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  1.  2
    A Walk with Goodstein and Ackermann.David Fernández-Duque & Andreas Weiermann - 2024 - Notre Dame Journal of Formal Logic 65 (2):181-201.
    Goodstein’s theorem states that certain sequences based on exponential notation for the natural numbers are always finite. The result is independent of Peano arithmetic and is a prototypical example of a proof of termination by transfinite induction. A variant based instead on the Ackermann function has more recently been proposed by Arai, Fernández-Duque, Wainer, and Weiermann, and instead is independent of the more powerful theory ATR0. However, this result is contingent on rather elaborate normal forms for natural numbers based on (...)
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  2.  4
    A Natural Deduction Calculus for S4.2.Simone Martini, Andrea Masini & Margherita Zorzi - 2024 - Notre Dame Journal of Formal Logic 65 (2):127-150.
    We propose a natural deduction calculus for the modal logic S4.2. The system is designed to match as much as possible the structure and the properties of the standard system of natural deduction for first-order classical logic, exploiting the formal analogy between modalities and quantifiers. The system is proved sound and complete with respect to (w.r.t.) the standard Hilbert-style formulation of S4.2. Normalization and its consequences are obtained in a natural way, with proofs that closely follow the analogous ones for (...)
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  3.  3
    Sets Completely Separated by Functions in Bishop Set Theory.Iosif Petrakis - 2024 - Notre Dame Journal of Formal Logic 65 (2):151-180.
    Within Bishop Set Theory, a reconstruction of Bishop’s theory of sets, we study the so-called completely separated sets, that is, sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We introduce the notion of a global family of completely separated sets over an index-completely separated set, and we describe its Sigma- and Pi-set. The free completely separated set on a given set is also presented. Purely set-theoretic versions of the classical Stone–Čech theorem (...)
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  4.  9
    Wittgensteinian Predicate Logic and Compositionality.Kai F. Wehmeier - 2024 - Notre Dame Journal of Formal Logic 65 (2):113-125.
    I investigate whether Wittgenstein’s “weakly exclusive” Tractarian semantics (as reconstructed by Rogers and Wehmeier) is compositional. In both Tarskian and Wittgensteinian semantics, one has the choice of either working exclusively with total variable assignments or allowing partial assignments; the choice has no bearing on the compositionality of Tarskian semantics, but turns out to make a difference in the Wittgensteinian case. Some philosophical ramifications of this observation are discussed.
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  5.  10
    Boolean-Valued Models of Set Theory with Urelements.Xinhe Wu & Bokai Yao - 2024 - Notre Dame Journal of Formal Logic 65 (2):203-227.
    We explore Boolean-valued models of set theory with a class of urelements. In an existing construction, which we call UB, every urelement is its own B-name. We prove the fundamental theorem of UB in the context of ZFUR (i.e., ZF with urelements formulated with Replacement). In particular, UB is shown to preserve Replacement and hence ZFUR. Moreover, UB can both destroy axioms, such as the DCω1-scheme, and recover axioms, such as the Collection Principle. One drawback of UB is that it (...)
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  6.  13
    Tableaux and Interpolation for Propositional Justification Logics.Meghdad Ghari - 2024 - Notre Dame Journal of Formal Logic 65 (1):81-112.
    We present tableau proof systems for the annotated version of propositional justification logics, that is, justification logics which are formulated using annotated application operators. We show that the tableau systems are sound and complete with respect to Mkrtychev models, and some tableau systems are analytic and provide a decision procedure for the annotated justification logics. We further show Craig’s interpolation property and Beth’s definability theorem for some annotated justification logics.
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  7.  14
    Modal Model Theory.Joel David Hamkins & Wojciech Aleksander Wołoszyn - 2024 - Notre Dame Journal of Formal Logic 65 (1):1-37.
    We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) (...)
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  8.  3
    A Problem for Relative-Sameness Semantics.James Milford - 2024 - Notre Dame Journal of Formal Logic 65 (1):39-53.
    In 2008, Graff Fara presented relative-sameness semantics, a semantics for a first-order modal and temporal language with the explicit aim of being able to render true certain contingent/temporary identity claims (relative to certain contexts). Graff Fara achieves this aim by abandoning a straightforward analysis of de re modal/temporal claims in terms of identity. Instead, such a claim is analyzed in terms of her relative-sameness relations (which need not be the identity relation), with the relevant relative-sameness relations in play determined by (...)
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  9.  4
    Logics of True Belief.Yuanzhe Yang - 2024 - Notre Dame Journal of Formal Logic 65 (1):55-80.
    In epistemic logic, the beliefs of an agent are modeled in a way very similar to knowledge, except that they are fallible. Thus, the pattern of an agent’s true beliefs is an interesting subject to study. In this paper, we conduct a systematic study on a novel modal logic with the bundled operator ⊡ϕ:=□ϕ∧ϕ as the only primitive modality, where ⊡ captures the notion of true belief. With the help of a novel notion of ⊡-bisimulation, we characterize the expressivity of (...)
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