68 found
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  1.  15
    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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  2. On Vagueness, Truth Values and Fuzzy Logics.Petr Hájek - 2009 - Studia Logica 91 (3):367-382.
    Some aspects of vagueness as presented in Shapiro’s book Vagueness in Context [23] are analyzed from the point of fuzzy logic. Presented are some generalizations of Shapiro’s formal apparatus.
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  3.  6
    A Complete Many-Valued Logic with Product-Conjunction.Petr Hájek, Lluis Godo & Francesc Esteva - 1996 - Archive for Mathematical Logic 35 (3):191-208.
    A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Gödel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.
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  4.  8
    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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  5.  32
    On Łukasiewicz's Four-Valued Modal Logic.Josep Maria Font & Petr Hájek - 2002 - Studia Logica 70 (2):157-182.
    ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.
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  6.  89
    Ontological Proofs of Existence and Non-Existence.Petr Hájek - 2008 - Studia Logica 90 (2):257-262.
    Caramuels’ proof of non-existence of God is compared with Gödel’s proof of existence.
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  7.  40
    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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  8.  11
    On Arithmetic in the Cantor-Łukasiewicz Fuzzy Set Theory.Petr Hájek - 2005 - Archive for Mathematical Logic 44 (6):763-782.
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  9. Experimental Logics and Π03 Theories.Petr Hájek - 1977 - Journal of Symbolic Logic 42 (4):515 - 522.
  10.  13
    Rational Pavelka Predicate Logic is a Conservative Extension of Łukasiewicz Predicate Logic.Petr Hájek, Jeff Paris & John Shepherdson - 2000 - Journal of Symbolic Logic 65 (2):669-682.
    Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0, 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.
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  11.  37
    Fuzzy Logic and Arithmetical Hierarchy III.Petr Hájek - 2001 - Studia Logica 68 (1):129-142.
    Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies and satisfiable sentences as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.
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  12.  5
    On Witnessed Models in Fuzzy Logic II.Petr Hájek - 2007 - Mathematical Logic Quarterly 53 (6):610-615.
    First the expansion of the Łukasiewicz logic by the unary connectives of dividing by any natural number is studied; it is shown that in the predicate case the expansion is conservative w.r.t. witnessed standard 1-tautologies. This result is used to prove that the set of witnessed standard 1-tautologies of the predicate product logic is Π2-hard.
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  13.  11
    Fuzzy Logic.Petr Hajek - 2008 - Stanford Encyclopedia of Philosophy.
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  14.  3
    A Note on the First‐Order Logic of Complete BL‐Chains.Petr Hájek & Franco Montagna - 2008 - Mathematical Logic Quarterly 54 (4):435-446.
    In [10] it is claimed that the set of predicate tautologies of all complete BL-chains and the set of all standard tautologies coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL-chain B satisfies all standard BL-tautologies iff for any transfinite sequence of elements of B, the condition ∧i ∈ I = 2 holds in B.
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  15.  43
    A New Small Emendation of Gödel's Ontological Proof.Petr Hájek - 2002 - Studia Logica 71 (2):149 - 164.
  16.  21
    Fuzzy Logic and Arithmetical Hierarchy, II.Petr Hájek - 1997 - Studia Logica 58 (1):129-141.
    A very simple many-valued predicate calculus is presented; a completeness theorem is proved and the arithmetical complexity of some notions concerning provability is determined.
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  17.  17
    Monadic Fuzzy Predicate Logics.Petr Hájek - 2002 - Studia Logica 71 (2):165-175.
    Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models.
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  18.  1
    Formal Systems of Fuzzy Logic and Their Fragments.Petr Cintula, Petr Hájek & Rostislav Horčík - 2007 - Annals of Pure and Applied Logic 150 (1):40-65.
    Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider (...)
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  19. On Witnessed Models in Fuzzy Logic.Petr Hájek - 2007 - Mathematical Logic Quarterly 53 (1):66-77.
    Witnessed models of fuzzy predicate logic are models in which each quantified formula is witnessed, i.e. the truth value of a universally quantified formula is the minimum of the values of its instances and similarly for existential quantification. Systematic theory of known fuzzy logics endowed with this semantics is developed with special attention paid to problems of arithmetical complexity of sets of tautologies and of satisfiable formulas.
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  20.  3
    The Logic of Π1-Conservativity.Petr Hájek & Franco Montagna - 1990 - Archive for Mathematical Logic 30 (2):113-123.
    We show that the modal prepositional logicILM (interpretability logic with Montagna's principle), which has been shown sound and complete as the interpretability logic of Peano arithmetic PA (by Berarducci and Savrukov), is sound and complete as the logic ofπ 1-conservativity over eachbE 1-sound axiomatized theory containingI⌆ 1 (PA with induction restricted tobE 1-formulas). Furthermore, we extend this result to a systemILMR obtained fromILM by adding witness comparisons in the style of Guaspari's and Solovay's logicR (this will be done in a (...)
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  21.  48
    Knowledge-Driven Versus Data-Driven Logics.Didier Dubois, Petr Hájek & Henri Prade - 2000 - Journal of Logic, Language and Information 9 (1):65--89.
    The starting point of this work is the gap between two distinct traditions in information engineering: knowledge representation and data - driven modelling. The first tradition emphasizes logic as a tool for representing beliefs held by an agent. The second tradition claims that the main source of knowledge is made of observed data, and generally does not use logic as a modelling tool. However, the emergence of fuzzy logic has blurred the boundaries between these two traditions by putting forward fuzzy (...)
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  22.  14
    On Sequences of Degrees of Constructibility.Bohuslav Balcar & Petr Hájek - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (19-24):291-296.
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  23.  6
    On Witnessed Models in Fuzzy Logic III - Witnessed Gödel Logics.Petr Hájek - 2010 - Mathematical Logic Quarterly 56 (2):171-174.
    Gödel logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise different (...)
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  24.  21
    Mathematical Fuzzy Logic – What It Can Learn From Mostowski and Rasiowa.Petr Hájek - 2006 - Studia Logica 84 (1):51-62.
    Important works of Mostowski and Rasiowa dealing with many-valued logic are analyzed from the point of view of contemporary mathematical fuzzy logic.
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  25.  5
    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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  26.  15
    Über Die Gültigkeit Des Fundierungsaxioms in Speziellen Systemen Der Mengentheorie.Petr Vopênka & Petr Hájek - 1963 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9 (12-15):235-241.
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  27.  4
    Special Issue: Methods for Investigating Self-Referential Truth Edited by Volker Halbach Volker Halbach/Editorial Introduction 3.Petr Hájek, Arithmetical Hierarchy Iii, Gerard Allwein & Wendy MacCaull - 2001 - Studia Logica 68:421-422.
  28.  27
    Embedding Logics Into Product Logic.Matthias Baaz, Petr Hájek, David Švejda & Jan Krajíček - 1998 - Studia Logica 61 (1):35-47.
    We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic.
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  29.  5
    The Logic ofII 1-Conservativity Continued.Petr Hájek & Franco Montagna - 1992 - Archive for Mathematical Logic 32 (1):57-63.
    It is shown that the propositional modal logic IRM (interpretability logic with Montagna's principle and with witness comparisons in the style of Guaspari's and Solovay's logicR) is sound and complete as the logic ofII 1-conservativity over each∑ 1-sound axiomatized theory containingI∑ 1. The exact statement of the result uses the notion of standard proof predicate. This paper is an immediate continuation of our paper [HM]. Knowledge of [HM] is presupposed. We define a modal logic, called IRM, which includes both ILM (...)
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  30.  17
    Ein dem fundierungsaxiom äquivalentes axiom.Petr Hájek & Antonín Sochor - 1964 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 10 (13-17):261-263.
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  31.  16
    Contributions to the Theory of Semisets I. Relations of the Theory of Semisets to the Zermelo-Fraenkel Set Theory.Petr Hájek - 1972 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 18 (16-18):241-248.
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  32.  16
    Modelle Der Mengenlehre, in Denen Mengen Gegebener Gestalt Existieren.Petr Hájek - 1965 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 11 (2):103-115.
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  33.  15
    Die durch die schwach inneren relationen gegebenen modelle der mengenlehre.Petr Hájek - 1964 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 10 (9-12):151-157.
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  34.  2
    On Sequences of Degrees of Constructibility (Solution of Friedman'S Problem 75).Bohuslav Balcar & Petr Hájek - 1978 - Mathematical Logic Quarterly 24 (19‐24):291-296.
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  35.  3
    A Note on the Notion of Truth in Fuzzy Logic.Petr Hájek & John Shepherdson - 2001 - Annals of Pure and Applied Logic 109 (1-2):65-69.
    In fuzzy predicate logic, assignment of truth values may be partial, i.e. the truth value of a formula in an interpretation may be undefined . A logic is supersound if each provable formula is true in each interpretation in which the truth value of is defined. It is shown that among the logics given by continuous t-norms, Gödel logic is the only one that is supersound; all others are not supersound. This supports the view that the usual restriction of semantics (...)
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  36.  16
    A Note on the Normal Form of Closed Formulas of Interpretability Logic.Petr Hájek & Vítězslav Švejdar - 1991 - Studia Logica 50 (1):25 - 28.
    Each closed (i.e. variable free) formula of interpretability logic is equivalent in ILF to a closed formula of the provability logic G, thus to a Boolean combination of formulas of the form n.
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  37.  8
    Modelle Der Mengenlehre, in Denen Mengen Gegebener Gestalt Existieren.Petr Hájek - 1965 - Mathematical Logic Quarterly 11 (2):103-115.
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  38.  14
    Godel's Ontological Proof and Its Variants.Petr Hájek - 2011 - In Matthias Baaz (ed.), Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press. pp. 307.
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  39.  1
    Über Die Gültigkeit Des Fundierungsaxioms in Speziellen Systemen Der Mengentheorie.Petr Vopênka & Petr Hájek - 1963 - Mathematical Logic Quarterly 9 (12‐15):235-241.
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  40.  24
    Some (Non)Tautologies of Łukasiewicz and Product Logic.Petr Hájek - 2010 - Review of Symbolic Logic 3 (2):273-278.
    The paper presents a particular example of a formula which is a standard tautology of Łukasiewicz but not its general tautology; an example of a model in which the formula is not true is explicitly constructed. Analogous example of a formula and its model is given for product logic.
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  41.  9
    Arithmetical Interpretations of Dynamic Logic.Petr Hájek - 1983 - Journal of Symbolic Logic 48 (3):704-713.
    An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping f satisfying the following: (1) f associates with each formula A of DPL a sentence f(A) of Peano arithmetic (PA) and with each program α a formula f(α) of PA with one free variable describing formally a supertheory of PA; (2) f commutes with logical connectives; (3) f([α] A) is the sentence saying that f(A) is provable in the theory f(α); (4) for each axiom A of DPL, f(A) is (...)
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  42.  7
    Ten Questions and One Problem on Fuzzy Logic.Petr Hájek - 1999 - Annals of Pure and Applied Logic 96 (1-3):157-165.
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  43.  4
    Review of Leonard Bolc and Piotr Borowik: Many-Valued Logics: 1. Theoretical Foundations, Berlin: Springer, 1991. [REVIEW]Petr Hajek & Richard Zach - 1994 - Journal of Applied Non-Classical Logics 4 (2):215-220.
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  44.  5
    Fuzzy Inference as Deduction.Lluís Godo & Petr Hájek - 1999 - Journal of Applied Non-Classical Logics 9 (1):37-60.
  45.  5
    Ein dem fundierungsaxiom äquivalentes axiom.Petr Hájek & Antonín Sochor - 1964 - Mathematical Logic Quarterly 10 (13‐17):261-263.
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  46.  9
    On Recursion Theory in I∑.Petr Hájek & Antonín Kučera - 1989 - Journal of Symbolic Logic 54 (2):576 - 589.
    It is shown that the low basis theorem is meaningful and provable in I∑ 1 and that the priority-free solution to Post's problem formalizes in this theory.
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  47.  2
    Review: Patrick Grim, Gary Mar, Paul St. Denis, The Philosophical Computer. Exploratory Essays in Philosophical Computer Modeling. [REVIEW]Petr Hájek - 2000 - Bulletin of Symbolic Logic 6 (3):347-349.
  48.  3
    Review: Erwin Engeler, Peter Lauchli, Ronald Peikert, Berechnungstheorie fur Informatiker; Arnold Oberschelp, Rekursionstheorie; Walter Felscher, Berechenbarkeit. Rekursive und Programmierbare Funktionen. [REVIEW]Petr Hajek - 1996 - Journal of Symbolic Logic 61 (2):699-701.
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  49.  3
    Gerla Giangiacomo. Fuzzy Logic—Mathematical Tools for Approximate Reasoning. Trends in Logic—Studia Logica Library 11. Kluwer Academic Publishers, 2001, Xii+ 269 Pp. [REVIEW]Petr Hájek - 2003 - Bulletin of Symbolic Logic 9 (4):510-511.
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  50.  1
    Grim Patrick, Mar Gary, and Denis Paul St.. The Philosophical Computer. Exploratory Essays in Philosophical Computer Modeling. With the Group for Logic and Formal Semantics. The MIT Press, Cambridge, Mass., and London, 1998, Viii + 323 Pp. + CD-ROM. [REVIEW]Petr Hájek - 2000 - Bulletin of Symbolic Logic 6 (3):347-349.
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