Before his death in March, 1976, A. H. Lightstone delivered the manu­ script for this book to Plenum Press. Because he died before the editorial work on the manuscript was completed, I agreed (in the fall of 1976) to serve as a surrogate author and to see the project through to completion. I have changed the manuscript as little as possible, altering certain passages to correct oversights. But the alterations are minor; this is Lightstone's book. H. B. Enderton vii Preface This is a treatment of the predicate calculus in a form that serves as a foundation for nonstandard analysis. Classically, the predicates and variables of the predicate calculus are kept distinct, inasmuch as no variable is also a predicate; moreover, each predicate is assigned an order, a unique natural number that indicates the length of each tuple to which the predicate can be prefixed. These restrictions are dropped here, in order to develop a flexible, expressive language capable of exploiting the potential of nonstandard analysis. To assist the reader in grasping the basic ideas of logic, we begin in Part I by presenting the propositional calculus and statement systems. This provides a relatively simple setting in which to grapple with the some­ times foreign ideas of mathematical logic. These ideas are repeated in Part II, where the predicate calculus and semantical systems are studied.

I. Statement Systems and Propositional Calculus.- 1. Statement Systems.- 2. Propositional Calculus.- 3. Provable Wffs.- 4. Substitution Theorems.- 5. Duality.- 6. Deducibility and Completeness.- II. Semantical Systems and Predicate Calculus.- 7. Semantical Systems.- 8. Predicate Calculus.- 9. Provable Wffs.- 10. Substitution Theorems.- 11. Duality.- 12. Deducibility and Completeness.- III. Applications.- 13. Nonstandard Analysis.- 14. Normal Semantical Systems.- 15. Axiomatic Set Theory.- 16. Complete Theories.- Symbol Index.