This is a wide-ranging 2004 book about arguments for and against beliefs in God. The arguments for the belief are analysed in the first six chapters and include ontological arguments from Anselm to Gödel, the cosmological arguments of Aquinas and Leibniz, and arguments from evidence for design and miracles. The next two chapters consider arguments against belief. The last chapter examines Pascalian arguments for and against belief in God. There are discussions of Cantorian problems for omniscience, of challenges to divine (...) omnipotence, and of the compatibility of everlasting complete knowledge of the world with free-will. There are appendices that present formal proofs in a system for quantified modal logic, a theory of possible worlds, notes on Cantorian set theory, and remarks concerning non-standard hyperreal numbers. This book will be a valuable resource for philosophers of religion and theologians and will interest logicians and mathematicians as well. (shrink)

Kurt Gödel’s version of the ontological argument was shown by J. Howard Sobel to be defective, but some plausible modifications in the argument result in a version which is immune to Sobel’s objection. A definition is suggested which permits the proof of some of Godel’s axioms.

Though Frege was interested primarily in reducing mathematics to logic, he succeeded in reducing an important part of logic to mathematics by defining relations in terms of functions. By contrast, Whitehead & Russell reduced an important part of mathematics to logic by defining functions in terms of relations (using the definite description operator). We argue that there is a reason to prefer Whitehead & Russell's reduction of functions to relations over Frege's reduction of relations to functions. There is an interesting (...) system having a logic that can be properly characterized in relational but not in functional type theory. This shows that relational type theory is more general than functional type theory. The simplification offered by Church in his functional type theory is an over-simplification: one can't assimilate predication to functional application.<br>. (shrink)

A semantic embedding of quantified conditional logic in classical higher-order logic is utilized for reducing cut-elimination in the former logic to existing results for the latter logic. The presented embedding approach is adaptable to a wide range of other logics, for many of which cut-elimination is still open. However, special attention has to be payed to cut-simulation, which may render cut-elimination as a pointless criterion.

It is a commonplace that in the development of modern logic towards its actual shape at least two directions or traditions have to be distinguished. These traditions may be called, following the mo...

We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We give examples to demonstrate that these embeddings provide an effective framework for computational investigations of various non-classical logics. We report some experiments using the higher-order automated theorem prover LEO-II.