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  1. Petr Cintula, Chris Fermüller, Lluis Godo & Petr Hájek (eds.) (forthcoming). Logical Models of Reasoning with Vague Information.
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  2. Petr Cintula, Christian Fermuller, Lluis Godo & Petr Hajek (eds.) (forthcoming). Reasoning Under Vagueness. College Publications.
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  3. Petr Hájek (2011). Godel's Ontological Proof and Its Variants. In Matthias Baaz (ed.), Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press. 307.
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  4. Petr Hájek (2010). On Witnessed Models in Fuzzy Logic III - Witnessed Gödel Logics. Mathematical Logic Quarterly 56 (2):171-174.
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  5. Petr Hájek (2010). Some (Non)Tautologies of Łukasiewicz and Product Logic. 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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  6. Petr Hájek (2009). Arithmetical Complexity of Fuzzy Predicate Logics—a Survey II. Annals of Pure and Applied Logic 161 (2):212-219.
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  7. Petr Hájek (2009). On Vagueness, Truth Values and Fuzzy Logics. 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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  8. Petr Hajek, Fuzzy Logic. Stanford Encyclopedia of Philosophy.
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  9. Petr Hájek (2008). Ontological Proofs of Existence and Non-Existence. 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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  10. Petr Hájek & Franco Montagna (2008). A Note on the First‐Order Logic of Complete BL‐Chains. Mathematical Logic Quarterly 54 (4):435-446.
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  11. Petr Cintula, Petr Hájek & Rostislav Horčík (2007). Formal Systems of Fuzzy Logic and Their Fragments. Annals of Pure and Applied Logic 150 (1):40-65.
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  12. Petr Hájek (2007). On Witnessed Models in Fuzzy Logic II. Mathematical Logic Quarterly 53 (6):610-615.
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  13. Petr Hájek (2006). Mathematical Fuzzy Logic – What It Can Learn From Mostowski and Rasiowa. 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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  14. Petr Hájek (2006). Priest Graham. An Introduction to Non-Classical Logic. Cambridge University Press, 2001, Xxi+ 242 Pp. [REVIEW] Bulletin of Symbolic Logic 12 (2):294-295.
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  15. Petr Hájek & Petr Cintula (2006). On Theories and Models in Fuzzy Predicate Logics. 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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  16. Petr Hájek (2005). On Arithmetic in the Cantor-Łukasiewicz Fuzzy Set Theory. Archive for Mathematical Logic 44 (6):763-782.
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  17. Petr Hájek (2004). Luca Vigano. Labelled Non-Classical Logics, With a Foreword by Gabbay Dov, Kluwer Academic Publishers, 2000, 291 Pp. [REVIEW] Bulletin of Symbolic Logic 10 (1):107-108.
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  18. Petr Hájek (2003). Gerla Giangiacomo. Fuzzy Logic—Mathematical Tools for Approximate Reasoning. Trends in Logic—Studia Logica Library 11. Kluwer Academic Publishers, 2001, Xii+ 269 Pp. [REVIEW] Bulletin of Symbolic Logic 9 (4):510-511.
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  19. Josep Maria Font & Petr Hájek (2002). On Łukasiewicz's Four-Valued Modal Logic. 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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  20. Petr Hájek (2002). A New Small Emendation of Gödel's Ontological Proof. Studia Logica 71 (2):149 - 164.
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  21. Petr Hájek (2002). Monadic Fuzzy Predicate Logics. 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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  22. Matthias Baaz, Petr Hájek, Franco Montagna & Helmut Veith (2001). Complexity of T-Tautologies. Annals of Pure and Applied Logic 113 (1-3):3-11.
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  23. Petr Hájek (2001). Fuzzy Logic and Arithmetical Hierarchy III. 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 (identically true sentences) and satisfiable sentences (sentences true in at least one interpretation) 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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  24. Petr Hájek, Arithmetical Hierarchy Iii, Gerard Allwein & Wendy MacCaull (2001). Special Issue: Methods for Investigating Self-Referential Truth Edited by Volker Halbach Volker Halbach/Editorial Introduction 3. Studia Logica 68:421-422.
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  25. Petr Hájek & John Shepherdson (2001). A Note on the Notion of Truth in Fuzzy Logic. Annals of Pure and Applied Logic 109 (1-2):65-69.
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  26. Didier Dubois, Petr Hájek & Henri Prade (2000). Knowledge-Driven Versus Data-Driven Logics. 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 rules as (...)
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  27. Francesc Esteva, Lluís Godo, Petr Hájek & Mirko Navara (2000). Residuated Fuzzy Logics with an Involutive Negation. 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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  28. P. Grim, G. Mar, P. St Denis & Petr Hajek (2000). REVIEWS-The Philosophical Computer. Bulletin of Symbolic Logic 6 (3):347-348.
     
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  29. Petr Hájek (2000). Review: Patrick Grim, Gary Mar, Paul St. Denis, The Philosophical Computer. Exploratory Essays in Philosophical Computer Modeling. [REVIEW] Bulletin of Symbolic Logic 6 (3):347-349.
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  30. Petr Hájek, Jeff Paris & John Shepherdson (2000). Rational Pavelka Predicate Logic is a Conservative Extension of Łukasiewicz Predicate Logic. 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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  31. Petr Hájek, Jeff Paris & John Shepherdson (2000). The Liar Paradox and Fuzzy Logic. 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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  32. Lluís Godo & Petr Hájek (1999). Fuzzy Inference as Deduction. Journal of Applied Non-Classical Logics 9 (1):37-60.
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  33. Petr Hájek (1999). Ten Questions and One Problem on Fuzzy Logic. Annals of Pure and Applied Logic 96 (1-3):157-165.
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  34. Matthias Baaz, Petr Hájek, David Švejda & Jan Krajíček (1998). Embedding Logics Into Product Logic. 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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  35. Petr Hájek (1997). Fuzzy Logic and Arithmetical Hierarchy, II. 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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  36. G. F. R. Ellis, Solomon Feferman, Daniel Isaacson, Boris A. Kushner, Petr Hájek & Jirı Zlatuška (1996). Brno, Czech Republic, August 25–29, 1996. Bulletin of Symbolic Logic 2 (4).
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  37. Petr Hajek (1996). Review: Erwin Engeler, Peter Lauchli, Ronald Peikert, Berechnungstheorie fur Informatiker; Arnold Oberschelp, Rekursionstheorie; Walter Felscher, Berechenbarkeit. Rekursive und Programmierbare Funktionen. [REVIEW] Journal of Symbolic Logic 61 (2):699-701.
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  38. Petr Hájek (1996). Engeler Erwin and Läuchli Peter. Berechnungstheorie für Informatiker. With assistance from Ronald Peikert. Leitfäden und Monographien der Informatik. BG Teubner, Stuttgart 1988, 120 pp. Oberschelp Arnold. Rekursionstheorie. BI Wissenschaftsverlag, Mannheim, Leipzig, Vienna, and Zürich, 1993, 339 pp. Felscher Walter. Berechenbarkeit. Rekursive und programmierbare Funktionen. Springer-Lehrbuch. Springer-Verlag, Berlin, Heidelberg, New York, etc., 1993, xi+ 478 pp. [REVIEW] Journal of Symbolic Logic 61 (2):699-701.
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  39. Petr Hájek, Lluis Godo & Francesc Esteva (1996). A Complete Many-Valued Logic with Product-Conjunction. 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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  40. Petr Hajek & Richard Zach (1994). Review of Leonard Bole and Piotr Borowik: Many-Valued Logics: 1. Theoretical Foundations, Berlin: Springer, 1991. [REVIEW] Journal of Applied Non-Classical Logics 4 (2):215-220.
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  41. Petr Hájek & Franco Montagna (1992). The Logic ofII 1-Conservativity Continued. 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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  42. Petr Hájek & Vítězslav Švejdar (1991). A Note on the Normal Form of Closed Formulas of Interpretability Logic. 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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  43. Petr Hájek & Franco Montagna (1990). The Logic of Π1-Conservativity. 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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  44. Petr Hajek & Antonin Kucera (1989). On Recursion Theory in $Isum_1$. Journal of Symbolic Logic 54 (2):576-589.
    It is shown that the low basis theorem is meaningful and provable in $I\sum_1$ and that the priority-free solution to Post's problem formalizes in this theory.
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  45. Petr Hájek & Antonín Kučera (1989). On Recursion Theory in I∑. 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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  46. Petr Hajek & Franco Montagna (1988). The Logic of 77,-Conservativity. Archive for Mathematical Logic 30:113-123.
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  47. Petr Hájek (1983). Arithmetical Interpretations of Dynamic Logic. 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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  48. Bohuslav Balcar & Petr Hájek (1978). On Sequences of Degrees of Constructibility (Solution of Friedman'S Problem 75). Mathematical Logic Quarterly 24 (19‐24):291-296.
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  49. Petr Hájek (1977). Experimental Logics and Π03 Theories. Journal of Symbolic Logic 42 (4):515 - 522.
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  50. Petr Hájek (1972). Contributions to the Theory of Semisets I. Relations of the Theory of Semisets to the Zermelo‐Fraenkel Set Theory. Mathematical Logic Quarterly 18 (16‐18):241-248.
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