《Introduction to Logical Theory》

Chapter 1．LOGICAL APPRAISAL1

1．Logical appraisal；and other kinds1

Ⅰ．INCONSISTENCY2

2．Words of logical appraisal have connected meanings2

3．Contradicting oneself2

4．Statements，not sentences，are inconsistent with one another3

5．Incompatible predicates5

6．Negation7

7．Definition9

8．Linguistic rules and logical relations9

Ⅱ．REASONING12

9．Arguing，proving，inferring：validity12

10．Not all＇valid steps＇are steps in reasoning13

Ⅲ．THE LOGICIAN＇S SECOND-ORDER VOCABULARY15

11．Use to be made of logician＇s higher-order words15

12．Contraries and contradictories16

13．Entailment and inconsistency19

14．Logically necessary statements；entailment and necessity21

15．A problem24

16．Logical equivalence；subcontrariety；necessary and sufficient conditions24

Chapter 2．FORMAL LOGIC26

1．The formal logician is not a list-maker26

Ⅰ．GENERALITY．THE USE OF FORMULAE26

2．The generality of logicians＇entailment-statements26

3．The use of formulae27

4．Entailment between sentences and between formulae31

5．The range of values of a variable32

6．＇Inconsistent＇，＇logically necessary＇applied to formulae33

7．Misinterpretations of＇？＇35

8．The point of using＇？＇39

Ⅱ．FORM40

9．The limitations implied by＇formal＇．Rules for representative patterns40

10．Formal analogies and verbal frameworks43

11．A formal analogy without a framework：transitivity45

12．Logical constants47

13．Logical forms of statements explained in terms of formal analogies49

14．Logical form and logicians＇formulae52

15．Mistakes about logical form53

Ⅲ．SYSTEM56

16．The logical ideal of system，and its effects56

17．Methods of systematization58

18．Abstract systems and their interpretation61

Chapter 3．TRUTH-FUNCTIONS64

Ⅰ．TRUTH TABLES64

1．Formation-rules64

2．The meanings of the symbols of the system65

3．Use of the tables in determining truth-conditions70

4．Use of the tables for establishing logical relations and testing formulae71

5．Some laws of the system74

Ⅱ．TRUTH-FUNCTIONAL CONSTANTS AND ORDINARY WORDS78

6．The customary identifications78

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-FUNCTIONS93

12．＇？＇and＇entails＇93

13．The stroke-function96

14．The deductive system of truth-functions97

15．Examples of derivations100

Chapter 4．CLASSES：AN ALTERNATIVE INTERPRETATION OF THE TARULAR SYSTEM102

1．The system as an abstract exercise102

2．An alternative interpretation．Class-expressions and class-constants103

3．Class-statement formulae107

4．The use of the tables to establish logical rules about class formulae111

5．Parallels with ordinary speech．Re-expression of rules as entailments between class-membership formulae119

6．Limitations of this use of the mechanism of the tables120

Chapter 5．PREDICATIVE FORMULAE AND QUANTIFIERS125

Ⅰ．THE ELEMENTS OF THE PREDICATIVE SYSTEM125

1．Relations between the class and truth-functional systems125

2．The paraphrasing of class-statement formulae in a new notation129

3．Proofs in the new notation134

4．Multiple quantification and relational predicates138

Ⅱ．THE PREDICATIVE SYSTEM AND ORDINARY SPEECH：PRELIMINARIES143

5．The distinction between individual and predicative expressions：referring and describing143

6．Claims made on behalf of the symbolism of the system146

7．＇(？)＇and tenses150

Chapter 6．SUBJECTS，PREDICATES，AND EXISTENCE152

Ⅰ．THE TRADITIONAL SYSTEM OF CATEGORICAL PROPOSITIONS152

1．Formulae152

2．Laws155

3．Systematization160

Ⅱ．THE ORTHODOX CRITICISMS OF THE SYSTEM163

4．The dilemma of existence163

5．The detail of the dilemma165

Ⅲ．SUBJECTS AND PREDICATES170

6．A formalistic solution170

7．The realistic solution：presupposition and entailment173

8．Class-membership and class-inclusion．Subject-predicate statements179

9．The relevance of the analysis of general statements as conjunctions of singular statements183

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

11．＂＇Exists＇is not a predicate＂190

12．Limitations of the traditional system192

Chapter 7．GENERAL STATEMENTS AND RELATIONS195

Ⅰ．GENERAL STATEMENTS195

1．The truth-conditions of general subject-predicate statements195

2．Different kinds of general sentences195

3．Law-statements198

4．The impossibility of a neat classification of general sentences201

Ⅱ．RELATIONS202

5．Transitivity202

6．Symmetry204

7．Relations and their converses204

8．The strained use of＇relation＇205

9．Attempts to＇reduce＇relational inferences207

10．The logical forms of arguments209

Chapter 8．TWO KINDS OF LOGIC211

Ⅰ．FORMAL LOGIC：APPLICATIONS AND LIMITATIONS211

1．Entailment-rules and referring rules211

2．Logically ideal sentences214

3．The use of modern logical symbolism in systems of necessary truth and natural law217

Ⅱ．TYPE-DIFFERENCES AND FORMATION-RULES224

4．Formation-rules and rules of grammar224

5．Non-grammatical type-restrictions226

6．The metaphysical belief in basic types227

Ⅲ．THE LOGIC OF LANGUAGE230

7．The fluidity of language230

8．The logic of ordinary speech231

Chapter 9．INDUCTIVE REASONING AND PROBABILITY233

Ⅰ．SUPPORT AND PROBABILITY233

1．Non-deductive reasoning233

2．Degrees of support：the use of＇probable＇，etc237

3．Support and relative frequencies241

4．Support and numerical chances242

5．Degrees of support for generalizations244

6．No precise rules for the assessment of evidence247

Ⅱ．THE＇JUSTIFICATION＇OF INDUCTION248

7．The doubt and its source248

8．The desire for a supreme premise of inductions251

9．The attempt to find a mathematical justification252

10．The senselessness of the demand256

11．Any successful method of finding things out must be inductively supported258

12．The belief that the soundness of induction requires a＇presupposition＇may rest upon the confused conflation of two questions260

INDEX264