Introduction to Symbolic Logic and Its Applications

Clear, comprehensive, intermediate introduction to logical languages, applications of symbolic logic to physics, mathematics, biology.

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
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  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