عنوان مقاله
منطق فازی ریاضی چیست؟
فهرست مطالب
چکیده
مقدمه
مسئله
تکنولوژی
حل کننده
نتیجه گیری
بخشی از مقاله
منطق مسندی فازی پیوسته بر مبنای نرم t
همان گونه که معمولاً مشخص است، نرمt به یک عملیات دوتایی * در فاصله حقیقی واحد [0,1] اتلاق می شود که در هر شناسه جابجایی پذیر، وابسته و افزاینده بوده و دارای عناصر 0 و 1 می باشد. باقیمانده آن عملیات دو تایی در فاصله [0,1] می باشد که به صورت x→y=max{z|x*z≤y} تعریف شده است. نرمt در صورتی دارای باقیمانده می باشد که از سمت چپ پیوسته باشد.
کلمات کلیدی:
1. Origin, motivation, task We mention some milestones of the development of fuzzy logic understood as a branch of mathematical (symbolic) logic. Clearly, the story begins by Zadeh’s first paper [40] on fuzzy sets. The term “fuzzy logic” is not used; but Zadeh mentions Kleene three-valued logic (just in passing). Goguen’s 1968–1969 paper [17] speaks on logic of inexact concepts but the term “fuzzy logic” occurs there (on p. 359; is this the first occurrence of the term in the literature?) The paper is very general, introduces algebras called closg, very near to algebras presently called residuated lattices, as algebras of truth functions of connectives for many-valued logics of inexact concepts, As an example he presents the unit real interval [0, 1] with product and its residuum (Goguen implication), thus a particular t-norm algebra (not speaking on t-norms). Zadeh has written several papers on fuzzy logic; an early paper is his “Fuzzy logic and approximate reasoning” [41] from 1975 (reprinted in [30]), where he uses connectives of Łukasiewicz logic min, max, 1 − x, Łukasiewicz implication—but not strong conjunction. Note that what we call Łukasiewicz strong (or bold) conjunction or Łukasiewicz t-norm (the t-norm whose residuum is Łukasiewicz implication) was never explicitly used by Łukasiewicz. The first explicit use of this conjunction in the context of Zadeh’s fuzzy logic appears to be the paper [16] by Giles. 1 In Zadeh’s understanding, fuzzy logic uses some many-valued logic but works with fuzzy truth values and his linguistic variables. Zadeh (and the majority of researchers up to today, including the present author) understands fuzzy logic as truth functional, i.e. having some truth functions for connectives determining the truth value of a compound The author was partly supported by Grant A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and partly by the Institutional Research Plan AV0Z10300504. E-mail address: hajek@cs.cas.cz. 1 But Giles refers to papers by Chang and Klaua from mid-sixties using the bold conjunction in studying some generalized set theories over Łukasiewicz or similar logics. 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.10.004 598 P. Hájek / Fuzzy Sets and Systems 157 (2006) 597 –603 formula constructed using a connective uniquely from the truth values of the formula(s) to which the connective is applied. This makes fuzzy logic different from any probability theory (probability is evidently not truth functional). A general, not necessarily truth-fun
