Programming metalogics with a fixpoint type

This University of Cambridge Computer Laboratory Technical Report is a revised version of my Ph.D. thesis. The report is essentially the same as my original thesis which was completed in July 1991, with corrections and alterations as suggested by my Ph.D. examiners, Dr. J.M.E. Hyland, Department of Pure Mathematics and Mathematical Statistics, Cambridge, England and Prof. E. Moggi, Dip. di Matematica, Univ. di Genova, Italy. I have also made some modi cations which I personally think improves the presentation of my original thesis. Any errors which remain are my sole responsibility. Summary of