Alfred Tarski . Life and Logic Reviewed by Hourya Benis Sinaceur

Alfred Tarski (1901–1983) is one of the two greatest logicians of the twentieth century, the other being Kurt Gödel (1906–1978). Each began his career in Europe, respectively in Warsaw and Vienna, and came to America shortly before the Second World War. In contrast to the otherworldly Gödel, Tarski was ambitious and practical. He strove for, and succeeded at, building a school of logic at the University of California, Berkeley, that attracted students and distinguished researchers from all over the world. Tarski was the leader of the " semantic turn " in mathematical logic. This means that he achieved a shift from a view focused on formal systems, axioms, and rules of deduction to a view focusing on the relations between formal systems and their possible interpretations by usual mathematical theories such as real numbers or Cartesian geometry. Hence he gave precise definitions of semantic concepts that had been used informally before. The most important of those concepts are truth, satisfiability, and definability of a formula; logical consequence; and model. Tarski was also the champion of the trend towards reconstructing logical notions by mathematical means. For instance, his decision method for elementary algebra and geometry is indeed a generalization of a Sturm's algorithm for counting the real roots of a polynomial. Tarski was eager to bring to the fore new connections between mathematics and logic and to show how mathematized concepts of logic can help to solve mathematical problems. Thus, Tarski initiated the shift from the foundational aims for which various branches of modern logic were originally developed to heuristic aims for which a new branch would turn out to be especially efficient. Namely, Tarski is the father of model theory, the results and tools of which are nowadays commonly used in various mathematical disciplines (algebra, analysis, geometry, computer science, etc.). Tarski's views and achievements have also changed the way we think about the nature, scope, and aims of logic. The rich and detailed biography of Tarski written by Anita and Solomon Fefer-man show us the roots and the full extent of this change. The Fefermans' biography is an enthralling success story of a self-confident, enterprising, untiring, and entrepreneurial scientist, and a rich and scrupulous account of the numerous achievements accomplished by this powerful logician and his colleagues in philosophy of logic, semantics, set theory, decision procedures, universal algebra, algebraic logic, axiomatic geometry, topology, and model theory. It is particularly remarkable …