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Chapter 1．LOGICAL APPRAISAL 1

1．Logical appraisal；and other kinds 1

Ⅰ．INCONSISTENCY 2

2．Words of logical appraisal have connected meanings 2

3．Contradicting oneself 2

4．Statements，not sentences，are inconsistent with one another 3

5．Incompatible predicates 5

6．Negation 7

7．Definition 9

8．Linguistic rules and logical relations 9

Ⅱ．REASONING 12

9．Arguing，proving，inferring：validity 12

10．Not all＇valid steps＇are steps in reasoning 13

Ⅲ．THE LOGICIAN＇S SECOND-ORDER VOCABULARY 15

11．Use to be made of logician＇s higher-order words 15

12．Contraries and contradictories 16

13．Entailment and inconsistency 19

14．Logically necessary statements；entailment and necessity 21

15．A problem 24

16．Logical equivalence；subcontrariety；necessary and sufficient conditions 24

Chapter 2．FORMAL LOGIC 26

1．The formal logician is not a list-maker 26

Ⅰ．GENERALITY．THE USE OF FORMULAE 26

2．The generality of logicians＇entailment-statements 26

3．The use of formulae 27

4．Entailment between sentences and between formulae 31

5．The range of values of a variable 32

6．＇Inconsistent＇，＇logically necessary＇applied to formulae 33

7．Misinterpretations of＇？＇ 35

8．The point of using＇？＇ 39

Ⅱ．FORM 40

9．The limitations implied by＇formal＇．Rules for representative patterns 40

10．Formal analogies and verbal frameworks 43

11．A formal analogy without a framework：transitivity 45

12．Logical constants 47

13．Logical forms of statements explained in terms of formal analogies 49

14．Logical form and logicians＇formulae 52

15．Mistakes about logical form 53

Ⅲ．SYSTEM 56

16．The logical ideal of system，and its effects 56

17．Methods of systematization 58

18．Abstract systems and their interpretation 61

Chapter 3．TRUTH-FUNCTIONS 64

Ⅰ．TRUTH TABLES 64

1．Formation-rules 64

2．The meanings of the symbols of the system 65

3．Use of the tables in determining truth-conditions 70

4．Use of the tables for establishing logical relations and testing formulae 71

5．Some laws of the system 74

Ⅱ．TRUTH-FUNCTIONAL CONSTANTS AND ORDINARY WORDS 78

6．The customary identifications 78

7．＇~＇and＇not＇ 79

8．＇·＇and＇and＇ 79

9．＇？＇and＇if＇ 82

10．＇？＇and＇if and only if＇ 90

11．＇v＇and＇or＇ 90

Ⅲ．TRUTH-FUNCTIONAL CONSTANTS AND LOGICAL RELATIONS．THE DEDUCTIVE SYSTEM OF TRUTH-FUNCTIONS 93

12．＇？＇and＇entails＇ 93

13．The stroke-function 96

14．The deductive system of truth-functions 97

15．Examples of derivations 100

Chapter 4．CLASSES：AN ALTERNATIVE INTERPRETATION OF THE TARULAR SYSTEM 102

1．The system as an abstract exercise 102

2．An alternative interpretation．Class-expressions and class-constants 103

3．Class-statement formulae 107

4．The use of the tables to establish logical rules about class formulae 111

5．Parallels with ordinary speech．Re-expression of rules as entailments between class-membership formulae 119

6．Limitations of this use of the mechanism of the tables 120

Chapter 5．PREDICATIVE FORMULAE AND QUANTIFIERS 125

Ⅰ．THE ELEMENTS OF THE PREDICATIVE SYSTEM 125

1．Relations between the class and truth-functional systems 125

2．The paraphrasing of class-statement formulae in a new notation 129

3．Proofs in the new notation 134

4．Multiple quantification and relational predicates 138

Ⅱ．THE PREDICATIVE SYSTEM AND ORDINARY SPEECH：PRELIMINARIES 143

5．The distinction between individual and predicative expressions：referring and describing 143

6．Claims made on behalf of the symbolism of the system 146

7．＇(？)＇and tenses 150

Chapter 6．SUBJECTS，PREDICATES，AND EXISTENCE 152

Ⅰ．THE TRADITIONAL SYSTEM OF CATEGORICAL PROPOSITIONS 152

1．Formulae 152

2．Laws 155

3．Systematization 160

Ⅱ．THE ORTHODOX CRITICISMS OF THE SYSTEM 163

4．The dilemma of existence 163

5．The detail of the dilemma 165

Ⅲ．SUBJECTS AND PREDICATES 170

6．A formalistic solution 170

7．The realistic solution：presupposition and entailment 173

8．Class-membership and class-inclusion．Subject-predicate statements 179

9．The relevance of the analysis of general statements as conjunctions of singular statements 183

10．Singular statements beginning with＇the＇and＇a＇ 184

11．＂＇Exists＇is not a predicate＂ 190

12．Limitations of the traditional system 192

Chapter 7．GENERAL STATEMENTS AND RELATIONS 195

Ⅰ．GENERAL STATEMENTS 195

1．The truth-conditions of general subject-predicate statements 195

2．Different kinds of general sentences 195

3．Law-statements 198

4．The impossibility of a neat classification of general sentences 201

Ⅱ．RELATIONS 202

5．Transitivity 202

6．Symmetry 204

7．Relations and their converses 204

8．The strained use of＇relation＇ 205

9．Attempts to＇reduce＇relational inferences 207

10．The logical forms of arguments 209

Chapter 8．TWO KINDS OF LOGIC 211

Ⅰ．FORMAL LOGIC：APPLICATIONS AND LIMITATIONS 211

1．Entailment-rules and referring rules 211

2．Logically ideal sentences 214

3．The use of modern logical symbolism in systems of necessary truth and natural law 217

Ⅱ．TYPE-DIFFERENCES AND FORMATION-RULES 224

4．Formation-rules and rules of grammar 224

5．Non-grammatical type-restrictions 226

6．The metaphysical belief in basic types 227

Ⅲ．THE LOGIC OF LANGUAGE 230

7．The fluidity of language 230

8．The logic of ordinary speech 231

Chapter 9．INDUCTIVE REASONING AND PROBABILITY 233

Ⅰ．SUPPORT AND PROBABILITY 233

1．Non-deductive reasoning 233

2．Degrees of support：the use of＇probable＇，etc 237

3．Support and relative frequencies 241

4．Support and numerical chances 242

5．Degrees of support for generalizations 244

6．No precise rules for the assessment of evidence 247

Ⅱ．THE＇JUSTIFICATION＇OF INDUCTION 248

7．The doubt and its source 248

8．The desire for a supreme premise of inductions 251

9．The attempt to find a mathematical justification 252

10．The senselessness of the demand 256

11．Any successful method of finding things out must be inductively supported 258

12．The belief that the soundness of induction requires a＇presupposition＇may rest upon the confused conflation of two questions 260

INDEX 264