Calculus of classes

The traditional name, going back to G. Boole, for the branch of mathematical logic studying the logic of classes. The calculus of classes corresponds, in fact, to propositional logic in which the subject-predicate structure of elementary propositions is also considered (that is, elementary propositions of the form "the element $x$ has the property $P$"); moreover, with each predicate (property) $P$ there is associated the class of elements in the domain of study that possess this property. The calculus of classes was conceived as the mathematical equivalent of Aristotle's syllogistics. However, this is not the case, because the empty set and singletons, which are admissible in the calculus of classes, were not considered by Aristotle. Usually, the calculus of classes is not singled out as an independent branch of mathematical logic, since all its expressive possibilities are covered by the calculus of one-place predicates (which in its turn is a decidable fragment of narrow predicate calculus, see Logical calculus). Aristotle's syllogistics have been formalized in an adequate fashion by J. Lukasiewicz [4].

References

[1]	D. Hilbert, W. Ackerman, "Grundzüge der theoretischen Logik" , Dover, reprint (1946)
[2]	L. Couturat, "L'algèbre de la logique" , Gauthier-Villars (1905)
[3]	M. Wajsberg, "Ein erweiterter Klassenkalkül" Monatsh. Math. Phys. , 40 (1933) pp. 113–126
[4]	J. Lukasiewicz, "Aristotle's syllogistic from the standpoint of modern formal logic" , Clarendon Press (1951)
[5]	S. Yanovskaya, "The logic of classes" , Philosophical Encyclopaedia , 3 , Moscow (1964) pp. 224–226 (In Russian)