Systems of Formal Logic
The present work constitutes an effort to approach the subject of symbol ic logic at the elementary to intermediate level in a novel way. The book is a study of a number of systems, their methods, their rela tions, their differences. In pursuit of this
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L. H. HACKSTAFF
SYSTEMS OF FORMAL LOGIC
D. REIDEL PUBLISHING COMPANY / DORDRECHT-HOLLAND
ISBN-13: 978-94-010-3549-1 e- ISBN-13: 978-94-010-3547-7 001: 10.1007/978-94-010-3547-7
1966 Softcover reprint of the hardcover 1st edition 1966 All rights reserved No part of this book may be reproduced in any form, by print, photoprint, microfilm or any other means, without permission from the publisher
PREFACE
The present work constitutes an effort to approach the subject of symbolic logic at the elementary to intermediate level in a novel way. The book is a study of a number of systems, their methods, their relations, their differences. In pursuit of this goal, a chapter explaining basic concepts of modern logic together with the truth-table techniques of definition and proof is first set out. In Chapter 2 a kind of ur-Iogic is built up and deductions are made on the basis of its axioms and rules. This axiom system, resembling a propositional system of Hilbert and Bernays, is called P +, since it is a positive logic, i.e., a logic devoid of negation. This system serves as a basis upon which a variety of further systems are constructed, including, among others, a full classical propositional calculus, an intuitionistic system, a minimum propositional calculus, a system equivalent to that of F. B. Fitch (Chapters 3 and 6). These are developed as axiomatic systems. By means of adding independent axioms to the basic system P +, the notions of independence both for primitive functors and for axiom sets are discussed, the axiom sets for a number of such systems, e.g., Frege's propositional calculus, being shown to be non-independent. Equivalence and non-equivalence of systems are discussed in the same context. The deduction theorem is proved in Chapter 3 for all the axiomatic propositional calculi in the book. Eight propositional logics are worked out in varying detail and a number of others are remarked upon. Besides axiomatic systems, the book presents exposition and deductions in a number of systems of natural deduction (Chapters 4 and 6). It is shown how systems of natural deduction, equivalent with axiomatic systems previously developed, may be constructed, e.g. the classical propositional calculus, the calculi of intuitionism, of Fitch, etc. If it is deemed desirable to introduce systems of natural deduction prior to axiom systems, Chapter 4 has been so written that it may be introduced v
SYSTEMS OF FORMAL LOGIC
immediately following Chapter I without much disaccommodation. In Chapter 7 a functional calculus of the first order is constructed as an axiomatic system and a number of theorems are proved. The system resembles that of Russell's functional calculus of 1908, but it is restricted to one-place predicates and is built up on the basis of a propositional calculus distinct from Russell's system. In Chapter 8 this system is extended and some results of the second order functional calculus with identity are proved. In this chapter two methods of proving results in the calculus of classes are set out. Chapter 9
