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 purposes" has seemed within the Reidel sequence "Mathematics and Its Applications". The papers during this quantity encompass the invited lectures brought on the convention, part 7: Non-Classical Logics and Modelling, in addition to a few chosen papers which supply an advent to the philosophy, method and to the lite rature of the large and interesting box of vagueness, imprecision and uncertainty. The contributed papers seemed within the quantity of photo-offset preprints disbursed on the convention. it truly is our desire that the papers current a great pattern with recognize to the heritage, the formalism and perform of this region of study so far as we know it this present day. because the topic "Vagueness" touches many elements of human considering, the contributions were made up of a wide spectrum starting from philo~ophy via natural arithmetic to likelihood thought and mathematical economics, for that reason 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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Math. Anal. Appl. 85, 543-565. 15. : 1982, 'Characterization of fuzzy measures constructed by means of triangular norms', J. Math. Anal. Appl. 86, 345-358. 16. : 'Fuzzy measures assuming their values in the set of fuzzy numbers', J. Math. Anal. , in press. 17. , Lowen, R. : 1981, 'Fuzzy probability measures', Fuzzy Sets and Systems 5, 21-30. 18. P. : 1982, 'Correspondence between fuzzy measures and classical measures', Fuzzy Sets and Systems 7, 57-70. 19. : 1955, 'On the axiomatic treatment of probability' Colloq.

Appl. 75, 330-339. 14. : 1982, 'Construction of fuzzy a-algebras using triangular norms', J. Math. Anal. Appl. 85, 543-565. 15. : 1982, 'Characterization of fuzzy measures constructed by means of triangular norms', J. Math. Anal. Appl. 86, 345-358. 16. : 'Fuzzy measures assuming their values in the set of fuzzy numbers', J. Math. Anal. , in press. 17. , Lowen, R. : 1981, 'Fuzzy probability measures', Fuzzy Sets and Systems 5, 21-30. 18. P. : 1982, 'Correspondence between fuzzy measures and classical measures', Fuzzy Sets and Systems 7, 57-70.

We can turn now to axiomatic theories of proximity and dominance, and we start with proximity. The theory P of proximity is the following list of ~ioms (where [RE] stands for reflexivity, [SY] for symmetry and [TR] for transitivity). ;;;; ~ c] Thus is an equivalence relation in L but [~] is a pseudo-metric in A. Our pseudo-metric reduces to the discrete pseudo-metric if we only allow the values 0 and 1, in which case we have a classical logic and we can replace [TR] by ~(x~y"y~z) ~ x ~ z. This last axiom translates in A to the u1trametric inequality, max([a~b],[b ~ c]) ;;;, [a~c] (see Shepard [1974]), which holds for the discrete metric but not for an arbitrary metric.