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-logic 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.
Author(s): L.H. Hackstaff
Edition: Softcover reprint of the original 1st ed. 1966
Publisher: Springer
Year: 2011
Language: English
Pages: 372
Preface
Table of Contents
Chapter 1 Introduction: Some Concepts and Definitions
1.0 Arguments and Argument Forms
1.1 Symbolic Logic and its Precursors
1.2 Symbolization
1.3 Logical Functors and Their Definitions
1.4 Tests of Validity Using Truth-tables
1.5 Proof and Derivation
1.6 The Axiomatic Method
1.7, Interpreted and Uninterpreted Systems
1.8 The Hierarchy of Logical Systems
1.9 The Systems of the Present Book
1.10 Abbreviations
Chapter 2 The System P_+
2.1 Summary
2.2 Rules of Formation of P_+
2.3 Rules of Transformation of P_+
2.4 Axioms of P_+
2.5 Definitions of P_+
2.6 Deductions in P_+
Chapter 3 Standard Systems with Negation (P_LT, P_LT', P_LTF, P_PM)
3.1 Summary
3.2 Rules of Formation of P_LT
3.3 Rules of Transformation of P_LT
3.4 Axioms of P_LT
3.5 Definitions of P_LT
3.6 Deductions in P_LT
3.7 The Deduction Theorem
3.8 The System P_LT
3.9 Independence of Functors and Axioms
Chapter 4 The System P_ND. Systems of Natural Deduction
4.1 Summary
4.2 The Bases of the System PNn
4.3 Proof and Derivation Techniques in P_ND
4.4 Rules of Formation of PNn
4.5 The Structure of Proofs in PNn
4.6 Rules of Transformation of P_ND
4.7 Proofs and Theorems of the System P_ND
4.8 Theorems of the Full System P_ND
4.9 A Decision Procedure for the System P_ND
4.10 A Reduction of P_ND
Chapter 5 The Consistency and Completeness of Formal Systems
5.1 Summary
5.2 The Consistency of P_LT
5.3 The Completeness of P_LT
5.4 Metatheorems on P_+
Chapter 6 Some Non-Standard Systems of Propositional Logic
6.1 Summary
6.2 What is a Non-Standard System?
6.3 The Intuitionistic System and the Fitch Calculus (P_I and P_y)
6.4 Rules of Formation of P_I
6.5 Rules of Transformation of P_I
6.6 Axioms of P_I
6.7 Definitions of P_I
6.8 Deductions in P_I
6.9 The Propositional Logic of F.B. Fitch
6.10 The Johansson Minimum Calculus
Chapter 7 The Lower Functional Calculus
7.1 Summary and Remarks
7.2 Rules of Formation of LF_LT
7.3 Transformation of LF_LT
7.4 Axioms of LF_LT
7.5 Definitions of LF_LT
7.6 Some Applications and Illustrations
7.7 Rules of Transformation of LF_LT
7.8 Axioms of LF_LT
7.9 The Propositional Calculus and LF_LT
7.10 Deductions in LF_LT
Chapter 8 An Extension of LF_LT' and Some Theorems of the Higher Functional System. The Calculus of Classes
8.1 Summary and Modification of the Formation Rules of LF_LT
8.2 The Lower Functional Calculus with Identity
8.3 Quantification over Predicate Variables. The System 2F_LT
8.4 Abstraction and the Boolean Algebra
8.5 The Boolean Algebra and Propositional Logic
Chapter 9 The Logical Paradoxes
9.1 Self Membership
9.2 The Russell Paradox
9.3 Order Distinctions, Levels of Language, and the Semantic Paradoxes
9.4 The Consistency of LF_LT
9.5 The Decision Problem
9.6 Consistency and Decision in Higher Functional Systems
Chapter 10 Non-Standard Functional Systems
10.1 Summary
10.2 Intuitionistic and Johansson Functional Logics
10.3 The Fitch Functional Calculus of the First Order with Identity (LFi;)
Bibliography
Index
