Developments in logic: Carnap, Gödel, and Tarski

Analytic philosophy and modern logic are intimately connected, both historically and systematically. Thinkers such as Frege, Russell, and Wittgenstein were major contributors to the early development of both; and the fruitful use of modern logic in addressing philosophical problems was, and still is, definitive for large parts of the analytic tradition. More specifically, Frege's analysis of the concept of number, Russell's theory of descriptions, and Wittgenstein's notion of tautology have long been seen as paradigmatic pieces of philosophy in this tradition. This close connection remained beyond what is now often called "early analytic philosophy", i.e., the tradition's first phase. In the present chapter I will consider three thinkers who played equally important and formative roles in analytic philosophy's second phase, the period from the 1920s to the 1950s: Rudolf Carnap, Kurt Gödel, and Alfred Tarski. Undoubtedly, Gödel and Tarski were two of the greatest logicians of the twentieth century, indeed of all time. Their influence on mathematical logic can hardly be exaggerated. However, they also exerted a significant influence on philosophy, especially analytic philosophy, as we will see. Carnap, in turn, is widely recognized a