This book is addressed primarily to researchers specializing in mathemat­ ical logic. It may also be of interest to students completing a Masters Degree in mathematics and desiring to embark on research in logic, as well as to teachers at universities and high schools, mathematicians in general, or philosophers wishing to gain a more rigorous conception of deductive reasoning. The material stems from lectures read from 1962 to 1968 at the Faculte des Sciences de Paris and since 1969 at the Universities of Provence and Paris-VI. The only prerequisites demanded of the reader are elementary combinatorial theory and set theory. We lay emphasis on the semantic aspect of logic rather than on syntax; in other words, we are concerned with the connection between formulas and the multirelations, or models, which satisfy them. In this context considerable importance attaches to the theory of relations, which yields a novel approach and algebraization of many concepts of logic. The present two-volume edition considerably widens the scope of the original [French] one-volume edition (1967: Relation, Formule logique, Compacite, Completude). The new Volume 1 (1971: Relation et Formule logique) reproduces the old Chapters 1, 2, 3, 4, 5 and 8, redivided as follows: Word, formula (Chapter 1), Connection (Chapter 2), Relation, operator (Chapter 3), Free formula (Chapter 4), Logicalformula,denumer­ able-model theorem (L6wenheim-Skolem) (Chapter 5), Completeness theorem (G6del-Herbrand) and Interpolation theorem (Craig-Lyndon) (Chapter 6), Interpretability of relations (Chapter 7).

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1/Local Isomorphism and Logical Formula; Logical Restriction Theorem.- 1.1. (k,p)-Isomorphism.- 1.2. (k,p)-Equivalence.- 1.3. Characteristic of a Logical Formula. Relations Between (k,p) -Isomorphism and Logical Formula.- 1.4. Logical Extension and Logical Restriction; Logical Restriction Theorem.- 1.5. Examples of Finitely-Axiomatizable and Non-Finitely-Axiomatizable Multirelations.- 1.6. (k,p)-Interpretability.- 1.7. Homogeneous and Logically Homogeneous Multirelations.- 1.8. Rigid and Logically Rigid Multirelations.- Exercises.- 2/Logical Convergence; Compactness, Omission and Interpretability Theorems.- 2.1. Logical Convergence.- 2.2. Compactness Theorem.- 2.3. Omission Theorem.- 2.4. Interpretability Theorem.- 2.5. Every Injective Logical Operator is Invertible.- Exercise.- 3/Elimination of Quantifiers.- 3.1. Absolute Eliminant.- 3.2. (k,p)-Eliminant.- 3.3. Elimination Algorithms for the Chain of Rational Numbers and the Chain of Natural Numbers.- 3.4. Positive Dense Sum; Elimination of Quantifiers over the Sum of Rational or Real Numbers.- 3.5. Positive Discrete Divisible Sum; Elimination of Quantifiers over the Sum of Natural Numbers.- 3.6. Real Field; Elimination of Quantifiers over the Sum and Product of Algebraic Numbers or Real Numbers.- Exercises.- 4/Extension Theorems.- 4.1. Restrictive Sequence; (k,p)-Isomorphism and (k,p)-Identimorphism.- 4.2. Application to Logical Restriction.- 4.3. Projection Filter.- 4.4. Logical Extension Theorems.- 4.5. Theorem on Common Logical Extensions.- 4.6. Logical Morphism and Logical Embedding.- Exercises.- 5/Theories and Axiom Systems.- 5.1. Theory: Consistency; Intersection of Theories.- 5.1 Axiom System. Class of Models; Union-Theory, Finitely-Axiomatizable Theory, Saturated Theory.- 5.3. Complement of a Theory.- 5.4. Categoricity.- 5.5. Model-Saturated Theory.- Exercises.- 6/Pseudo-Logical Class; Interpretability of Theories; Expansion of a Theory; Axiomatizability.- 6.1. Pseudo-Logical Class.- 6.2. Interpretability of Theories.- 6.3. Canonical Expansion, Semantic Expansion, and Other Expansions.- 6.4. Axiomatizable Multirelations and Theories.- 6.5. Free Expansion.- Exercises.- 7/Ultraproduct.- 7.1. Family of Multirelations, Ultrafilter, Induced Logical Equivalence Class; Ultraproduct and Ultrapower; Maximal Case.- 7.2. Logical Equivalence Implies the Existence of Isomorphic Ultrapowers.- 7.3. Characterization of Logical Classes.- 7.4. Normal Ultraproduct; Definitions and Examples.- 7.5. Normal Ultraproducts and Logical Equivalence.- Exercises.- 8/Forcing.- 8.1. Generic Predicate; System: (+)-Forced and (?)-Forced Formulas.- 8.2. Elementary Properties.- 8.3. Forcing with Constraints.- 8.4. General Relation.- 8.5. Forcing and Deduction; Theory Forced by a Generic Predicate.- Exercises.- 9/Isomorphisms and Equivalences in Relation to the Calculus of Infinitely Long Formulas with Finite Quantifiers.- 9.1. ?-Isomorphism and ?-Equivalence.- 9.2. ?-Isomorphism and ?-Equivalence; Karpian Families.- 9.3. Automorphic Rank of a Multirelation.- 9.4. Multirelations with Denumerable Bases and ?-Isomorphisms.- 9.5. ?-Extension and ?-Interpretability.- 9.6. Infinite Logical Calculi and their Relation to Local Isomorphisms and Equivalences.- Proof of Lemmas Needed to Prove J. Robinson's Theorem.- Closure of a Relation.- References.