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.
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.
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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.
Product Details
| ISBN-13: | 9780486143491 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 06/14/2012 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 272 |
| File size: | 3 MB |
Read an Excerpt
Introduction to Symbolic Logic and its Applications
By Rudolf Carnap, William H. Meyer, John Wilkinson
Dover Publications, Inc.
Copyright © 1958 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-14349-1
CHAPTER 1
The simple language A
1. THE PROBLEM OF SYMBOLIC LOGIC
1a. The purpose of symbolic language. Symbolic logic (also called mathematical logic or logistic) is the modern form of logic developed in the last hundred years. This book presents a system of symbolic logic, together with illustrations of its use. Such a system is not a theory (i.e. a system of assertions about objects), but a language (i.e. a system of signs and of rules for their use). We will so construct this symbolic language that into it can be translated the sentences of any given theory about any objects whatever, provided only that some signs of the language have received determinate interpretations such that the signs serve to designate the basic concepts of the theory in question. So long as we remain in the domain of pure logic (i.e. so long as we are concerned with building this language, and not with its application and interpretation respecting a given theory), the signs of our language remain uninterpreted. Strictly speaking, what we construct is not a language but a schema or skeleton of a language: out of this schema we can produce at need a proper language (conceived as an instrument of communication) by interpretation of certain signs.
Part Two of this book sees a variety of such interpretations, and the symbolic formulation (axiomatically, for the most part) of theories from various domains of science. All this is applied logic.Part One of the book attends to pure logic: here we describe the structure of the symbolic language by specifying its rules. In the present Chapter A, the first of the three chapters comprising Part One, we describe a simple symbolic language A containing the following sorts of signs (to be explained later): sentential constants and variables, individual constants and variables, predicate constants and variables of various levels and types, functor constants and variables, sentential connectives, and quantifiers. The third chapter, Chapter C, presents a more comprehensive language C. In Chapter B a symbolic language B is represented both as a syntactical system and as a semantical system.
If certain scientific elements—concepts, theories, assertions, derivations, and the like—are to be analyzed logically, often the best procedure is to translate them into the symbolic language. In this language, in contrast to ordinary word-language, we have signs that are unambiguous and formulations that are exact: in this language, therefore, the purity and correctness of a derivation can be tested with greater ease and accuracy. A derivation is counted as pure when it utilizes no other presuppositions than those specifically enumerated. A derivation in a word-language often involves presuppositions which were not made explicitly, but which entered unnoticed. Numerous examples of this are afforded by the history of geometry, especially in connection with attempts to derive Euclid's axiom of parallels from his other axioms.
A further advantage of using artificial symbols in place of words lies in the brevity and perspicuity of the symbolic formulas. Frequently a sentence that requires many lines in a word-language (and whose perspicuity is consequently slight) can be represented symbolically in a line or less. Brevity and perspicuity facilitate manipulation and comparison and inference to an extraordinary degree. The twin advantages of exactness and brevity appear also in the usual mathematical notations. Had the mathematician been confined to words and denied the use of numerals and other special symbols, the development of mathematics to its present high level would have been not merely more difficult, but psychologically impossible. To appreciate this point, one need only attempt to translate into the word-language e.g. so elementary a formula as "(x + y)3 = x3 + 3x2y + 3xy2 + y3" ("The third power of the sum of two arbitrary numbers equals the sum of the following summands: ..."). The symbolic method gives mathematics an advantage in its investigation of numbers, numerical functions, etc.; symbolic logic seeks this same advantage in full generality for its treatment of concepts of any kind.
In the course of constructing our symbolic language systems, it frequently happens that a new precisely-defined concept is introduced in place of one which is familiar but insufficiently precise. Such a new concept is called an explicatum of the old one, and its introduction an explication. (The concept to be explicated is sometimes called the explicandum.) E.g. the concept of L-truth (to be defined technically later (5b) on the basis of exact rules) is an explicatum of the concept of logical or necessary truth, which is defined with insufficient exactness despite its frequent occurrence in philosophy and traditional logic. Again, the concept of the inductive cardinal numbers (37c) is an explicatum for the concept of finite number that has been widely used in mathematics, logic and philosophy, but never exactly defined prior to Frege. [For a more complete exposition of the methods of explication and the requirements an adequate explicatum must meet, see Carnap [Probability], Chapter I.]
1b. The development of symbolic logic. Symbolic logic was founded around the middle of the last century and carried on into the present more by mathematicians than philosophers (cf. references to the literature, 57). The reason for this lies in the historical fact that during the past century mathematicians became increasingly more conscious of the need to reexamine and reconstruct the foundations of the whole edifice of mathematics. Finding the traditional (i.e. aristotelian-scholastic) logic a totally inadequate instrument for this purpose, the mathematicians set about to develop a system of logic that was at once more appropriate, more accurate and more comprehensive.
The resulting new symbolic logic (especially in the systems of Frege, Whitehead-Russell, and Hilbert) clearly evinced a suitability to the first task set it, viz. to provide a basis for the reconstruction of mathematics (arithmetic, analysis, function theory, and the infinitesimal calculus). Further, in its logic of relations the new symbolic logic developed an abstract theory of arbitrary order-forms, and thereby created the possibility of representing and logically analyzing theories in which relations play an essential role, e.g. the various geometries, physical theories (especially in reference to space and time), epistemology and, latterly, even certain branches of biology. This development was a particularly significant advance beyond traditional logic. For traditional logic had neglected relations almost completely and hence proved entirely useless in connection with the axiomatic method (e.g. in geometry) that has become so important in recent decades. Still another merit of symbolic logic—minor, but nonetheless valuable—is that it achieved the complete solution of certain contradictions, viz. the so-called logical antinomies (cf. 21c), whose analysis and elimination were beyond the reach of the old logic.
For literature on matters treated here, see the references, 57. In the text of this book, citations of the literature are phrased with the help of abbreviated titles in square brackets; cf. the bibliography, 56. ('[P.M.]' is used without author names for: Whitehead and Russell, Principia mathematical; and similarly for several of my own works.)
Regarding terminology. In the domain of symbolic logic the expressions "algebraic logic", "algebra of logic", etc., were employed at an earlier date but are no longer customary today. In addition to "symbolic logic" and "mathematical logic", the designation "logistics" is often used, especially on the European continent; it is short and permits the formulation of the adjective "logistic". The word "logistics" originally signified the art of reckoning, and was proposed by Couturat, Itelson and Lalande independently in 1904 as a name for symbolic logic (according to the assertion of Ziehen, Lehrbuch der Logik, p. 173, note 1, and Meinong, Die Stellung der Gegenstandstheorie, p. 115).
Concerning results of the new symbolic logic in comparison with traditional logic, cf. Russell [World], Chap. II; Carnap [Neue Logik]; Menger [Logic]. On the special importance of the logic of relations, cf. Russell, ibid.
Concerning the reconstruction of mathematics on the basis of the new logic, cf. the basic older works: Frege [Grundlagen] and [Grundgesetze]; Peano [Formulaire]; as chief work, [P.M.]; and also Russell [Introduction]; a more recent work: Hilbert and Bernays [Grundlagen]; for an easy presentation of the basic ideas: Carnap, "Die Mathematik als Zweig der Logik", Blätter f. dt. Philos. 4, 1930; Carnap, "Die logizistische Grundlegung der Mathematik", Erkenntnis 2, 1931.
2. INDIVIDUAL CONSTANTS AND PREDICATES
2a. Individual constants and predicates. The theoretical treatment of any domain of objects consists in setting up sentences concerning the objects of the domain (sentences ascribing certain properties and relations to the objects in question), and in establishing rules according to which other sentences can be derived from those given. The basic objects treated of in a given language system are called the individuals of the system; and their totality, the domain of individuals (briefly, the domain) of the system. This domain is sometimes called the universe of discourse. To form sentences concerning the individuals of a given domain there must first of all be available in the language two kinds of signs: 1. names for the individuals of the domain—we call these individual constants; 2. designations for the properties and relations predicated of the individuals—we call these predicates.
For individual constants we use the letters 'a ', 'b', 'c', 'd', 'e'. E.g. if our language were to be applied to the domain comprising the heavenly bodies, 'a' might perhaps designate the sun, 'b' the moon, etc. Again, if the domain were a certain group of people, 'a' might be taken as an abbreviation for 'Charles Smith', 'b' for 'John Miller', etc. So long as our considerations are purely logical, we shall not trouble ourselves as to what special domain of individuals our language might be applied, and what particular individuals of that domain might be designated by 'a', 'b', etc. It is only when we move away from pure logic (i.e. from consideration of the skeleton language to be constructed in what follows) that we speak of the interpretation of the separate individual constants and predicates. We do this last e.g. in the second part of this book, where several systems are presented as applications; we do it also in the first part, in connection with illustrative examples.
For predicates we use the letters 'P', 'Q', 'R', 'S', 'T'. In connection with illustrative applications, we also use for predicates various letter groups with first letter capitalized (cf. the examples in 2c below); these letter groups are based on words of the word-language.
E.g. in a certain application 'P' might designate the property Spherical. [I prefer this mode of expression to the more elaborate turn of phrase "the property of being spherical". Similarly, I write "the property Prime Number", "the property Odd", etc. Again, I use "the class Spherical" in place of "the class of spherical individuals"; and analogously, "the class Blue", etc. And again, I say "the relation Greater" rather than "the relation that obtains between x and y when x is greater than y"; and similarly "the relation Similar", "the relation Father", etc.] Now suppose that, in addition to designating the property Spherical by 'P', we take 'a' to designate the sun and 'b' to designate the moon. Then in our symbolic language we write the sentence 'P (a)' for "the sun is spherical". Similarly, 'P (b)' is the translation into our symbolic language of the English sentence "the moon is spherical". To give a symbolic translation of the sentence "the sun is greater than the moon", we need a sign for the relation Greater. Taking 'R' for this relation, we write 'R (a,b)' as our symbolic translation of "the sun is greater than the moon". Again, if a and b are persons (i.e. 'a' and 'b' are interpreted as personal names), and 'S' is taken to designate the relation Similar, then 'S (a,b)' means "a is similar to b". Likewise, we can translate the sentence "a is jealous of b with respect to c" into 'T (a, b, c)' if we use 'T' to designate the triadic or three-place relation Jealous.
In the sentences 'P (a)' and 'R (b,c)', the 'a' and 'b' and 'c' are called argument-expressions. Further, 'b' is said to stand in the first argument-position, 'c' in the second. We say 'P' is a one-place (or monadic) predicate, and 'R' a two-place (or dyadic) predicate. Generally, a predicate is said to be n-adic (or n place, or of degree n) in case it has n argument-positions. Predicates of degree higher than two can be introduced whenever they are needed in connection with a given domain of objects. We say that 'P (a)' is a sentence-completion or full-sentence of the predicate 'P'; similarly, 'R (b, c)' is a sentence-completion of 'R'. The examples given here illustrate the use of single letters as predicates and argument-expressions, but not such a use of letter groups (this occurs in 2c) and compound expressions. When single letters are so used we usually omit parentheses and commas, and write simply 'Pa', 'Rab', 'Tabc', etc.
Regarding terminology. 1. In ordinary word language there is no word which comprehends both properties and relations. Since such a word would serve a useful purpose, let us agree in what follows that the word "attribute" shall have this sense. Thus a one-place attribute is a property, and a two-place (or a many-place) attribute is a relation. 2. Similarly, it is useful to have a comprehensive term for the designations of one- and many-place attributes. For this, let us follow Hilbert and use the word "predicate ". (Heretofore, this word has been confined mostly to properties or to designations of them, and has not included many-place attributes or predicates.) Thus a one-place predicate is a sign for a one-place attribute (i.e. for a property); and in general an n -place predicate is a sign for an n -place attribute. 3. Let us always distinguish clearly between signs and what is designated. Failure to observe this distinction has in the past occasioned much confusion in logic and in philosophy generally (cf. [Syntax] 42). In speaking about an expression, let us always put the expression in quotation marks or use some special designation for it, e.g. a German letter as in 21a. We make but one exception to this practice: we omit quotation marks in case the expression stands on a line either alone or with a designating number or letter; see e.g. our enumeration of the formulas in T8-2. Suppose e.g. 'Pa' is taken as a symbolic translation of "a is old"; then we say: "P (but not: 'P') is a one-place attribute, viz. the property Old; this attribute is designated by a one-place predicate 'P'". Similarly, we say: "the two-place relation R exists between such and such persons", "the two-place predicate 'R' occurs in such and such a sentence". And similarly: "the individual a ...", "the name 'a' ...".
2b. Sentential constants. It is often burdensome to work with sentences that are entirely written out like 'Pa' or 'Rbc', especially if they are even longer or are repeated frequently in the same connection. We therefore use on occasion the letters 'A', 'B', 'C' as abbreviations for any sentences whatever of the symbolic language. These letters are called sentential constants (or: propositional constants). E.g. in a certain case 'A' might be taken as an abbreviation for 'Pa'; as soon as 'P' and 'a' are interpreted, 'A' is also interpreted. In our use of a sentential constant we will for the most part leave open what particular sentence it stands for as an abbreviation.
2c. Illustrative predicates. To facilitate framing examples in connection with the further construction of our symbolic language system, we list here various predicates, functors (cf. 18) and individual constants for particular domains of individuals.
(Continues...)
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Table of Contents
PART ONE System of symbolic logicChapter 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