# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Theory of truth degrees of propositions in the logic system ${L}_{n}^{*}$. (English) Zbl 1104.03014
Summary: By means of the infinite product of uniformly distributed probability spaces of cardinality $n$ the concept of truth degrees of propositions in the $n$-valued generalized Łukasiewicz propositional logic system ${L}_{n}^{*}$ is introduced. It is proved that the set consisting of the truth degrees of all formulas is dense in [0, 1], and a general expression of truth degrees of formulas as well as a deduction rule of truth degrees is then obtained. Moreover, similarity degrees among formulas are proposed and a pseudo-metric is defined therefrom on the set of formulas, and hence a possible framework suitable for developing approximate reasoning theory in $n$-valued generalized Łukasiewicz propositional logic is established.
##### MSC:
 03B50 Many-valued logic 03B52 Fuzzy logic; logic of vagueness 68T37 Reasoning under uncertainty
##### Keywords:
truth degree; similarity degree; approximate reasoning
##### References:
 [1] Zadeh L A. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Systems, Man and Cybernet, 1973, 1: 28–44 · Zbl 0273.93002 · doi:10.1109/TSMC.1973.5408575 [2] Dubois D, Prade H. Fuzzy sets in approximate reasoning. Fuzzy Sets and Systems, 1991, 40(1): 143–202 · Zbl 0722.03017 · doi:10.1016/0165-0114(91)90050-Z [3] Pavelka J. On Fuzzy Logic I, II, III, Zeitschr. f. Math. Logik u. Grundlagen d. Math, 1979, 25: 45–52; 119–134; 447–464 [4] Ying M S. The fundamental theorem of ultroproduct in Pavelka’s logic. Z Math Logic Grundlagen Math, 1992, 38: 197–201 · Zbl 0798.03021 · doi:10.1002/malq.19920380115 [5] Xu Y, Qin K Y, Liu J, et al. L-valued propositional logic Lvpl. Information Sciences, 1999, 144: 205–235 [6] Xu Y, Liu J, Song Z M, et al. On semantics of L-valued first order logic Lvfl. Int J General Systems, 2000, 29(1): 53–79 · Zbl 0953.03028 · doi:10.1080/03081070008960924 [7] Wang G J. On the logic foundation of fuzzy reasoning. Information Sciences, 1999, 117(1): 47–88 · Zbl 0939.03031 · doi:10.1016/S0020-0255(98)10103-2 [8] Hajek P. Metamathematics of Fuzzy Logic. Dordrecht: Kluwer Academic Publishers, 1998 [9] Wang G J. Non-classic Logic and Approximate Reasoning (in Chinese). Beijing: Science Press, 2000 [10] Ying M S. A logic for approximate reasoning. J Symbolic Logic, 1994, 59: 830–837 · Zbl 0809.03015 · doi:10.2307/2275910 [11] Wang G J, Chin K S, Dang C Y. A unified approximate reasoning theory suitable for propositional calculus system L* and predicate calculus system K*. Sci China Ser F-Inf Sci, 2005, 48(1): 1–14 · Zbl 1182.03057 · doi:10.1360/04yf0094 [12] Wang G J, Wang W. Logical metric spaces (in Chinese), Acta Math Sin, 2001, 44(1): 159–168 [13] Wang G J, Fu L, Song J S. Theory of truth degrees of propositions in two-valued logic. Sci China Ser A-Math, 2002, 45(9): 1106–1116 [14] Halmos P R. Measure Theory, New York: Spring-Verlag, 1974 [15] Adams EW. A Primer of Probability Logic, Stanford: CSLI Publications, 1998