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Introduction to Symbolic Logic and Its Applications


Introduction to Symbolic Logic and Its Applications Cover


Synopses & Reviews

Publisher Comments:

A clear, comprehensive, and rigorous treatment develops the subject from elementary concepts to the construction and analysis of relatively complex logical languages. It then considers the application of symbolic logic to the clarification and axiomatization of theories in mathematics, physics, and biology. Hundreds of problems, examples, and exercises. 1958 edition.


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

Table of Contents

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


  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


  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


Symbols of the symbolic language of the metalanguage

Product Details

Carnap, Rudolf
Dover Publications
Logic, symbolic and mathematical
Modern mathematics
General Mathematics
General Philosophy
symbolic language
introduction to logic
Philosophy of logic
Edition Description:
Trade Paper
Publication Date:
8.5 x 5.38 in 0.63 lb

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Humanities » Philosophy » General
Humanities » Philosophy » Logic
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Clear, comprehensive, intermediate introduction to logical languages, applications of symbolic logic to physics, mathematics, biology.
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