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Universal Logic cover

Universal Logic

Ross Brady

The classical logic of Frege and Russell dominated formal logic in the 20th century. But a new type of weak relevant logic may prove itself to be better equipped to present new solutions to persisting paradoxes.

Universal Logic conceptualizes a new weak quantified relevant logic where the main inference connective is understood as `meaning containment'. This logic is intended to analyze naïve set/class theories. The volume begins with an overview of classical logic and relevant logic, and discusses the limitations of both types of logic in analyzing certain paradoxes. A summary on the history of logic segues into the author's introduction of his new logic modeled on the properties of set-theoretic containment. This book is the first to demonstrate how the main set-theoretic and semantic paradoxes can be solved in a systematic way, which is conceptualized independently of the paradoxes themselves.

Ross Brady is a senior lecturer in philosophy at La Trobe University in Australia.

Contents

  • Preface
  • 1 Philosophical Support for the Logic DJdQ
    • 1.1 Classical Logic
    • 1.2 Some Problems with Classical Logic
    • 1.3 Relevant Logic
    • 1.4 Some Problems with Relevant Logic
    • 1.5 Setting up the Semantics of Meaning and Containment
    • 1.6 Constraints on DJsupdQ
    • 1.7 The Addition of Classical Sentences
    • 1.8 The Solution of Paradoxes

  • 2 Semantics for the Logic DJdQ
    • 2.1 The Routley-Meyer Truth Functional Semantics
    • 2.2 The Semantics of Meaning Containment
    • 2.3 Quantified Content Semantics

  • 3 Proof for the Logic DJdQ
    • 3.1 Natural Deduction Systems
    • 3.2 Gentzen Systems

  • 4 Properties of the Logic DJdQ
    • 4.1 Rules and Derived Rules
    • 4.2 Meta-Completeness Properties
    • 4.3 Depth Relevance
    • 4.4 The Addition of Classical Formulae

  • 5 Philosophical Support for the Theories of Classes and Sets
    • 5.1 The Native Theory of Classes
    • 5.2 Classicality and the Theory of Sets
    • 5.3 Indeterminability
    • 5.4 Consistent Dialectical Approaches
    • 5.5 Non-ad hoc Solution to the Set-Theoretic Paradoxes

  • 6 Simple Consistency of the Class Theory
    • 6.1 The Simple Consistency Proof
    • 6.2 The Practical Maximality of TNdQ
    • 6.3 Non-triviality of Dialectical Class Theory

  • 7 Simple Consistency of the Class Theory Combined with the Set Theory and Other Theories
    • 7.1 Combining the Class Theory and Set Theory
    • 7.2 The Simple Consistency of the Combined Theory CST
    • 7.3 The Simple Consistency of CST, Together with Other Mathematical Theories

  • 8 Simple Consistency of the Higher-Order Predicate Logic
    • 8.1 The Predicate Logic
    • 8.2 Simple Consistency of the Predicate Logic
    • 8.3 Solution of the Semantic Paradoxes

  • 9 Formal Development of the Axiomatic Theories
    • 9.1 The Class Theory
    • 9.2 The Set Theory
    • 9.3 The Combined Theory of Classes and Sets
    • 9.4 Arithmetical Theories
    • 9.5 The Higher-Order Predicate Logic

  • 10 Conclusion
  • References
  • Index

2/1/2006

ISBN (Paperback): 1575862565 (9781575862569)
ISBN (Electronic): 1575869039 (9781575869032)

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