There is one set of postulates whose position in the theory is unique. This is the set of pos tulates for symbolic logic (q.v.). Symbolic or mathematical logic consists in the criteria and methods of logical and mathematical reasoning in all fields. It is consequently in essence an individual system, not a type of systems. Con sequently, if it can be treated by the postulate method at all, its postulates are genuine indi vidual propositions, and not propositional forms. It is by no means obvious that any finite set of postulates will suffice for symbolic logic, and the attempt of Whitehead and Russell to find such a set breaks down in certain de tails, yet is sufficiently close to complete suc cess to give promise of even better results in the future.
In the course of any attempt to formulate postulates for logic, two propositions appear which are not indispensable for the develop ment of the simpler and many of the more coin plicated propositions in logic and mathematics, but are continually involved in many processes of reasoning that were formerly accepted with out question. There are the proposition that
there are an infinite number of things in the world, and the proposition that given an aggre gate A of aggregates B, some aggregate C exists containing one member of each of the B's, and only one. This latter is essentially due to Zer melo. See ASSEMBLAGES, GENERAL THEORY OF.
There is no book devoted to postulate theory at large, although almost any of the mathematical works of the writers already named or of Huntington, Veblen, Sheffer, Peano, Vailati or of many other writers of the last quarter century approach mathematical problems with this attitude in mind. By far the largest part of the literature is contained in the American Journal of Mathematics, the Transactions of the American Mathematical Society and the Annals of Mathematics for the past 15 years. For the postulates of logic consult A. N. Whitehead and B. Russell, (Pnn cipia Mathematica' (1910-).
Consult the various articles on mathematical disciples for their peculiarpostulates, and see especially LOGIC, SYMBOLIC; ASSEMBLAGES, GEN ERAL THEORY OF.
Nonerr WIENER.