Why Relations are not Identical to their Graphs

Abstract

In mathematics, the identification of relations with their graphs, proposed by Giuseppe Peano, is generally accepted. However, this identification is refuted by the extreme example of 0-ary relations. The paper contains the proof under question. Therefore, we develop an alternative theory of relations, which is built within the logic of functions (functional logic, or function logic). On the basis of the primary, undefined logical notions of a particular (atomic object), representation (the ambiguous specifying), and a sequence, we define the notions of a(n ambiguous) function and a set, after which we define relations as those laws that uniquely define (determine, specify) individual functions. So a relation is in general case the representation (of an object) by some function, which we call forming one for this relation. The degenerate case of a relation is the case of the representation by a paricular (without a function). The general case of a function is an umbiguous function, i.e., a partial multimap rather than a map. Representation is the ulimate generalization of equality, so we define equality and uniformity (sameness) via representation as its partial cases. In functional logic, we use the notion of a finite sequence as the simplest case of the general notion of a sequence. Each function has two different defining relations: the obverse one and the reverse one. Every more than 0-ary function has two graphs: the obverse one (the graph of its obverse relation) and the inverse one (the graph of its reverse relation). The relations between relations themselves and their graphs are determined by postulate V of Frege's formal arithmetic (Grundgesetze System), which (postulate) is a theorem in the logic of functions. In the paper, this theorem is proved in its most general formulation.