M. Fitting 
Types, Tableaus, and Godel’s God [PDF ebook] 

Godel’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 Godel argument is fully formalized. Objections to the Godel argument are examined, including one due to Howard Sobel showing Godel’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 Godel’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).