Abstract
We discuss the present state of mathematical fuzzy logic in narrow sense, its extension - fuzzy logic in broader sense (FLb) as a logic of natural human reasoning and also some related theories, e.g., the fuzzy transform. We argue that these are good theories with potential to be very practical.
The research was supported by the project MSM 6198898701 of the MŠMT ČR.
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References
Běhounek, L., Cintula, P.: Fuzzy class theory. Fuzzy Sets and Systems 154, 34–55 (2005)
Běhounek, L., Cintula, P.: From fuzzy logic to fuzzy mathematics: A methodological manifesto. Fuzzy Sets and Systems 157(5), 642–646 (2006)
Cignoli, R.L.O., D’ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht (2000)
Cintula, P., Hájek, P., Horčík, R.: Formal systems of fuzzy logic and their fragments. Annals of Pure and Applied Logic 150, 40–65 (2007)
Davis, E., Morgenstern, L.: Introduction:progress in formal commonsense reasoning. Artifical Intelligence 153, 1–12 (2004)
De Cooman, G., Kerre, E., Cappelle, B., Ruan, D., Vanmassenhove, F.: On the extension of classical propositional logic by means of a triangular norm. Int. J. of Intelligent Systems 5, 307–322 (1990)
Di Martino, F., Loia, V., Perfilieva, I., Sessa, S.: An image coding/decoding method based on direct and inverse fuzzy transforms. Int. Journ. of Appr. Reasoning 48, 110–131 (2008)
Dubois, D., Prade, H.: What are fuzzy rules and how to use them. Fuzzy Sets and Systems 84, 169–185 (1996)
Dvořák, A., Novák, V.: Fuzzy logic deduction with crisp observations. Soft Computing 8, 256–263 (2004)
Dvořák, A., Holčapek, M.: L-fuzzy quantifiers of the type 〈1〉 determined by measures. Fuzzy Sets and Systems 160, 3425–3452 (2009)
Dvořák, A., Novák, V.: Formal theories and linguistic descriptions. Fuzzy Sets and Systems 143, 169–188 (2004)
Dvořák, A., Novák, V.: Towards automatic modeling of economic texts. Mathware & Soft Computing XIV(3), 217–231 (2007)
Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124, 271–288 (2001)
Fitting, M.: Intensional logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2006)
Flaminio, T., Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning 50, 138–152 (2009)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices. In: An Algebraic Glimpse At Substructural Logics. Elsevier, Amsterdam (2007)
Goguen, J.A.: The logic of inexact concepts. Synthese 19, 325–373 (1968-1969)
Gottwald, S.: A Treatise on Many-Valued Logics. Research Studies Press Ltd., Baldock (2001)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)
Hájek, P., Cintula, P.: On theories and models in fuzzy predicate logics. Journal of Symbolic Logic 71(3), 863–880 (2006)
Holčapek, M.: Monadic L-fuzzy quantifiers of the type 〈1n, 1〉. Fuzzy Sets and Systems 159, 1811–1835 (2008)
Jenei, S., Kerre, E.: Convergence of residuated operators and connective stability. Fuzzy Sets and Systems 114, 411–415 (2000)
Kennedy, C.: Vagueness and grammar: The semantics of relative and absolute gradable adjectives. Linguistics and Philosophy 30(3), 1–45 (2007)
Kerre, E., De Cock, M.: A comparative study of the behaviour of some popular fuzzy implication operators on the generalized modus ponens. In: Martinez, J. (ed.) Fuzzy Logic for the Management of Uncertainty, pp. 281–296. John Wiley and Sons, New York (1992)
Kerre, E., De Cock, M.: Linguistic modifiers: an overview. In: Martinez, J. (ed.) Fuzzy Logic and Soft Computing, pp. 69–86. Kluwer Academic, Boston (1999)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)
Lewin, K.: Field theory in social science: Selected theoretical papers. Harper & Row, New York (1951)
Mamdani, E., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. of Man-Machine Studies 7, 1–13 (1975)
Materna, P.: Concepts and Objects, Acta Philosophica Fennica 63, Helsinki (1998)
Materna, P.: Conceptual Systems. Logos Verlag, Berlin (2004)
McCarthy, J.: Programs with common sense. In: Proc. of the Teddington Conf. on Mechanization of Thought Processes, Her Majesty’s Stationary Office, London (1959)
Murinová, P., Novák, V.: A formal theory of generalized intermediate syllogisms. Fuzzy Sets and Systems
Novák, V.: On the syntactico-semantical completeness of first-order fuzzy logic I, II. Kybernetika 26, 47–66, 134–154 (1990)
Novák, V.: Towards formalized integrated theory of fuzzy logic. In: Bien, Z., Min, K. (eds.) Fuzzy Logic and Its Applications to Engineering, Information Sciences, and Intelligent Systems, pp. 353–363. Kluwer, Dordrecht (1995)
Novák, V.: Fuzzy relation equations with words. In: Nikravesh, M., Zadeh, L., Korotkikh, V. (eds.) Fuzzy Partial Differential Equations and Relational Equations, pp. 167–185. Springer, Berlin (2004)
Novák, V.: On fuzzy type theory. Fuzzy Sets and Systems 149, 235–273 (2005)
Novák, V.: Perception-based logical deduction. In: Reusch, B. (ed.) Computational Intelligence, Theory and Applications, pp. 237–250. Springer, Berlin (2005)
Novák, V.: Perception-based logical deduction as alternative approximate reasoning method. In: Proc. Int. Conf. FUZZ-IEEE 2005, Reno, USA (May 2005)
Novák, V.: Fuzzy logic theory of evaluating expressions and comparative quantifiers. In: Proc. 11th Int. Conf. IPMU, Paris, Éditions EDK, Les Cordeliers, Paris, vol. 2 (July 2006)
Novák, V.: Which logic is the real fuzzy logic? Fuzzy Sets and Systems 157, 635–641 (2006)
Novák, V.: Mathematical fuzzy logic in modeling of natural language semantics. In: Wang, P., Ruan, D., Kerre, E. (eds.) Fuzzy Logic – A Spectrum of Theoretical & Practical Issues, pp. 145–182. Elsevier, Berlin (2007)
Novák, V.: A comprehensive theory of trichotomous evaluative linguistic expressions. Fuzzy Sets and Systems 159(22), 2939–2969 (2008)
Novák, V.: A formal theory of intermediate quantifiers. Fuzzy Sets and Systems 159(10), 1229–1246 (2008), doi:10.1016/j.fss.2007.12.008
Novák, V.: EQ-algebra-based fuzzy type theory and its extensions. Logic Journal of the IGPL (to appear, 2010)
Novák, V., Lehmke, S.: Logical structure of fuzzy IF-THEN rules. Fuzzy Sets and Systems 157, 2003–2029 (2006)
Novák, V., Perfilieva, I. (eds.): Discovering the World With Fuzzy Logic. Studies in Fuzziness and Soft Computing, vol. 57. Springer, Heidelberg (2000)
Novák, V., Perfilieva, I.: On the semantics of perception-based fuzzy logic deduction. International Journal of Intelligent Systems 19, 1007–1031 (2004)
Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer, Boston (1999)
Novák, V., Štěpnička, M.U., Dvořák, A., Perfilieva, I., Pavliska, V., Vavříčková, L.: Analysis of seasonal time series using fuzzy approach. Int. Journal of General Systems 39, 305–328 (2010)
Pavelka, J.: On fuzzy logic I, II, III, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 25, 45–52, 119–134, 447–464 (1979)
Perfilieva, I.: Fuzzy logic normal forms for control law representation. In: Verbruggen, H., Zimmermann, H.-J., Babusha, R. (eds.) Fuzzy Algorithms for Control, pp. 111–125. Kluwer, Boston (1999)
Perfilieva, I.: Logical approximation. Soft Computing 7(2), 73–78 (2002)
Perfilieva, I.: Fuzzy transform: Application to reef growth problem. In: Demicco, R.B., Klir, G.J. (eds.) Fuzzy Logic in Geology, pp. 275–300. Academic Press, Amsterdam (2003)
Perfilieva, I.: Fuzzy function as an approximate solution to a system of fuzzy relation equations. Fuzzy Sets and Systems 147, 363–383 (2004)
Perfilieva, I.: Normal forms in BL-algebra of functions and their contribution to universal approximation. Fuzzy Sets and Systems 143, 111–127 (2004)
Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets and Systems 157, 993–1023 (2006)
Perfilieva, I.: Fuzzy transforms: A challenge to conventional transforms. In: Hawkes, P.W. (ed.) Advances in Images and Electron Physics, vol. 147, pp. 137–196. Elsevier Academic Press, San Diego (2007)
Perfilieva, I., De Meyer, H., De Baets, B., Plskova, D.: Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform. In: Proc. of WCCI 2008, IEEE Int. Conf. on Fuzzy Systems, Hong Kong (2008)
Perfilieva, I., Lemhke, S.: Correct models of fuzzy if–then rules are continuous. Fuzzy Sets and Systems 157, 3188–3197 (2006)
Perfilieva, I., Nosková, L.: System of fuzzy relation equations with \(\inf\rightarrow\) composition: complete set of solutions. Fuzzy Sets and Systems 159, 2256–2271 (2008)
Perfilieva, I., Novák, V.: System of fuzzy relation equations as a continuous model of IF-THEN rules. Information Sciences 177, 3218–3227 (2007)
Perfilieva, I., Novák, V., Dvořák, A.: Fuzzy transform in the analysis of data. Int. Journ. of Appr. Reasoning 48, 36–46 (2008)
Peterson, P.: Intermediate Quantifiers. Logic, linguistics, and Aristotelian semantics, Ashgate, Aldershot (2000)
Plšková, D.: Fuzzy transform in geological applications. Journal of Electrical Engineering 57, 43–46 (2006)
Sanchez, E.: Resolution of composite fuzzy relation equations. Information and Control 30, 38–48 (1976)
Sgall, P., Hajičová, E., Panevová, J.: The Meaning of the Sentence in Its Syntactic and Pragmatic Aspects. D. Reidel, Dordrecht (1986)
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)
Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. on Systems, Man, and Cybernetics SMC 3, 28–44 (1973)
Zadeh, L.A.: A rationale for fuzzy control, Trans. ASME, Ser. G, J. Dynamic. Systems, Measurement and Control 94, 3–4 (1974)
Zadeh, L.A.: Precisiated natural language. AI Magazine 25, 74–91 (2004)
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Novák, V., Perfilieva, I. (2010). Mathematical Fuzzy Logic: A Good Theory for Practice. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds) 35 Years of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16629-7_3
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DOI: https://doi.org/10.1007/978-3-642-16629-7_3
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