1. On Mathematical Proof (Polish) (1962), by Witold A Pogorzelski and Jerzy Słupecki.
The Journal of Symbolic Logic 31 (2) (1966), 284.
The aim of this book, which is written in an easy and readable style, is to acquaint the reader with the propositional and predicate calculi and with the main ideas of the theory of deductive systems (in the sense of Tarski). The table of contents gives an adequate description of the topics presented in the book: Chapter I: §1. Some notions and theorems of set theory, §2. Propositional calculus, §3. The calculus of quantifiers, §4. Identity; Chapter II: §1. On the axiomatic method, §2. The notion of proof, §3. Fundamental properties of proofs; Chapter III: §1. The notion of system, §2. Axiomatizable sets, §3. Consistent systems, §4. Complete sets, §5. Independent sets, §6. On the notion of interpretation.
1.2. Review by: Tadeusz Batóg.
Studia Logica: An International Journal for Symbolic Logic 14 (1963), 348-350.
This book provides an elementary introduction to the general methodology of deductive systems and is designed - as indicted in the introduction by its authors - for the reader whose knowledge of mathematics does not go beyond high school. It consists of the following three chapters: I. Introduction II. Proof and its properties, III. Mathematical systems and their essential characteristics.
Les Études philosophiques, Nouvelle Série 18 (4) (1963), 480-481.
Professor Słupecki (Wrocław, Poland), student of the two founders and heads of the Warsaw School of Logic (Polish School): Jan Łukasiewicz (1878-1956) and Stanisław Leśniewski (1886-1939), is a logician well known not only to his colleagues but also to mathematicians. He has just published a manual of mathematical logic and set theory, written in collaboration with his disciple, Borkowski. Inaccessible to the [French] reader of Philosophical Studies because of the language in which it is written, this manual is mentioned here only because of the problems it poses and which can also be found in France. The manual is intended for use by students of philosophy. It introduces them to the calculus of propositions, to the calculus of quantifiers, in the indispensable elements of the calculus of relations, and in the general theory of sets to which more than half the book is devoted. Should students of philosophy be taught mathematical logic and set theory? This is the first problem with the manual of Słupecki and Borkowski poses, solving it at the same time. For without hesitation it is no longer possible when one becomes aware, following the developments of the authors to the end, of the fact that one of the eternal philosophical problems around in the Middle Ages which led to its quarrel was that of the existence of sets. Now, if this is the case - and the fact is beyond doubt - a certain knowledge of the theory of sets, clearly exposed according to the methods and techniques peculiar to contemporary mathematics, proves indispensable to the student of philosophy who does not want to remain unfamiliar with one of the most fundamental philosophical problems. On the other hand, the manual of Słupecki and Borkowski does not universally apply the axiomatic method used in works of this kind, but that of the natural deduction given pride of place by Gentzen and Jaskowski. This poses a second problem. Whenever mathematical logic is taught to students of philosophy or to students of mathematics, which of these two methods should be followed? As a result of long years of university teaching, the authors of this book provide evidence for the great didactic value of natural deduction.
Mathematical reviews MR0228323.
The book is primarily intended for students of philosophy. The first part is devoted to classical propositional calculus and first order functional calculus. These systems are presented using a natural deduction method. The metatheory of propositional calculus is studied in some detail, while for functional calculus the authors limit themselves to stating some results of independence, consistency and completeness. The second part is devoted to a very elementary exposition of the theory of sets. Finally, the appendix (30 pages) briefly analyzes the paradoxes of set theory, semantic categories, type theory, logical definitions of certain set concepts, axiomatic set theory, arithmetic theory of sets and the philosophical aspects of the concept of a set.
Mathematical reviews MR0476367.
This elementary introduction to logic and set theory is intended to help mathematics students in the teachers' training colleges in Poland. It comprises only some basic notions and subjects that are considered now to be a rather traditional part of logic and set theory. The book is divided into two parts. Part I includes an informal description of logic and set theory ... . Part II presents logic as a formalized deductive system and contains a detailed description of the propositional and predicate calculi.