This book, written by one of philosophy's preeminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. Jaakko 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. 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.