THE XVIII CONFERENCE ON FAMEMS AND THE IV WORKSHOP ON HILBERT’S SIXTH PROBLEM,KRASNOYARSK,SIBERIA,RUSSIA, 2019 Two levels of fuzziness in the theory of co∼events Oleg Yu. Vorobyev Institute of mathematics and computer science Siberian Federal University Krasnoyarsk mailto:oleg.yu.vorobyev@gmail.com http://www.sfu-kras.academia.edu/OlegVorobyev http://olegvorobyev.academia.edu Abstract: The co∼event theory [1] allows you to distinguish between two levels of what is commonly called fuzziness in modern fuzzy mathematics. This is the fuzziness that Zadeh [2, 3], the creator of the theory of fuzzy sets, had in mind. Moreover, the one level of fuzziness, which usually serves as the subject of study of modern fuzzy mathematics, is studied by one of the dual halves of the co∼event theory: the believability theory and another level of fuzziness is the subject of research of the co∼event theory as such. In this work, we strictly define both levels of fuzziness, examine their properties, relationships, differences, and interpretations in practical applications. Keywords: Probability, event, set of events, Kolmogorov’s axiomatics, eventology, event-probability distribution, fuzzy mathematics, fuzzy logic, fuzzy set theory, multivariate statistics, false fuzziness, ket-fuzziness, bra- ket-fuzziness. MSC: 60A05, 60A10, 60A86, 62A01, 62A86, 62H10, 62H11, 62H12, 68T01, 68T27, 81P05, 81P10, 91B08, 91B10, 91B12, 91B14, 91B30, 91B42, 91B80, 93B07, 94D05 The external level of fuzziness characterizes a co∼event (as a measurable binary relation) that is a result of a experienced-random experiment, the internal level characterizes a fuzzy co∼event (as a measurable fuzzy binary relation) that is a result of a fuzzy experienced-random experiment. The external level of fuzziness is generated by differences in the experience of the set of observers who participate in the experienced random experiment. The internal level of fuzziness is generated by instabilities in the experience of each observer who participate in the experienced random experiment. The fuzziness of the external level is only a redefinition of the generally accepted concept of fuzziness within the framework of the co∼event theory. In the terminology of co∼event theory [1], the fuzziness of the external level is determined and equivalent to the imbelievability of a set of observers in their own experiences. Let ⟨Ω, , B|Ω, , P⟩ be a bra-ket-space where B is a believability measure on ⟨| and P is a probability measure on |⟩. A measurable binary relation R ⊆⟨Ω|Ω⟩ with the element-set-labeling ⟨X R | S X R ⟩ is defined by the set of membership relations «x ∈ X», i.e., by the indicator function 1X(x) ∈ [0, 1] on ⟨X R | S X R ⟩ that characterizes the co∼event R. The co∼event R has the believability distribution {bx : x ∈ X R } on the bra-space ⟨Ω| and the probability distribution {p(X): X ∈ S X R } on the ket-space |Ω⟩. Moreover, the believability measure B induces the function {b(X): X ∈ S X R } where b(X)= ∑ x∈X bx on the ket-space |Ω⟩ which is a believability of experiences of observers that the X-observation happened and 1 − b(X) is a believability of experiences of observers that the X-observation did not happen, i.e. imbelievability that X-observation happened. Namely, this imbelievability is a measure modeling the fuzziness of the external level in the theory of co∼events which corresponds to the concept of fuzziness in the sense of Zadeh [2, 3]. But the fuzziness of the internal level is a new concept, the definition of which became possible only after the newly introduced notion of a fuzzy event as a measurable fuzzy binary relation. A measurable fuzzy binary relation ˜ R with the element-set-labeling ⟨X R | S X R ⟩ is defined by the set of membership degrees of relations «x ∈ X», i.e., by the membership function RX(x) ∈ [0, 1] on ⟨X R | S X R ⟩ that characterizes the fuzzy co∼event ˜ R for x ∈ X R ,X ∈ S X R where each set-label X ∈ S X R is determined by the membership function by the formula: X = {x ∈ X R : RX(x) ≥ 1/2}. The membership function of an ordinary co∼event R ⊆⟨Ω|Ω⟩ with the same element-set-labeling ⟨X R | S X R ⟩ coincides with its indicator on ⟨X R | S X R ⟩: RX(x) = 1X(x)= { 1, x ∈ X, 0, x ̸∈ X. (2.1) c ○ 2019 O.Yu.Vorobyev This is an open-access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited. Oleg Vorobyev (ed.), Proc. of the XVIII FAMEMS’2019, Krasnoyarsk: Siberian Federal University Press