This is a description of TPS, a theorem proving system for classical type theory (Church’s typed λ-calculus). TPS has been designed to be a general research tool for manipulating wffs of first- and higher-order logic, and searching for proofs of such wffs interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. Examples of theorems which TPS can prove completely automatically are given to illustrate certain aspects of TPS’s behavior and problems of theorem proving in higher-order logic. 7

Consideration of the question of meaning in the framework of linguistics often requires an allusion to sets and other higher-order notions. The traditional approach to representing and reasoning about meaning in a computational setting has been to use knowledge representation systems that are either based on first-order logic or that use mechanisms whose formal justifications are to be provided after the fact. In this paper we shall consider the use of a higher-order logic for this task. We first present a version of definite clauses (positive Horn clauses) that is based on this logic. Predicate and function variables may occur in such clauses and the terms in the language are the typed -terms. Such term structures have a richness that may be exploited in representing meanings. We also describe a higher-order logic programming language, called Prolog, which represents programs as higher-order definite clauses and interprets them using a depth-first interpreter. A virtue of this languag...

This is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed l-calculus). TPS can be used interactively or automatically, or in a combination of these modes. An important feature of TPS is the ability to translate between expansion proofs and natural deduction proofs. CATEGORY: Demonstration 1. Introduction This presentation is a demonstration of TPS, a theorem proving system for classical type theory (Church's typed 3 l-calculus [14]) which has been under development at Carnegie Mellon University for a number of years. TPS is based on an approach to automated theorem proving called the mating method [2], which is essentially the same as the connection method developed independently by Bibel [13]. The mating method does not require reduction to clausal form. TPS handles two sorts of proofs, natural deduction proofs and expansion proofs. Natural deduction proofs are human-readable formal proofs. An example of such a proof which was produced aut...