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  1.  11
    Hierarchical Multiverse of Sets.Ahmet Çevik - 2023 - Notre Dame Journal of Formal Logic 64 (4):545-570.
    In this article, I develop a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the notion of “degrees of intentionality” of theories. The presented view is articulated for the purpose of reconciling epistemological realism and the multiverse theory of sets so as to preserve a considerable amount of epistemic objectivity when working with the multiverse theory. I give some arguments in favor of a hierarchical picture of the multiverse in which theories or models are thought (...)
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  2.  3
    There Are No Minimal Effectively Inseparable Theories.Yong Cheng - 2023 - Notre Dame Journal of Formal Logic 64 (4):425-439.
    This paper belongs to the research on the limit of the first incompleteness theorem. Effectively inseparable (EI) theories can be viewed as an effective version of essentially undecidable (EU) theories, and EI is stronger than EU. We examine this question: Are there minimal effectively inseparable theories with respect to interpretability? We propose tEI, the theory version of EI. We first prove that there are no minimal tEI theories with respect to interpretability (i.e., for any tEI theory T, we can effectively (...)
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  3.  8
    Topic-Theoretic Extensions of Analytic Implication.Thomas Macaulay Ferguson - 2023 - Notre Dame Journal of Formal Logic 64 (4):471-493.
    Like many intensional logics, William Parry’s logic of analytic implication PAI admits extensions determined by imposing semantic conditions on its account of modality. PAI is unique, however, in its allowing a second dimension—a topic-theoretic dimension—along which extensions can be defined. The recent introduction by Francesco Berto of topic-sensitive intentional modals (TSIMs)—which disagree with PAI on this type of condition—provide further motivations to examine such topic-theoretic extensions. In this paper, we introduce, motivate, and characterize a number of such extensions of PAI, (...)
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  4.  6
    Products of Classes of Finite Structures.Vince Guingona, Miriam Parnes & Lynn Scow - 2023 - Notre Dame Journal of Formal Logic 64 (4):441-469.
    We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this article do not yield (...)
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  5.  3
    Topologizing Interpretable Groups in p-Adically Closed Fields.Will Johnson - 2023 - Notre Dame Journal of Formal Logic 64 (4):571-609.
    We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is (...)
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  6.  4
    A Note on Torsion Modules with Pure Embeddings.Marcos Mazari-Armida - 2023 - Notre Dame Journal of Formal Logic 64 (4):407-424.
    We study Martsinkovsky–Russell torsion modules with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative Σ-pure-injective modules extends the classical characterization of Gruson and Jenson as well as Zimmermann. We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a (...)
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  7.  14
    Identity and the Cognitive Value of Logical Equations in Frege’s Foundational Project.Matthias Schirn - 2023 - Notre Dame Journal of Formal Logic 64 (4):495-544.
    In this article, I first analyze and assess the epistemological and semantic status of canonical value-range equations in the formal language of Frege’s Grundgesetze der Arithmetik. I subsequently scrutinize the relation between (a) his informal, metalinguistic stipulation in Grundgesetze I, Section 3, and (b) its formal counterpart, which is Basic Law V. One point I argue for is that the stipulation in Section 3 was designed not only to fix the references of value-range names, but that it was probably also (...)
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  8.  8
    An Invitation to Extension Domination.Kyle Gannon & Jinhe Ye - 2023 - Notre Dame Journal of Formal Logic 64 (3):253-280.
    Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its typesetting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion (e.g., approximations by formulas, closure under localizations, convex combinations). We also prove a few preservation theorems (...)
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  9. Substitutional Validity for Modal Logic.Marco Grossi - 2023 - Notre Dame Journal of Formal Logic 64 (3):291-316.
    In the substitutional framework, validity is truth under all substitutions of the nonlogical vocabulary. I develop a theory where □ is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition.
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  10.  16
    Geometric Modal Logic.Brice Halimi - 2023 - Notre Dame Journal of Formal Logic 64 (3):377-406.
    The purpose of this paper is to generalize Kripke semantics for propositional modal logic by geometrizing it, that is, by considering the space underlying the collection of all possible worlds as an important semantic feature in its own right, so as to take the idea of accessibility seriously. The resulting new modal semantics is worked out in a setting coming from Riemannian geometry, where Kripke semantics is shown to correspond to a particular case, namely, the discrete one. Several correspondence results, (...)
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  11.  24
    Weak Kleene and Other Weak Logics of Conditionals.Jeremiah Joven Joaquin - 2023 - Notre Dame Journal of Formal Logic 64 (3):281-290.
    This paper presents a weak Kleene approach to conditionals that preserves some salient formal features of conditionals, particularly their interdefinability with Boolean logical connectives. I argue that such an approach fares better than other proposed weak logics of conditionals in this regard. In particular, it fares better than the logics proposed by Cooper, Cantwell, Farrell, De Finetti, Égré, Rossi, and Sprenger.
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  12.  4
    Lower Bounds of Sets of P-points.Borisa Kuzeljevic, Dilip Raghavan & Jonathan L. Verner - 2023 - Notre Dame Journal of Formal Logic 64 (3):317-327.
    We show that MAκ implies that each collection of Pc-points of size at most κ which has a Pc-point as an RK upper bound also has a Pc-point as an RK lower bound.
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  13.  11
    C. I. Lewis’s Intensional Semantics.Edwin Mares - 2023 - Notre Dame Journal of Formal Logic 64 (3):329-352.
    This paper begins with a discussion of C. I. Lewis’s theory of meaning in his book, An Analysis of Knowledge and Valuation (1946) and his pragmatic theory of analyticity and necessity. I bring this theories together with some remarks that he makes in an appendix to the second edition of Symbolic Logic to construct an algebraic semantics for his logics S2 and S3. These logics and their semantics are compared and evaluated with regard to how well they implement Lewis’s theories (...)
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  14.  8
    Definability of Boolean Functions in Kripke Semantics.Naosuke Matsuda - 2023 - Notre Dame Journal of Formal Logic 64 (3):363-376.
    A set F of Boolean functions is said to be functionally complete if every Boolean function is definable by combining functions in F. Post clarified when a set of Boolean functions is functionally complete (with respect to classical semantics). In this paper, by extending Post’s theorem, we clarify when a set of Boolean functions is functionally complete with respect to Kripke semantics.
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  15.  8
    Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures.Jaruwat Rodbanjong & Athipat Thamrongthanyalak - 2023 - Notre Dame Journal of Formal Logic 64 (3):353-362.
    Here we work in an arbitrary o-minimal expansion of a divisible ordered abelian group. We say that a definable ring is definably semiprime if squares of nontrivial two-sided ideals definable in the expansion are nontrivial. We prove a definable version of Wedderburn–Artin theorem and give a characterization of definably semiprime rings.
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  16.  6
    A Local Version of the Slaman–Wehner Theorem and Families Closed Under Finite Differences.Marat Faizrahmanov - 2023 - Notre Dame Journal of Formal Logic 64 (2):197-203.
    The main question of this article is whether there is a family closed under finite differences (i.e., if A belongs to the family and B=∗A, then B also belongs to the family) that can be enumerated by any noncomputable c.e. degree, but which cannot be enumerated computably. This question was formulated by Greenberg et al. (2020) in their recent work in which families that are closed under finite differences, close to the Slaman–Wehner families, are deeply studied.
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  17.  10
    Games on Base Matrices.Vera Fischer, Marlene Koelbing & Wolfgang Wohofsky - 2023 - Notre Dame Journal of Formal Logic 64 (2):247-251.
    We show that base matrices for P(ω)∕fin of regular height larger than h necessarily have maximal branches that are not cofinal. The same holds for base matrices of height h if tSpoiler
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  18.  8
    Witt Vectors and Separably Closed Fields with Higher Derivations.Daniel Max Hoffmann - 2023 - Notre Dame Journal of Formal Logic 64 (2):173-184.
    The main scope of this short article is to provide a modification of the axioms given by Messmer and Wood for the theory of separably closed fields of positive characteristic and finite imperfectness degree. As their original axioms failed to meet natural expectations, a new axiomatization was given (i.e., Ziegler’s one), but the new axioms do not follow Messmer and Wood’s initial idea. Therefore, we aim to give a correct axiomatization that is more similar to the original one and that, (...)
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  19.  17
    Spiritus Asper versus Lambda: On the Nature of Functional Abstraction.Ansten Klev - 2023 - Notre Dame Journal of Formal Logic 64 (2):205-223.
    The spiritus asper as used by Frege in a letter to Russell from 1904 bears resemblance to Church’s lambda. It is natural to ask how they relate to each other. An alternative approach to functional abstraction developed by Per Martin-Löf some thirty years ago allows us to describe the relationship precisely. Frege’s spiritus asper provides a way of restructuring a unary function name in Frege’s sense such that the argument place indicator occurs all the way to the right. Martin-Löf’s alternative (...)
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  20.  18
    Deduction Theorem in Congruential Modal Logics.Krzysztof A. Krawczyk - 2023 - Notre Dame Journal of Formal Logic 64 (2):185-196.
    We present an algebraic proof of the theorem stating that there are continuum many axiomatic extensions of global consequence associated with modal system E that do not admit the local deduction detachment theorem. We also prove that all these logics lack the finite frame property and have exactly three proper axiomatic extensions, each of which admits the local deduction detachment theorem.
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  21.  21
    Dialetheias and Numbers Distinct from Themselves.Uwe Petersen - 2023 - Notre Dame Journal of Formal Logic 64 (2):239-246.
    According to Priest, a proof can be distinct from itself in the same way that a number can. Priest does not specify any such number, so the present little note aims at filling this lacuna by providing a plain arithmetical code of a dialetheia similar to but simpler than the one presented in our recent work and thereby a natural number distinct from itself.
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  22.  6
    A Family of dp-Minimal Expansions of (Z;+).Chieu-Minh Tran & Erik Walsberg - 2023 - Notre Dame Journal of Formal Logic 64 (2):225-238.
    We consider structures of the form (Z;+,C), where C is an additive cyclic order on (Z;+). We show that such structures are dp-minimal and in this way produce a continuum-size family of dp-minimal expansions of (Z;+) such that no two members of the family define the same subsets of Z.
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  23.  7
    Unitary Representations of Locally Compact Groups as Metric Structures.Itaï Ben Yaacov & Isaac Goldbring - 2023 - Notre Dame Journal of Formal Logic 64 (2):159-172.
    For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how nondegenerate ∗-representations of a general ∗-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and nondegenerate ∗-representations of L1(G). We relate the notion (...)
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  24.  9
    Embeddings between Partial Combinatory Algebras.Anton Golov & Sebastiaan A. Terwijn - 2023 - Notre Dame Journal of Formal Logic 64 (1):129-158.
    Partial combinatory algebras (pcas) are algebraic structures that serve as generalized models of computation. In this article, we study embeddings of pcas. In particular, we systematize the embeddings between relativizations of Kleene’s models, of van Oosten’s sequential computation model, and of Scott’s graph model, showing that an embedding between two relativized models exists if and only if there exists a particular reduction between the oracles. We obtain a similar result for the lambda calculus, showing in particular that it cannot be (...)
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  25.  25
    Bicontextualism.Lorenzo Rossi - 2023 - Notre Dame Journal of Formal Logic 64 (1):95-127.
    Can one quantify over absolutely everything? Absolutists answer positively, while relativists answer negatively. Here, I focus on the absolutism versus relativism debate in the framework of theories of truth, where relativism becomes a form of contextualism about truth predications. Contextualist theories of truth provide elegant and uniform solutions to the semantic paradoxes while preserving classical logic. However, they interpret harmless generalizations (such as “everything is self-identical”) in less than absolutely comprehensive domains, thus systematically misconstruing them. In this article, I show (...)
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  26.  28
    Core Gödel.Neil Tennant - 2023 - Notre Dame Journal of Formal Logic 64 (1):15-59.
    This study examines how the Gödel phenomena are to be treated in core logic. We show in formal detail how one can use core logic in the metalanguage to prove Gödel’s incompleteness theorems for arithmetic even when classical logic is used for logical closure in the object language.
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  27.  5
    A Completeness Proof for a Regular Predicate Logic with Undefined Truth Value.Antti Valmari & Lauri Hella - 2023 - Notre Dame Journal of Formal Logic 64 (1):61-93.
    We provide a sound and complete proof system for an extension of Kleene’s ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of “is defined” is extended to terms and formulas via a straightforward recursive algorithm. The “is defined” formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the (...)
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  28.  4
    The Complexity of Decomposability of Computable Rings.Huishan Wu - 2023 - Notre Dame Journal of Formal Logic 64 (1):1-14.
    This article studies the complexity of decomposability of rings from the perspective of computability. Based on the equivalence between the decomposition of rings and that of the identity of rings, we propose four kinds of rings, namely, weakly decomposable rings, decomposable rings, weakly block decomposable rings, and block decomposable rings. Let R be the index set of computable rings. We study the complexity of subclasses of computable rings, showing that the index set of computable weakly decomposable rings is m-complete Σ10 (...)
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