In the book M. E. Szabo, Algebra of Proofs, Studies in Logic and the Foundations of Mathematics, Vol. 88, North-Holland, 1978. the author, according to a 1980 review by Carlo Cellucci published in Mathematical Reviews (80b:03097), [...] studies the algebraic properties of the proof theory of intuitionistic first-order logic in a categorical setting. [...] Following the Introduction (Chapter I), there are twelve additional chapters, in which the author studies twelve theories of varied linguistic and deductive strength. The theories are divided into two main types: the monoidal type, in which theories based on the common algebraic properties of conjunction and disjunction are investigated, and the Cartesian type, in which conjunction and disjunction have their proper meanings. In every chapter the author follows the same scheme. He first constructs a category of a certain type as an algebraic model for the class of formal proofs being considered. Then he proves a completeness theorem to the effect that the arrows of the constructed category can be represented by formal proofs in a Gentzen-style sequent calculus with cut elimination. In the propositional cases the algorithmic character of the cut-elimination process is used to provide an effective description of the arrows of the category constructed and to develop decision procedures, in the form of Church-Rosser theorems, for the commutativity of the finite diagrams of these categories. In the last chapter, the author also shows how to accommodate quantifiers in the calculus of adjoints and describes the topos-theoretic setting required in order to develop the proof theory of intuitionistic first-order logic. The book itself contains a wealth of technical detail that includes many dozens of claims whose proofs are not worked out in detail. I'm writing to ask whether anyone has any knowledge about the degree to which this work was refereed and/or whether the results have been verified independently. I'm appealing especially to those who work in categorical logic or those interested in automatic proof verification in category theory, both of which groups, it would seem, should be interested in Szabo's work. -- Todd Wilson A smile is not an individual Computer Science Department product; it is a co-product. California State University, Fresno -- Thich Nhat Hanh