zamora - 2020 - Notre Dame Journal of Formal Logic 61 (2):297-316.
    We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.
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  18.  19
    The Formalities of Temporaryism Without Presentness.Fabrice Correia & Sven Rosenkranz - 2020 - Notre Dame Journal of Formal Logic 61 (2):181-202.
    Temporaryism—the view that not always everything always exists—comes in two main versions: presentism and expansionism. Both versions of the view are commonly formulated using the notion of being present, which we, among others, find problematic. Expansionism is also sometimes accused of requiring extraordinary conceptual tools for its formulation. In this paper, we put forward systematic characterizations of presentism and expansionism which involve neither the notion of being present nor unfamiliar conceptual tools. These characterizations are full-blown logics, each logic comprising an (...)
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  19.  4
    Distality for the Asymptotic Couple of the Field of Logarithmic Transseries.Allen Gehret & Elliot Kaplan - 2020 - Notre Dame Journal of Formal Logic 61 (2):341-361.
    We show that the theory Tlog of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP, this provides a new proof that Tlog is NIP.
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  20.  25
    Formal Notes on the Substitutional Analysis of Logical Consequence.Volker Halbach - 2020 - Notre Dame Journal of Formal Logic 61 (2):317-339.
    Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. (...)
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  21.  8
    Uniformly Bounded Arrays and Mutually Algebraic Structures.Michael C. Laskowski & Caroline A. Terry - 2020 - Notre Dame Journal of Formal Logic 61 (2):265-282.
    We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.
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  22.  1
    Effective Domination and the Bounded Jump.Keng Meng Ng & Hongyuan Yu - 2020 - Notre Dame Journal of Formal Logic 61 (2):203-225.
    We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: We prove that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility. We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use (...)
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  23.  5
    A Note on Strongly Almost Disjoint Families.Guozhen Shen - 2020 - Notre Dame Journal of Formal Logic 61 (2):227-231.
    For a set M, let |M| denote the cardinality of M. A family F is called strongly almost disjoint if there is an n∈ω such that |A∩B|<n for any two distinct elements A, B of F. It is shown in ZF (without the axiom of choice) that, for all infinite sets M and all strongly almost disjoint families F⊆P(M), |F|<|P(M)| and there are no finite-to-one functions from P(M) into F, where P(M) denotes the power set of M.
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  24.  1
    On Amalgamation in NTP $_{2}$ Theories and Generically Simple Generics.Pierre Simon - 2020 - Notre Dame Journal of Formal Logic 61 (2):233-243.
    We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.
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  25.  3
    An Effective Analysis of the Denjoy Rank.Linda Westrick - 2020 - Notre Dame Journal of Formal Logic 61 (2):245-263.
    We analyze the descriptive complexity of several Π11-ranks from classical analysis which are associated to Denjoy integration. We show that VBG, VBG∗, ACG, and ACG∗ are Π11-complete, answering a question of Walsh in case of ACG∗. Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most α steps of the transfinite process of Denjoy totalization: if |⋅| is the Π11-rank naturally associated to VBG, VBG∗, or ACG∗, and if α<ωck1, then {F∈C(I):|F|≤α} is Σ02α-complete. These (...)
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  26.  5
    A Lindström Theorem for Intuitionistic Propositional Logic.Guillermo Badia - 2020 - Notre Dame Journal of Formal Logic 61 (1):11-30.
    We show that propositional intuitionistic logic is the maximal abstract logic satisfying a certain form of compactness, the Tarski union property, and preservation under asimulations.
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  27.  13
    Questions and Dependency in Intuitionistic Logic.Ivano Ciardelli, Rosalie Iemhoff & Fan Yang - 2020 - Notre Dame Journal of Formal Logic 61 (1):75-115.
    In recent years, the logic of questions and dependencies has been investigated in the closely related frameworks of inquisitive logic and dependence logic. These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core. In this paper, we broaden the scope of these investigations by studying questions and dependency in the context of intuitionistic logic. We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models. (...)
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  28.  9
    Short Proofs for Slow Consistency.Anton Freund & Fedor Pakhomov - 2020 - Notre Dame Journal of Formal Logic 61 (1):31-49.
    Let Con↾x denote the finite consistency statement “there are no proofs of contradiction in T with ≤x symbols.” For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con↾n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con)↾n. In contrast, we show that Peano arithmetic has polynomial proofs of Con)↾n, (...)
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  29.  8
    The Logic of Turing Progressions.Eduardo Hermo Reyes & Joost J. Joosten - 2020 - Notre Dame Journal of Formal Logic 61 (1):155-180.
    Turing progressions arise by iteratedly adding consistency statements to a base theory. Different notions of consistency give rise to different Turing progressions. In this paper we present a logic that generates exactly all relations that hold between these different Turing progressions given a particular set of natural consistency notions. Thus, the presented logic is proven to be arithmetically sound and complete for a natural interpretation, named the formalized Turing progressions interpretation.
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  30.  2
    Canonization of Smooth Equivalence Relations on Infinite-Dimensional $Mathsf{E}_{0}$-Large Products.Vladimir Kanovei & Vassily Lyubetsky - 2020 - Notre Dame Journal of Formal Logic 61 (1):117-128.
    We propose a canonization scheme for smooth equivalence relations on Rω modulo restriction to E0-large infinite products. It shows that, given a pair of Borel smooth equivalence relations E, F on Rω, there is an infinite E0-large perfect product P⊆Rω such that either F⊆E on P, or, for some ℓ<ω, the following is true for all x,y∈P: xEy implies x=y, and x↾=y↾ implies xFy.
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  31.  15
    A Remark on Probabilistic Measures of Coherence.Sergi Oms - 2020 - Notre Dame Journal of Formal Logic 61 (1):129-140.
    In recent years, some authors have proposed quantitative measures of the coherence of sets of propositions. Such probabilistic measures of coherence are, in general terms, functions that take as their argument a set of propositions and yield as their value a number that is supposed to represent the degree of coherence of the set. In this paper, I introduce a minimal constraint on PMC theories, the weak stability principle, and show that any correct, coherent, and complete PMC cannot satisfy it. (...)
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  32.  8
    Splittings and Disjunctions in Reverse Mathematics.Sam Sanders - 2020 - Notre Dame Journal of Formal Logic 61 (1):51-74.
    Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with two RM-phenomena, namely, splittings and disjunctions. As to splittings, there are some examples in RM of theorems A, B, C such that A↔, that is, A can be split into two (...)
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  33.  3
    Pseudofiniteness in Hrushovski Constructions.Ali N. Valizadeh & Massoud Pourmahdian - 2020 - Notre Dame Journal of Formal Logic 61 (1):1-10.
    In a relational language consisting of a single relation R, we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation R plays a crucial role in this context. When R is ternary, by extending the methods recently developed by Brody and Laskowski, we interpret 〈Q+,<〉 in the 〈K+,≤∗〉-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in an earlier work by Evans (...)
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  34.  3
    The Complexity of Radicals and Socles of Modules.Huishan Wu - 2020 - Notre Dame Journal of Formal Logic 61 (1):141-153.
    This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of Z-modules, where the radical of a Z-module M is defined as the intersection of pM={px:x∈M} with p taken from all primes. It shows that ACA0 is equivalent to the existence of radicals of Z-modules over RCA0. We then study socles of modules over commutative rings with identity. The socle of an R-module M is the largest semisimple submodule of (...)
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