Jean-Pierre Jouannaud has deeply influenced, and is still influencing, research in Informatics, through the many important results he has produced in various research fields and through the generations of scholars he has educated. He has played a leading role in field of rewriting and its technology, advancing the research areas of unification, rewriting and completion modulo, conditional rewriting, termination proofs, modular properties, and automated proofs by induction in rewrite theories. Following his strong interest for algebraic specification languages and their efficient implementation, Jean-Pierre Jouannaud has contributed to the design, semantics and implementation of OBJ2 and has recently augmented the Maude rewriting logic system with an essential feature: membership equational logic. He was a member of the CNU (National University Council) and a member of the CNRS national committee. He is now heading with great success the computer science laboratory at the Ecole Polytechnique thus showing his exceptional qualities as a research team manager and supervisor of students.

Probabilistic techniques are increasingly being employed in computer programs and systems because they can increase efficiency in sequential algorithms, enable otherwise nonfunctional distribution applications, and allow quantification of risk and safety in general. This makes operational models of how they work, and logics for reasoning about them, extremely important.

Fact finding in judicial proceedings is a dynamic process. This collection of papers considers whether computational methods or other formal logical methods developed in disciplines such as artificial intelligence, decision theory, and probability theory can facilitate the study and management of dynamic evidentiary and inferential processes in litigation. The papers gathered here have several epicenters, including (i) the dynamics of judicial proof, (ii) the relationship between artificial intelligence or formal analysis and "common sense," (iii) the logic of factual inference, including (a) the relationship between causality and inference and (b) the relationship between language and factual inference, (iv) the logic of discovery, including the role of abduction and serendipity in the process of investigation and proof of factual matters, and (v) the relationship between decision and inference.

During the university reform of the 1970s, the classical Faculty of Science of the venerable Ludwig-Maximilians-Universitat in Munich was divided into five smaller faculties. One was for mathematics, the others for physics, chemistry and pharmaceutics, biology, and the earth sciences. Nevertheless, in order to maintain an exchange of ideas between the various disciplines and so as not to permit the complete undermining of the original notion of "universitas,,,l the Carl-Friedrich-von-Siemens Foundation periodically invites the pro­ fessors from the former Faculty of Science to a luncheon gathering. These are working luncheons during which recent developments in the various disciplines are presented by means of short talks. The motivation for such talks does not come, in the majority of cases, from the respective subject itself, but from another discipline that is loosely affiliated with it. In this way, the controversy over the modern methods used in the proof of the Four-Color Theorem had also spread to disciplines outside of mathematics. I, as a trained algebraic topologist, was asked to comment on this. Naturally, I was acquainted with the Four-Color 1 A Latin word meaning the whole of something, a collective entirety. Vll viii Preface Problem but, up to that point, had never intensively studied it. As an outsider,2 I dove into the material, not so much to achieve any scientific progress with it but to make this already achieved objective more understandable.

This volume is presented as a companion study to my translation of Galileo's MS 27, Galileo's Logical Treatises, which contains Galileo's appropriated questions on Aristotle's Posterior Analytics - a work only recently transcribed from the Latin autograph. Its purpose is to acquaint an English-reading audience with the teaching in those treatises. This is basically a sixteenth-century logic of discovery and of proof about which little is known in the present day, yet one that arguably guided the most significant research program of the seventeenth century. Despite its historical and systematic importance, the teaching is difficult to explain to the modern reader. Part of the problem stems from the fragmentary nature of the manuscript in which it is preserved, part from the contents of the teaching itself, which requires a considerable propadeutic for its comprehension. A word of explanation is thus required to set out the structure of the volume and to detail the editorial decisions that underlie its organization. Two major manuscript studies have advanced the cause of scholarship on Galileo within the past two decades. The first relates to Galileo's experimental activity at Padua prior to his discoveries with the telescope that led to the publication of his Sidereus nuncius in 1610. Much of this activity has been uncovered by Stillman Drake in analyses of manuscript fragments associated with the composition of Galileo's Two New Sciences, fragments now bound in a codex identified as MS 72 in the collection of Galileiana at the Biblioteca Nazionale Centrale in Florence.

During the last few years, we have seen quite spectacular progress in the area of approximation algorithms: for several fundamental optimization problems we now actually know matching upper and lower bounds for their approximability. This textbook-like tutorial is a coherent and essentially self-contained presentation of the enormous recent progress facilitated by the interplay between the theory of probabilistically checkable proofs and aproximation algorithms. The basic concepts, methods, and results are presented in a unified way to provide a smooth introduction for newcomers. These lectures are particularly useful for advanced courses or reading groups on the topic.

This book was originally intended to be the second edition of the book "Beweis­ theorie" (Grundlehren der mathematischen Wissenschaften, Band 103, Springer 1960), but in fact has been completely rewritten. As well as classical predicate logic we also treat intuitionistic predicate logic. The sentential calculus properties of classical formal and semiformal systems are treated using positive and negative parts of formulas as in the book "Beweistheorie". In a similar way we use right and left parts of formulas for intuitionistic predicate logic. We introduce the theory of functionals of finite types in order to present the Gi:idel interpretation of pure number theory. Instead of ramified type theory, type-free logic and the associated formalization of parts of analysis which we treated in the book "Beweistheorie", we have developed simple classical type theory and predicative analysis in a systematic way. Finally we have given consistency proofs for systems of lI~-analysis following the work of G. Takeuti. In order to do this we have introduced a constni'ctive system of notation for ordinals which goes far beyond the notation system in "Beweistheorie".

This book is a collection of essays that offer original logical and philosophical investigations into the century-long endeavor to understand paradoxes. It bridges the gap between the two most prominent traditions in the analysis of paradoxes: the truth-theoretic and proof-theoretic approaches. The truth-theoretic tradition stems from Alfred Tarski's solution to the semantic paradoxes, while the proof-theoretic tradition dates back to Dag Prawitz's analysis of set-theoretic paradoxes in terms of structural proof theory. Rather than viewing these traditions as competing perspectives, this volume advocates for the idea that a deeper understanding of paradoxes requires insights from both truth-theoretic and proof-theoretic conceptions of language and meaning. Although the collection does not aim to be exhaustive, it seeks to highlight the vast scope of the subject and its deep connections to various fields of inquiry. The essays are organized into four sections: the first focuses on methodology, the second and third examine paradoxes through the conventional lenses of logical investigation—semantics and syntax—, and the fourth presents a selection of paradoxes that extend beyond the interplay between syntax and semantics, exploring other dimensions of human rationality.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge.

Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.

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