This volume contains seventeen essays in the history of modern mathematical logic. The first nine are concerned with the completeness of various logical calculi. The second five essays deal with the completeness of classical first-order predicate logic. One essay deals with the history of Cantor's definition of set, another with the set-theoretical reduction of the concept of relation, and a final essay is devoted to a survey of various meanings of the concept of completeness of formalized deductive theories. The essays were first presented in the national conferences of the Thematic Group for the History of Logic organized by the Department of Logic of the Polish Academy of Sciences in 1966-1971. The Advanced Reasoning Forum is pleased to make available this exact reprint of the original volume first published in 1973 by the Polish Academy of Sciences and the Department of Logic of Jagiellonian University, edited by Stanislaw J. Surma. * * * * The essays are: 1. Emil Post's doctoral dissertation (Stanislaw J. Surma); 2. A historical survey of the significant methods of proving post's theorem about the completeness of the classical propositional calculus (Stanislaw J. Surma); 3. A survey of the results and methods of investigations of the equivalential propositional calculus (Stanislaw J. Surma); 4. A uniform method of proof of the completeness theorem for the equivalential propositional calculus and for some of its extensions (Stanislaw J. Surma); 5. Kolmogorov and Glivenko's papers about intuitionistic logic (Jacek K. Kabzinski); 6. Jaskowski's matrix criterion for the intuitionistic propositional calculus (Stanislaw J. Surma); 7. Axiomatization of the implicational Gödel's matrices by Kalmar's method (Andrzej Wronski); 8. A contribution to the history of the investigations into the intermediate propositional calculi (Andrzej Wronski); 9. On Ackermann's rigorous implication (Jan Wolenski); 10. Kurt Gödel's doctoral dissertation (Jan Zygmunt); 11. A survey of the methods of proof of the Gödel-Malcev's completeness theorem (Jan Zygmunt); 12. The concept of the Lindenbaum algebra: its genesis (Stanislaw J. Surma); 13. On the old and new methods of interpreting quantifiers (Andrzej Wronski); 14. L. Rieger's logical achievement (Wladyslaw Szczech); 15. The development of Cantor's definition of set (Jerzy Perzanowski); 16. On the origins of the set-theoretical concept of relation (Piotr Kossowski); 17. A survey of various concepts of completeness of the deductive theories (Stanislaw J. Surma).