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  • The Power of Naive Truth
    Rev. Symb. Log. (IF 0.75) Pub Date : 2020-07-06
    HARTRY FIELD

    Nonclassical theories of truth that take truth to be transparent have some obvious advantages over any classical theory of truth (which must take it as nontransparent on pain of inconsistency). But several authors have recently argued that there’s also a big disadvantage of nonclassical theories as compared to their “external” classical counterparts: proof-theoretic strength. While conceding the relevance

    更新日期:2020-07-06
  • A NOTE ON THE SEQUENT CALCULI ${{{\bf G3}[{\bf mic}]}}^{=}$
    Rev. Symb. Log. (IF 0.75) Pub Date : 2020-07-02
    FRANCO PARLAMENTO; FLAVIO PREVIALE

    We show that the replacement rule of the sequent calculi ${\bf G3[mic]}^= $ in [8] can be replaced by the simpler rule in which one of the principal formulae is not repeated in the premiss.

    更新日期:2020-07-02
  • CALCULIZING CLASSICAL INFERENTIAL EROTETIC LOGIC
    Rev. Symb. Log. (IF 0.75) Pub Date : 2020-06-29
    MORITZ CORDES

    This paper contributes to the calculization of evocation and erotetic implication as defined by Inferential Erotetic Logic (IEL). There is a straightforward approach to calculizing (propositional) erotetic implication which cannot be applied to evocation. First-order evocation is proven to be uncalculizable, i.e. there is no proof system, say FOE, such that for all $X, Q$ : X evokes Q iff there is

    更新日期:2020-06-29
  • VOLUNTARY IMAGINATION: A FINE-GRAINED ANALYSIS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2020-06-29
    ILARIA CANAVOTTO; FRANCESCO BERTO; ALESSANDRO GIORDANI

    We study imagination as reality-oriented mental simulation (ROMS): the activity of simulating nonactual scenarios in one’s mind, to investigate what would happen if they were realized. Three connected questions concerning ROMS are: What is the logic, if there is one, of such an activity? How can we gain new knowledge via it? What is voluntary in it and what is not? We address them by building a list

    更新日期:2020-06-29
  • WITTGENSTEIN’S ELIMINATION OF IDENTITY FOR QUANTIFIER-FREE LOGIC
    Rev. Symb. Log. (IF 0.75) Pub Date : 2020-06-25
    TIMM LAMPERT; MARKUS SÄBEL

    One of the central logical ideas in Wittgenstein’s Tractatus logico-philosophicus is the elimination of the identity sign in favor of the so-called “exclusive interpretation” of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier

    更新日期:2020-06-25
  • DIFFERENCE-MAKING CONDITIONALS AND THE RELEVANT RAMSEY TEST
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-12-12
    HANS ROTT

    This article explores conditionals expressing that the antecedent makes a difference for the consequent. A ‘relevantised’ version of the Ramsey Test for conditionals is employed in the context of the classical theory of belief revision. The idea of this test is that the antecedent is relevant to the consequent in the following sense: a conditional is accepted just in case (i) the consequent is accepted

    更新日期:2019-12-12
  • SOME MODEL-THEORETIC RESULTS ON THE 3-VALUED PARACONSISTENT FIRST-ORDER LOGIC QCIORE
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-12-09
    MARCELO E. CONIGLIO; G.T. GOMEZ-PEREIRA; MARTÍN FIGALLO

    The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is

    更新日期:2019-12-09
  • NON–WELL-FOUNDED DERIVATIONS IN THE GÖDEL-LÖB PROVABILITY LOGIC
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-11-26
    DANIYAR SHAMKANOV

    We consider Hilbert-style non–well-founded derivations in the Gödel-Löb provability logic GL and establish that GL with the obtained derivability relation is globally complete for algebraic and neighbourhood semantics.

    更新日期:2019-11-26
  • NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-11-18
    WILFRIED SIEG; PATRICK WALSH

    Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal

    更新日期:2019-11-18
  • A LEARNING-THEORETIC CHARACTERISATION OF MARTIN-LÖF RANDOMNESS AND SCHNORR RANDOMNESS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-11-04
    FRANCESCA ZAFFORA BLANDO

    Numerous learning tasks can be described as the process of extrapolating patterns from observed data. One of the driving intuitions behind the theory of algorithmic randomness is that randomness amounts to the absence of any effectively detectable patterns: it is thus natural to regard randomness as antithetical to inductive learning. Osherson and Weinstein [11] draw upon the identification of randomness

    更新日期:2019-11-04
  • STRONG COMPLETENESS OF MODAL LOGICS OVER 0-DIMENSIONAL METRIC SPACES
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-10-24
    ROBERT GOLDBLATT; IAN HODKINSON

    We prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.

    更新日期:2019-10-24
  • LOGIC AND TOPOLOGY FOR KNOWLEDGE, KNOWABILITY, AND BELIEF
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-10-09
    ADAM BJORNDAHL; AYBÜKE ÖZGÜN

    In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge 35. Building on Stalnaker’s core insights, and using frameworks developed in 11 and 3, we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking)

    更新日期:2019-10-09
  • A NOTE ON THE CUT-ELIMINATION PROOF IN “TRUTH WITHOUT CONTRA(DI)CTION”
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-10-09
    ANDREAS FJELLSTAD

    This note shows that the permutation instructions presented by Zardini (2011) for eliminating cuts on universally quantified formulas in the sequent calculus for the noncontractive theory of truth IKTω are inadequate. To that purpose the note presents a derivation in the sequent calculus for IKTω ending with an application of cut on a universally quantified formula which the permutation instructions

    更新日期:2019-10-09
  • MATHEMATICAL RIGOR AND PROOF
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-10-04
    YACIN HAMAMI

    Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated

    更新日期:2019-10-04
  • THREE CHARACTERIZATIONS OF STRICT COHERENCE ON INFINITE-VALUED EVENTS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-10-04
    TOMMASO FLAMINIO

    This article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of

    更新日期:2019-10-04
  • DISTANCES BETWEEN FORMAL THEORIES
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-10-04
    MOHAMED KHALED; GERGELY SZÉKELY; KOEN LEFEVER; MICHÈLE FRIEND

    In the literature, there have been several methods and definitions for working out whether two theories are “equivalent” (essentially the same) or not. In this article, we do something subtler. We provide a means to measure distances (and explore connections) between formal theories. We introduce two natural notions for such distances. The first one is that of axiomatic distance, but we argue that

    更新日期:2019-10-04
  • THE DEVELOPMENT OF GÖDEL’S ONTOLOGICAL PROOF
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-09-20
    ANNIKA KANCKOS; TIM LETHEN

    Gödel’s ontological proof is by now well known based on the 1970 version, written in Gödel’s own hand, and Scott’s version of the proof. In this article new manuscript sources found in Gödel’s Nachlass are presented. Three versions of Gödel’s ontological proof have been transcribed, and completed from context as true to Gödel’s notes as possible. The discussion in this article is based on these new

    更新日期:2019-09-20
  • SUBSTITUTION IN RELEVANT LOGICS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-09-17
    TORE FJETLAND ØGAARD

    This essay discusses rules and semantic clauses relating to Substitution—Leibniz’s law in the conjunctive-implicational form $s\dot{ = }t \wedge A\left( s \right) \to A\left( t \right)$ —as these are put forward in Priest’s books In Contradiction and An Introduction to Non-Classical Logic: From If to Is. The stated rules and clauses are shown to be too weak in some cases and too strong in others. New

    更新日期:2019-09-17
  • RESTRICTED INTERPOLATION AND LACK THEREOF IN STIT LOGIC
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-09-13
    GRIGORY K. OLKHOVIKOV

    We consider the propositional logic equipped with Chellas stit operators for a finite set of individual agents plus the historical necessity modality. We settle the question of whether such a logic enjoys restricted interpolation property, which requires the existence of an interpolant only in cases where the consequence contains no Chellas stit operators occurring in the premise. We show that if action

    更新日期:2019-09-13
  • FRACTIONAL SEMANTICS FOR CLASSICAL LOGIC
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-09-10
    MARIO PIAZZA; GABRIELE PULCINI

    This article presents a new (multivalued) semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total

    更新日期:2019-09-10
  • DE ZOLT’S POSTULATE: AN ABSTRACT APPROACH
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-09-02
    EDUARDO N. GIOVANNINI; EDWARD H. HAEUSLER; ABEL LASSALLE-CASANAVE; PAULO A. S. VELOSO

    A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”)

    更新日期:2019-09-02
  • CUT AND GAMMA I: PROPOSITIONAL AND CONSTANT DOMAIN R
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-08-29
    YALE WEISS

    The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility

    更新日期:2019-08-29
  • THE ELIMINATION OF ATOMIC CUTS AND THE SEMISHORTENING PROPERTY FOR GENTZEN’S SEQUENT CALCULUS WITH EQUALITY
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-08-13
    FRANCO PARLAMENTO; FLAVIO PREVIALE

    We study various extensions of Gentzen’s sequent calculus obtained by adding rules for equality. One of them is singled out as particularly natural and shown to satisfy full cut elimination, namely, also atomic cuts can be eliminated. Furthermore we tell apart the extensions that satisfy full cut elimination from those that do not and establish a strengthened form of the nonlenghtening property of

    更新日期:2019-08-13
  • ON MORITA EQUIVALENCE AND INTERPRETABILITY
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-08-13
    PAUL ANH MCELDOWNEY

    In a recent article, Barrett & Halvorson (2016) define a notion of equivalence for first-order theories, which they call “Morita equivalence.” To argue that Morita equivalence is a reasonable measure of “theoretical equivalence,” they make use of the claim that Morita extensions “say no more” than the theories they are extending. The goal of this article is to challenge this central claim by raising

    更新日期:2019-08-13
  • THE LARGE STRUCTURES OF GROTHENDIECK FOUNDED ON FINITE-ORDER ARITHMETIC
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-08-02
    COLIN MCLARTY

    The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure

    更新日期:2019-08-02
  • EFFECTIVE INSEPARABILITY, LATTICES, AND PREORDERING RELATIONS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-07-12
    URI ANDREWS; ANDREA SORBI

    We study effectively inseparable (abbreviated as e.i.) prelattices (i.e., structures of the form $L = \langle \omega , \wedge , \vee ,0,1,{ \le _L}\rangle$ where ω denotes the set of natural numbers and the following four conditions hold: (1) $\wedge , \vee$ are binary computable operations; (2) ${ \le _L}$ is a computably enumerable preordering relation, with $0{ \le _L}x{ \le _L}1$ for every x; (3)

    更新日期:2019-07-12
  • PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-06-17
    SERGEY DROBYSHEVICH; HEINRICH WANSING

    We present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic

    更新日期:2019-06-17
  • NONREPRESENTABLE RELATION ALGEBRAS FROM GROUPS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-06-13
    HAJNAL ANDRÉKA; ISTVÁN NÉMETI; STEVEN GIVANT

    A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

    更新日期:2019-06-13
  • ANOTHER LOOK AT THE SECOND INCOMPLETENESS THEOREM
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-06-10
    ALBERT VISSER

    In this article we study proofs of some general forms of the Second Incompleteness Theorem. These forms conform to the Feferman format, where the proof predicate is fixed and the representation of the set of axioms varies. We extend the Feferman framework in one important point: we allow the interpretation of number theory to vary.

    更新日期:2019-06-10
  • INCOMPLETENESS VIA PARADOX AND COMPLETENESS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-05-23
    WALTER DEAN

    This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result

    更新日期:2019-05-23
  • SOME OBSERVATIONS ON TRUTH HIERARCHIES: A CORRECTION
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-05-20
    PHILIP D. WELCH

    A correction is needed to our paper: to the definition contained within the statement of Lemma 1.5 and thus arguments around it in §3.

    更新日期:2019-05-20
  • GENERIC LARGE CARDINALS AS AXIOMS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-05-14
    MONROE ESKEW

    We argue against Foreman’s proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.

    更新日期:2019-05-14
  • HEREDITARILY STRUCTURALLY COMPLETE POSITIVE LOGICS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-04-23
    ALEX CITKIN

    Positive logics are $\{ \wedge , \vee , \to \}$ -fragments of intermediate logics. It is clear that the positive fragment of $Int$ is not structurally complete. We give a description of all hereditarily structurally complete positive logics, while the question whether there is a structurally complete positive logic which is not hereditarily structurally complete, remains open.

    更新日期:2019-04-23
  • STANDARD BAYES LOGIC IS NOT FINITELY AXIOMATIZABLE
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-03-22
    ZALÁN GYENIS

    In the article [2] a hierarchy of modal logics has been defined to capture the logical features of Bayesian belief revision. Elements in that hierarchy were distinguished by the cardinality of the set of elementary propositions. By linking the modal logics in the hierarchy to the modal logics of Medvedev frames it has been shown that the modal logic of Bayesian belief revision determined by probabilities

    更新日期:2019-03-22
  • CERTAIN MODERN IDEAS AND METHODS: “GEOMETRIC REALITY” IN THE MATHEMATICS OF CHARLOTTE ANGAS SCOTT
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-03-05
    JEMMA LORENAT

    Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived

    更新日期:2019-03-05
  • DE MORGAN INTERPRETATION OF THE LAMBEK–GRISHIN CALCULUS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-02-26
    MICHAEL KAMINSKI; NISSIM FRANCEZ

    We present an embedding of the Lambek–Grishin calculus into an extension of the nonassociative Lambek calculus with negation. The embedding is based on the De Morgan interpretation of the dual Grishin connectives.

    更新日期:2019-02-26
  • BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-02-22
    SERGEI P. ODINTSOV; STANISLAV O. SPERANSKI

    We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to

    更新日期:2019-02-22
  • BUNDER’S PARADOX
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-02-06
    MICHAEL CAIE

    Systems of illative logic are logical calculi formulated in the untyped λ-calculus supplemented with certain logical constants.1 In this short paper, I consider a paradox that arises in illative logic. I note two prima facie attractive ways of resolving the paradox. The first is well known to be consistent, and I briefly outline a now standard construction used by Scott and Aczel that establishes this

    更新日期:2019-02-06
  • SEVENTEENTH-CENTURY SCHOLASTIC SYLLOGISTICS. BETWEEN LOGIC AND MATHEMATICS?
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-01-28
    MIROSLAV HANKE

    The seventeenth century can be viewed as an era of (closely related) innovation in the formal and natural sciences and of paradigmatic diversity in philosophy (due to the coexistence of at least the humanist, the late scholastic, and the early modern tradition). Within this environment, the present study focuses on scholastic logic and, in particular, syllogistic. In seventeenth-century scholastic

    更新日期:2019-01-28
  • VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-01-26
    T. MORASCHINI; J. G. RAFTERY; J. J. WANNENBURG

    The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4,

    更新日期:2019-01-26
  • A FULLY CLASSICAL TRUTH THEORY CHARACTERIZED BY SUBSTRUCTURAL MEANS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2019-01-04
    FEDERICO MATÍAS PAILOS

    We will present a three-valued consequence relation for metainferences, called CM, defined through ST and TS, two well known substructural consequence relations for inferences. While ST recovers every classically valid inference, it invalidates some classically valid metainferences. While CM works as ST at the inferential level, it also recovers every classically valid metainference. Moreover, CM can

    更新日期:2019-01-04
  • INEFFABILITY AND REVENGE
    Rev. Symb. Log. (IF 0.75) Pub Date : 2018-12-27
    CHRIS SCAMBLER

    In recent work Philip Welch has proven the existence of ‘ineffable liars’ for Hartry Field’s theory of truth. These are offered as liar-like sentences that escape classification in Field’s transfinite hierarchy of determinateness operators. In this article I present a slightly more general characterization of the ineffability phenomenon, and discuss its philosophical significance. I show the ineffable

    更新日期:2018-12-27
  • RELEVANCE FOR THE CLASSICAL LOGICIAN
    Rev. Symb. Log. (IF 0.75) Pub Date : 2018-11-06
    ETHAN BRAUER

    Although much technical and philosophical attention has been given to relevance logics, the notion of relevance itself is generally left at an intuitive level. It is difficult to find in the literature an explicit account of relevance in formal reasoning. In this article I offer a formal explication of the notion of relevance in deductive logic and argue that this notion has an interesting place in

    更新日期:2018-11-06
  • PEIRCE’S CALCULI FOR CLASSICAL PROPOSITIONAL LOGIC
    Rev. Symb. Log. (IF 0.75) Pub Date : 2018-10-29
    MINGHUI MA; AHTI-VEIKKO PIETARINEN

    This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted by PC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to present PC as a sequent calculus. The

    更新日期:2018-10-29
  • A NOTE ON THE EXISTENCE OF RATIFIABLE ACTS
    Rev. Symb. Log. (IF 0.75) Pub Date : 2018-10-15
    JOSEPH Y. HALPERN

    Sufficient conditions are given under which ratifiable acts exist.

    更新日期:2018-10-15
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