Provability in Logic

Description

Most of the investigations contained in this essay were made for a course in logic, which I gave during the spring term of 1955 at the University of Stockholm. My intention at that time was to present a technique of logical proof that would be easier to master than those usually encountered in textbooks on logic. Thus, the essay may be regarded as having a kind of pedagogical aim. It is hoped that this aim is not overshadowed by the technical character of my exposition. Professor Anders Wedberg of the University of Stockholm has kindly read an early draft of the essay, and I am greatly indebted to him for his valuable criticism. Mr. Robert Miller has obligingly read the text with an eye to proper English usage. I dedicate this booklet to my wife. Without her aid and encouragement it would never have been written—in any event, it would never have been prepared for the press. In this essay I shall describe a 'Gentzen-type' proof procedure which is effective in this sense and which supplies the need for simplicity. I shall prove its completeness by means of arguments which are new in some respects and which involve a new turn to the notion of validity. Elementary logic may be extended with various modal notions. Such an extension has raised well-known difficulties connected with the interpretation of the resulting modal logic—in fact, a fully satisfactory interpretation is yet to be found. In the last sections I intend to give a new interpretation which yields an acceptable foundation for modal logic. I shall conclude the essay with a few comments on provability in modal logic.