One of the preeminent problems confronting logicians is that of constructing a system of logic which will be adequate for mathematics. By a system's being adequate for mathematics, we mean that all mathematical theorems in general use can be deduced within the system. Several distinct logical systems, all having this end in view, have been proposed. Among these perhaps the best known are the systems referred to as “Principia Mathematica” and “set theory.” In both of these systems (we refer to the revised and simplified versions) there is a nucleus of propositions which can be derived by using only the axioms and rules of the restricted predicate calculus. However, if anything like adequacy for mathematics is to be expected, additional primitives and axioms must be added to the restricted predicate calculus. It is in their treatment of the additional primitive ε, denoting class or set membership, that the above-mentioned systems differ.

In addition to these two, a third and a stronger system has been proposed by W. V. Quine in his paper New foundations for mathematical logic. It is with this system of Quine's that our work is concerned and of which we now give a brief description.