Swedish logician and philosopher Dag Prawitz and his distinguished contributions to philosophical and mathematical logic are the focus of this book. Gerhard Gentzen, in his PhD thesis published in 1934, introduced the seminal idea of structural proof theory, as contrasted to the rather involved axiomatic method promoted by David Hilbert. Although each style has its followers and defenders, structural proof theory (or general proof theory) offers a more methodological approach to axiomatization and proofs, as, for instance, by means of the notion of analytic proofs.

In Gentzen’s natural deduction calculus, analytic proofs are those in which no formula occurs simultaneously as the principal premise of an elimination rule and as the conclusion of an introduction rule. In Gentzen’s sequent calculus, the analytic proofs are those that do not use the cut rule. The first version of the famous normal form theorems, one of the most basic results in proof theory, was established by Gentzen for the calculi of sequents in the 1930s. More than 30 years after Gentzen, the PhD thesis of Dag Prawitz renewed interest in the subject and contributed to an enormous impetus in proof theory and proof-theoretic semantics with consequences in logic, theoretical computer science, and linguistics.

The notion of proof-theoretic semantics (the name was coined by Peter Schroeder-Heister) is related to the idea that the meaning of a term should be explained by reference to the way it is used in a language, or that the meaning of the logical constants can be specified in terms of rules of inference governing them. Proof-theoretic semantics was born together with natural deduction, with the remarks by Gentzen that the introduction rules in his calculus of natural deduction define the logical operators, and the elimination rules can be obtained as a consequence of these definitions.

This book celebrates Prawitz’s contributions to proof theory and proof-theoretic semantics and their consequences. Eighteen chapters by distinguished logicians study aspects of Prawitz’s work and address sophisticated topics on constructibility, motivations for normalization of proofs, and proof-theoretic harmony, including a chapter where Dag Prawitz himself explains his ideas on the epistemic force of deductive inferences, analyzing the problem of evidence and meaning: how can an inference make us know something we did not know before? According to Prawitz, this is a neglected problem in contemporary logic. Even if Aristotle recognized the role of the “perfect syllogisms,” syllogisms that are self-evidently (or transparently) true (Barbara Celarent provides examples), today’s logic does not yet have a theory for what makes an inference legitimate.

Another chapter by Prawitz unveils his biography, with delicious details of his childhood in Sweden, his period as a farmer breeding sheep, and his experience as a president of the Rolf Schock Foundation. The last chapter provides a selection of 83 papers published by Prawitz since 1960, plus seven books and special issues edited in scientific journals. Dag Prawitz was the recipient of the 2008 Gad Rausing Prize for Outstanding Humanistic Research of the Royal Academy of Letters for his contributions to the theory of evidence, a fact modestly not mentioned in his scientific autobiography.

This is an excellent book, celebrating not only Prawitz’s career, but also a movement in the contrary direction of W. V. O. Quine’s views against the so-called (somehow prejudicially) “deviant” logics, and I cannot forbear from congratulating the editor for the distinctive choice of topics and for the general tone of the book.