Mathematical Logic
Textbooks in Mathematics Series

This graduate level text on first-order logic highlights the importance of this area  as well as the abundance of results and some applications. The best-known of textbooks originated in an earlier era, and despite  frequent updating by their authors, they reflect a general view and a particular approach that is less adequate today. The addition of "metatheory" clarifies that this is not a textbook in  which the emphasis is on the basics such as formalizing English sentences and learning the use of  one or another calculus. This textbook takes a fresh look at the current state of  first-order  logic, and integrates newer results with a reevaluated stock of earlier ones.

Chapter 1. Language and interpretation of first-order logic

Chapter 2. Proof systems: sequent calculus, tableaux, axiomatic calculus

Chapter 3. Propositional logic (as a restriction of rst-order logic), truth tables, disjunctive and conjunctive normal forms, prenex normal forms

Chapter 4. Resolution calculus and its applications; equivalence of proof calculi

Chapter 5. Core metatheorems I: Soundness and completeness proofs (including separate proofs for propositional logic and di erent constructions for the quanti cational case)

Chapter 6. Core metatheorems II: Compactness, upward and downward L•owenheim{Skolem theorems, Lindstr •om's theorem

Chapter 7. Core metatheorems III: Craig's interpolation theorem, Robinson's consistency theorem, Beth's de nability theorem

Chapter 8. Core metatheorems IV: Undecidability, the impact of the metatheorems

Chapter 9. Expressibility and de nability (variations on the set of logical connectives and the set of logical operators; choosing and modifying the non-logical vocabulary)

Chapter 10. Algebraizations (Boolean algebra for propositional logic, cylindric algebra and polyadic algebra for rst-order logic)

Chapter 11. Mathematical theories within first-order logic (varieties: semi-groups, groups, etc.; ordered structures; arithmetic; set theory)

Chapter 12. Decidability (propositional logic, classes of quanti cational formulas speci ed by quanti er prex, by shape of formulas)

Chapter 13. Complexity (satis ability problem, validity problem)

Chapter 14. Categorial view (category of proofs, quanti ers as adjoint functors)