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Book description

Professor Merrie Bergmann presents an accessible introduction to the subject of many-valued and fuzzy logic designed for use on undergraduate and graduate courses in non-classical logic. Bergmann discusses the philosophical issues that give rise to fuzzy logic - problems arising from vague language - and returns to those issues as logical systems are presented. For historical and pedagogical reasons, three-valued logical systems are presented as useful intermediate systems for studying the principles and theory behind fuzzy logic. The major fuzzy logical systems - Lukasiewicz, Gödel, and product logics - are then presented as generalisations of three-valued systems that successfully address the problems of vagueness. A clear presentation of technical concepts, this book includes exercises throughout the text that pose straightforward problems, that ask students to continue proofs begun in the text, and that engage students in the comparison of logical systems.


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Ackermann, Robert. 1967. Introduction to Many-Valued Logics. London: Routledge & Kegan Paul.
Aguzzoli, S., and Ciabattoni, A.. 2000. “Finiteness in Infinite-Valued Łukasiewicz Logic.” Journal of Logic, Language, and Information 9, pp. 5–29.
Baaz, Matthias. 1996. “Infinite-Valued Gödel Logics with 0–1-Projections and Relativizations.” In ed. Hájek, Petr, Gödel '96: Logical Foundations of Mathematics, Computer Science and Physics – Kurt Gödel's Legacy. New York: Springer, pp. 23–33.
Baaz, Matthias, Fermüller, Christian G., and Zach, Richard. 1993. “Systematic Construction of Natural Deduction Systems for Many-Valued Logics.” Proceedings of the 23rd International Symposium on Multiple Valued Logic. Los Alamitos, CA: IEEE Computer Society Press, pp. 208–213.
Baaz, Mathias, Hájek, Petr, Kraníček, Jan, and Švejda, David. 1998. “Embedding Logics into Product Logic.” Studia Logica 61, pp. 35–47.
Baaz, M., and Zach, R.. 1998. “Compact Propositional Gödel Logics.” Proceedings of the 28th International Symposium on Multiple-Valued Logic. Los Alamitos, CA: IEEE Computer Society Press, pp. 108–113.
Balbes, Raymond, and Dwinger, Philip. 1974. Distributive Lattices. Columbia: University of Missouri Press.
Beall, J. C., and Colyvan, Mark. 2001. “Heaps of Gluts and Hyde-ing the Sorities.” Mind 110, pp. 401–408.
Beall, J. C., and Fraassen, Bas C.. 2003. Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic. New York: Oxford University Press.
Behounek, Libor. “A Model of Higher-Order Vagueness in Higher-Order Fuzzy Logic.”
Bergmann, Merrie, Moor, James H., and Nelson, Jack. 2004. The Logic Book, 4th ed. New York: McGraw-Hill.
Black, Max. 1937. “Vagueness: An Exercise in Logical Analysis.” Philosophy of Science 4, pp. 427–455.
Bochvar, D. A. 1937. “Ob odnom Tréhznačnom Isčislénii i égo Priménénii k Analizu Paradoksov Klassičéskogo Rasširénnogo Funkcional'nogo Isčisléniá.” Matématčéskij Sbornik 4 (46), pp. 287–308. (English translation by Merrie Bergmann, “On a Three-Valued Calculus and Its Application to the Analysis of the Paradoxes of the Classical Extended Functional Calculus.” History and Philosophy of Logic 2, 1981, pp. 87–112.)
Bolc, Leonard, and Borowik, Piotr. 1992. Many-Valued Logics. I: Theoretical Foundations. New York: Springer-Verlag.
Chang, C. C. 1958a. “Proof of an Axiom of Łukasiewicz.” Transactions of the American Mathematical Society 87, 55–56.
Chang, C. C. 1958b. “Algebraic Analysis of Many Valued Logics.” Transactions of the American Mathematical Society 88, pp. 476–490.
Chang, C. C. 1959. “A New Proof of the Completeness of the Łukasiewicz Axioms.” Transactions of the American Mathematical Society 93, pp. 74–80.
Church, Alonzo. 1936. “A Note on the Entscheidungsproblem.” Journal of Symbolic Logic 1, pp. 40–41.
Cignoli, Roberto L. O., D'Ottaviano, Itala M. L., and Mundici, Daniele. 2000. Algebraic Foundations of Many-Valued Reasoning. Boston: Kluwer.
Delong, Howard. 1970. A Profile of Mathematical Logic. Reading, MA: Addison-Wesley.
Dilworth, R. P., and Ward, M.. 1939. “Residuated Lattices.” Transactions of the American Mathematical Society 45, pp. 335–354.
Dunn, J. Michael, and Hardegree, Gary M.. 2001. Algebraic Methods in Philosophical Logic. New York: Oxford University Press.
Edgington, Dorothy. 1999. “Vagueness by Degrees.” In eds. Keefe, Rosanna and Smith, Peter, Vagueness: A Reader. Cambridge, MA: MIT Press, pp. 294–316.
Esteva, Francesc, Godo, Lluis, Hájek, Petr, and Navara, Mirko. 2000. “Residuated Fuzzy Logics with an Involutive Negation.” Archive for Mathematical Logic 39, pp. 103–124.
Esteva, Francesc, Godo, Lluis, and Montagna, Franco. 2001. “The ŁΠ and ŁΠ½ Logics: Two Complete Fuzzy Systems Joining Lukasiewicz and Product Logics.” Archive for Mathematical Logic 40, pp. 39–67.
Fine, Kit. 1975. “Vagueness, Truth and Logic.” Synthese 30, pp. 265–300.
Frege, Gottlob. 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Verlag von Louis Nebert.
Gödel, Kurt. 1932. “Zum Intuitionistischen Aussagenkalkül.” Anzeiger der Akademie der Wissenschaften Wien, mathematisch, naturwissenschaftliche Klasse 69, pp. 65–66.
Goguen, Joseph A. 1967. “L-Fuzzy Sets.” Journal of Mathematical Analysis and Applications 18, pp. 145–174.
Goguen, Joseph A. 1968–1969. “The Logic of Inexact Concepts.” Synthese 19, pp. 325–373.
Goldberg, H., LeBlanc, H., and Weaver, G.. 1974. “A Strong Completeness Theorem.” Notre Dame Journal of Formal Logic 15 (2), pp. 325–331.
Gottwald, Siegfried. 2001. A Treatise on Many-Valued Logics. Philadelphia: Research Studies Press.
Gottwald, Siegfried, and Petr Hájek. 2005. “Triangular Norm-based Mathematical Fuzzy Logics.” In eds. Klement, E. P. and Mesiar, R., Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Numbers. New York: Elsevier, pp. 275–299.
Haack, Susan. 1979. “Do We Need Fuzzy Logic?International Journal of Man-Machine Studies 11, pp. 437–445.
Hájek, Petr. 1995a. “Fuzzy Logic and Arithmetical Hierarchy.” Fuzzy Sets and Systems 73 (3), pp. 359–363.
Hájek, Petr. 1995b. “Fuzzy Logic from the Logical Point of View.” In eds. Bartošek, M., Staudek, J., and Wiedermann, J., SOFSEM '95: Theory and Practice of Informatics; Lecture Notes in Computer Science 1012. New York: Springer-Verlag, pp. 31–49.
Hájek, Petr. 1997. “Fuzzy Logic and Arithmetical Hierarchy II.” Studia Logica 58, pp. 129–141.
Hájek, Petr. 1998a. “Basic Fuzzy Logic and BL-algebras.” Soft Computing 2, pp. 124–128.
Hájek, Petr. 1998b. Metamathematics of Fuzzy Logic. Boston: Kluwer.
Hájek, Petr. 2001. “On Very True.” Fuzzy Sets and Systems 124, pp. 329–333.
Hájek, Petr, Paris, Jeff, and Shepherdson, John. 2000. “Rational Pavelka Predicate Logic Is a Conservative Extension of Łukasiewicz Predicate Logic.” The Journal of Symbolic Logic 65 (2), pp. 669–682.
Heck, Richard G. 1993. “A Note on the Logic of (Higher-Order) Vagueness.” Analysis 53, pp. 201–208.
Hirota, K., ed. 1993. Industrial Applications of Fuzzy Technology (translated by H. Solomon). New York: Springer-Verlag.
Hunter, Geoffrey. 1971. Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. Los Angeles: University of California Press.
Hyde, Dominic. 1997. “From Heaps and Gaps to Heaps of Gluts.” Mind N. S. 106, pp. 641–660
Kearns, John T. 1979. “The Strong Completeness of a System for Kleene's Three-Valued Logic.” Zeitschrift für mathematische Logik und Grundlagen der Mathematik 25, pp. 61–68.
Keefe, Rosanna, and Smith, Peter, eds. 1997. Vagueness: A Reader. Cambridge, MA: MIT Press.
Kleene, Stephen C. 1938. “On a Notation for Ordinal Numbers.” The Journal of Symbolic Logic 3, pp. 150–155.
Klir, George J., and Yuan, Bo. 1995. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Saddle River, NJ: Prentice Hall.
Lakoff, George. 1973. “Hedges: A Study in Meaning Criteria,” Journal of Philosophical Logic 2, pp. 459–508.
LeBlanc, Hugues. 1977. “A Strong Completeness Theorem for 3-Valued Logic: Part II.” Notre Dame Journal of Formal Logic 18 (1), pp. 107–116.
Lee, R. C. T., and Chang, C.-L.. 1971. “Some Properties of Fuzzy Logic.” Information and Control 19, pp. 417–431.
Łukasiewicz, Jan. 1930. “Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls.” Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie 23, ⅽⅼ. ⅲ, pp. 51–77. (English translation by H. Weber, “Philosophical Remarks on Many-Valued Systems of Propositional Logic.” In ed. Storrs McCall, Polish Logic: 1920–1939, New York: Oxford University Press, 1967, pp. 40–65.)
Łukasiewicz, Jan. 1934. “Z historii logiki zdań.” Przeglad Filozoficzny 37, pp. 417–437. (English translation by Storrs McCall, “On the History of the Logic of Propositions.” In ed. Storrs McCall, Polish Logic 1920–1939, New York: Oxford University Press, 1967, pp. 66–87.)
Łukasiewicz, J., and Tarski, A.. 1930. “Untersuchungen über den Aussagenkalkül.” Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie 23, ⅽⅼ. ⅲ, pp. 39–50. (English translation by J. H. Woodger, “Investigations into the Sentential Calculus.” In Alfred Tarski, Logic, Semantics, Metamathematics: Papers from 1923 to 1938, 2nd ed., Indianapolis: Hackett Publishing Co., 1983, pp. 38–59.
Machina, Kenton F. 1976. “Truth, Belief, and Vagueness.” Journal of Philosophical Logic 5, pp. 47–78.
MacLane, Saunders, and Birkhoff, Garrett. 1999. Algebra, 3rd ed. Providence, RI: American Mathematical Society.
Mangani, P. 1973. “Su Certe Algebre Connesse con Logiche a Piú Valori (On Certain Algebras Related to Many-Valued Logics).” Bollettino dell'Unione Matematica Italiana (Series 4) 8, pp. 68–78.
Martin, Robert L., ed. 1970. Paradox of the Liar. New Haven, CT: Yale University Press.
Martin, Robert L., ed. 1984. Recent Essays on Truth and the Liar Paradox. New York: Oxford University Press.
McNaughton, Robert. 1951. “A Theorem about Infinite-Valued Sentential Logic.” Journal of Symbolic Logic 16, pp. 1–13.
Menger, K. 1942. “Statistical Metrics.” Proceedings of the National Academy of Sciences in the USA 8, pp. 535–537.
Meredith, C. A. 1928. “The Dependence of an Axiom of Łukasiewicz.” Transactions of the American Mathematical Society 87, p. 54.
Minari, Pierluigi. 2003. “A Note on Łukasiewicz's Three-Valued Logic.”
Morgan, Charles G., and Pelletier, Francis J.. 1977. “Some Notes Concerning Fuzzy Logics.” Linguistics and Philosophy 1, pp. 79–97.
Novák, Vilém. 1990. “On the Syntactico-Semantical Completeness of First-Order Fuzzy Logic” (Parts I and II). Kybernetika 26, pp. 47–66, 134–154.
Novák, Vilém. 2001. “Antonyms and Linguistic Quantifiers in Fuzzy Logic.” Fuzzy Sets and Systems 124, pp. 335–351.
Novák, Vilém, Perfilieva, Irina, and Močkoř, Jiří. 1999. Mathematical Principles of Fuzzy Logic. Boston: Kluwer.
Pavelka, J. M. 1979. “On Fuzzy Logic” (Parts I, II, and III). Zeitschrift für mathematische Logik und Grundlagen der Mathematik 25, pp. 45–52, 119–134, 447–464.
Pedrycz, Witold, and Gomide, Fernando. 1998. An Introduction to Fuzzy Sets: Analysis and Design. Cambridge, MA: MIT Press.
Pogorzelski, W. A. 1964. “The Deduction Theorem for Łukasiewicz Many-Valued Propositional Calculi.” Studia Logica 15, pp. 7–23.
Priest, Graham. 2001. An Introduction to Non-Classical Logic. Cambridge: Cambridge University Press, p. 215.
Quine, W. V. O. 1960. Word and Object. Cambridge, MA: MIT Press.
Rescher, Nicholas. 1969. Many-Valued Logic. New York: McGraw-Hill.
Robinson, J. A. 1965. “A Machine-Oriented Logic Based on the Resolution Principle.” Journal of the Association for Computational Machinery 12, pp. 23–41.
Rose, Alan, and Rosser, J. Barkley. 1958. “Fragments of Many-Valued Statement Calculi.” Transactions of the American Mathematical Society 87 (1), pp. 1–53.
Ruspini, Enrique H., Bonissone, Piero P., and Predrycz, Witold, eds. 1998. Handbook of Fuzzy Computation. Philadelphia: Institute of Physics Publishing.
Russell, Bertrand. 1923. “Vagueness.” Australasian Journal of Philosophy 1, pp. 84–92.
Scarpellini, B. 1962. “Die Nichtaxiomatisierbarkeit des undenlichwertigen Prädikatenkalküls von Łukasiewicz.” Journal of Symbolic Logic 27, pp. 159–170.
Słupecki, Jerzy. 1936. “Der volle dreivertige Aussagenkalkül.” Comptes rendus des séances de la Société des sciences et des lettres de Varsovie 3 (29), pp. 9–11. (English translation by Storrs McCall, “The Full Three-Valued Propositional Calculus.” In ed. Storrs McCall, Polish Logic: 1920–1939. New York: Oxford University Press, 1967, pp. 335–337.)
Smullyan, Raymond M. 1968. First-Order Logic. New York: Springer-Verlag.
Stoll, Robert R. 1961. Set Theory and Logic. San Francisco: W. H. Freeman.
Takeuti, Gaisi, and Titani, Satoko. 1984. “Intuitionistic Fuzzy Logic and Intuitionistic Fuzzy Set Theory.” The Journal of Symbolic Logic 49, pp. 851–866.
Tarski, Alfred. 1936. “Der Wahrheitsbegriff in den formalisierten Sprachen.” Studia Philosophica 1, pp. 261–405.
Turksen, I. B. 1991. “Measurement of Membership Functions and Their Acquisition.” Fuzzy Sets and Systems 40, pp. 5–38.
Fraassen, Bas C. 1966. “Singular Terms, Truth-Value Gaps, and Free Logic.” Journal of Philosophy 63, pp. 481–495.
Wajsberg, Mordchaj. 1931. “Aksjomatyzacja trójwartościowego rachunku zdań.” Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie 24, ⅽⅼ. ⅲ, pp. 126–145. (English translation by B. Gruchman and S.McCall, “Axiomatization of the Three-Valued Propositional Calculus.” In ed. Storrs McCall, Polish Logic: 1920–1939. New York: Oxford University Press, 1967, pp. 264–284.)
Wright, Crispin. 1987. “Further Reflections on the Sorites Paradox.” Philosophical Topics 15, pp. 227–290.
Wu, Olivia. 2003. “Rice Goes Digital Cooked the Fuzzy Logic Way: Side-by-Side Tests Show Appliance Makes a Difference.” San Francisco Chronicle, December 10, p. E1.
Zadeh, Lotfi A. 1965. “Fuzzy Sets.” Information and Control 8, pp. 338–353.
Zadeh, Lotfi A. 1975. “Fuzzy Logic and Approximate Reasoning.” Synthese 30, pp. 407–428.


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