The prehistory of infinitary logic: 1885-1955 (Q2769634)

Infinitary logic admits either infinitely long formulas or rules of inference with infinitely long premises. The paper under review surveys the stages of development of infinitary logic of the first kind up to the mid-1950s when its continuous development began with L. Henkin's look for an extended first-order logic analogous to A. Tarski's \(\omega\)-dimensional cylindric algebras. A. Tarski himself used subequently infinitely long formulas in logic despite of his earlier sceptical attitude.NEWLINENEWLINENEWLINEThe earliest precursors of this tradition can be found in the beginnings of the algebra of logic, some hints already in \textit{G.\ Boole}'s pioneering ``Mathematical analysis of logic'' [Cambridge and London (1847)], but especially in C. S. Peirce's and E. Schröder's definition of quantifiers as infinite conjunctions or disjunctions. This tradition ended with the work of L. Löwenheim and T. Skolem. Some of its ideas were, however, taken up by D. Hilbert (1904/05) and C. I. Lewis (1918).NEWLINENEWLINENEWLINEA different tradition is seen in conceptions starting with some infinitary aspects of L. Wittgenstein's idea that every proposition is a truth-function of atomic propositions. These truth-functions operate, in effect, on arguments whose number is variable (and perhaps infinite). The infinitary nature of Wittgenstein's proposal was recognized by F. P. Ramsey and refuted by K. Gödel.NEWLINENEWLINENEWLINEFurther conceptions presented are the anti-Skolemian (``Against the tradition'', p. 116) infinitary logic of E. Zermelo (1930/35), O. Helmer's suggestion (1938), R. Carnap's syntactical approach towards transfinite junctives (1943), A. Robinson's use of infinitary results to show the existence of non-Archimedean ordered fields (1951), and Russian contributions by P. S. Novikov and D. A. Bochvar (1939/43).NEWLINENEWLINENEWLINEIn sum, the author succeeds in showing the richness of the prehistory of infinitary logic.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00023].