Abstract

The two main accomplishments of this thesis are that it provides the first adequate semantics for Hilbert's epsilon-operator and that it describes a general semantics for term forming operators more flexible than any in the literature. ;The epsilon-operator was introduced by David Hilbert in the 1920s as a term forming operator in first order logic. The semantics so far available for epsilon has been designed for classical two-valued logic, and has required that additional extensionality assumptions be made. This thesis provides complete semantics for epsilon in classical extensional, classical non-extensional, Boolean valued, and intuitionistic first order systems. The natural step to generalizing the technique used in the epsilon case to get a general theory of term forming operators which handles the non-extensional and non-classical cases is then taken. ;The thesis proceeds as follows. Chapter One gives a historical discussion of term forming operators. A brief, self-contained presentation of the untyped lambda-calculus, which illustrates the inevitable differences between lambda and any possible operator in first order logic, follows. A chapter is devoted to solving the syntactical difficulties involved in introducing a variable binding term forming operator to standard languages for first order logic. The semantics for epsilon, and in the intuitionistic case also for another of Hilbert's creatures, tau, takes up the next several chapters. The discussion includes several new completeness and soundness results, and some new results about the extra strength these operators add to intuitionistic logic, including some new independence results. The final chapter includes an argument to the effect that the results earlier in the thesis show that we need a more general theory of term forming operators than any in the literature, and indicates the shape such a theory should take