This Book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions commonly made in logic, the foundations of mathematics, and metaphysics are in need of change. It is therefore a book of critical importance to logical theory and the philosophy of mathematics. Jaako Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous impossibility results by Godel and Tarski that have dominated the field for the past sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with all problems concerning the existence of sets and other higher-order entities. Hintikka's new logic is highly original and will prove appealing to logicians, philosophers of mathematics, and mathematicians concerned with the foundations of the discipline.