A Precis of Mathematical Logic

Description

This précis is intended both as an elementary textbook or companion on which a course of lectures for beginners may be based and as a background for the application of symbolic logic to non-mathematical subjects. In both respects, it has a certain affinity with Carnap's well-known Abriss der Logistik [Springer, Berlin, 1929], consciously so. Thus the main sections of the book deal with the calculi of propositions, of predicates, of classes, and of relations. However, while Carnap's book is based almost exclusively on the logic of Principia Mathematica, the present work aims at giving even the beginner an idea of more recent developments. Prominence is given to the symbolism of Lukasiewicz, and there are sections on Gentzen's calculus of deduction and on the formalisation of syllogistic logic and brief references to many-valued and combinatorial logics. Considerable emphasis is placed on precise syntactic and semantic definitions. A beginner might find it difficult to appreciate some of these without the guidance of an instructor. The reviewer has the following suggestions to offer for use in subsequent editions. 1) By the introduction of only minor modifications in the present book it should be possible to exhibit clearly Bernays' complete system of axioms for the lower functional calculus (restricted calculus of predicates) as given in Hilbert and Ackermann [Grundzüge der theoretischen Logik, Springer, Berlin, 1928]. 2) The brief and apparently innocuous section 12.1 in which quantified sentences are interpreted as infinite conjunctions or disjunctions really lets the devil in through a back-door. The author is of course well aware of the problems associated with the introduction of the symbol ⋯ (et cetera) into a formal calculus. However, for the sake of a beginner, it may be better either to omit this section or to discuss the difficulty in more detail. 3) The sentence following 20.11 is out of place. This is clearly a printer's error.
Summing up, this is a very useful little book. Its appearance is timely especially in view of the recent revival of interest in symbolic logic in the French-speaking world. Reviewed by A. Robinson

Innocent-Marie Bocheński's A History of Formal Logic cited in Deely's edition of John of St. Thomas (João Poinsot, O.P.)'s Tractatus de Signis

"What does the term 'mathematical logic' mean?"

p. 1 (PDF p. 9):

0.2. Logic and mathematics. Mathematical logic is called 'mathematical' because of its origin, since it has been developed particularly with the aim of examining the foundations of this science. There is moreover a certain external resemblance between its formulas and those of mathematics. Certain logicians also claim that mathematics is only a part of logic, although this opinion is far from receiving general approval. However, mathematical logic does not consider either numbers or quantities as such, but any objects whatsoever.

p. 66 (PDF p. 75): Logic of relations "Developed originally for the foundations of mathematics, it has gone beyond this science to embrace the whole of knowledge."