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  1. (1 other version)LOGIC TEACHING IN THE 21ST CENTURY.John Corcoran - manuscript
    We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...)
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  2. Inferential Quantification and the ω-rule.Constantin C. Brîncuş - 2024 - In Antonio Piccolomini D'Aragona (ed.), Perspectives on Deduction: Contemporary Studies in the Philosophy, History and Formal Theories of Deduction. Springer Verlag. pp. 345--372.
    Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...)
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  3. Inferential Quantification and the ω-Rule.Constantin C. Brîncuş - 2024 - In Antonio Piccolomini D'Aragona (ed.), Perspectives on Deduction: Contemporary Studies in the Philosophy, History and Formal Theories of Deduction. Springer Verlag. pp. 345-372.
    Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...)
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  4. Representing multiply de re epistemic modal statements.Cem Şişkolar - 2024 - Linguistics and Philosophy 47 (2):211-237.
    I review Ninan’s Hundred Tickets case pertaining to quantification into epistemic modal contexts, and his counterpart theoretic way to address it (Ninan, Philos Rev, 2018). Ninan’s solution employs a ‘counterpart relation’ parameter intended to reflect how the domain of quantification is thought of in a context. This approach theoretically rules out the possibility of contexts where different ways of thinking about the domain can be deployed through different quantificational noun phrases. I bring out the case of the multiply de re (...)
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  5. Against Fregean Quantification.Bryan Pickel & Brian Rabern - 2023 - Ergo: An Open Access Journal of Philosophy 9 (37):971-1007.
    There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...)
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  6. The Maximization of Chaos.Ilexa Yardley - 2023 - Https://Medium.Com/the-Circular-Theory/.
  7. Symbolic Logic.Rebeka Ferreira - 2022 - Gig Φ Philosophy.
    Basic Concepts in Logic Identifying & Evaluating Arguments Valid Argument Forms Complex Arguments Propositional Logic: Symbols & Translation Truth Tables: Statements Classifying & Comparing Statements.
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  8. Paraconsistent Logic Programming in Three and Four-Valued Logics.Kleidson Êglicio Carvalho da Silva Oliveira - 2022 - Bulletin of Symbolic Logic 28 (2):260-260.
  9. The myth of occurrence-based semantics.Bryan Pickel & Brian Rabern - 2021 - Linguistics and Philosophy 44:813-837.
    The principle of compositionality requires that the meaning of a complex expression remains the same after substitution of synonymous expressions. Alleged counterexamples to compositionality seem to force a theoretical choice: either apparent synonyms are not synonyms or synonyms do not syntactically occur where they appear to occur. Some theorists have instead looked to Frege’s doctrine of “reference shift” according to which the meaning of an expression is sensitive to its linguistic context. This doctrine is alleged to retain the relevant claims (...)
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  10. Binding bound variables in epistemic contexts.Brian Rabern - 2021 - Inquiry: An Interdisciplinary Journal of Philosophy 64 (5-6):533-563.
    ABSTRACT Quine insisted that the satisfaction of an open modalised formula by an object depends on how that object is described. Kripke's ‘objectual’ interpretation of quantified modal logic, whereby variables are rigid, is commonly thought to avoid these Quinean worries. Yet there remain residual Quinean worries for epistemic modality. Theorists have recently been toying with assignment-shifting treatments of epistemic contexts. On such views an epistemic operator ends up binding all the variables in its scope. One might worry that this yields (...)
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  11. Context and Coherence: The Logic and Grammar of Prominence.Una Stojnić - 2021 - Oxford, UK: Oxford University Press.
    Natural languages are riddled with context-sensitivity. One and the same string of words can express many different meanings on occasion of use, and yet we understand one another effortlessly, on the fly. How do we do so? What fixes the meaning of context-sensitive expressions, and how are we able to recover the meaning so effortlessly? -/- This book offers a novel response: we can do so because we draw on a broad array of subtle linguistic conventions that determine the interpretation (...)
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  12. Team Semantics for Interventionist Counterfactuals: Observations vs. Interventions.Fausto Barbero & Gabriel Sandu - 2020 - Journal of Philosophical Logic 50 (3):471-521.
    Team semantics is a highly general framework for logics which describe dependencies and independencies among variables. Typically, the dependencies considered in this context are properties of sets of configurations or data records. We show how team semantics can be further generalized to support languages for the discussion of interventionist counterfactuals and causal dependencies, such as those that arise in manipulationist theories of causation. We show that the “causal teams” we introduce in the present paper can be used for modelling some (...)
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  13. Binding, Compositionality, and Semantic Values.Michael Glanzberg & Jeffrey C. King - 2020 - Philosophers' Imprint 20.
    In this paper, we defend a traditional approach to semantics, that holds that the outputs of compositional semantics are propositional, i.e. truth conditions. Though traditional, this view has been challenged on a number of fronts over the years. Since classic work of Lewis, arguments have been offered which purport to show that semantic composition requires values that are relativized, e.g. to times, or other parameters that render them no longer propositional. Focusing in recent variants of these arguments involving quantification and (...)
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  14. Philosophy and Model Theory.Tim Button & Sean P. Walsh - 2018 - Oxford, UK: Oxford University Press. Edited by Sean Walsh & Wilfrid Hodges.
    Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...)
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  15. Quantification and Epistemic Modality.Dilip Ninan - 2018 - Philosophical Review 127 (4):433-485.
    This essay introduces a puzzle about the interaction between quantifiers and epistemic modals. The puzzle motivates the idea that whether an object satisfies an epistemically modalized predicate depends on the mode of presentation of the domain of quantification. I compare two ways of implementing this idea, one using counterpart theory, the other using Aloni's 'conceptual covers' theory, and then provides some evidence in favor of the former.
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  16. (1 other version)Reviving the parameter revolution in semantics.Bryan Pickel, Brian Rabern & Josh Dever - 2018 - In Derek Ball & Brian Rabern (eds.), The Science of Meaning: Essays on the Metatheory of Natural Language Semantics. Oxford: Oxford University Press. pp. 138-171.
    Montague and Kaplan began a revolution in semantics, which promised to explain how a univocal expression could make distinct truth-conditional contributions in its various occurrences. The idea was to treat context as a parameter at which a sentence is semantically evaluated. But the revolution has stalled. One salient problem comes from recurring demonstratives: "He is tall and he is not tall". For the sentence to be true at a context, each occurrence of the demonstrative must make a different truth-conditional contribution. (...)
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  17. The proper treatment of variables in predicate logic.Kai F. Wehmeier - 2018 - Linguistics and Philosophy 41 (2):209-249.
    In §93 of The Principles of Mathematics, Bertrand Russell observes that “the variable is a very complicated logical entity, by no means easy to analyze correctly”. This assessment is borne out by the fact that even now we have no fully satisfactory understanding of the role of variables in a compositional semantics for first-order logic. In standard Tarskian semantics, variables are treated as meaning-bearing entities; moreover, they serve as the basic building blocks of all meanings, which are constructed out of (...)
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  18. Translating Non-classical Logics into Classical Logic by Using Hidden Variables.Juan C. Agudelo-Agudelo - 2017 - Logica Universalis 11 (2):205-224.
    Dyadic semantics is a sort of non-truth-functional bivalued semantics introduced in Caleiro et al. Logica Universalis, Birkhäuser, Basel, pp 169–189, 2005). Here we introduce an algorithmic procedure for constructing conservative translations of logics characterised by dyadic semantics into classical propositional logic. The procedure uses fresh propositional variables, which we call hidden variables, to represent the indeterminism of dyadic semantics. An alternative algorithmic procedure for constructing conservative translations of any finite-valued logic into classical logic is also introduced. In this alternative procedure (...)
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  19. A unified non monstrous semantics for third person pronouns.Fabio Del Prete & Sandro Zucchi - 2017 - Semantics and Pragmatics 10.
    It is common practice in formal semantics to assume that the context specifies an assignment of values to variables and that the same variables that receive contextually salient values when they occur free may also be bound by quantifiers and λs. These assumptions are at work to provide a unified account of free and bound uses of third person pronouns, namely one by which the same lexical item is involved in both uses. One way to pursue this account is to (...)
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  20. Does Semantic Relationism Solve Frege's Puzzle?Bryan Pickel & Brian Rabern - 2017 - Journal of Philosophical Logic 46 (1):97-118.
    In a series of recent works, Kit Fine, 605–631, 2003, 2007) has sketched a novel solution to Frege’s puzzle. Radically departing from previous solutions, Fine argues that Frege’s puzzle forces us to reject compositionality. In this paper we first provide an explicit formalization of the relational semantics for first-order logic suggested, but only briefly sketched, by Fine. We then show why the relational semantics alone is technically inadequate, forcing Fine to enrich the syntax with a coordination schema. Given this enrichment, (...)
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  21. Logic in the Tractatus.Max Weiss - 2017 - Review of Symbolic Logic 10 (1):1-50.
    I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably (...)
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  22. A Counterexample to Variabilism.Mihnea D. I. Capraru - 2016 - Analysis 76 (1):26-29.
    Recent literature contains influential arguments for variabilism, the view that we should understand proper names as analogues not of constants but of variables. In particular, proper names are said to sometimes take semantic values that are not referential but purely general. I present a counter-example to this view.
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  23. The Antinomy of the Variable: A Tarskian Resolution.Bryan Pickel & Brian Rabern - 2016 - Journal of Philosophy 113 (3):137-170.
    Kit Fine has reawakened a puzzle about variables with a long history in analytic philosophy, labeling it “the antinomy of the variable”. Fine suggests that the antinomy demands a reconceptualization of the role of variables in mathematics, natural language semantics, and first-order logic. The difficulty arises because: (i) the variables ‘x’ and ‘y’ cannot be synonymous, since they make different contributions when they jointly occur within a sentence, but (ii) there is a strong temptation to say that distinct variables ‘x’ (...)
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  24. On the concepts of function and dependence.André Bazzoni - 2015 - Principia: An International Journal of Epistemology 19 (1):01-15.
    This paper briefly traces the evolution of the function concept until its modern set theoretic definition, and then investigates its relationship to the pre-formal notion of variable dependence. I shall argue that the common association of pre-formal dependence with the modern function concept is misconceived, and that two different notions of dependence are actually involved in the classic and the modern viewpoints, namely effective and functional dependence. The former contains the latter, and seems to conform more to our pre-formal conception (...)
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  25. The Constituents of the Propositions of Logic.Kevin C. Klement - 2015 - In Donovan Wishon & Bernard Linsky (eds.), Acquaintance, Knowledge, and Logic: New Essays on Bertrand Russell's The Problems of Philosophy. Stanford: CSLI Publications. pp. 189–229.
    In he Problems of Philosophy and other works of the same period, Russell claims that every proposition must contain at least one universal. Even fully general propositions of logic are claimed to contain “abstract logical universals”, and our knowledge of logical truths claimed to be a species of a priori knowledge of universals. However, these views are in considerable tension with Russell’s own philosophy of logic and mathematics as presented in Principia Mathematica. Universals generally are qualities and relations, but if, (...)
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  26. The Epsilon Calculus.Jeremy Avigad & Richard Zach - 2014 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Stanford, CA: The Metaphysics Research Lab.
    The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms (...)
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  27. A Note on Logical Truth.Corine Besson - 2014 - Logique Et Analyse 57 (227):309-331.
    Classical logic counts sentences such as ‘Alice is identical with Alice’ as logically true. A standard objection to classical logic is that Alice’s self-identity, for instance, is not a matter of logic because the identity of particular objects is not a matter of logic. For this reason, many philosophers argue that classical logic is not the right logic, and that it should be abandoned in favour of free logic — logic free of existential commitments with respect to singular terms. In (...)
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  28. Relational Hidden Variables and Non-Locality.Samson Abramsky - 2013 - Studia Logica 101 (2):411-452.
    We use a simple relational framework to develop the key notions and results on hidden variables and non-locality. The extensive literature on these topics in the foundations of quantum mechanics is couched in terms of probabilistic models, and properties such as locality and no-signalling are formulated probabilistically. We show that to a remarkable extent, the main structure of the theory, through the major No-Go theorems and beyond, survives intact under the replacement of probability distributions by mere relations.
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  29. Structured Variables.B. Halimi - 2013 - Philosophia Mathematica 21 (2):220-246.
    Drawing on Russell's substitutional theory, this paper examines the notion of ‘structured variable’, in order to compare Russell's and Tarski's conceptions of variables. The framework of syntactic fibrations, coming from categorical logic, is used as a common setting. The main objective of this paper is to make sense of the notion of structured variable beyond the context of Russell's theory, to question the Tarskian way of understanding what it is to be a possible value for a variable, and to bring (...)
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  30. Monsters in Kaplan’s logic of demonstratives.Brian Rabern - 2013 - Philosophical Studies 164 (2):393-404.
    Kaplan (1989a) insists that natural languages do not contain displacing devices that operate on character—such displacing devices are called monsters. This thesis has recently faced various empirical challenges (e.g., Schlenker 2003; Anand and Nevins 2004). In this note, the thesis is challenged on grounds of a more theoretical nature. It is argued that the standard compositional semantics of variable binding employs monstrous operations. As a dramatic first example, Kaplan’s formal language, the Logic of Demonstratives, is shown to contain monsters. For (...)
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  31. Propositions and compositionality.Juhani Yli-Vakkuri - 2013 - Philosophical Perspectives 27 (1):526-563.
  32. Tractarian First-Order Logic: Identity and the N-Operator.Brian Rogers & Kai F. Wehmeier - 2012 - Review of Symbolic Logic 5 (4):538-573.
    In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we (...)
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  33. The solo numero paradox.István Aranyosi - 2011 - American Philosophical Quarterly 48 (4):347.
    Leibniz notoriously insisted that no two individuals differ solo numero, that is, by being primitively distinct, without differing in some property. The details of Leibniz’s own way of understanding and defending the principle –known as the principle of identity of indiscernibles (henceforth ‘the Principle’)—is a matter of much debate. However, in contemporary metaphysics an equally notorious and discussed issue relates to a case put forward by Max Black (1952) as a counter-example to any necessary and non-trivial version of the principle. (...)
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  34. (1 other version)The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper.John Corcoran & José Miguel Sagüillo - 2011 - History and Philosophy of Logic 32 (4):359-374.
    This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple (...)
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  35. Scope and binding.Anna Szabolcsi - 2011 - In von Heusinger, Maienborn & Portner (eds.), Semantics: An International Handbook of Natural Language Meaning, Vol. 2. de Gruyter Mouton.
    The first part of this article (Sections 1–5) focuses on the classical notions of scope and binding and their formal foundations. It argues that once their semantic core is properly understood, it can be implemented in various different ways: with or without movement, with or without variables. The second part (Sections 6–12) takes up the empirical issues that have redrawn the map in the past two decades. It turns out that scope is not a primitive. Existential scope and distributive scope (...)
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  36. Object.Henry Laycock - 2010 - Stanford Encyclopedia of Philosophy.
    In The Principles of Mathematics, Russell writes: Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words unit, individual and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. (...)
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  37. On Ramsey's 'Silly Delusion' Regarding Tractatus 5.53.Kai Wehmeier - 2009 - In Giuseppe Primiero (ed.), Acts of Knowledge: History, Philosophy and Logic. College Publications.
    We investigate a variant of the variable convention proposed at Tractatus 5.53ff for the purpose of eliminating the identity sign from logical notation. The variant in question is what Hintikka has called the strongly exclusive interpretation of the variables, and turns out to be what Ramsey initially (and erroneously) took to be Wittgenstein's intended method. We provide a tableau calculus for this identity-free logic, together with soundness and completeness proofs, as well as a proof of mutual interpretability with first-order logic (...)
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  38. Wittgensteinian Tableaux, Identity, and Co-Denotation.Kai F. Wehmeier - 2008 - Erkenntnis 69 (3):363-376.
    Wittgensteinian predicate logic (W-logic) is characterized by the requirement that the objects mentioned within the scope of a quantifier be excluded from the range of the associated bound variable. I present a sound and complete tableaux calculus for this logic and discuss issues of translatability between Wittgensteinian and standard predicate logic in languages with and without individual constants. A metalinguistic co-denotation predicate, akin to Frege’s triple bar of the Begriffsschrift, is introduced and used to bestow the full expressive power of (...)
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  39. The but not all: A partitive account of plural definite descriptions.Berit Brogaard - 2007 - Mind and Language 22 (4):402–426.
    A number of authors in favor of a unitary account of singular descriptions have alleged that the unitary account can be extrapolated to account for plural definite descriptions. In this paper I take a closer look at this suggestion. I argue that while the unitary account is clearly onto something right, it is in the end empirically inadequate. At the end of the paper I offer a new partitive account of plural definite descriptions that avoids the problems with both the (...)
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  40. A Theory of Bondage.Nathan Salmon - 2006 - Philosophical Review 115 (4):415-448.
  41. It is raining (somewhere).François Recanati - 2005 - Linguistics and Philosophy 30 (1):123-146.
    The received view about meteorological predicates like ‘rain’ is that they carry an argument slot for a location which can be filled explicitly or implicitly. The view assumes that ‘rain’, in the absence of an explicit location, demands that the context provide a specific location. In an earlier article in this journal, I provided a counter-example, viz. a context in which ‘it is raining’ receives a location-indefinite interpretation. On the basis of that example, I argued that when there is tacit (...)
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  42. Descriptions and Situations.Francois Recanati - 2004 - In Marga Reimer & Anne Bezuidenhout (eds.), Descriptions and beyond. New York: Oxford University Press. pp. 15-40.
  43. Wittgensteinian Predicate Logic.Kai F. Wehmeier - 2004 - Notre Dame Journal of Formal Logic 45 (1):1-11.
    We investigate a rst-order predicate logic based on Wittgenstein's suggestion to express identity of object by identity of sign, and difference of objects by difference of signs. Hintikka has shown that predicate logic can indeed be set up in such a way; we show that it can be done nicely. More specically, we provide a perspicuous cut-free sequent calculus, as well as a Hilbert-type calculus, for Wittgensteinian predicate logic and prove soundness and completeness theorems.
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  44. Binding On the Fly: Cross-Sentential Anaphora in Variable— Free Semantics.Anna Szabolcsi - 2003 - In R. Oehrle & J. Kruijff (eds.), resource sensitivity, binding, and anaphora. kluwer. pp. 215--227.
    Combinatory logic (Curry and Feys 1958) is a “variable-free” alternative to the lambda calculus. The two have the same expressive power but build their expressions differently. “Variable-free” semantics is, more precisely, “free of variable binding”: it has no operation like abstraction that turns a free variable into a bound one; it uses combinators—operations on functions—instead. For the general linguistic motivation of this approach, see the works of Steedman, Szabolcsi, and Jacobson, among others. The standard view in linguistics is that reflexive (...)
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  45. Variables.Joshua Bruce Dever - 1998 - Dissertation, University of California, Berkeley
    Variables is a project at the intersection of the philosophies of language and logic. Frege, in the Begriffsschrift, crystalized the modern notion of formal logic through the first fully successful characterization of the behaviour of quantifiers. In Variables, I suggest that the logical tradition we have inherited from Frege is importantly flawed, and that Frege's move from treating quantifiers as noun phrases bearing word-world connection to sentential operators in the guise of second-order predicates leaves us both philosophically and technically wanting.
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  46. (3 other versions)Relational Belief.Nathan Salmón - 1995 - In Paolo Leonardi & Marco Santambrogio (eds.), Metaphysics, Mathemeatics, and Meaning. Cambridge University Press. pp. 206-228.
  47. (3 other versions)Relational Belief.Nathan Salmon - 1995 - In Paolo Leonardi & Marco Santambrogio (eds.), On Quine: New Essays. New York, NY, USA: Cambridge University Press. pp. 206-228.
  48. (1 other version)Variable-Binders as Functors.Achille C. Varzi - 1995 - Poznan Studies in the Philosophy of the Sciences and the Humanities 40:303-19.
    This work gives an extended presentation of the treatment of variable-binding operators adumbrated in [3:1993d]. Illustrative examples include elementary languages with quantifiers and lambda-equipped categorial languages. Some remarks are also offered to illustrate the philosophical import of the resulting picture. Particularly, a certain conception of logic emerges from the account: the view that logics are true theories in the model-theoretic sense, i.e. the result of selecting a certain class of models as the only “admissible” interpretation structures (for a given language).
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  49. (1 other version)Variable-Binders as Functors.Achille C. Varzi - 1995 - In Vito Sinisi & Jan Woleński (eds.), The heritage of Kazimierz Ajdukiewicz. Rodopi. pp. 303.
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  50. Bound variables in syntax (Are there any?).Anna Szabolcsi - 1987 - In J. Groenendijk, F. Veltman & M. Stokhof (eds.), Sixth Amsterdam Colloquium Proceedings. Univ of Amserdam.
    Current theories of grammar handle both extraction and anaphorization by introducing variables into syntactic representations. Combinatory categorial grammar eliminates variables corresponding to gaps. Using the combinator W, the paper extends this approach to anaphors, which appear to act as overt bound variables. [Slightly extended version in Bartsch et al 1989.].
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