The paper is the Part III of the large research, dedicated to both revision of the system of basic logical categories and generalization of modern predicate logic to functional logic. We determinate and contrapose modern fregean logistics and proposed by the author ultrafregean logistics, next we descibe values and arguments of functions, arguments of relations, relations themselves, sets (classes) and subsets (subclasses) as derivative categories (concepts) of ultrafregean logistics. Logictics is a part of metalogic, independent of semantics. Fregean logictics is a metalogical theory, based on the quadruple <particular (individual), predicate, equality, sequence>; it generates predicate logic. Ultrafregean logictics is based on the quadruple <particular (individual), function, representation, sequence>, where the notion a function is a generalization of the notion of a predicate and the notion of representation is a generalization of the notion of equality; this logictics generates functional logic. For the completely correct denotation of the functional values we need the chorchian symbolics with parenthesis. Predicates are usually identified with relations. A relation is the derived and even definable category of ultrafregean logictics. Namely, relations are representations by functions (of one of their arguments). We show that Frege could realy establish this definition and the notion (category) of representation but, unfortunately, rejected this course of thought. Next, we show that every n-ary relation can be solved for some its argument via some (n-1)-ary function. A set, or class, is a derived and not definable category of ultrafregean logictics. The universal way to introduce the sets is Frege’s abstraction principle. We formulate this principle for functional logic and show that the notion of a set is a quantified notion, so there is the dual existential notion of a nonempty subset, involved by the same abstraction principle.
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