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  1. A Taxonomy for Set-Theoretic Potentialism.Davide Sutto - 2024 - Philosophia Mathematica:1-28.
    Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. (...)
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  • (1 other version)Mathematical Modality: An Investigation in Higher-order Logic.Andrew Bacon - 2024 - Journal of Philosophical Logic 53 (1):131-179.
    An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show that contingency about the (...)
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  • Divergent Potentialism: A Modal Analysis With an Application to Choice Sequences.Ethan Brauer, Øystein Linnebo & Stewart Shapiro - 2022 - Philosophia Mathematica 30 (2):143-172.
    Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those (...)
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  • 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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  • Arithmetic is Necessary.Zachary Goodsell - 2024 - Journal of Philosophical Logic 53 (4).
    (Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are applied to answer a question (...)
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  • 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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