SOUND AND COMPLETE TRANSLATIONSFROM SORTED HIGHER - ORDER LOGICINTO SORTED FIRST - ORDER LOGICManfred

Extending existing calculi by sorts is a strong means for improving the deductive power of rst-order theorem provers. Since many mathematical facts can be more easily expressed in higher-order logic { aside the greater power of higher-order logic in principle {, it is desirable to transfer the advantages of sorts in the rst-order case to the higher-order case. One possible method for automating higher-order logic is the translation of problem formulations into rst-order logic and the usage of rst-order theorem provers. For a certain class of problems this method can compete with proving theorems directly in higher-order logic as for instance with the TPS theorem prover of Peter Andrews or with the Nuprl proof development environment of Robert Constable. There are translations from unsorted higher-order logic based on Church's simple theory of types into many-sorted rst-order logic, which are sound and complete with respect to a Henkin-style general models semantics. In this paper we extend corresponding translations to translations of order-sorted higher-order logic into order-sorted rst-order logic, thus we are able to utilize corresponding rst-order theorem prover for proving higher-order theorems. We do not use any-expressions, therefore we have to add so-called comprehension axioms, which a priori make the procedure well-suited only for essentially rst-order theorems. However, in practical applications of mathematics many theorems are essentially rst-order and as it seems to be the case, the comprehension axioms can be mastered too.