Birkhauser Basel, 2014.

**ACM CR categories:**

F.4.1 Mathematical logic

A.0 Biographies/autobiographies

I.2.3 Deduction and theorem proving - metatheory

K.3.2 Logic education

K.2 History of computing - People

This collection of papers, along with biographical information, on the logician Leon Henkin includes reminiscences from students of Henkin and of Alfred Tarski about the help provided by Henkin. The account by Benjamin Wells (a student of Tarski for 20 years) is worthy of a movie script with its roller coaster ride of twists and turns. Solomon Feferman (a student of Tarski) mentions how he was stuck on problems regarding cylindric algebras before Henkin moved him into provability and interpretability of theories.

I was pleasantly surprised to learn of the variety of Henkin's interests. His student Diane Resek writes about doing a dissertation on cylindric algebras while collaborating with Henkin on a math education project. She suggests that Henkin used the drive among young people at that time to be a change agent in American society, to use his graduate students' training to make a real difference. He persuaded the Berkeley math faculty to allow graduate students financial support for working in elementary school math classrooms equal to what they would get as teaching assistants in university courses. His student Nitsa Movshovits-Hadar describes five different presentations of the rational numbers that they had worked out for school students. The Committee on Student Diversity and Academic Development of the Berkeley Academic Senate awarded the first Citation for Distinguished Service to Leon Henkin, and the award is now named after him.

There are articles on Henkin's work in the areas of cylindric algebra, provability, type theory, including extensions of type theory to fuzzy and hybrid logics. Cylindric algebra, developed by Henkin, Donald Monk and Tarski, is the algebraic counterpart of first-order logic, like Boolean algebra is for propositional logic. There are five papers on different aspects of Henkin's now-famous model construction technique for completeness of logics. For example, Robert Goldblatt discusses using the Rasiowa-Sikorski lemma instead of Henkin's Lindenbaum construction in a proof for probabilistic modal logic.

I feel this rich discussion on completeness proofs makes the book of interest
to teachers of logic. We learn that in his logic course, Henkin would also
teach a proof by Herbrand. Until I read this book I did not know how
Gödel proved the completeness theorem for first-order logic,
as we are so used to Henkin's proof method. Enrique Alonso argues that
only after the Henkin proof appeared in Mendelson's textbook in 1964 [1]
(and then other textbooks) did it become routine for logic textbooks
to give completeness proofs, and papers proposing new logics were expected
to discuss their completeness. Unusually, the book has a contribution on
the reviews written by Henkin for the *Journal of Symbolic Logic*
(where he was for many years Reviews Editor).

The book is rounded out by a wonderful survey on Henkin's technique of generalizing models using subsets of valuations, written by Hajnal Andréka, Johan van Benthem, Nick Bezhanishvili, and István Németi. From a suitable viewpoint, they even provide a generalization of first-order logic, which is decidable. I would like to compliment the editors on putting together a very readable volume in all aspects: technical, educational and biographical.

Unfortunately, the publishers have not done a very good job of binding the volume. I could not read two papers because several pages were missing. A third article had the last page of references missing.

References

[1] Mendelson, E. *Introduction to mathematical logic*.
Van Nostrand, New York, NY, 1964.