N.A. Vasil'ev's Logical Ideas and the Categorical Semantics of Many-Valued Logic

Here we suggest a formal using of N.A. Vasil'ev's logical ideas in categorical logic: the idea of "accidental" assertion is formalized with topoi and the idea, of the notion of nonclassical negation, that is not based on incompatibility, is formalized in special cases of monoidal categories. For these cases, the variant of the law of "excluded n-th" suggested by Vasil'ev instead of the tertium non datur is obtained in some special cases of these categories. The paraconsistent law suggested by Vasil'ev is also demonstrated with linear and tensor logics but in a form weaker than lie supposed. As we have, in fact, many truth-values in linear logic and topos logic, the admissibility of the traditional notion of inference in the categorical interpretation of linear and intuitionistic proof theory is discussed.