55 found
Sort by:
  1. Petr Cintula, Chris Fermüller, Lluis Godo & Petr Hájek (eds.) (forthcoming). Logical Models of Reasoning with Vague Information.
    No categories
    Translate to English
    |
     
    My bibliography  
     
    Export citation  
  2. Petr Cintula, Christian Fermuller, Lluis Godo & Petr Hajek (eds.) (forthcoming). Reasoning Under Vagueness. College Publications.
    Translate to English
    |
     
    My bibliography  
     
    Export citation  
  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.
    No categories
    Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  4. Petr Hájek (2010). On Witnessed Models in Fuzzy Logic III - Witnessed Gödel Logics. 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 (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  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.
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  6. Petr Hájek (2009). Arithmetical Complexity of Fuzzy Predicate Logics—a Survey II. Annals of Pure and Applied Logic 161 (2):212-219.
    Results on arithmetical complexity of important sets of formulas of several fuzzy predicate logics are surveyed and some new results are proven.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  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.
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  8. Petr Hajek, Fuzzy Logic. Stanford Encyclopedia of Philosophy.
    Direct download  
     
    My bibliography  
     
    Export citation  
  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.
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  10. Petr Hájek & Franco Montagna (2008). A Note on the First‐Order Logic of Complete BL‐Chains. 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.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  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.
    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 (...)
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  12. Petr Hájek (2007). On Witnessed Models in Fuzzy Logic II. 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.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  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.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  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.
    No categories
    Direct download  
     
    My bibliography  
     
    Export citation  
  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 (...)
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  16. Petr Hájek (2005). On Arithmetic in the Cantor-Łukasiewicz Fuzzy Set Theory. Archive for Mathematical Logic 44 (6):763-782.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  17. Petr Hájek, Luis Valdés-Villanueva & Dag Westerståhl (eds.) (2005). Logic, Methodology and Philosophy of Science. College Publications.
    No categories
     
    My bibliography  
     
    Export citation  
  18. 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.
    No categories
    Direct download  
     
    My bibliography  
     
    Export citation  
  19. 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.
    Direct download  
     
    My bibliography  
     
    Export citation  
  20. 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.
    Direct download (8 more)  
     
    My bibliography  
     
    Export citation  
  21. Petr Hájek (2002). A New Small Emendation of Gödel's Ontological Proof. Studia Logica 71 (2):149 - 164.
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  22. 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.
    Direct download (8 more)  
     
    My bibliography  
     
    Export citation  
  23. Matthias Baaz, Petr Hájek, Franco Montagna & Helmut Veith (2001). Complexity of T-Tautologies. 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.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  24. 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 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.
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  25. 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.
    Direct download  
     
    My bibliography  
     
    Export citation  
  26. 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.
    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 (...)
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  27. 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 (...)
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  28. 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, (...)
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  29. P. Grim, G. Mar, P. St Denis & Petr Hajek (2000). REVIEWS-The Philosophical Computer. Bulletin of Symbolic Logic 6 (3):347-348.
     
    My bibliography  
     
    Export citation  
  30. 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.
    Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  31. 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.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  32. 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.
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  33. Lluís Godo & Petr Hájek (1999). Fuzzy Inference as Deduction. Journal of Applied Non-Classical Logics 9 (1):37-60.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  34. Petr Hájek (1999). Ten Questions and One Problem on Fuzzy Logic. Annals of Pure and Applied Logic 96 (1-3):157-165.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  35. 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.
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  36. 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.
    Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  37. 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).
    Direct download  
     
    My bibliography  
     
    Export citation  
  38. 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.
    Translate to English
    | Direct download (2 more)  
     
    My bibliography  
     
    Export citation  
  39. 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.
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  40. 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.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  41. 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.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
  42. 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 (...)
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  43. 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.
    Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  44. 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 (...)
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  45. 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.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  46. 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.
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  47. Petr Hajek & Franco Montagna (1988). The Logic of 77,-Conservativity. Archive for Mathematical Logic 30:113-123.
     
    My bibliography  
     
    Export citation  
  48. 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 (...)
    Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  49. 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.
    No categories
    Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  50. Petr Hájek (1977). Experimental Logics and Π03 Theories. Journal of Symbolic Logic 42 (4):515 - 522.
    Direct download (4 more)  
     
    My bibliography  
     
    Export citation  
1 — 50 / 55