ISBN:
9789401035477
Language:
English
Pages:
Online-Ressource (372p)
,
online resource
Edition:
Springer eBook Collection. Humanities, Social Sciences and Law
Parallel Title:
Erscheint auch als
Parallel Title:
Erscheint auch als
Parallel Title:
Erscheint auch als
Keywords:
Philosophy (General)
;
Logic
Abstract:
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 -- 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+ -- 3 Standard Systems with Negation (PLT, PLT’, PLTF, PPM) -- 3.1 Summary -- 3.2 Rules of Formation of PLT -- 3.3 Rules of Transformation of PLT -- 3.4 Axioms of PLT -- 3.5 Definitions of PLT -- 3.6 Deductions in PLT -- 3.7 The Deduction Theorem -- 3.8 The System PLT’ -- 3.9 Independence of Functors and Axioms -- 4 The System PND. Systems of Natural Deduction -- 4.1 Summary -- 4.2 The Bases of the System PND -- 4.3 Proof and Derivation Techniques in PND -- 4.4 Rules of Formation of PND -- 4.5 The Structure of Proofs in PND -- 4.6 Rules of Transformation of PND -- 4.7 Proofs and Theorems of the System PND -- 4.8 Theorems of the Full System PND -- 4.9 A Decision Procedure for the System PND -- 4.10 A Reduction of PND -- 5 The Consistency and Completeness of Formal Systems -- 5.1 Summary -- 5.2 The Consistency of PLT’ -- 5.3 The Completeness of PLT’ -- 5.4 Metatheorems on P+ -- 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 (PI and PF) -- 6.4 Rules of Formation of PI -- 6.5 Rules of Transformation of PI -- 6.6 Axioms of PI -- 6.7 Definitions of PI -- 6.8 Deductions in PI -- 6.9 The Propositional Logic of F.B. Fitch -- 6.10 The Johansson Minimum Calculus -- 7 The Lower Functional Calculus -- 7.1 Summary and Remarks -- 7.2 Rules of Formation of LFLT’ -- 7.3 Transformation of LFLT’ -- 7.4 Axioms of LFLT’ -- 7.5 Definitions of LFLT’ -- 7.6 Some Applications and Illustrations -- 7.7 Rules of Transformation of LFLT’ -- 7.8 Axioms of LFLT’ -- 7.9 The Propositional Calculus and LFLT’ -- 7.10 Deductions in LFLT’ -- 8 An Extension of LFLT’ and Some Theorems of the Higher Functional System. The Calculus of Classes -- 8.1 Summary and Modification of the Formation Rules of LFLT’ -- 8.2 The Lower Functional Calculus with Identity -- 8.3 Quantification over Predicate Variables. The System 2FLT’= -- 8.4 Abstraction and the Boolean Algebra -- 8.5 The Boolean Algebra and Propositional Logic -- 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 LFLT’ -- 9.5 The Decision Problem -- 9.6 Consistency and Decision in Higher Functional Systems -- 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 (LFFF=).
Abstract:
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 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 Ber nays, is called P +, since it is a positive logic, i. e. , a logic devoid of nega tion. This system serves as a basis upon which a variety of further sys tems are constructed, including, among others, a full classical proposi tional calculus, an intuitionistic system, a minimum propositional calcu lus, 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.
Description / Table of Contents:
1 Introduction: Some Concepts and Definitions1.0 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 -- 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+ -- 3 Standard Systems with Negation (PLT, PLT’, PLTF, PPM) -- 3.1 Summary -- 3.2 Rules of Formation of PLT -- 3.3 Rules of Transformation of PLT -- 3.4 Axioms of PLT -- 3.5 Definitions of PLT -- 3.6 Deductions in PLT -- 3.7 The Deduction Theorem -- 3.8 The System PLT’ -- 3.9 Independence of Functors and Axioms -- 4 The System PND. Systems of Natural Deduction -- 4.1 Summary -- 4.2 The Bases of the System PND -- 4.3 Proof and Derivation Techniques in PND -- 4.4 Rules of Formation of PND -- 4.5 The Structure of Proofs in PND -- 4.6 Rules of Transformation of PND -- 4.7 Proofs and Theorems of the System PND -- 4.8 Theorems of the Full System PND -- 4.9 A Decision Procedure for the System PND -- 4.10 A Reduction of PND -- 5 The Consistency and Completeness of Formal Systems -- 5.1 Summary -- 5.2 The Consistency of PLT’ -- 5.3 The Completeness of PLT’ -- 5.4 Metatheorems on P+ -- 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 (PI and PF) -- 6.4 Rules of Formation of PI -- 6.5 Rules of Transformation of PI -- 6.6 Axioms of PI -- 6.7 Definitions of PI -- 6.8 Deductions in PI -- 6.9 The Propositional Logic of F.B. Fitch -- 6.10 The Johansson Minimum Calculus -- 7 The Lower Functional Calculus -- 7.1 Summary and Remarks -- 7.2 Rules of Formation of LFLT’ -- 7.3 Transformation of LFLT’ -- 7.4 Axioms of LFLT’ -- 7.5 Definitions of LFLT’ -- 7.6 Some Applications and Illustrations -- 7.7 Rules of Transformation of LFLT’ -- 7.8 Axioms of LFLT’ -- 7.9 The Propositional Calculus and LFLT’ -- 7.10 Deductions in LFLT’ -- 8 An Extension of LFLT’ and Some Theorems of the Higher Functional System. The Calculus of Classes -- 8.1 Summary and Modification of the Formation Rules of LFLT’ -- 8.2 The Lower Functional Calculus with Identity -- 8.3 Quantification over Predicate Variables. The System 2FLT’= -- 8.4 Abstraction and the Boolean Algebra -- 8.5 The Boolean Algebra and Propositional Logic -- 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 LFLT’ -- 9.5 The Decision Problem -- 9.6 Consistency and Decision in Higher Functional Systems -- 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 (LFFF=).
DOI:
10.1007/978-94-010-3547-7
URL:
Volltext
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