No . 5 , “ Problems of Theoretical Logic ”

In the period 1914–1922, a significant amount of work on logic was carried out in Göttingen. Beginning in 1914, Hilbert and his students, in particular Heinrich Behmann, were studying Whitehead and Russell’s Principia Mathematica, which resulted in Behmann’s dissertation (1918, see Mancosu 1999). In the Fall of 1917, Paul Bernays joined Hilbert in Göttingen and from then on worked as his assistant, concentrating on logic and the foundations of mathematics. Hilbert taught several courses on these topics in Göttingen, and in each case, Bernays was actively involved in the preparation of the content itself as well as in the preparation of typewritten notes, which Bernays often amended and extended. The first course in this series, Principles of Mathematics (1917/18), was analyzed by Sieg (1999); it contained the first formal presentation of a firstorder calculus and formed the basis of Hilbert and Ackermann’s 1928 textbook Principles of Theoretical Logic (Hilbert and Ackermann, 1928). In 1920, this was followed by Logical Calculus, and in 1921/22 and 1922/23 by two courses on the foundations of mathematics (the latter co-taught with Bernays). At around the same time, Bernays (1918, 1926) studied the propositional calculus in detail and carried out proofs of completeness and decidability; these results were already contained in less explicit form in the 1917/18 lecture notes (see Zach 1999). Around 1920, Behmann began to work on the decision problem, and give a positive solution for firstand second-order monadic logic (1922). This result also follows from a paper by Löwenheim (1915), which, however, seems to have been unknown in the Hilbert school until the mid-1920s. Bernays himself contributed to this line of research, extending a result on a decidable class of first-order formulas which Moses Schönfinkel had obtained around 1923 (see Bernays and Schönfinkel 1928). During this time, he also continued to improve on the axiomatization of propositional and first-order logic which Hilbert and he began in 1917. Bernays’s “Problems of Theoretical Logic” is a survey of this early work on formal logic in Hilbert’s school. The occasion of this paper was the Versammlung Deutscher Philologen und Schulmänner in Göttingen in September 1927, a biennial meeting of German high-school teachers. As would be appropriate for the occasion, Bernays’s paper is not directed at specialists, but at a philosophically educated lay audience. It is intended to dispel the apparently then prevalent prejudice that mathematical logic is mere ‘idle play.’ To this end, Bernays shows how the machinery of propositional connectives and quantifiers