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Fuzzy logic

Stanford Encyclopedia of Philosophy (2008)

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  1. Axiomatization of Crisp Gödel Modal Logic.Ricardo Oscar Rodriguez & Amanda Vidal - 2021 - Studia Logica 109 (2):367-395.
    In this paper we consider the modal logic with both $$\Box $$ and $$\Diamond $$ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra $$[0,1]_G$$. We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most (...)
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  • Nonassociative substructural logics and their semilinear extensions: Axiomatization and completeness properties: Nonassociative substructural logics.Petr Cintula, Rostislav Horčík & Carles Noguera - 2013 - Review of Symbolic Logic 6 (3):394-423.
    Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural logics, a form (...)
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  • Zum intuitionistischen aussagenkalkül.K. Gödel - 1932 - Anzeiger der Akademie der Wissenschaften in Wien 69:65--66.
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  • Standard Gödel Modal Logics.Xavier Caicedo & Ricardo O. Rodriguez - 2010 - Studia Logica 94 (2):189-214.
    We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this (...)
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  • Modal Definability Based on Łukasiewicz Validity Relations.Bruno Teheux - 2016 - Studia Logica 104 (2):343-363.
    We study two notions of definability for classes of relational structures based on modal extensions of Łukasiewicz finitely-valued logics. The main results of the paper are the equivalent of the Goldblatt-Thomason theorem for these notions of definability.
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  • Fuzzy Logics in Theories of Vagueness.Nicholas J. J. Smith - 2015 - In Petr Cintula, Christian Fermüller & Carles Noguera (eds.), Handbook of Mathematical Fuzzy Logic - Volume 3. College Publications.
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  • Distinguished algebraic semantics for t -norm based fuzzy logics: Methods and algebraic equivalencies.Petr Cintula, Francesc Esteva, Joan Gispert, Lluís Godo, Franco Montagna & Carles Noguera - 2009 - Annals of Pure and Applied Logic 160 (1):53-81.
    This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit (...)
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  • Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras.Vincenzo Marra & Luca Spada - 2013 - Annals of Pure and Applied Logic 164 (3):192-210.
    We prove that the unification type of Łukasiewicz logic and of its equivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardiʼs algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background (...)
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  • The $L\Pi$ and $L\Pi\frac{1}{2}$ logics: two complete fuzzy systems joining Łukasiewicz and Product Logics. [REVIEW]Francesc Esteva, Lluís Godo & Franco Montagna - 2001 - Archive for Mathematical Logic 40 (1):39-67.
    In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called $L\Pi$ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from $L \Pi$ by the adding of a constant symbol and of a defining axiom for $\frac{1}{2}$ , called $L \Pi\frac{1}{2}$ . We show that $L \Pi \frac{1}{2}$ contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Gödel's Fuzzy Logic, Takeuti and Titani's (...)
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  • Residuated fuzzy logics with an involutive negation.Francesc Esteva, Lluís Godo, Petr Hájek & Mirko Navara - 2000 - Archive for Mathematical Logic 39 (2):103-124.
    Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, (...)
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  • Structural Completeness in Fuzzy Logics.Petr Cintula & George Metcalfe - 2009 - Notre Dame Journal of Formal Logic 50 (2):153-182.
    Structural completeness properties are investigated for a range of popular t-norm based fuzzy logics—including Łukasiewicz Logic, Gödel Logic, Product Logic, and Hájek's Basic Logic—and their fragments. General methods are defined and used to establish these properties or exhibit their failure, solving a number of open problems.
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  • A Treatise on Many-Valued Logic.Siegfried Gottwald - 2001 - Research Studies Press.
    A growing interest in many-valued logic has developed which to a large extent is based on applications, intended as well as already realised ones. These applications range from the field of computer science, e.g. in the areas of automated theorem proving, approximate reasoning, multi-agent systems, switching theory, and program verification, through the field of pure mathematics, e.g. in independence of consistency proofs, in generalized set theories, or in the theory of particular algebraic structures, into the fields of humanities, linguistics and (...)
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  • Residuated Lattices: An Algebraic Glimpse at Substructural Logics.Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski & Hiroakira Ono - 2007 - Elsevier.
    This is also where we begin investigating lattices of logics and varieties, rather than particular examples.
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  • Substructural Fuzzy Logics.George Metcalfe & Franco Montagna - 2007 - Journal of Symbolic Logic 72 (3):834 - 864.
    Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are (...)
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  • On Theories and Models in Fuzzy Predicate Logics.Petr Hájek & Petr Cintula - 2006 - Journal of Symbolic Logic 71 (3):863 - 880.
    In the last few decades many formal systems of fuzzy logics have been developed. Since the main differences between fuzzy and classical logics lie at the propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this text we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate logics. First, we generalize the completeness theorem, then we use it to get results on conservative extensions of theories (...)
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  • Vagueness and Degrees of Truth.Nicholas J. J. Smith - 2008 - Oxford, England: Oxford University Press.
    In VAGUENESS AND DEGREES OF TRUTH, Nicholas Smith develops a new theory of vagueness: fuzzy plurivaluationism. -/- A predicate is said to be VAGUE if there is no sharply defined boundary between the things to which it applies and the things to which it does not apply. For example, 'heavy' is vague in a way that 'weighs over 20 kilograms' is not. A great many predicates -- both in everyday talk, and in a wide array of theoretical vocabularies, from law (...)
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  • Fuzzy logic and approximate reasoning.L. A. Zadeh - 1975 - Synthese 30 (3-4):407-428.
    The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as a (...)
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  • The logic of inexact concepts.J. A. Goguen - 1969 - Synthese 19 (3-4):325-373.
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  • Intuitionistic fuzzy logic and intuitionistic fuzzy set theory.Gaisi Takeuti & Satoko Titani - 1984 - Journal of Symbolic Logic 49 (3):851-866.
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  • The liar paradox and fuzzy logic.Petr Hájek, Jeff Paris & John Shepherdson - 2000 - Journal of Symbolic Logic 65 (1):339-346.
    Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying "x is true" and satisfying the "dequotation schema" $\varphi \equiv \text{Tr}(\bar{\varphi})$ for all sentences φ? This problem is investigated in the frame of Lukasiewicz infinitely valued logic.
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  • Weakly Implicative (Fuzzy) Logics I: Basic Properties. [REVIEW]Petr Cintula - 2006 - Archive for Mathematical Logic 45 (6):673-704.
    This paper presents two classes of propositional logics (understood as a consequence relation). First we generalize the well-known class of implicative logics of Rasiowa and introduce the class of weakly implicative logics. This class is broad enough to contain many “usual” logics, yet easily manageable with nice logical properties. Then we introduce its subclass–the class of weakly implicative fuzzy logics. It contains the majority of logics studied in the literature under the name fuzzy logic. We present many general theorems for (...)
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  • Metamathematics of Fuzzy Logic.Petr Hájek - 1998 - Dordrecht, Boston and London: Kluwer Academic Publishers.
    This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. It aims to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named ‘fuzzy inference’ can be naturally understood as logical deduction. It is for mathematicians, logicians, computer scientists, specialists in artificial intelligence and knowledge engineering, and developers of fuzzy logic.
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  • Giles’s Game and the Proof Theory of Łukasiewicz Logic.Christian G. Fermüller & George Metcalfe - 2009 - Studia Logica 92 (1):27 - 61.
    In the 1970s, Robin Giles introduced a game combining Lorenzen-style dialogue rules with a simple scheme for betting on the truth of atomic statements, and showed that the existence of winning strategies for the game corresponds to the validity of formulas in Łukasiewicz logic. In this paper, it is shown that ‘disjunctive strategies’ for Giles’s game, combining ordinary strategies for all instances of the game played on the same formula, may be interpreted as derivations in a corresponding proof system. In (...)
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  • A proof of standard completeness for Esteva and Godo's logic MTL.Sándor Jenei & Franco Montagna - 2002 - Studia Logica 70 (2):183-192.
    In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo''s logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.
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  • A non-classical logic for physics.Robin Giles - 1974 - Studia Logica 33 (4):397 - 415.
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  • A propositional calculus with denumerable matrix.Michael Dummett - 1959 - Journal of Symbolic Logic 24 (2):97-106.
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  • ‎Proof Theory for Fuzzy Logics.George Metcalfe, Nicola Olivetti & Dov M. Gabbay - 2008 - Dordrecht, Netherland: Springer.
    Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers (...)
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  • Lindström theorems in graded model theory.Guillermo Badia & Carles Noguera - 2021 - Annals of Pure and Applied Logic 172 (3):102916.
    Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of maximality of first-order logics in terms of (...)
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  • (1 other version)Fuzzy Models of First Order Languages.A. di Nola & G. Gerla - 1986 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24):331-340.
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  • Fuzzy logic and fuzzy set theory.Gaisi Takeuti & Satoko Titani - 1992 - Archive for Mathematical Logic 32 (1):1-32.
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  • Three complexity problems in quantified fuzzy logic.Franco Montagna - 2001 - Studia Logica 68 (1):143-152.
    We prove that the sets of standard tautologies of predicate Product Logic and of predicate Basic Logic, as well as the set of standard-satisfiable formulas of predicate Basic Logic are not arithmetical, thus finding a rather satisfactory solution to three problems proposed by Hájek in [H01].
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  • Logic with truth values in a linearly ordered Heyting algebra.Alfred Horn - 1969 - Journal of Symbolic Logic 34 (3):395-408.
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  • Fuzzy Logic and Mathematics: A Historical Perspective.Radim Bělohlávek, Joseph W. Dauben & George J. Klir - 2017 - Oxford, England and New York, NY, USA: Oxford University Press. Edited by Joseph Warren Dauben & George J. Klir.
    The term "fuzzy logic," as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic---the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediate truth values between true and false are allowed, which are interpreted as degrees of truth. This opens a (...)
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  • Vagueness as closeness.Nicholas J. J. Smith - 2005 - Australasian Journal of Philosophy 83 (2):157 – 183.
    This paper presents and defends a definition of vagueness, compares it favourably with alternative definitions, and draws out some consequences of accepting this definition for the project of offering a substantive theory of vagueness. The definition is roughly this: a predicate 'F' is vague just in case for any objects a and b, if a and b are very close in respects relevant to the possession of F, then 'Fa' and 'Fb' are very close in respect of truth. The definition (...)
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  • (1 other version)Die nichtaxiomatisierbarkeit Des unendlichwertigen prädikatenkalküls Von łukasiewicz.Bruno Scarpellini - 1962 - Journal of Symbolic Logic 27 (2):159-170.
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  • (1 other version)Axiomatization of the infinite-valued predicate calculus.Louise Schmir Hay - 1963 - Journal of Symbolic Logic 28 (1):77-86.
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  • (1 other version)A theorem about infinite-valued sentential logic.Robert McNaughton - 1951 - Journal of Symbolic Logic 16 (1):1-13.
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  • Triangular Norms.Erich Peter Klement, Radko Mesiar & Endre Pap - 2000 - Dordrecht, Netherland: Springer.
    This book discusses the theory of triangular norms and surveys several applied fields in which triangular norms play a significant part: probabilistic metric spaces, aggregation operators, many-valued logics, fuzzy logics, sets and control, and non-additive measures together with their corresponding integrals. It includes many graphical illustrations and gives a well-balanced picture of theory and applications. It is for mathematicians, computer scientists, applied computer scientists and engineers.
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  • Randomized game semantics for semi-fuzzy quantifiers.C. G. Fermuller & C. Roschger - 2014 - Logic Journal of the IGPL 22 (3):413-439.
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  • Algebraic proof theory for substructural logics: cut-elimination and completions.Agata Ciabattoni, Nikolaos Galatos & Kazushige Terui - 2012 - Annals of Pure and Applied Logic 163 (3):266-290.
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  • Fuzzy logic: Mathematical tools for approximate reasoning.Giangiacomo Gerla - 2003 - Bulletin of Symbolic Logic 9 (4):510-511.
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  • Complexity of t-tautologies.Matthias Baaz, Petr Hájek, Franco Montagna & Helmut Veith - 2001 - Annals of Pure and Applied Logic 113 (1-3):3-11.
    A t-tautology is a propositional formula which is a tautology in all fuzzy logics defined by continuous triangular norms. In this paper we show that the problem of recognizing t-tautologies is coNP complete, and thus decidable.
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