Summary
1 Introduction 13
1.1 What is logic? 13
1.2 Reasoning and Inference 14
1.3 Arguments 18
1.4 Sentences, Propositions, Statements 25
2 Logic and arguments 33
2.1 Validity and form 33
2.2 Validity and Correctness 38
2.3 Deduction and induction 42
2.4 Logic and the process of inference 46
2.5 A little history 47
3 Preliminaries 53
3.1 Languages 53
3.2 Artificial Languages 55
3.3 Use and mention 57
3.4 Object language and metalanguage 62
3.5 The use of variables 63
4 Sets 65
4.1 Characterization of sets 65
4.2 Special sets 68
4.3 Relationships between sets 71
4.4 Set operations 73
4.5 Properties and relations 77
4.6 Functions 79
4.7 Infinite sets 82
5 The classical propositional calculation 89
5.1 Logic 89
5.2 Introducing CPC 92
5.3 Sentential letters and atomic formulas 96
5.4 Operator and molecular formulas 99
5.5 Punctuation Signals 106
6 Propositional interpretations 115
6.1 Meaning and Truth 115
6.2 Basic ideas 120
6.3 Functions of truth 125
6.4 Values 133
7 Tautologies and Tautological Consequence 139
7.1 Truth table 139
7.2 Tautologies, contradictions and contingencies 145
7.3 Tautological implication and equivalence 149
8 Syntax of predicate calculus (I) 157
8.1 Introducing the CQC 157
8.2 Some characteristics of classical logic 162
8.3 Individual Symbols 163
8.4 Predicate Constants and atomic formulas 167
8.5 Operators and molecular formulas 175
8.6 Quantifiers and general formulas 178
9 Syntax of predicate calculation (II)187
9.1 First-order languages 187
9.2 Categorical Propositions 197
9.3 Multiple quantification 206
10 Structures and Truth 213
10.1 The semantic value of expressions 213
10.2 Structures 216
10.3 Truth 224
10.4 Definition of truth 236
11 Validity and logic consequence 247
11.1 Validity 247
11.2 Logic consequence (semantics) 252
11.3 Some properties of ë 257
11.4 Validity of arguments 259
12 Semantic tableaux 263
12.1 Proof procedures 263
12.2 Examples of tableaux 267
12.3 Rules for molecular formulas 273
12.4 Logic consequence 277
12.5 Quantifiers 280
12.6 Invalidity 289
12.7 CQC indecisiveness 293
13 Axiomatic systems and formal systems 297
13.1 Mathematicians and the truth 297
13.2 Geometry 299
13.3 Formal Systems 303
13.4 Lewis Carroll doublets 304
14 Natural deduction (I) 307
14.1 Introducing the Natural Deduction 307
14.2 Direct rules of inference 314
14.3 Making a deduction 318
14.4 Hypothetical rules of inference 325
14.5 Deriving Strategies 332
15 Natural deduction (II) 339
15.1 Derived Rules 339
15.2 Rules for quantifiers 343
15.3 A derived rule for quantifiers 357
15.4 Theorems 358
15.5 Syntactic consequence and semantic consequence 360
16 Identity and function symbols 365
16.1 Identity 365
16.2 Function symbols 380
16.3 Logical consequence in CQC = f 390
16.4 Semantic tableaux for CQC = f 392
16.5 Natural Deduction in CQC = f 398
17 Formalized Theories 405
17.1 Conceptualizations 405
17.2 A theory of blocks 410
17.3 Formalized Arithmetic 421
18 Non-classical Logic 435
18.1 What is classical logic? 435
18.2 Non-classical Logic 440
18.3 Alethic modal logic 444
18.4 Other modal logics 460
18.5 Alternative logics 462
18.6 Most recent history 477
Appendix A – Notions on the theory of the syllogism 483
A.1 Categoric propositions 483
A.2 The traditional square of opposition 486
A.3 Categorical syllogisms 490
A.4 Validity of syllogisms 497
A.5 Venn-Euler diagrams 508
A.6 Validity and existence 517
Bibliographic references 523