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  1. The Embedding Theorem: Its Further Developments and Consequences. Part 1.Alexei Y. Muravitsky - 2006 - Notre Dame Journal of Formal Logic 47 (4):525-540.
    We outline the Gödel-McKinsey-Tarski Theorem on embedding of Intuitionistic Propositional Logic Int into modal logic S4 and further developments which led to the Generalized Embedding Theorem. The latter in turn opened a full-scale comparative exploration of lattices of the (normal) extensions of modal propositional logic S4, provability logic GL, proof-intuitionistic logic KM, and others, including Int. The present paper is a contribution to this part of the research originated from the Gödel-McKinsey-Tarski Theorem. In particular, we show that the lattice ExtInt (...)
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  • Proof analysis in intermediate logics.Roy Dyckhoff & Sara Negri - 2012 - Archive for Mathematical Logic 51 (1):71-92.
    Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...)
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  • Advances in Natural Deduction: A Celebration of Dag Prawitz's Work.Luiz Carlos Pereira & Edward Hermann Haeusler (eds.) - 2012 - Dordrecht, Netherland: Springer.
    This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...)
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  • Leo Esakia on Duality in Modal and Intuitionistic Logics.Guram Bezhanishvili (ed.) - 2014 - Dordrecht, Netherland: Springer.
    This volume is dedicated to Leo Esakia's contributions to the theory of modal and intuitionistic systems. Consisting of 10 chapters, written by leading experts, this volume discusses Esakia’s original contributions and consequent developments that have helped to shape duality theory for modal and intuitionistic logics and to utilize it to obtain some major results in the area. Beginning with a chapter which explores Esakia duality for S4-algebras, the volume goes on to explore Esakia duality for Heyting algebras and its generalizations (...)
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  • Gödel's functional interpretation and its use in current mathematics.Ulrich Kohlenbach - 2008 - Dialectica 62 (2):223–267.
  • Gödel's Functional Interpretation and its Use in Current Mathematics.Ulrich Kohlenbach - 2008 - Dialectica 62 (2):223-267.
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  • Proof Theory and Algebra in Logic.Hiroakira Ono - 2019 - Singapore: Springer Singapore.
    This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate (...)
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  • Ruth Barcan Marcus.Roberta Ballarin - 2024 - Stanford Encyclopedia of Philosophy.
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  • Modal Logics Between Propositional and First Order.Melvin Fitting - unknown
    One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, but it is not exactly propositional either. For propositional logics such as K, T and D, adding such machinery produces (...)
     
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  • Proof of a conjecture of Roman Suszko.Stanislaw Zachorowski - 1975 - Studia Logica 34 (3):253 - 256.
  • Decidability of quantified propositional intuitionistic logic and s4 on trees of height and arity ≤ω.Richard Zach - 2004 - Journal of Philosophical Logic 33 (2):155-164.
    Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers ∀p, ∃p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most ω, the resulting logics are decidable. (...)
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  • Self-referentiality of Brouwer–Heyting–Kolmogorov semantics.Junhua Yu - 2014 - Annals of Pure and Applied Logic 165 (1):371-388.
    The Gödel–Artemov framework offered a formalization of the Brouwer–Heyting–Kolmogorov semantics of intuitionistic logic via classical proofs. In this framework, the intuitionistic propositional logic IPC is embedded in the modal logic S4, S4 is realized in the Logic of Proofs LP, and LP has a provability interpretation in Peano Arithmetic. Self-referential LP-formulas of the type ‘t is a proof of a formula ϕ containing t itself’ are permitted in the realization of S4 in LP, and if such formulas are indeed involved, (...)
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  • On non-self-referential fragments of modal logics.Junhua Yu - 2017 - Annals of Pure and Applied Logic 168 (4):776-803.
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  • From Interior Algebras to Unital ℓ-Groups: A Unifying Treatment of Modal Residuated Lattices.William Young - 2015 - Studia Logica 103 (2):265-286.
    Much work has been done on specific instances of residuated lattices with modal operators . In this paper, we develop a general framework that subsumes three important classes of modal residuated lattices: interior algebras, Abelian ℓ-groups with conuclei, and negative cones of ℓ-groups with nuclei. We then use this framework to obtain results about these three cases simultaneously. In particular, we show that a categorical equivalence exists in each of these cases. The approach used here emphasizes the role played by (...)
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  • Basic Intuitionistic Conditional Logic.Yale Weiss - 2019 - Journal of Philosophical Logic 48 (3):447-469.
    Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas’ important classical system CK. I give ICK both worlds (...)
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  • In the shadows of the löwenheim-Skolem theorem: Early combinatorial analyses of mathematical proofs.Jan von Plato - 2007 - Bulletin of Symbolic Logic 13 (2):189-225.
    The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation is (...)
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  • The defective conditional in mathematics.Mathieu Vidal - 2014 - Journal of Applied Non-Classical Logics 24 (1-2):169-179.
    This article focuses on defective conditionals ? namely indicative conditionals whose antecedents are false and whose truth-values therefore cannot be determined. The problem is to decide which formal connective can adequately represent this usage. Classical logic renders defective conditionals true whereas traditional mathematics dismisses them as irrelevant. This difference in treatment entails that, at the propositional level, classical logic validates some sentences that are intuitively false in plane geometry. With two proofs, I show that the same flaw is shared by (...)
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  • Recenzje.Seweryna Łuszczewska-Romahnowa - 1964 - Studia Logica 15 (1):319-333.
  • A semantical study of constructible falsity.Richmond H. Thomason - 1969 - Mathematical Logic Quarterly 15 (16-18):247-257.
  • The Gödel-McKinsey-Tarski embedding for infinitary intuitionistic logic and its extensions.Matteo Tesi & Sara Negri - 2023 - Annals of Pure and Applied Logic 174 (8):103285.
  • Constructive theories through a modal lens.Matteo Tesi - forthcoming - Logic Journal of the IGPL.
    We present a uniform proof-theoretic proof of the Gödel–McKinsey–Tarski embedding for a class of first-order intuitionistic theories. This is achieved by adapting to the case of modal logic the methods of proof analysis in order to convert axioms into rules of inference of a suitable sequent calculus. The soundness and the faithfulness of the embedding are proved by induction on the height of the derivations in the augmented calculi. Finally, we define an extension of the modal system for which the (...)
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  • B-varieties with normal free algebras.Bronis?aw Tembrowski - 1989 - Studia Logica 48 (4):555 - 564.
    The starting point for the investigation in this paper is the following McKinsey-Tarski's Theorem: if f and g are algebraic functions (of the same number of variables) in a topological Boolean algebra (TBA) and if C(f)C(g) vanishes identically, then either f or g vanishes identically. The present paper generalizes this theorem to B-algebras and shows that validity of that theorem in a variety of B-algebras (B-variety) generated by SCI B -equations implies that its free Lindenbaum-Tarski's algebra is normal. This is (...)
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  • Constructing a continuum of predicate extensions of each intermediate propositional logic.Nobu-Yuki Suzuki - 1995 - Studia Logica 54 (2):173 - 198.
    Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 (...)
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  • Systemy S4 I S5 Lewisa a spójnik identyczności.Roman Suszko & Wiesława Żandarowska - 1971 - Studia Logica 29 (1):169-177.
  • Identity connective and modality.Roman Suszko - 1971 - Studia Logica 27 (1):7-39.
  • Mass Problems and Intuitionism.Stephen G. Simpson - 2008 - Notre Dame Journal of Formal Logic 49 (2):127-136.
    Let $\mathcal{P}_w$ be the lattice of Muchnik degrees of nonempty $\Pi^0_1$ subsets of $2^\omega$. The lattice $\mathcal{P}$ has been studied extensively in previous publications. In this note we prove that the lattice $\mathcal{P}$ is not Brouwerian.
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  • Competitive equilibrium with intuitionistic agents.Jack Douglas Stecher - 2011 - Synthese 181 (S1):49 - 63.
    This paper studies an economy whose agents perceive their consumption possibilities subjectively, and whose preferences are defined on what they subjectively experience, rather than on those alternatives that are objectively present. The model of agents' perceptions is based on intuitionistic logic. Roughly, this means that agents reason constructively: a solution to a problem exists only if there is a construction by which the problem can be solved. The theorems that can be proved determine how an agent perceives a set of (...)
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  • Intermediate logics and factors of the Medvedev lattice.Andrea Sorbi & Sebastiaan A. Terwijn - 2008 - Annals of Pure and Applied Logic 155 (2):69-85.
    We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them.
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  • Modal counterparts of Medvedev logic of finite problems are not finitely axiomatizable.Valentin Shehtman - 1990 - Studia Logica 49 (3):365 - 385.
    We consider modal logics whose intermediate fragments lie between the logic of infinite problems [20] and the Medvedev logic of finite problems [15]. There is continuum of such logics [19]. We prove that none of them is finitely axiomatizable. The proof is based on methods from [12] and makes use of some graph-theoretic constructions (operations on coverings, and colourings).
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  • 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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  • Why classical logic is privileged: justification of logics based on translatability.Gerhard Schurz - 2021 - Synthese 199 (5-6):13067-13094.
    In Sect. 1 it is argued that systems of logic are exceptional, but not a priori necessary. Logics are exceptional because they can neither be demonstrated as valid nor be confirmed by observation without entering a circle, and their motivation based on intuition is unreliable. On the other hand, logics do not express a priori necessities of thinking because alternative non-classical logics have been developed. Section 2 reflects the controversies about four major kinds of non-classical logics—multi-valued, intuitionistic, paraconsistent and quantum (...)
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  • Meaning-Preserving Translations of Non-classical Logics into Classical Logic: Between Pluralism and Monism.Gerhard Schurz - 2021 - Journal of Philosophical Logic 51 (1):27-55.
    In order to prove the validity of logical rules, one has to assume these rules in the metalogic. However, rule-circular ‘justifications’ are demonstrably without epistemic value. Is a non-circular justification of a logical system possible? This question attains particular importance in view of lasting controversies about classical versus non-classical logics. In this paper the question is answered positively, based on meaning-preserving translations between logical systems. It is demonstrated that major systems of non-classical logic, including multi-valued, paraconsistent, intuitionistic and quantum logics, (...)
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  • Review. [REVIEW]Katia Saporiti & Uwe Scheffler - 1993 - Erkenntnis 38 (1):135-143.
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  • Expansions of Semi-Heyting Algebras I: Discriminator Varieties.H. P. Sankappanavar - 2011 - Studia Logica 98 (1-2):27-81.
    This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion (...)
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  • Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus.V. V. Rybakov - 1990 - Annals of Pure and Applied Logic 50 (1):71-106.
    Questions connected with the admissibility of rules of inference and the solvability of the substitution problem for modal and intuitionistic logic are considered in an algebraic framework. The main result is the decidability of the universal theory of the free modal algebra imageω extended in signature by adding constants for free generators. As corollaries we obtain: there exists an algorithm for the recognition of admissibility of rules with parameters in the modal system Grz, the substitution problem for Grz and for (...)
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  • Ideological innocence.Daniel Rubio - 2022 - Synthese 200 (5):1-22.
    Quine taught us the difference between a theory’s ontology and its ideology. Ontology is the things a theory’s quantifiers must range over if it is true, Ideology is the primitive concepts that must be used to state the theory. This allows us to split the theoretical virtue of parsimony into two kinds: ontological parsimony and ideological parsimony. My goal is help illuminate the virtue of ideological parsimony by giving a criterion for ideological innocence—a rule for when additional ideology does not (...)
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  • BI‐Modal Logic, Double‐Closure Algebras, and Hilbert Space.Jean E. Rubin - 1962 - Mathematical Logic Quarterly 8 (3‐4):305-322.
  • Willem Blok and Modal Logic.W. Rautenberg, M. Zakharyaschev & F. Wolter - 2006 - Studia Logica 83 (1):15-30.
    We present our personal view on W.J. Blok's contribution to modal logic.
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  • Conservative Translations Revisited.J. Ramos, J. Rasga & C. Sernadas - 2023 - Journal of Philosophical Logic 52 (3):889-913.
    We provide sufficient conditions for the existence of a conservative translation from a consequence system to another one. We analyze the problem in many settings, namely when the consequence systems are generated by a deductive calculus or by a logic system including both proof-theoretic and model-theoretic components. We also discuss reflection of several metaproperties with the objective of showing that conservative translations provide an alternative to proving such properties from scratch. We discuss soundness and completeness, disjunction property and metatheorem of (...)
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  • On two problems of Harvey Friedman.Tadeusz Prucnal - 1979 - Studia Logica 38 (3):247 - 262.
    The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).
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  • The deducibilities of S.Jean Porte - 1981 - Journal of Philosophical Logic 10 (4):409 - 422.
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  • Propositional Quantification in the Topological Semantics for S.Philip Kremer - 1997 - Notre Dame Journal of Formal Logic 38 (2):295-313.
    Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems , , : given a Kripke frame, the quantifiers range over all the sets of possible worlds. is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub , (...)
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  • Synonymous logics.Francis Jeffry Pelletier & Alasdair Urquhart - 2003 - Journal of Philosophical Logic 32 (3):259-285.
    This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are "synonymous" or "translationally equivalent." We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to well-known examples. Some philosophical morals (...)
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  • Unified quantum logic.Mladen Pavičić - 1989 - Foundations of Physics 19 (8):999-1016.
    Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.
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  • Restricted Extensions of Implicational Calculi.Biswambhar Pahi - 1971 - Mathematical Logic Quarterly 17 (1):11-16.
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  • Bivalence: Meaning theory vs metaphysics.Peter Pagin - 1998 - Theoria 64 (2-3):157-186.
    This paper is an attack on the Dummett-Prawitz view that the principle of bivalence has a crucial double significance, metaphysical and meaning theoretical. On the one hand it is said that holding bivalence valid is what characterizes a realistic view, i.e. a view in metaphysics, and on the other hand it is said that there are meaning theoretical arguments against its acceptability. I argue that these two aspects are incompatible. If the failure of validity of bivalence depends on properties of (...)
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  • Interpretations of intuitionist logic in non-normal modal logics.Colin Oakes - 1999 - Journal of Philosophical Logic 28 (1):47-60.
    Historically, it was the interpretations of intuitionist logic in the modal logic S4 that inspired the standard Kripke semantics for intuitionist logic. The inspiration of this paper is the interpretation of intuitionist logic in the non-normal modal logic S3: an S3 model structure can be 'looked at' as an intuitionist model structure and the semantics for S3 can be 'cashed in' to obtain a non-normal semantics for intuitionist propositional logic. This non-normal semantics is then extended to intuitionist quantificational logic.
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  • The intensional side of algebraic-topological representation theorems.Sara Negri - 2017 - Synthese 198 (Suppl 5):1121-1143.
    Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic (...)
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  • Proofs and Countermodels in Non-Classical Logics.Sara Negri - 2014 - Logica Universalis 8 (1):25-60.
    Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class (...)
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  • Truth-value stipulations for the Von Wright system m′ and the Heyting system.Akira Nakamura - 1964 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 10 (9-12):173-183.
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