This book is one of the clearest, most comprehensive and rigorous introductions to modern symbolic logic available in any language. Professor Carnap, a world authority on symbolic logic, develops the subject from elementary concepts and simple exercises through the construction and analysis of a number of relatively complex logical languages. He then considers, in great detail, the application of symbolic logic to the clarification and axiomatization of various theories in mathematics, physics, and biology.
Such topics as the nature and use of constants and variables, predicates, sentential connectives, truth-tables, universal and existential sentences, definitions, identity, isomorphism, syntactical and semantical systems and the relations between them, the system of types, varieties of relations, linear order, special operators, structures and cardinal numbers, descriptions, finite and infinite concepts, continuity, thing languages, coordinate languages, axiom systems for set theory, arithmetic, geometry, space-time topology, biological concepts, and many other subjects, are covered in detail. The logic of relations is given a particularly extensive treatment. Hundreds of problems, examples, and exercises are included to give students practice in the techniques of symbolic logic and their usage.
PART ONE System of symbolic logic Chapter A. The simple language A 1. The problem of symbolic logic a. The purpose of symbolic language b. The development of symbolic logic 2 Individual constants and predicates a. Individual constants and predicates b. Sentential constants c. Illustrative predicates 3 Sentential connectives a. Descriptive and logical signs b. Connective signs c. Omission of parentheses d. Exercises 4. Truth-tables a. Truth-tables b. Truth-conditions and meaning 5. L-concepts a. Tautologies b. Range and L-truth 6. L-implication and L-equivalence a. L-implication and L-equivalence b. Content c. Classes of sentences d. Examples and exercises 7. Sentential variables a. Variables and sentential formulas b. Sentential variables 8. Sentential formulas that are tautologies a. Conditional formulas that are tautologies b. Interchangeability c. Biconditional formulas that are tautologies d. Derivations 9. Universal and existential sentences a. Individual variables and quantifiers b. Multiple quantification c. Universal conditionals d. Translation from the word-langage 10. Predicate variables a. Predicate variables b. Intensions and extensions 11. Value-assignments 12. Substitutions a. Substitutions for sentential variables b. Substitutions for individual variables c. Substitutions for predicate variables d. Theorems on substitutions e. Example and exercises 13. Theorems on quantifiers 14. L-true formulas with quantifiers a. L-true conditionals b. L-true biconditionals c. Exercises 15. Definitions a. Interchangeability b. Definitions c. Examples 16. Predicates of higher levels a. Predicates and predicate variables of different levels b. Raising levels c. Examples and exercises 17. Identity. Cardinal numbers a. Identity b. Examples and exercises c. Cardinal numbers 18. Functors a. Functors. Domains of a relation b. Conditions permitting the introduction of functors 19. Ismorphism Chapter B. The language B 20. Semantical and syntactical systems 21. Rules of formation for language B a. The language B b. The system of types c. Russell's antinomy 21. Rules of formation for language B-continued d. Sentential formulas and sentences in B e. Definitions in B 22. Rules of transformation for language B a. Primitive sentence schemata b. Explanatory notes on the separate primitive sentences c. Rules of inference 23. Proofs and derivations in language B a. Proofs b. Derivations 24. Theorems on provability and derivability in language B a. General theorems for B b. Interchangeability 25. The semantical system for language B a. Value-assignments and evaluations b. Rules of designation c. Truth 26. Relations between syntactical and semantical systems a. Interpretation of a language b. On the possibility of a formalization of syntax and semantics Chapter C. The extended language C 27. The language C 28. Compound predicate expressions a. Predicate expressions b. Universality c. Class terminology d. Exercises 29. Identity. Extensionality a. Identity b. Regarding the types of logical constants c. Extensionality 30. Relative product. Powers of relations a. Relative product b. Powers of relations c. Supplementary remarks 31. Various kinds of relations a. Representations of relations b. "Symmetry, transitivity, reflexivity" c. Theorems about relations d. Linear order: series and simple order e. One-oneness 32. "Additional logical predicates, functors and connectives" a. The null class and the universal class b. Union class and intersection class c. Connections between relations and classes d. Theorems e. Enumeration classes 33. The ?-operator a. The ?-operator b. Rule for the ?-operator c. Definitions with the help of ?-expressions d. The R's of b 34. "Equivalence classes, structures, cardinal numbers" a. Equivalence relations and equivalence classes b. Structures c. Cardinal numbers d. Structural properties 35. Individual descriptions a. Descriptions b. Relational descriptions 36. Heredity and ancestral relations a. Heredity b. Ancestral relations c. R-families 37. Finite and infinite a. Progressions b. Sum and predecessor relation c. Inductive cardinal numbers d. Reflexive classes e. Assumption of infinity 38. Continuity a. "Well-ordered relations, dense relations, rational orders" b. Dedekind continuity and Cantor continuity PART TWO Application of symbolic logic Chapter D. Forms and methods of the construction of languages 39. Thing languages a. Things and their slices b. Three forms of the thing language; language form I c. Language form II d. Language form III 40. Coordinate languages a. Coordinate language with natural numbers b. Recursive definitions c. Coordinate language with integers d. Real numbers 41. Quantitative concepts a. Quantitative concepts in thing languages b. Formulation of laws c. Quantitative concepts in coordinate languages 42. The axiomatic method a. Axioms and theorems b. Formalization and symbolization; interpretations and models c. "Consitency, completeness, monomorphism" d. The explicity concept e. Concerning the axiom systems (ASs) in Part Two of this book Chapter E. Axiom systems (ASs) for set theory and arithmetic 43. AS for set theory a. The Zermelo-Fraenkel AS b. The axiom of restriction &nbs 53. AS involving biological concepts a. Division and fusion b. "Hierarchies, cells, organisms" 54. AS for kinship relations a. Biological concepts of kinship b. Legal concepts of kinship Appendix 55. Problems in the application of symbolic logic a. Set theory and arithmetic b. Geometry c. Physics d. Biology 56. Bibliography 57. General guide to the literature Index Symbols of the symbolic language of the metalanguage
