By Jiři Bečvář (auth.), Heinz J. Skala, S. Termini, E. Trillas (eds.)

The moment international convention on arithmetic on the provider of guy was once held on the Universidad Politecnica de Las Palmas, Canary Islands, Spain, June 28 to July three, 1982. the 1st quantity of the court cases of the convention, entitled "Functional Equations-Theory and functions" has seemed within the Reidel sequence "Mathematics and Its Applications". The papers during this quantity include the invited lectures brought on the convention, part 7: Non-Classical Logics and Modelling, in addition to a few chosen papers which provide an advent to the philosophy, technique and to the lite rature of the extensive and interesting box of vagueness, imprecision and uncertainty. The contributed papers seemed within the quantity of photo-offset preprints allotted on the convention. it's our desire that the papers current an outstanding pattern with recognize to the historical past, the formalism and perform of this zone of analysis so far as we know it at the present time. because the topic "Vagueness" touches many elements of human pondering, the contributions were made up of a wide spectrum starting from philo~ophy via natural arithmetic to likelihood conception and mathematical economics, accordingly the cautious reader should still locate a few new insights right here. In end, the editors are looking to thank all authors who've contributed to this quantity; the publishers of "Commenta tiones Mathematicae Universitatis Carolinae" for permission to reprint the paper "Fuzziness and Fuzzy Equality", Commentationes Mathematicae Universitatis Carolinae 23 (1982), 249-267, and D. Reidel for pleasant cooperation.

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The next result establishes a canonical relationship between plausibility measures on L and a class of Borel probability measures on r(L) (see Hohle [7,8]). 3 Theorem. Let L be a ®-fuzzy a-algebra and m: L - [0,1] be a mapping. ) u ({wlw(V J..! 1) {wlw(J..! )=1}))=0. n Proof. The implication (ii) ~ (i) is evident. 2). 3) o i=1 1 [\I iE {1, ... ,n}: Il :;;;Il. and 3 i E {1, ... ,n}: J..! 1). m to a a-additive probability measure ~ on the a-algebra a(A) generated by A. Since a(A) coincides with the a-algebra of Baire subsets of r(L), TIm can be extended to a Tr-regular Borel probability measure urn on r(L).

1957, 'Bemerkungen zum Komprehensionsaxiom', Zeitschr. math. Logik Grundlagen Math. 3, 1-17. : 1960, 'A set theory based on a certain 3-valued logic', Mathematica Scand. 8, 127-136. D. Thesis, Tokyo, Tokyo Institute of Technology. : 1959, 'Partially ordered recursive arithmetics', Mathematica Scand. 7, 305-320. Wechler, W. and V. L. ), Information Processing 1974 North-Holland Publ. , Amsterdam, pp. 657-660. : 1965, 'Fuzzy sets', Inform. and Control 8, 338-353. : 1971, 'Similarity relations and fuzzy orderings' , Information Sci.

The theory PSD of proximity and strict dominance, in the new language L, consists of the axioms of P together with the following list of deductions (where [ST] stands for strict- transitivity, [IR] for irreflexivity and [se] for strict connectedness): [sTh x ~ y [sTh x ~ y \- (z [IR] x ~ y \- ,(x [se] ,(x ~ Z -+ x ~ z) x z ~ y) \- (y r ~ ~ y), ,(y -+ y) ~ x) \- x ~ y. A structure A for the new L is a model of these four deduction expressions if for every E we can find 81, 82 , 8 3 , 84, such that for all a,b,c in the domain A of A [ST)l [a ~ b] < 8 1 => [a ~ d - [b ~ d < [ST) [a ~ b] < 8 2 => [c ~ b] - [c ~ a] < E 2 E b] < 8 3 => 1 - [a ~ b] < E [IR] [a [se] max{l-[a ~ b],l-[b ~ a]j < 8" => [a ~ b] < E.