Proof theory, one of the two main directions of logic, has been mostly concentrated on pure logic. There have been systematic reasons to think that such a limitation of proof theory to pure logic is inevitable, but about twelve years ago, we found what appears to be a very natural way of extending the proof theory of pure logic to cover also axiomatic theories. How this happens, and how extensive of our method is, is explained in this book. We have written it so that, in principle, no preliminary knowledge of proof theory or even of logic is necessary.

The book can be profitably read by students and researchers in philosophy, mathematics, and computer science. The emphasis is on the presentation of a method, divided into four parts of increasing difficulty and illustrated by any examples. No intricate constructions or specialized techniques appear in these; all methods of proof analysis for axiomatic theories are developed by analogy to methods familiar from pure logic, such as normal forms, subformula properties, and rules of proof that support root-first proof search. The book can be used as a basis for a second course in logic, with emphasis on proof systems and their applications, and with the basics of natural deduction and sequent calculus for pure logic covered in Part I, Chapter 2, and Part II, Chapter 6.