By Vilém Novák, Irina Perfilieva, J. Mockor

*Mathematical ideas of Fuzzy Logic* presents a scientific learn of the formal idea of fuzzy good judgment. The ebook relies on logical formalism demonstrating that fuzzy good judgment is a well-developed logical thought. It contains the speculation of sensible structures in fuzzy common sense, offering an evidence of what could be represented, and the way, through formulation of fuzzy good judgment calculi. It additionally provides a extra normal interpretation of fuzzy good judgment in the setting of different right different types of fuzzy units stemming both from the topos concept, or perhaps generalizing the latter.

This publication provides fuzzy good judgment because the mathematical concept of vagueness in addition to the idea of common sense human reasoning, in response to using average language, the distinguishing characteristic of that's the vagueness of its semantics.

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**Example text**

B . c. (b) The distributivity of . with respect to +. The method of the proof will be the same as above. a· (b (a· b) + c) = a 1\ «b 1\ c') V (b' 1\ c)), + (a· c) = «a 1\ b) 1\ (a 1\ c)') V «a 1\ b)' 1\ (a 1\ c)) = = (a 1\ b 1\ a') V (a 1\ b 1\ c') V (a' 1\ a 1\ c) V (b' 1\ a 1\ c) = = a 1\ «b 1\ c') V (b' 1\ c)). 4 In a Boolean ring with lattice operations V and true: (a) a + a' = 1, 1\, the following identities are ALGEBRAIC STRUCTURES FOR LOGICAL CALCULI 21 = (a + b') . b, a V b = a . b' + b, (b) a /\ b (c) (d) (a + b') .

Obviously, U ~ is a filter containing F. Then use Zorn's lemma. 0 The lemma below is a good example of the filter based technique. It has also it's own importance in the proof of the first representation theorem. , be an MV-algebra and a E L, a", 1. , not containing a. PROOF: Let a ELand a", 1. By ~(a) we denote the set of all filters, which do not contain a. Obviously, {1} E ~(a) and therefore, ~(a) is nonempty. t. the ordinary inclusion. We show that F is prime. Suppose the opposite. Let b1 V b2 E F and b1 , b2 ¢ F.

Hence, VxEK X ~ a -t b. e. a -t b E K, which gives a -t b ~ VxEK x. (b )-(e) are proved analogously. (f) and (g) are the consequences of the isotonicity of -t in the second argument and antitonicity in the first one. 0 If the index set I is finite then the properties (c)-(g) of this lemma hold in every residuated lattice. A special case of residuated lattices are BL-algebras 1 , which have been introduced by P. Hajek to develop a kernel logical calculus, which would be included in various kinds of many- valued logical calculi.

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