Universal logic / Ross Brady.

Format:

Book

Author/Creator:

Brady, Ross, 1943-

Series:

CSLI lecture notes ; no. 109.

CSLI lecture notes ; no. 109

Language:

English

Subjects (All):

Logic, Symbolic and mathematical.

Physical Description:

xii, 346 pages : illustrations ; 24 cm.

Place of Publication:

Stanford, Calif. : CSLI Publications, [2006]

Summary:

Throughout the twentieth century, the classical logic of Frege and Russell dominated the field of formal logic. But, as Ross Brady argues here, persistent logical paradoxes may be better solved by a new type of weak relevant logic. Brady's universal logic, intended to analyze naive set/class theories, is modeled on the properties of set-theoretic containment and driven by an inference connective understood as "meaning containment."

Universal Logic begins with an overview of classical and strong relevant logic and discusses the limitations of both in analyzing many historically significant paradoxes, including Russell's Paradox and the Heterologicality Paradox. By formulating a weak quantified relevant logic, this is the first text to demonstrate how such set-theoretic and semantic paradoxes can be solved in a systematic way that is conceptualized independently of the paradoxes themselves. An important part of this system is the two-sorted treatment of classicality, where the Law of Excluded Middle and Disjunctive Syllogism apply to classical sentences.

Contents:

1 Philosophical Support for the Logic DJ[superscript d]Q 1

1.1 Classical Logic 1

1.2 Some Problems with Classical Logic 2

1.3 Relevant Logic 4

1.4 Some Problems with Relevant Logic 11

1.5 Setting up the Semantics of Meaning Containment 13

1.6 Constraints on DJ[superscript d]Q 28

1.7 The Addition of Classical Sentences 39

1.8 The Solution of Paradoxes 45

2 Semantics for the Logic DJ[superscript d]Q 51

2.1 The Routley-Meyer Truth-Functional Semantics 51

2.2 The Semantics of Meaning Containment 62

2.3 Quantified Content Semantics 65

3 Proof Theory for the Logic DJ[superscript d]Q 81

3.1 Natural Deduction Systems 81

3.2 Gentzen Systems 93

4 Properties of the Logic DJ[superscript d]Q 141

4.1 Rules and Derived Rules 141

4.2 Meta-Completeness Properties 155

4.3 Depth Relevance 162

4.4 The Addition of Classical Formulae 165

5 Philosophical Support for the Theories of Classes and Sets 171

5.1 The Naive Theory of Classes 172

5.2 Classicality and the Theory of Sets 180

5.3 Indeterminability 190

5.4 Consistent and Dialectical Approaches 191

5.5 Non-ad hoc Solution to the Set-Theoretic Paradoxes 194

6 Simple Consistency of the Class Theory 197

6.1 The Simple Consistency Proof 197

6.2 The Practical Maximality of TN[superscript d]Q 240

6.3 Non-triviality of Dialectical Class Theory 242

7 Simple Consistency of the Class Theory Combined with the Set Theory and Other Theories 247

7.1 Combining the Class Theory and the Set Theory 247

7.2 The Simple Consistency of the Combined Theory CST 255

7.3 The Simple Consistency of CST, Together with Other Mathematical Theories 266

8 Simple Consistency of the Higher-Order Predicate Logic 275

8.1 The Predicate Logic 276

8.2 Simple Consistency of the Predicate Logic 282

8.3 Solution of the Semantic Paradoxes 293

9 Formal Development of the Axiomatic Theories 299

9.1 The Class Theory 299

9.2 The Set Theory 306

9.3 The Combined Theory of Classes and Sets 311

9.4 Arithmetical Theories 312

9.5 The Higher-Order Predicate Logic 318.

Notes:

Includes bibliographical references (pages 327-337) and index.

ISBN:

1575862557

1575862565

OCLC:

44117970

Online:

Publisher description