Robin Hirsch - Logic: by quantifiers or by algebra.

Modern logic is supported by two pillars: algebraic logic and Frege's quantifier logic. In the former approach, instead of explicit quantification, you have algebraic objects and operations. To handle unary relations (or sets) you get boolean algebra and to handle binary relations you get relation algebra. This includes an operation for "composition of binary relations" which is an implicit form of quantification.

In the twentieth century, first-order logic has become the dominant approach. But the interaction between the two approaches continues to be fruitful. So, for example, Tarksi defined cylindric algebra as the algebraic counter-part to first-order logic.

A consequence of some recent algebraic results about relation algebra reducts of cylindric algebras, is roughly, that in a Hilbert system you can prove more with n+1 variables than you can with n. Further more, if |-_{m,n} denotes a certain n-variable proof system of m-variable formulae, there is no finite, m-variable schema \Sigma that "turns |-_{m,n} into |-_{m,n+1}".