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logic diagram[′läj·ik ‚dī·ə‚gram]
a graphic (geometric, more precisely, topological) apparatus of mathematical logic.
The concept of logic diagrams, already known in the Middle Ages, was developed by G. W. Leibniz. However, its first sufficiently detailed and substantiated formulation was given by L. Euler in 1768 in his Lettres à une princesse d’Allemagne, in which he examined what are now called Euler circles. Since then it has become customary to depict relationships between classes (extensions of concepts) by means of systems of mutually intersecting circles (or other simply connected regions). The (set-theoretic) union of the regions representing the classes corresponds to the union of the classes, the intersection of the regions corresponds to the intersection of the classes, and the complement with respect to a “standard” enclosing region (for example, a rectangle) corresponds to the complement (with respect to the universal class). There is a correspondence between the relation of inclusion between the classes represented and the analogous relation between their representations (here, the cases when the enclosing class coincides with the enclosed class and when it is substantially broader than the latter are not distinguished).
The concept of logic diagrams was subsequently developed and perfected. It acquired a particularly clear form in the works of J. Venn. (The British mathematician C. Dodgson, known as a children’s writer under the pseudonym L. Carroll, also proposed an original method for constructing logic diagrams.) The apparatus of Venn diagrams is based on the idea, which is central to the algebra of logic, of the analysis of logical functions into “constituents.” This makes it possible to solve by a uniform method a number of problems in propositional logic and the logic of one-place predicates. These include a survey of conclusions from given premises, the solution of logical equations (for any finite number of variables), and other problems, up to a simple and elegant solution of the decision problem. The apparatus of logic diagrams has also been extended to the classical calculus of many-place predicates and has proved to be an extremely convenient tool for solving a number of problems arising in the application of mathematical logic to automata theory.
REFERENCESCouturat, L. Algebra logiki. Odessa, 1909. (Translated from French.)
Kuzichev, A. S. Diagrammy Venna. Istoriia i primeneniia. Moscow, 1968. (See references.)
Venn, J. Symbolic Logic, 2nd ed. London-New York, 1894.
IU. A. GASTEV