Types, Tableaus, and Gödel’s God

1. Front Matter

   Pages i-xv

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2. Classical Logic

   1. Front Matter

      Pages 1-1

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   2. Classical Logic—Syntax

      • Melvin Fitting

      Pages 3-10

   3. Classical Logic—Semantics

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      Pages 11-32

   4. Classical Logic—Basic Tableaus

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      Pages 33-42

   5. Soundness And Completeness

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      Pages 43-68

   6. Equality

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      Pages 69-76

   7. Extensionality

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      Pages 77-79

3. Modal Logic

   1. Front Matter

      Pages 81-81

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   2. Modal Logic Syntax And Semantics

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      Pages 83-104

   3. Modal Tableaus

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      Pages 105-114

   4. Miscellaneous Matters

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      Pages 115-130

4. Ontological Arguments

   1. Front Matter

      Pages 131-131

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   2. Gödel’s Argument, Background

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      Pages 133-143

   3. Gödel’s Argument, Formally

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      Pages 145-172

5. Back Matter

   Pages 173-181

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Gödel's modal ontological argument is the centrepiece of an extensive examination of intensional logic. First, classical type theory is presented semantically, tableau rules for it are introduced, and the Prawitz/Takahashi completeness proof is given. Then modal machinery is added, semantically and through tableau rules, to produce a modified version of Montague/Gallin intensional logic. Extensionality, rigidity, equality, identity, and definite descriptions are investigated. Finally, various ontological proofs for the existence of God are discussed informally, and the Gödel argument is fully formalized. Objections to the Gödel argument are examined, including one due to Howard Sobel showing Gödel's assumptions are so strong that the modal logic collapses. It is shown that this argument depends critically on whether properties are understood intensionally or extensionally.
Parts of the book are mathematical, parts philosophical. A reader interested in (modal) type theory can safelyskip ontological issues, just as one interested in Gödel's argument can omit the more mathematical portions, such as the completeness proof for tableaus. There should be something for everybody (and perhaps everything for somebody).

• corpus
• identity
• issue
• logic
• modal logic
• semantic
• semantics
• sound

• Lehman College and the Graduate Center, City University of New York, USA

  Melvin Fitting

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