I. Statement Systems and Propositional Calculus.- 1. Statement Systems.- 1.1. Statement Systems.- 1.2. Language of a Statement System.- 1.3. Names for Swffs.- 2. Propositional Calculus.- 2.1. Well-Formed Formulas.- 2.2. Parentheses.- 2.3. Main Connective of Wffs.- 2.4. Names for Wffs; Principal Connective of Name.- 2.5. Valuations.- 3. Provable Wffs.- 3.1. ?-Interpreters.- 3.2. True Wffs.- 3.3. Proofs and Provable Wffs.- 3.4. Rules of Inference.- 3.5. Equivalent Wffs.- 4. Substitution Theorems.- 4.1. Subwffs, Components, and Wff-Builders.- 4.2. Substitution Theorem for Wffs.- 5. Duality.- 5.1. Normal Form.- 5.2. Syntactical Transforms.- 5.3. Normal Transforms.- 5.4. Duality.- 6. Deducibility and Completeness.- 6.1. More Provable Wffs.- 6.2. Conjunctive Normal Form.- 6.3. Completeness.- 6.4. Deducibility.- 6.5. Consistent Sets and Contradictory Sets.- 6.6. Maximal-Consistent Sets.- 6.7. Strong Completeness Theorem.- II. Semantical Systems and Predicate Calculus.- 7. Semantical Systems.- 7.1. Relational Systems.- 7.2. Semantical Systems.- 7.3. Language of a Semantical System.- 7.4. Extensions; Elementary Extensions.- 8. Predicate Calculus.- 8.1. Well-Formed Formulas.- 8.2. Parentheses.- 8.3. Main Connective of Wffs.- 8.4. Names for Wffs; Principal Connective of Name.- 8.5. Syntactical Transforms.- 8.6. Interchange and Substitution Transforms.- 8.7. Valuations.- 9. Provable Wffs.- 9.1. ?-Interpreters.- 9.2. True Wffs.- 9.3. Proofs and Provable Wffs.- 9.4. Equivalent Wffs.- 9.5. Rules of Inference.- 9.6. A Fact about the Interchange Transform.- 10. Substitution Theorems.- 10.1. Subwffs, Components, and Wff-Builders.- 10.2. Substitution Theorem for Wffs.- 11. Duality.- 11.1. Normal Form.- 11.2. Normal Transforms.- 11.3. Duality.- 11.4. Prenex Normal Form.- 12. Deducibility and Completeness.- 12.1. Deducibility.- 12.2. Consistent Sets and Contradictory Sets.- 12.3. Strong Completeness Theorem.- 12.4. Maximal-Consistent Sets.- 12.5. ?-Complete Sets.- 12.6. Proof of the Strong Completeness Theorem.- III. Applications.- 13. Nonstandard Analysis.- 13.1. Extended Natural Number System.- 13.2. Extended Real Number System.- 13.3. Properties of ?.- 13.4. Paradoxes.- 13.5. The Limit Concept.- 13.6. Continuity; Uniform Continuity.- 13.7. Principles of Permanence.- 14. Normal Semantical Systems.- 14.1. Equality Relations.- 14.2. Normal Semantical Systems.- 14.3. Löwenheim-Skolem Theorem.- 14.4. Theories.- 15. Axiomatic Set Theory.- 15.1. Introduction.- 15.2. Axiom of Extensionality.- 15.3. Axiom Scheme of Replacement.- 15.4. Axiom of Power Set.- 15.5. Axiom of Sum Set.- 15.6. Axiom of Infinity.- 15.7. Nonstandard Set Theory.- 15.8. Axiom of Regularity.- 15.9. Axiom of Choice.- 16. Complete Theories.- 16.1. Vaught's Test.- 16.2. Diagrammatic Sets.- 16.3. Simplifying the Concept of a Model.- 16.4. Robinson's Test.- Symbol Index.

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.