# Part A. Introduction, Basic Theory and Examples ## Chapter I. Model-Theoretic Logics: Background and Aims J. Barwise 1. Logics Embodying Mathematical Concepts . . . . . . . . . . . 3 2. Abstract Model Theory . . . . . . . . . . . . . . . . . . . 13 3. Conclusion .........22 ## Chapter II. Extended Logics: The General Framework H.D. Ebbinghaus 1. General Logics . . . . . . . . . . . . . . . . . . . . . . . 26 2. Examples of Principal Logics . . . . . . . . . . . . . . . . . 33 3. Comparing Logics . . . . . . . . . . . . . . . . . . . . . . 41 4. Lindström Quantifiers . . . . . . . . . . . . . . . . . . . . 49 5. Compactness and Its Neighbourhood . . . . . . . . . . . . . 59 6. Löwenheim-Skolem Properties . . . . . . . . . . . . . . . . 64 7. Interpolation and Definability . . . . . . . . . . . . . . . . 68 ## Chapter III. Characterizing Logics J. Flum 1. Lindström's Characterizations of First-Order Logic . . . . . . . 78 2. Further Characterizations of $\mathscr{L}_{\omega \omega}$. . . . . . . . . . . . . . . 91 3. Characterizing $\mathscr{L}_{\infty \omega \omega}$. . . . . . . . . . . . . . . . . . . . . 104 4. Characterizing Cardinality Quantifiers . . . . . . . . . . . . . 110 5. A Lindström-Type Theorem for Invariant Sentences . . . . . . . 115 # Part B. Finitary Languages with Additional Quantifiers ## Chapter IV. The Quantifier "There Exist Uncountably Many" and Some of Its Relatives M. Kaufmann 1. Introduction to $\mathscr{L}\left(Q_\alpha\right)$. . . . . . . . . . . . . . . . . . . . 124 2. A Framework for Reducing to First-Order Logic . . . . . . . . 127 3. $\mathscr{L}\left(Q_1\right)$ and $\mathscr{L}_{\omega_1 \omega}\left(Q_1\right)$ : Completeness and Omitting Types Theorems . 4. Filter Quantifiers Stronger Than $Q_1$ : Completeness, Compactness, and Omitting Types 5. Extensions of $\mathscr{L}\left(Q_1\right)$ by Quantifiers Asserting the Existence of Certain Uncountable Sets 6. Interpolation and Preservation Questions . . . . . . . . . . . . . 155 7. Appendix (An Elaboration of Section 2) . . . . . . . . . . . . 173 ## Chapter V. Transfer Theorems and Their Applications to Logics J. H. Schmerl 1. The Notions of Transfer and Reduction . . . . . . . . . . . . 177 2. The Classical Transfer Theorems . . . . . . . . . . . . . . . 182 3. Two-Cardinal Theorems and the Method of Identities . . . . . . 188 4. Singular Cardinal-like Structures . . . . . . . . . . . . . . . 196 5. Regular Cardinal-like Structures . . . . . . . . . . . . . . . 198 6. Self-extending Models . . . . . . . . . . . . . . . . . . . . 202 7. Final Remarks . . . . . . . . . . . . . . . . . . . . . . . 208 ## Chapter VI. Other Quantifiers: An Overview D. Mundici 1. Quantifiers from Partially Ordered Prefixes . . . . . . . . . . . 212 2. Quantifiers for Comparing Structures . . . . . . . . . . . . . 217 3. Cardinality, Equivalence, Order Quantifiers and All That . . . . . 225 4. Quantifiers from Robinson Equivalence Relations . . . . . . . . 232 ## Chapter VII. Decidability and Quantifier-Elimination A. Baudisch, D. Seese, P. Tuschik and M. Weese 1. Quantifier-Elimination . . . . . . . . . . . . . . . . . . . . 236 2. Interpretations . . . . . . . . . . . . . . . . . . . . . . . 252 3. Dense Systems . . . . . . . . . . . . . . . . . . . . . . . 259 # Part C. Infinitary Languages ## Chapter VIII. $\mathscr{L}_{\omega_1 \omega}$ and Admissible Fragments M. Nadel ### Part I. Compactness Lost . . . . . . . . . . . . . . . . . . . 272 1. Introduction to Infinitary Logics . . . . . . . . . . . . . . . 272 2. Elementary Equivalence . . . . . . . . . . . . . . . . . . . 276 3. General Model-Theoretic Properties . . . . . . . . . . . . . . 278 4. "Harder" Model Theory . . . . . . . . . . . . . . . . . . . 284 ### Part II. Compactness Regained . . . . . . . . . . . . . . . 288 5. Admissibility . . . . . . . . . . . . . . . . . . . . . . . . 288 6. General Model-Theoretic Properties with Admissibility . . . . . . 296 7. "Harder" Model Theory with Admissibility . . . . . . . . . . 304 8. Extensions of $\mathscr{L}_{\omega_1 \omega}$ by Propositional Connectives . . . . . . . . 310 9. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 316 ## Chapter IX. Larger Infinitary Languages M. A. Dickmann 1. The Infinitary Languages $\mathscr{L}_{\kappa \lambda}$ and $\mathscr{L}_{\infty \lambda}$. . . . . . . . . . . . 317 2. Basic Model Theory: Counterexamples . . . . . . . . . . . . . 326 3. Basic Model Theory: The Löwenheim-Skolem Theorems . . . . . 338 4. The Back-and-Forth Method . . . . . . . . . . . . . . . . . 348 ## Chapter X. Game Quantification Ph. G. Kolaitis 1. Infinite Strings of Quantifiers . . . . . . . . . . . . . . . . . 365 2. Projective Classes and the Approximations of the Game Formulas . 378 3. Model Theory for Game Logics . . . . . . . . . . . . . . . . 395 4. Game Quantification and Local Definability Theory . . . . . . . 400 ## Chapter XI. Applications to Algebra P. C. Eкlof 1. Universal Locally Finite Groups . . . . . . . . . . . . . . . 424 2. Subdirectly Irreducible Algebras . . . . . . . . . . . . . . 426 3. Lefschetz's Principle . . . . . . . . . . . . . . . . . . . . 428 4. Abelian Groups . . . . . . . . . . . . . . . . . . . . . . 431 5. Almost-Free Algebras . . . . . . . . . . . . . . . . . . . . 434 6. Concrete Algebraic Constructions . . . . . . . . . . . . . . . 437 7. Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . 441 # Part D. Second-Order Logic ## Chapter XII. Definable Second-Order Quantifiers J. Baldwin 1. Definable Second-Order Quantifiers . . . . . . . . . . . . . . 446 2. Only Four Second-Order Quantifiers . . . . . . . . . . . . . 451 3. Infinitary Monadic Logic and Generalized Products . . . . . . . 465 4. The Comparison of Theories . . . . . . . . . . . . . . . . . 470 5. The Classification of Theories by Interpretation of Second-Order Quantifiers . . . . . . . . . . . . . . . . 472 6. Generalizations . . . . . . . . . . . . . . . . . . 476 ## Chapter XIII. Monadic Second-Order Theories 1. Monadic Quantification . . . . . . . . . . . . . . . . . . . 479 2. The Automata and Games Decidability Technique . . . . . . . . 482 3. The Model-Theoretic Decidability Technique . . . . . . . . . . 490 4. The Undecidability Technique . . . . . . . . . . . . . . . . 496 5. Historical Remarks and Further Results . . . . . . . . . . . . 501 # Part E. Logics of Topology and Analysis ## Chapter XIV. Probability Quantifiers H. J. Keisler 1. Logic with Probability Quantifiers . . . . . . . . . . . . . . . 509 2. Completeness Theorems . . . . . . . . . . . . . . . . . . . 520 3. Model Theory . . . . . . . . . . . . . . . . . . . . . . . 530 4. Logic with Conditional Expectation Operators . . . . . . . . . 544 5. Open Questions and Research Problems . . . . . . . . . . . . 555 ## Chapter XV. Topological Model Theory M. Ziegler 1. Topological Structures . . . . . . . . . . . . . . . . . . . . 557 2. The Interpolation Theorem . . . . . . . . . . . . . . . . . . 560 3. Preservation and Definability . . . . . . . . . . . . . . . . . 565 4. The Logic $\mathscr{L}_{\omega_1 \omega}^1$. . . . . . . . . . . . . . . . . . . . . . . 568 5. Some Applications . . . . . . . . . . . . . . . . . . . . . 570 6. Other Structures . . . . . . . . . . . . . . . . . . . . . . 575 ## Chapter XVI. Borel Structures and Measure and Category Logics C. I. Steinhorn 1. Borel Model Theory . . . . . . . . . . . . . . . . . .579 2. Axiomatizability and Consequences for Category and Measure Logics . . . . . . . . . . . . . . . . . . . . . . . 586 3. Completeness Theorems . . . . . . . . . . . . . . . . . . . 591 # Part F. Advanced Topics in Abstract Model Theory ## Chapter XVII. Set-Theoretic Definability of Logics 599 J. Väänänen 1. Model-Theoretic Definability Criteria . . . . . . . . . . . . . 600 2. Set-Theoretic Definability Criteria . . . . . . . . . . . . . . . 609 3. Characterizations of Abstract Logics . . . . . . . . . . . . . . 619 4. Other Topics . . . . . . . . . . . . . . . . . . . . . . . . 630 ## Chapter XVIII. Compactness, Embeddings and Definability J. A. Makowsky 1. Compact Logics . . . . . . . . . . . . . . . . . . . . . . 648 2. The Dependence Number . . . . . . . . . . . . . . . . . . 663 3. $\mathscr{L}$-Extensions and Amalgamation . . . . . . . . . . . . . . . 670 4. Definability . . . . . . . . . . . . . . . . . . . . . . . . 685 ## Chapter XIX. Abstract Equivalence Relations J. A. Makowsky and D. MundicI 1. Logics with the Robinson Property . . . . . . . . . . . . . . 719 2. Abstract Model Theory for Enriched Structures . . . . . . . . . 728 3. Duality Between Logics and Equivalence Relations . . . . . . . 730 4. Duality Between Embedding and Equivalence Relations . . . . . 736 5. Sequences of Finite Partitions, Global and Local Back-and-Forth Games . . . . . . . . . . . . . . . . . 740 ## Chapter XX. Abstract Embedding Relations J. A. Makowsky 1. The Axiomatic Framework . . . . . . . . . . . . . . . . . . 750 2. Amalgamation . . . . . . . . . . . . . . . . . . . . . . . 759 3. $\omega$-Presentable Classes . . . . . . . . . . . . . . . . . . . . 776 # Bibliography . . . . . . . . . . . . . . . . .793 D. S. Scott, D. C. McCarty, and J. F. Horty