Logic of Relations

a branch of logic dealing with the study of relations between different kinds of objects.

In natural languages relations are expressed by predicates in sentences that have more than one subject or one subject and one or several objects. Depending on the number of these subjects (or subjects and objects), we speak of binary (two-place, two-member), ternary (three-place, three-member), and, in general, n-ary (n-place, n-member) relations. The concept of a (many-place) predicate is used in formalized languages of mathematical logic as an analogue of the concept of a relation. The corresponding modern modification of the logic of relations is called the predicate logic.

In the language of set theory and of algebra, a class of ordered systems of n elements is called an n-place relation. For example, if the ordered pair (x, y) belongs to some relation R, it is said that x is related toy by R. The concepts of a domain of definition for a given relation (the set of the first elements of the pairs in it) and that of a range of values (the set of all their second elements) are defined for relations understood in this way and, as is done in set theory, the operations of union (sum) and intersection (product) of relations are introduced. In the resulting “algebra of relations” (a term also used as a synonym to the term “logic of relations”) the equivalence relation, which possesses the properties of reflexivity (for all x, xRx), symmetricity (xRy implies yRx)_y and transitivity (xRy and yRz implies xRz) plays the role of “unity.” Equality of numbers, similarity of polygons, parallelism of straight lines, and other relationships belong to this most important class of relations.

A second very important class of relations are ordering relations. These include “fair-ordered” relation, which is reflexive and transitive but nonsymmetric, and “well-ordered” relation, which is transitive but nonreflexive and nonsymmetric. The relations “not greater than” and “less than,” respectively, for numbers or segments are examples of those ordering relations. Many very important concepts in logic and mathematics, in particular, that of a function and an operation, are introduced in terms of relations using the symbolism of the algebra of relations.