A Precis of Mathematical Logic

By Joseph M. Bocheński; Otto Bird | Go to book overview

I
GENERAL PRINCIPLES

§ 0. INTRODUCTION

0.1. Notion and history. Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last century with the aid of an artificial notation and a rigorously deductive method. Leibniz ( 1646-1716) is generally recognized as the first mathematical logician; but it was George Boole ( 1815-1864) and Augustus De Morgan ( 1806-1878) who first presented systems in a form like those known today. Their work was taken up and furthered by C. S. Peirce ( 1839-1914), Gottlob Frege ( 1848-1925) and Giuseppe Peano ( 1858-1932), and then by Alfred North Whitehead and Bertrand Russell in their monumental work, Principia Mathematica ( 1910- 1913). Since then active schools of mathematical logic have arisen in numerous countries, especially in America, Germany, and Poland. Progress has been rapid and is still continuing.

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.

0.3. Applications. Mathematical logic has been successfully applied not only to mathematics and its foundations ( G. Frege, B. Russell, D. Hilbert, P. Bernays, H. Scholz, R. Carnap, Leśniewski, T. Skolem), but also to physics ( R. Carnap, A. Dittrich, B. Russell, C. E. Shannon, A. N. Whitehead, H. Reichenbach, P. Férier), to biology ( J. H. Woodger, A. Tarski), to psychology ( F. B. Fitch, C. G. Hempel), to law and morals

-1-

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A Precis of Mathematical Logic
Table of contents

Table of contents

  • Title Page v
  • Contents vii
  • Translator's Preface xi
  • I - GENERAL PRINCIPLES 1
  • II - THE LOGIC OF SENTENCES 9
  • III - THE LOGIC OF PREDICATES AND CLASSES 37
  • IV - THE LOGIC OF RELATIONS 66
  • Bibliography 94
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