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  1. A Theory of Necessities.Andrew Bacon & Jin Zeng - 2022 - Journal of Philosophical Logic 51 (1):151-199.
    We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about (...)
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  2. Engineering Existence?Lukas Skiba - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    This paper investigates the connection between two recent trends in philosophy: higher-orderism and conceptual engineering. Higher-orderists use higher-order quantifiers (in particular quantifiers binding variables that occupy the syntactic positions of predicates) to express certain key metaphysical doctrines, such as the claim that there are properties. I argue that, on a natural construal, the higher-orderist approach involves an engineering project concerning, among others, the concept of existence. I distinguish between a modest construal of this project, on which it aims at engineering (...)
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  3. Essence and Necessity.Andreas Ditter - forthcoming - Journal of Philosophical Logic:1-38.
    What is the relation between metaphysical necessity and essence? This paper defends the view that the relation is one of identity: metaphysical necessity is a special case of essence. My argument consists in showing that the best joint theory of essence and metaphysical necessity is one in which metaphysical necessity is just a special case of essence. The argument is made against the backdrop of a novel, higher-order logic of essence (HLE), whose core features are introduced in the first part (...)
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  4. Higher‐Order Metaphysics.Lukas Skiba - 2021 - Wiley: Philosophy Compass 16 (10):1-11.
    Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, propositions, and identity, (...)
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  5. Logic Works: A Rigorous Introduction to Formal Logic.Lorne Falkenstein, Scott Stapleford & Molly Kao - 2022 - New York: Routledge.
    Logic Works is a critical and extensive introduction to logic. It asks questions about why systems of logic are as they are, how they relate to ordinary language and ordinary reasoning, and what alternatives there might be to classical logical doctrines. It considers how logical analysis can be applied to carefully represent the reasoning employed in academic and scientific work, better understand that reasoning, and identify its hidden premises. Aiming to be as much a reference work and handbook for further, (...)
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  6. Modes of Truth: The Unified Approach to Truth, Modality, and Paradox.Carlo Nicolai & Johannes Stern (eds.) - 2021 - Routledge.
    The aim of this volume is to open up new perspectives and to raise new research questions about a unified approach to truth, modalities, and propositional attitudes. The volume's essays are grouped thematically around different research questions. The first theme concerns the tension between the theoretical role of the truth predicate in semantics and its expressive function in language. The second theme of the volume concerns the interaction of truth with modal and doxastic notions. The third theme covers higher-order solutions (...)
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  7. Inner Models From Extended Logics: Part 1.Juliette Kennedy, Menachem Magidor & Jouko Väänänen - 2020 - Journal of Mathematical Logic 21 (2):2150012.
    If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...
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  8. A Characterization of Σ11-Reflecting Ordinals.J. P. Aguilera - 2021 - Annals of Pure and Applied Logic 172 (10):103009.
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  9. The Logic of Logical Necessity.Andrew Bacon & Kit Fine - manuscript
    Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in (...)
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  10. Logical Combinatorialism.Andrew Bacon - 2020 - Philosophical Review 129 (4):537-589.
    In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the “book of the world.” This paper attempts to make good on this metaphor. To that end, a modality is introduced that, put informally, stands to propositions as logical truth stands to sentences. (...)
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  11. Higher-Order Metaphysics and the Tropes Versus Universals Dispute.Lukas Skiba - 2020 - Philosophical Studies 178 (9):2805-2827.
    Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue that higher-order realism does not in (...)
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  12. The FAN Principle and Weak König's Lemma in Herbrandized Second-Order Arithmetic.Fernando Ferreira - 2020 - Annals of Pure and Applied Logic 171 (9):102843.
    We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.
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  13. The Placeholder View of Assumptions and the Curry–Howard Correspondence.Ivo Pezlar - 2020 - Synthese (11):1-17.
    Proofs from assumptions are amongst the most fundamental reasoning techniques. Yet the precise nature of assumptions is still an open topic. One of the most prominent conceptions is the placeholder view of assumptions generally associated with natural deduction for intuitionistic propositional logic. It views assumptions essentially as holes in proofs, either to be filled with closed proofs of the corresponding propositions via substitution or withdrawn as a side effect of some rule, thus in effect making them an auxiliary notion subservient (...)
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  14. Randomness Notions and Reverse Mathematics.André Nies & Paul Shafer - 2020 - Journal of Symbolic Logic 85 (1):271-299.
    We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent (...)
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  15. Mechanizing Principia Logico-Metaphysica in Functional Type-Theory.Daniel Kirchner, Christoph Benzmüller & Edward N. Zalta - 2020 - Review of Symbolic Logic 13 (1):206-218.
    Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory that distinguishes between ordinary and abstract objects.This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is (...)
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  16. Elementary Inductive Dichotomy: Separation of Open and Clopen Determinacies with Infinite Alternatives.Kentaro Sato - 2020 - Annals of Pure and Applied Logic 171 (3):102754.
    We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; (...)
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  17. Substitution Structures.Andrew Bacon - 2019 - Journal of Philosophical Logic 48 (6):1017-1075.
    An increasing amount of twenty-first century metaphysics is couched in explicitly hyperintensional terms. A prerequisite of hyperintensional metaphysics is that reality itself be hyperintensional: at the metaphysical level, propositions, properties, operators, and other elements of the type hierarchy, must be more fine-grained than functions from possible worlds to extensions. In this paper I develop, in the setting of type theory, a general framework for reasoning about the granularity of propositions and properties. The theory takes as primitive the notion of a (...)
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  18. Radical Anti‐Disquotationalism.Andrew Bacon - 2018 - Philosophical Perspectives 32 (1):41-107.
    A number of `no-proposition' approaches to the liar paradox find themselves implicitly committed to a moderate disquotational principle: the principle that if an utterance of the sentence `$P$' says anything at all, it says that $P$ (with suitable restrictions). I show that this principle alone is responsible for the revenge paradoxes that plague this view. I instead propose a view in which there are several closely related language-world relations playing the `semantic expressing' role, none of which is more central to (...)
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  19. A Model of Second-Order Arithmetic Satisfying AC but Not DC.Sy-David Friedman, Victoria Gitman & Vladimir Kanovei - 2019 - Journal of Mathematical Logic 19 (1):1850013.
    We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. (...)
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  20. On the Mathematical and Foundational Significance of the Uncountable.Dag Normann & Sam Sanders - 2019 - Journal of Mathematical Logic 19 (1):1950001.
    We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] (...)
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  21. Minimum Models of Second-Order Set Theories.Kameryn J. Williams - 2019 - Journal of Symbolic Logic 84 (2):589-620.
    In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum (...)
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  22. M. R. Krom. Separation Principles in the Hierarchy Theory of Pure First-Order Logic. The Journal of Symbolic Logic, Vol. 28 No. 3 , Pp. 222–236.D. A. Clarke - 1966 - Journal of Symbolic Logic 31 (3):503.
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  23. Angelo Margaris. First Order Mathematical Logic. Blaisdell Publishing Company, Waltham, Massachusetts, Toronto, and London, 1967, X + 211 Pp. [REVIEW]A. H. Lightstone - 1972 - Journal of Symbolic Logic 37 (3):616.
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  24. Dag Prawitz. Hauptsatz for Higher Order Logic. The Journal of Symbolic Logic, Bd. 33 , S. 452–457. - Dag Prawitz. Completeness and Hauptsatz for Second Order Logic. Theoria , Bd. 33 , S. 246–258. - Moto-o Takahashi. A Proof of Cut-Elimination in Simple Type-Theory. Journal of the Mathematical Society of Japan, Bd. 19 , S. 399–410. [REVIEW]K. Schutte - 1974 - Journal of Symbolic Logic 39 (3):607-607.
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  25. Stewart Shapiro. Second-Order Languages and Mathematical Practice. The Journal of Symbolic Logic, Vol. 50 , Pp. 714–742. [REVIEW]Geoffrey Hellman - 1989 - Journal of Symbolic Logic 54 (1):291-293.
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  26. Nominalist Realism.Nicholas Jones - 2018 - Noûs:0-1.
    This paper explores the impact of quantification into predicate position on the metaphysics of properties, arguing that two familiar debates about properties are fundamentally altered by recasting them in a second-order setting. Two theories of properties are outlined, differing over whether the existence of properties is expressed using first-order or second-order quantifiers. It is argued that the second-order theory: provides good reason to regard debate about the locations of properties as contentless; resolves debate about whether properties are particulars or universals (...)
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  27. The Broadest Necessity.Andrew Bacon - 2018 - Journal of Philosophical Logic 47 (5):733-783.
    In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible (...)
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  28. The Logic of Opacity.Andrew Bacon & Jeffrey Sanford Russell - 2019 - Philosophy and Phenomenological Research 99 (1):81-114.
    We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models to show (...)
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  29. Second-Order Logic And Foundations Of Mathematics.Jouko V. "A. "An "Anen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
  30. Comparing Approaches To Resolution Based Higher-Order Theorem Proving.Christoph Benzmüller - 2002 - Synthese 133 (1):203-335.
    We investigate several approaches to resolution based automated theoremproving in classical higher-order logic (based on Church's simply typed-calculus) and discuss their requirements with respect to Henkincompleteness and full extensionality. In particular we focus on Andrews'higher-order resolution (Andrews 1971), Huet's constrained resolution (Huet1972), higher-order E-resolution, and extensional higher-order resolution(Benzmüller and Kohlhase 1997). With the help of examples we illustratethe parallels and differences of the extensionality treatment of these approachesand demonstrate that extensional higher-order resolution is the sole approach thatcan completely avoid additional (...)
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  31. Philosophy and Order in Logic.John J. Wellmuth - 1941 - Proceedings and Addresses of the American Philosophical Association 17:12.
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  32. Foundations Without Foundationalism: A Case for Second-Order Logic.Gila Sher - 1994 - Philosophical Review 103 (1):150.
  33. On Second-Order Characterizability.T. Hyttinen, K. Kangas & J. Vaananen - 2013 - Logic Journal of the IGPL 21 (5):767-787.
  34. Quasipolynomial Size Frege Proofs of Frankl’s Theorem on the Trace of Sets.James Aisenberg, Maria Luisa Bonet & Sam Buss - 2016 - Journal of Symbolic Logic 81 (2):687-710.
    We extend results of Bonet, Buss and Pitassi on Bondy’s Theorem and of Nozaki, Arai and Arai on Bollobás’ Theorem by proving that Frankl’s Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parametert, we prove that Frankl’s Theorem has polynomial size AC0-Frege proofs from instances of the pigeonhole principle.
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  35. Conservations of First-Order Reflections.Toshiyasu Arai - 2014 - Journal of Symbolic Logic 79 (3):814-825.
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  36. Geometrisation of First-Order Logic.Roy Dyckhoff & Sara Negri - 2015 - Bulletin of Symbolic Logic 21 (2):123-163.
    That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can (...)
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  37. Formalization of Reliability Block Diagrams in Higher-Order Logic.Waqar Ahmed, Osman Hasan & Sofiène Tahar - 2016 - Journal of Applied Logic 18:19-41.
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  38. Circle Graphs and Monadic Second-Order Logic.Bruno Courcelle - 2008 - Journal of Applied Logic 6 (3):416-442.
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  39. The Monadic Second-Order Logic of Graphs XV: On a Conjecture by D. Seese.Bruno Courcelle - 2006 - Journal of Applied Logic 4 (1):79-114.
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  40. A Higher-Order Solution to the Problem of the Concept Horse.Nicholas Jones - 2016 - Ergo: An Open Access Journal of Philosophy 3.
    This paper uses the resources of higher-order logic to articulate a Fregean conception of predicate reference, and of word-world relations more generally, that is immune to the concept horse problem. The paper then addresses a prominent style of expressibility problem for views of broadly this kind, versions of which are due to Linnebo, Hale, and Wright.
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  41. Relative Categoricity and Abstraction Principles.Sean Walsh & Sean Ebels-Duggan - 2015 - Review of Symbolic Logic 8 (3):572-606.
    Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...)
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  42. Order.Michael Berrill - 1966 - Perspectives in Biology and Medicine 9 (4):515-522.
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  43. First-Order Logic and First-Order Functions.Rodrigo A. Freire - 2015 - Logica Universalis 9 (3):281-329.
    This paper begins the study of first-order functions, which are a generalization of truth-functions. The concepts of truth-table and systems of truth-functions, both introduced in propositional logic by Post, are also generalized and studied in the quantificational setting. The general facts about these concepts are given in the first five sections, and constitute a “general theory” of first-order functions. The central theme of this paper is the relation of definition among notions expressed by formulas of first-order logic. We emphasize that (...)
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  44. Geometrisation of First-Order Logic.Roy Dyckhoff And Sara Negri - 2015 - Bulletin of Symbolic Logic 21 (2):123-163.
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  45. Second‐Order Logic and Set Theory.Jouko Väänänen - 2015 - Philosophy Compass 10 (7):463-478.
    Both second-order logic and set theory can be used as a foundation for mathematics, that is, as a formal language in which propositions of mathematics can be expressed and proved. We take it upon ourselves in this paper to compare the two approaches, second-order logic on one hand and set theory on the other hand, evaluating their merits and weaknesses. We argue that we should think of first-order set theory as a very high-order logic.
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  46. Second-Order Logic : Ontological and Epistemological Problems.Marcus Rossberg - unknown
    In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether second-order logic can count as logic in some proper sense of logic, and what epistemic status it occupies. More specifically, second-order logic is sometimes taken to be mathematical, a mere notational variant of some fragment of set theory. If this is the case, it might be argued that it does not have the (...)
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  47. [Standing Order : Staff Use Only].Niels Öffenberger & Albert Menne - 1982
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  48. R. I. G. HUGHES "A Philosophical Companion to First-Order Logic". [REVIEW]E. J. Lowe - 1994 - History and Philosophy of Logic 15 (2):255.
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  49. Higher-Order Logic.Johan van Benthem & Kees Doets - 1989 - Journal of Symbolic Logic 54 (3):1090-1092.
  50. Review of Montague, "Set Theory and Higher-Order Logic".Richard Mansfield - 1975 - Journal of Symbolic Logic 40 (3):459.
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