Logic of Classes
Logic of Classes
a branch of logic that deals primarily with classes (sets) of objects specified by properties that characterize them and that are common to all elements in a given class.
Within the framework of contemporary formal mathematical logic, the logic of classes can be understood as an extension or expansion of propositional logic in which “elementary propositions” are no longer considered indivisible wholes; rather, each proposition exhibits a subject-predicate form and may thus be considered on the level of content as a simple declarative sentence in which [grammatical] subjects and predicates are distinguished.
A second interpretation, differing in form but equivalent in substance to the first interpretation, treats the logic of classes as a case of predicate logic, that is, the logic of one-place predicates, or, more precisely, a logic that operates with the extensions of concepts and whose contents are expressed by corresponding one-place predicates. Finally, there exists another interpretation of the logic of classes that is isomorphic with the first two interpretations. This third interpretation deals with sets (classes) of objects, independently of any properties common to the elements of the sets, and operations on sets. In other words, the logic of classes in this case can be identified with the algebra of sets, in which arbitrary sets and ordinary set-theoretic operations are considered. The above-mentioned isomorphism between the algebra of propositions and the algebra of sets (logic of classes) is obtained by establishing a one-to-one correspondence between sets (classes) and propositions about the membership of a given element in a set, between the intersection of sets and the conjunction of the corresponding propositions, between the union and disjunction, and between complement and negation. By examining a realization of the logic of classes in a single-element domain, the question of truth (falsity) of formulas in the logic of classes can be reduced to the corresponding problems for propositional logic and, like it, the logic of classes thus turns out to be decidable. The decidability of one-place predicate logic is thereby easily established as well. Since, as has been shown, one-place predicate logic essentially coincides with the logic of classes, the latter is not usually regarded a special theory but rather an aspect of predicate logic.
IU. A. GASTEV