Logic of Relations

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.


References in periodicals archive ?
From Ioan Biris's analysis it clearly results that only with Hegel we can talk about an integrating rational model, a model supported by a dialectical logic of relations. With this statement, backed by arguments, he is closer to the philosophers who see in the Hegelian system a perfection of philosophy along the lines opened by Aristotle's question ti esti.
Russell said that that new logic of relations developed by Peirce and DeMorgan and put to work in Principia Mathematica gave thought wings, whereas Aristotelian categorical logic put it in fetters.
Seen in this way, the logic of relations between Turkey and Barzani is simpler.
What is important is that linguists who try to understand how lexical items and grammatical markers partition the world of meaning need to turn to non-linguistic facts, methods, and theories to understand how that world is structured: the physics of colors, the logic of relations, the conceptualization of spatial trajectories.
Basing his approach on the work of Bertalanffy, Buckley does not side with either the cybernetics or general system movements as he describes levels of integrations systems, organization and the logic of relations, information communication, entropy and life, behavior and reading, semiotics and purpose, homeostasis and evolution, self regulation and sociocultural systems, social control organizations, decision processes and group structures.
Hoerder's emphasis on community, the logic of relations, and importance of local economies to migration renders the absence of these connections and causes with culture-destroying forced movements all the more brutal and disconcerting.
Importantly, Peirce's reading of Mill's work in the logic of science was supplemented by reading Mill's antagonist, William Whewell, and Augustus De Morgan's essays on the logic of relations. Yet Smyth's interest is in Peirce's reading of the more obscure Richard Whatley, whose "text on argumentative rhetoric, rather than his text on logic," argues Smyth, proves most important in the study of the early essays.
A logician and philosopher of science, Peirce is best known for his work on the logic of relations and for his promotion of pragmatism as a method of research.
His topics include sources of Victorian mathematical idealism, Benjamin Peirce and the divinity of mathematics at Harvard, George Boole and the genesis of symbolic logic, and Augustus de Morgan and the logic of relations.
In "Hume" he claims that the Scottish philosopher is the first to break "with the constraining form of predicative judgment and makes possible an autonomous logic of relations" (p.
The remaining fifteen essays treat the following topics: the logic of ideas, the logic of relations, the logic of inference, modality, faculty psychology, and methodology in early modern philosophy.
Contemporary philosophers sometimes speak as if no one could have possessed the concept of a polyadic property prior to the nineteenth and twentieth centuries--as if conceiving of relations in this way only became possible with advent of a formal logic of relations and a logic of multiple quantification.(36) But surely this is mistaken.